168
A. Murli

A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

  • Upload
    others

  • View
    7

  • Download
    0

Embed Size (px)

Citation preview

Page 1: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli� �����������

� ��������������� ����� ������������������� "! ��#�����$

% �"��&' (�)�*�,+-�. /�*���"��

02130 46587:9<;>=?;>@BACA<58A<@$7D@BEFEHGIAC;FJK58EFLDMN;H587D@O=?;PL?7RQ?A<58S?EH@BTUGTU@B=?;HGV7:9#@�WXGVEFWX58EHGY9#58A<@

Z\[^]`_^aHbdc^ef]hg^ijaHk8bmln]h[^_poqk8aH[srtoqijb�_^aHijiuo8vwoxryaHk`oxrybjzFo2{\|^k8aHefbjzFo6}aY~�|^aHijij]2_^b��qoqij|�rtoqefa�i�� oqzHzH|��eyoxryaH�H�Fo2_^aHi�efbj�f|^i�rtoxry]�_^b�|^[�zFoqijzH]hij]Va�_^|^[^~�|^aYi�� ox�8_poqg^bjijb�rx}o��C_^aHi�efbj�f|^i�rtoxry]���ryaH�f�f]^��Boxryb�_^|^aB[�|^k8aHefbm��a6���aB�f|^c^c�]h��ry]2zt�^a6��w�fbuo�|^[�� oqc^c^ef]h�f�fbjk`oq�Hbj]h[^aN_^bm�d�^bji�c^efbjk8]Vc^ef]hg^ijaHk`ozt�^aY�fb�c�]h[^a2}aB~�|^aHijij]2_^b���rybjk`oqefa�iuo���g�]h[srx}oq�2_^bX��DzH]hk8a�oqc^c^ef]h�f�fbjk`oq�Hbj]h[^aB_^b��d��do�k`oq[^bjaHeyoVc^by}|�bj[sry|^b�ryb��qo�c�aHe�l�oqefaBzHb }]2�fb�gpoq�yo2�f|^i�zH]h[^zHa¡rfry]8_^b£¢¡¤¥¤y¦x¤y¢V§©¨t¨3¦xª¬«�­®¦x�¯�°s±#²�³n´�³nµ�²d°�¶p·�¶p·`¸�¹�º©º©µ»º©°�¼p½F½Hµ�¾�¿�À3µ�ÁÂ�§q­�Ã�«�Ä6Äp«�Å`¢¡¤y¦X�*¢yÆ�«�Ä�§Y¨¡«^§�§¥ÇsÇ�¤y¦3¨t¨¡ÃÈÅ8§3ÉHÃ�¦xÄm¢Y���Êd¨¡ÃKÆqÃ�Ët¢B¢¡¤¥¤y¦x¤y¢�§©¨t¨3¦xª¬«�­®¦YÃÈÄ��ª�§�ÌH«^§qÄp­�ÃÈ­�ͧ

Î�ÏÑÐ �`Ò���<Ð¥Ó

Ô b�c^|�}]Pefb�ryaH[^aHefa?zt�^aÕ����fbuo�|^[po>g^|^]h[po>oqc^c^ef]h�f�fbjk`oq�Hbj]h[^aD_^bN���faÎ�}aÕ���f|��8zHbjaH[sryaHk8aH[sryac^bjzHzH]hij]h�^�#�do�_^b��8zH]hi�rx}o�zH]h[^�fbj��ryaB]heyoV[^aHi�zt�^buoqefbjefa�zt�^aBzH]h�yo��fb�bj[sryaH[^_po�c�aHe2���f|��8zHbjaH[sryaHk8aH[sryac^bjzHzH]hij]h�^�

ÖP×KØfÙFÚ�Û^Ü�Ý2Þqß�Þqß#àFá»â ã¡äxå�áçæ3è�é�á ä©ãNê2ëPìfí©î ìyï¥ð¥ñXèVòê2ëPìfí©î ìyï¡ó¥ô©õCö�èø÷ ã¡é�ã\ùxú¥é�û�ápá äHû�è�é�èøèø÷ èø÷ ã¡é�ã\ù©é�á üøè�æ3ý©èâ á þÿé�è<æ3èfâ á ü�ú¥÷ á�å�ã¡ä©ã£ý��¡ýxú¥÷ ásè<é�áçå�ý©÷ û ú���ë6í©î í¡í¥ñxì��wìfí���tõ Ö

®[��haH[^aHeyoqija��qoqijaBiuoV�fa �h|^aH[srya����� è�é#ã¡é ú��xú¡ånû ú\á äHû�ý©á é�èøá ÷»å�á �¡ä©á �qâ�útû�ãNæ3è�÷^û�è�é�üøá ä©è������������! "�#$�xú¡å�ú¥äxæ3ã3â ápå�ýVâ ã¡äxå�áçæ3è�é ú&%�á ã¡ä©á�è�üøù©á é�áçâ('©èøãå�ý©÷ ÷*) èfå�ùqè�é�á è�ä�%fúøâ('©è<â áçú¡å�â ý©ä©ãøæ3á�ä©ã¡á+'xú©õ

,

Page 2: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

�Yº©µ���ºx³n°sÀ��¼>¶p·�¶p·8¸��d°�¾�¾�ÿ°�º©º©µ»º©°6¼p½F½Hµ�¾�¿�À3µ�Á ¢�Æq«p¢6Äp«�Å`¢¡¤¥Ã � ¢����^§qÄpÄm¦Dª�§?¨¡­®¢t¨t¨F§:Ç^§q¤¥­®¢6ÃÈÄp­®¢¡¤f§�¢6ªj¢`¨¡­®¢t¨t¨3¢VÇ�¤¥ÃÈÅ`¢� Ë¡Ã �¥¤y¢�Æ�¢tË¡ÃÈÅ8§qª¬Ã§qªÈªj¦x¤f§8¤¥Ãj¨¡«�ª¬­ §

Î/Ï Ð �:Ò ��CÐ����������wÓ

�Koqijaøc^ef]hc^efbja¡rx}oNln]hef[^bj�fzHaø|^[�� bj[sryaHefc^efa¡rtoq�Hbj]h[^a�~�|poq[sryb�rtoxryb��qo�_^aHiji�� aHefef]hefaøoq�f�f]hij|�ry]�a�~�|^bj[^_^b^zt�^buo©�efbj�fzHaq�^bj[6~�|poqijzt�^a�k8]�_^]^�^bjim�fb*�h[^b��pzFoxry]2_^aHiji�� aH�fc^efaH�f�fbj]h[^aw���f|��8zHbjaH[sryaHk8aH[sryaYc^bjzHzH]hij]h�^����<� aHefef]hefaYoq�f�f]hij|�ry]2[^]h[6�faHk8c^efaYgpoq��rto2oV�qoqij|�rtoqefaBiuo�g�]h[srx}o�_^b�|^[�� oqc^c^ef]h�f�fbjk`oq�Hbj]h[^aq�

ÖP×KØfÙFÚ�Û^Ü�Ý2Þqß��sß#àFèCå�á�â ã¡äxå�áçæ3è�é ú¥ä©ã ê � ë?ìfí©î ìyïCèBòê � ë?ìfí©î íHó�#3å�á+'xú � � ë � ê �"!$#ê � �Hë6í©î í&% #Fü�ú�ú¥äxâ('©è<å�èå�áhâ ã¡äxå�áçæ3è�é ú¥ä©ã<ê � ëDìfí¡í¡í¡í©î ì�ñ#èNòê � ë?ìfí¡í¡í¡í©î í&%dé�áçå�ý©÷ û ú � � ë � ê � ! òê � �¡ë6í©î í&%Fõ^ö ) è�é�é�ã¡é�èdú¡å�å�ã¡÷ ý3û�ã('èKæ3ý©ä*)Hý©è÷ ã�ånû�èfå�å�ãNá ä�è�äHû�é ú¥ü �©á^á»â�ú¡å�á*# �qè�äxâ(',+è�#�á äHû�ý©á û�á -tú¥üøè�äHû�è�#�ä©è�÷»å�èfâ ã¡äxæ3ã�â�ú¡å�ãN÷*) ú¥ù©ù©é�ãHå�å�á ü�ú&%�á ã¡ä©è ú¥ù©ùxú¥áçú\èfå�å�è�é�èüøá �¡÷ á ã¡é�è¡õ. äxú¥÷ ã��Hú¥üøè�äHû�è�#yý©äøè�é�é�ã¡é�è�æ3áqó¥í�üøá ÷çúKè�ý©é�ãKä©è�÷ ÷ è�ù©é�è/-Fáçå�á ã¡ä©á3æ3á©å�ùqèfå�úKú¥ä©äFýxúKæ3á©ý©ä+) á äxæ3ýxånû�é�áçú#â ã¡ä ý©ä þ�útû�û�ý©é útû�ãú¥ä©äFý©ãXæ3ápï¡ó¥í¡íøüøá ÷ á ã¡ä©á^æ3á�è�ý©é�ã0'èCù©é útû�áçâ�ú¥üøè�äHû�èCá é�é�á ÷ è/-tú¥äHû�è�#xüøè�äHû�é�è ÷ ã\ånû�èfå�å�ãXè�é�é�ã¡é�è 'xú\â è�é�û ú¥üøè�äHû�èCè1hè û�û�áä©ã¥û�è/-¡ã¡÷ áså�ý©÷+�©á ÷çú¥äxâ á ã�æ3á�ý©äxú£ù©áçâ�â ã¡÷çú£á üøù©é�èfå�ú þ�ú¥üøá ÷ áçú¥é�è�#3â ã¡ä�ý©äNþ�útû�û�ý©é útû�ã�ú¥ä©äFý©ãøæ3á�ï¥í¡í üøá ÷çúøè�ý©é�ãxõ Ö

®[�l�oxrfryb��£c�aHe8zH]hk8a}a�_^a2�p[^b�ry]^�øi�� aHefef]hefa�oq�f�f]hij|�ry]�[^]h[PrybjaH[^a�zH]h[sry]�_^aHijiuo �heyoq[^_^aH�H�Fo�_^aHi�qoqij]hefaV_po6oqc^c^ef]h�f�fbjk`oqefaq��Z\[k8]�_^]�c�aHe�l�oqefa�~�|^aH��ry]�}a2eyoqc^c�]he�rtoqefaVi�� aHefef]hefa2oq�f�f]hij|�ry]�o�Ð �<Ðç�]3���haHef]*���fzFoqiuoqefaH�2i�� aHefef]hefa�zH]h[^�fbj_^aHeyoq[^_^]�Ð �<Ð�zH]hk8a�|^[^b�rx}o�_^b�k8bj�f|^eyo��43 rtoqi"�p[^aY�fbdbj[sryef]�_^|^zHabjimzH]h[^zHa¡rfry]8_^b£¢¡¤¥¤y¦x¤y¢Y¤y¢¡ª�§q­�Ã65q¦x�¯�°s±#²�³n´�³nµ�²d°�¶p·87�·8¸�¹�º©º©µ»º©°�º©°�¾n¼�À©³69�µ�ÁÂ�§q­�Ãd«�ÄwÄp«�Å`¢¡¤y¦N�>¢yÆ�«�Ä�§�¨¡«^§`§¥ÇsÇ�¤y¦3¨t¨¡ÃÈÅ8§3ÉHÃ�¦xÄm¢Y���ÊK¨¡Ã#ÆqÃ�Ët¢�¢¡¤¥¤y¦x¤y¢�¤y¢¡ª�§q­�Ã65q¦VÃÈÄ����ª�§8ÌH«^§qÄp­�ÃÈ­�ͧ

Î0:^Ï Ð �:Ò ��CÐÐ �<Ð ; �=<Ï>�@?mÓ

ÖP×KØfÙFÚ�Û^Ü�Ý2ÞqßBA�ßDC�ã¡äxå�áçæ3è�é�áçú¥üøã\äFý©ãE-tú¥üøè�äHû�è£ê � ëPìfí©î ìyï�#�òê � ëPìfí©î íHó�#hê � ëPìfí¡í¡í¡í©î ì�ñ�èVòê � ë>ìfí¡í¡í¡í©î í&%è\â�ú¥÷çâ ã¡÷ áçú¥üøã �¡÷ á�è�é�é�ã¡é�á»é�è�÷çútû�á -Fá�â ã¡é�é�áçå�ùqã¡äxæ3è�äHû�ájõGFdû�û�è�ä©áçú¥üøãYâ ãHå6' H��4I�KJ í©î í¡í2L¡ð�ë í©î L¡ðKM`ìfí����èfæ �4I� Jí©î í¡í¡í¡í¡í2L¡ðKë�í©î L¡ð�M<ìfí��ON #¥â á ã*'è�#tâ ã¡üøè�è�é údù©é�è/-¡èfæ3á �©á ÷ è�#t÷*) è�é�é�ã¡é�èpé�è�÷çútû�á -¡ã4'è�ä©è�÷3å�èfâ ã¡äxæ3ãKâ�ú¡å�ãdüøã¡÷ û�ãKù©áP'ý ù©áçâ�â ã¡÷ ã�\àFámä©ã¥û�ámâ('©è�û ú¥÷ èXù©é�ã¡ù©é�á è ûE'ú�ä©ã¡ä�ù©ýQ'ã�èfå�å�è�é�è�á ä&-¡è�é�û�á û ú�#�ä©è�÷�å�è�äxå�ã�â('©èXä©ã¡äR'è(-¡è�é�ã�â('©è\å�è\æ3ý©èXäFý©üøè�é�áæ3á 1hè�é�áçå�â ã¡ä©ãXá äYüøã3æ3ý©÷ ã\ùqè�éKüøè�ä©ã�æ3ámìfí��OS�ú¥÷ ÷ ã¡é ú 'xú¥ä©ä©ãX÷çúNånû�èfå�å�ú�ùxú¥é�û�è á äHû�è�é úXè ÷ è£ånû�èfå�å�è ù©é�á üøèUT�â á þÿé�èæ3èfâ á ü�ú¥÷ á8V»ý©ä2èfå�è�üøù©á ãYå�á �¡ä©á �qâ�útû�á -¡ã�'è\â ãHånû�á û�ý©á û�ã�æ©úBê6ëPô©î í¡í¡í¡í¡í¡íNè8òê6ë ï3î ð¡ð¡ð¡ð¡ð¡ð©õ0W�ý©èfånû úBé�èfånû�é�á %�á ã¡ä©è�#â ã¡ü ý©ä*)Hý©è�#�'èdæ3ãE-Fý3û úKý©ä©áçâ�ú¥üøè�äHû�è�ú¥÷xâ é�á û�è�é�á ã#æ3á©é ú¥ù©ù©é�èfå�è�äHû ú&%�á ã¡ä©èmùqãHå�á %�á ã¡äxú¥÷ è�#¥â á ã*'è�ú¥÷©å�á �¡ä©á �qâ�útû�ã<â('©è 'xú¥ä©ä©ã÷ è â á þÿé�è æ3è�÷ ÷çú�é ú¥ù©ù©é�èfå�è�äHû ú&%�á ã¡ä©è#ùqãHå�á %�á ã¡äxú¥÷ è æ3á�ý©ä�äFý©üøè�é�ã #xèU)Hý©á äxæ3áX'èCý©äxú�é�èfånû�é�á %�á ã¡ä©è#þÿã¡é�ü�ú¥÷ è¡õ � è�é�û ú¥äHû�ã #á äXå�è �¡ý©á û�ãCå�áhú¡å�å�ý©üøè�éE'ú#â('©èdê\èXòê 'xú¥ä©ä©ã<÷ èKånû�èfå�å�èdâ á þÿé�è �xä©ã ú¥÷ ÷çúDTKYuü�ú<â á þÿé úCæ3èfâ á ü�ú¥÷ èdã��¡ä©á*)Hýxú¥÷ -¡ã¡÷ û ú#é�áçå�ý©÷ û�á� ê ! òêZ� �wìfí��OSCõ

Page 3: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# 1

â('©è�ä©è�÷»ù©é�á üøãxõ\àFá�ãHå�å�è�é -Fápâ('©è�ê � è�òê � 'xú¥ä©ä©ãB÷ è�ù©é�á üøèXæ3ý©èXâ á þÿé�èXå�á �¡ä©á �qâ�útû�á -¡è02£ý��¡ýxú¥÷ á»èfæ 'è��4I� �Pìfí�� � #üøè�äHû�é�è#ê � è�òê � 'xú¥ä©ä©ãø÷ è#ù©é�á üøèCâ á ä*)Hý©è<â á þÿé�èCå�á �¡ä©á �qâ�útû�á -¡èKý��¡ýxú¥÷ ásèfæ 'è �4I� �wìfí��%3¥õ Ö

®[��haH[^aHeyoqija��qoqijaBiuoV�fa �h|^aH[srya��

�Yº©µ���ºx³n°sÀ��¼>¶p·87�·8¸��d°�¾�¾�ÿ°�º©º©µ»º©°�º©°�¾n¼�À©³69�µ�Á ¢�Æq«p¢YÄp«�Å`¢¡¤¥Ã�� ¢`�� �^§qÄpÄm¦2ªj¢B¨¡­®¢t¨t¨3¢øÇ�¤¥ÃÈÅ`¢ OË¡Ã �¥¤y¢B¨¡Ã54qÄpà 6øËy§q­�Ã65q¢tÊ §qªÈªj¦x¤f§8¤¥Ãj¨¡«�ª¬­ §

Î0:^Ï Ð �:Ò ��CÐÐ �<Ð ����������7 � Ó

8�aHe�rtoq[sry]2i�� aHefef]hefaNefaHiuoxryb��h]Vk8bj�f|^eyoYiuo�_^bj��rtoq[^�Fo�ryeyo�_^|^aB[�|^k8aHefbmbj[`ryaHefk8bj[^b�_^bmzHb�lnefa��fb*�h[^b��pzFo©�ryb��ha�zH]hefefa¡rfryaq�pc^bj|�rfry]h��ry]2zt�^a�_^b�zHb�lnefaB_^aHzHbjk`oqijbmzH]hefefa¡rfryaq� Ô b�[^]qryb�zt�^aYiuoVefaHiuoq�Hbj]h[^a

Î0:^Ï Ð �`Ò���<ÐÐ �<Ð ����������7 � ; �h��� ?

[^]h[�bjk8c^ijbjzFoYzt�^a��Da���?oqg^g^buoq[^]�ijaB��ryaH�f�fa�c^efbjk8a )zHb�lnefaB�fb*�h[^b��pzFoxryb��haq�

ÖP×KØfÙFÚ�Û^Ü�Ý2Þqß 9�ßDC�ã¡äxå�áçæ3è�é útû�ásêNë�ó %Fî ï %<è�òê�ë6ó %Fî ô�#©å�á 'xú�:� I ë � ó %Fî ï % ! ó %Fî ô �ó %Fî ï % J í©î ó¡ï M�ìfí ��2 �wìfí ��2 ëDìfí �%3�; ��<

ü�ú£êBèBòêBä©ã¡ä 'xú¥ä©ä©ãø÷ è#ù©é�á üøè4)Hýxútû�û�é�ãøâ á þÿé�èCå�á �¡ä©á �qâ�útû�á -¡èKý��¡ýxú¥÷ á*#�'xú¥ä©ä©ã�ý��¡ýxú¥÷ á�å�ã¡÷ û ú¥äHû�ã£÷ è<ù©é�á üøè<æ3ý©è¡õ Ö

�doVc^ef]hc^efbja¡rx}o �h�>=�[^]h[�bj[^_^b���bj_^|poYbjimk`oq�f�fbjk8]Y�qoqij]hefa�_^b" "c�aHeøzH|^bm�qoqijaNiuoVefaHiuoq�Hbj]h[^a ; �h��� ?a�[^]h[6c^|�}]8c�aHe�rtoq[sry]8aH�f�faHefaYbj[s�haHe�ryb�rto�� Ô b�c^|�}]8bj[s�haHzHaY_^bjk8]h��ryeyoqefa�zt�^aY�faÎ0:^Ï Ð �:Ò ��CÐ

�<���������@?

oqijij]heyo:efbj�f|^i�rto�zt�^a8�*a���*�poq[^[^]Doqijk8aH[^]6ija2c^efbjk8a� �zHb�lnefa2�fb*�h[^b��pzFoxryb��ha2| �h|poqijb��do�cpoxrfry]�_^bryeyoq�fzH|^eyoqefaBiuoVefaH��ryefbj�Hbj]h[^aNln]hefk`oqijaN_^b�zH|^b��fb<}a�cpoqefiuoxry]�[^aHijiuo�[^]qrto ; =O?¥� ®[��fa �h|^b�ry]Vrtoqija�efaH��ryefbÿ��Hbj]h[^aB�yoqe©}o�ryeyoq�fzH|^eyoxrto��{\]h[�}a`c�]h�f�fbjg^bjija:o AmaHefk`oqefa`zt�^aq�Cbj[ �haH[^aHeyoqijaq�<i�� aHefef]hefa`efaHiuoxryb��h]D�fbuowc^by}|�a¡�8zFoqzHa6_^aHiji�� aHe �

ef]hefa�oq�f�f]hij|�ry]�B�bj[�l�oxrfryb�iuo`�fzHaHi�rto:_^aHiji�� |^[^]:]�_^aHiji�� oqi�ryef]8_^bjc�aH[^_^a�_poqiKcpoqe�rybjzH]hiuoqefaYc^ef]hg^ijaHk`o`bj[2 C�ã¡ä<÷*) èfå�ù©é�èfå�å�á ã¡ä©èDC�� E�F "�G�� H�I � J�C(��K*�ML "�å�á¡á äxæ3áçâ�ú¥ä©ã�û�ý3û�û�è�÷ è�â á þÿé�è�â ã¡üøù©é�èfå�è^û�é ú�÷çú�ù©é�á ü�ú�è»÷*) ý©÷ û�á ü�údâ á þÿé ú�ä©ã¡ääFý©÷ ÷çúøæ3è�÷�äFý©üøè�é�ãøâ ã¡äxå�áçæ3è�é útû�ã #Fá äxâ ÷ ýxå�è#÷çúøù©é�á ü�ú è<÷*) ý©÷ û�á ü�ú©õ . æNèfå�è�üøù©á ã #©â ã¡äxå�áçæ3è�é útû�áhê � ë�ñHí¡í¡í¡í¡í¡í�#¡ê � ëí©î í¡í¡í¡í¡í¡íHï¥í¡í�è�ê 2 ë6í©î í¥ñFó¥í©ìfí©ìfí�#tå�á'xú â('©è�ê � èKê � 'xú¥ä©ä©ã ý©äxú£å�ã¡÷çúCâ á þÿé ú å�á �¡ä©á �qâ�útû�á -túONÿé�áçå�ùqè û�û�á ü�ú¥üøè�äHû�èK÷çúâ á þÿé ú£ñøè<÷çúøâ á þÿé ú�ïQP�è<ê 2 'xú�å�è�á�â á þÿé�èCå�á �¡ä©á �qâ�útû�á -¡è�Nÿ÷ èCâ á þÿé�è#ñFó¥í©ìfí©ìRP®õ

Page 4: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

���

aH�yoqk8aq� 8�aHeXaH�faHk8c^bj]^�^~�|poq[^_^]8�fb�c^ef]�ha¡rfrto�bjimzH]h[sryef]hijij]�_^b�|^[�k8bj�f�fbjija�o�ij|^[ �so �hb�rfrtoxrto����fb�_^a¥��fbj_^aHeyo�zt�^a�i�� ]hg^bja¡rfryb��h]�c^efa2�p�f�yoxry]`�fbuo2eyo���hbj|^[sry]�zH]h[w|^[�aHefef]hefaYbj[^_^bjc�aH[^_^aH[srya�_poqijiuoV_^bj��rtoq[^�Fo_^aHiji�� ]hg^bja¡rfryb��h]6��ryaH�f�f]?_poqi c^|^[sry]D_^b iuoq[^zHbj]^� ®[�~�|^aH��ry]DzFoq�f]^�#bj[�l�oxrfryb���bj[sryaHefaH�f�yowc^ef]hc^efbj]wzt�^ai�� aHefef]hefaNoq�f�f]hij|�ry]YryeyoYijaNzH]�]hef_^bj[poxryaX_^aHimc^|^[sry]V_^b�bjk8cpoxrfry]YaN~�|^aHijija\_^aHiji�� ]hg^bja¡rfryb��h]Y�fbuo�rtoqija\_pozH]h[^�faH[srybjefa�iuoV_^bj��ryef|^�Hbj]h[^a�_^aHiji�� ]hg^bja¡rfryb��h]V��ryaH�f�f]^�Ô bK]h�f�faHe���b#bj[Q�p[^a�zt�^a�bj[Dk8]hi�rybdzFoq�fbK[^]h[?�fbK�po`bj[sryaHefaH�f�fa2o`zH]h[^]h�fzHaHefa2aH�yoxrfrtoqk8aH[sryaVbji��qo©�

ij]hefa�[�|^k8aHefbjzH]2_^aHiji�� aHefef]hefa ; oq�f�f]hij|�ry]2]2efaHiuoxryb��h]�?¥��k`o��fb�zHaHefzFo2_^bdoF�haHefa�bj[�ln]hefk`oq�Hbj]h[^b��f|^i��f|^]¦x¤fÆqÃÈÄm¢�ÆqÃD4q¤f§qÄ�Æ�¢yÉtÉ3§ � �®[ �haH[^aHeyoqijaq�^[^aHijiuoVefbj�f]hij|^�Hbj]h[^aN_^b�|^[�c^ef]hg^ijaHk`o�k8aH_^buoq[sryaBzFoqijzH]hiuoxry]hefa ; ¤¥Ãj¨3¦xª¬«hÉHÃ�¦xÄm¢VËt¦xÅ��Ç�«�­ §3ÉHÃ�¦xÄ�§qªj¢ ?¥�s[^]h[`zHb��fb�oq�fc�a¡rfrtoY_^b»zFoqijzH]hiuoqefaXiuoB�f]hij|^�Hbj]h[^a`��aH�yoxrfrtoq�^��bj[8~�|poq[sry]Vbjipc^ef]�zHaH�f�f]V_^b

efbj�f]hij|^�Hbj]h[^a zH]hk8c�]he�rto\�faHk8c^efaøi�� bj[sryef]�_^|^�Hbj]h[^a _^b�aHefef]hefb�� �Koqijb�aHefef]hefb��f]h[^]Bbj[sryefbj[^�faHzFoqk8aH[sryaøija¥��soxryb�oN~�|^aH��ry]�c^ef]�zHaH�f�f]�B�zHb }]Nbjk8c^ijbjzFoXzt�^a�aH�f�fb^[^]h[Vc�]h�f�f]h[^]BaH�f�faHefaXa¡��b�rtoxryb��xk`oN�f]hij]�zH]h[sryef]hijiuoxryba���rybjk`oxryb��^�faYc�]h�f�fbjg^bjijaB~�|poq[sryb�rtoxryb��qoqk8aH[sryaq�^c�aHe\aH�faHk8c^bj]`oxrfryeyoF�haHef�f]`k`o���hbj]heyoq�Hbj]h[^b���3�rtoqijac^ef]hc�]h�fb�ry]`}aN_^]3�haHef]h�f]2zHb�rtoqefaNiuo�lneyoq�faN_^aHi 8<ef]ql ���N� 8Koqefija¡rfrF��c^ef]qlnaH�f�f]hefa�aHk8aHefb�ry]Voqiji�� |^[^b��haHef�fb�rx}o_^aHijiuo��øoqijb�ln]hef[^buo����£aHe�qaHija�h�»oqc^cpoqef�yo�[^aHi Ô 3Bv { ��� Ô ���h]hi��������p{Y� 1 �^_^aHi�{\]3�haHkVg^efa = �O������N� 8Koqefija¡rfrF�sefb�lnaHefaH[^_^]h�fbpoq_`oqijzH|^[^a�~�|^aH��rybj]h[^bp_^bpzFoqijzH]hij]�[�|^k8aHefbjzH]YoBc^efaHzHbj�fbj]h[^a �p[^b�rto���o AmaHefk`ozt�^a\ijaX_^b��8zH]hi�rx}oB�f]h[^]��fc�aH�f�f]V�f|^g�_^]hijaq��c^bj|�rfry]h��ry]Yzt�^aNc^ef]qln]h[^_^a ; ���¬����� �Nv�2�f| ��haH��rybj]h[`bj�<ry�poxrry�^aB_^b��8zH|^i�rybjaH�øbj[�k`oxryefb��:zH]hk8c^|�rtoxrybj]h[^�øk`o�:[^]qrXg�a8Æ�¢t¢®Ç`g^|�rXry�^a�6oqefa�¨¡«��H­�ªj¢¡� � ?¥� Ô b��f|^_��_^b���bj_po�bji�c^ef]�zHaH�f�f]:_^b�efbj�f]hij|^�Hbj]h[^aBzH]hk8c^|�rtoq�Hbj]h[poqijaB_^b�|^[wc^ef]hg^ijaHk`o�[^aHijijaBl�oq�fb�ln]h[^_poqk8aH[srtoqijb�fa �h|^aH[sryb ;�� b*�h|^eyo �h��� ?&��h��_^aH�fzHefbj�Hbj]h[^a�_^aHi�c^ef]hg^ijaHk`oVk8aH_^buoq[sryaB|^[Å`¦FÆ�¢¡ªÈªj¦���Ç�¤y¦��Hªj¢¡Å8§! �Å8§q­®¢¡Å8§q­�Ã�Ët¦ B=��Xoqc^c^ef]h�f�fbjk`oq�Hbj]h[^aB_^aHidk8]�_^aHijij]Vk`oxryaHk`oxrybjzH]�k8aH_^buoq[srya�|^[Å`¦FÆ�¢¡ªÈªj¦���Ç�¤y¦��Hªj¢¡Å8§! 2Äp«�Å`¢��¤¥Ã�Ët¦ B

����_^aH�fzHefbj�Hbj]h[^a�_^aHi�k8]�_^aHijij]V[�|^k8aHefbjzH]2k8aH_^buoq[sryaB|^[�§qª 4�¦x¤¥ÃÈ­�Å`¦ B" �BÃÈÅNÇ�ªj¢¡Å`¢¡Äp­ §3ÉHÃ�¦xÄm¢C_^aHiji�� oqi*�h]hefb�ryk8]�bj[:|^[^]��fc�aHzHb��pzH]2oqkVg^bjaH[sryaN_^bmaHiuoqg�]heyoq�Hbj]h[^a ; ¨3¦�¥­�# §$�¤y¢ ?¥�3

% Ù'&�(�Ü*)¡Ü�Ý$(�Ù�ÞqßBA�ß,+.-,/10�Ü*(�Ù20�Ü435/167(80�Ù�)�)�6$9: I I5;=< " F.>�F "(� "dê�?�� Ý5/10�Ü*(�Ù@0�Ü435/167(80�Ù�)�)�6BA�S G "

� ê �¡ëDC$M A S A � �FE C �wì < Nnì¡õ ïQP��>�L "�AHG"� ���(�RG " � � I5;=< " F �I��*>�I" ��"� �G��>GK "J< � ��F��MKK< " K*�5CL> CL>�I�G�� ��" F ��KM>ON �MIQP ��FK*�5CL> ��F "SR G "DT�ë�í=RF>�L0L " F.>A�S�ë�ì!R.G�� � �MF�G� CL?�"øêHG" ��"� ! �T >�F � �MI" ��"� ! �T ; I �MK=G��UWV "OL � " �YX[Z�U \J] G�� � �MF�G� ��I�CL?�" CL?�"£ê^G" � �_>�F � �MI"UTBRG`>�K K*�MI K " I��" I��> � �(�RG " � �"I5;=< " F �I��*>�I"U

C�ã¡äCû ú¥÷ è�æ3è��xä©á %�á ã¡ä©è�#¡å�è�å�á©â ã¡äxå�áçæ3è�é úKú¡æCèfå�è�üøù©á ã�AYë?ìfí�#té�áçå�ý©÷ û ú#â('©èba<ë6í©î í¡í¡í¡í¡í¡í¡í¡í¡í¡íHó¡ï¥ð Nÿé ú&���¡á ã#æ3á7cmã�'©éæ3è�÷ ÷*) útû�ã¡üøã�æ3á�áçæ3é�ã��¡è�ä©ã #Fá ä�â ü P 'xúøã¡é æ3á ä©è<æ3á �¡é ú¥äxæ3è % %fú\ìfí�� �Md #xãE-&-¡è�é�ãK'èCæ3á�ã¡é æ3á ä©è ! ìfí�#3è�e�ë�ô¡í¡í¡í¡í¡í¡í¡í¡íN -¡è�÷ ã3â á ûE'úøæ3è�÷ ÷çú£÷ ýxâ è<á äNügftå�èfâ0P 'xú£ã¡é æ3á ä©è<æ3á+�¡é ú¥äxæ3è % %fúXìfí�h&#3ãE-&-¡è�é�ãK'è<æ3á�ã¡é æ3á ä©è<ð©õ

Page 5: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# �O�

��º©µ��K¾n°���¼�º©°�¼p¾n°

��µ �d°�¾�¾nµ���¼�À3°���¼�À©³��hµ

��µ �d°�¾�¾nµ�²�¿���°�ºx³��hµ

¼p¾�»µ»ºx³ÈÀ��µ ½Hµ���À �V¼^º©°

����������� �

� � � � � � � � � � ���� � � � � � � � � � ���

����������� �

��

� b*�h|^eyo �h���� Ô zt�^aHk`o2_^aHi�c^ef]�zHaH�f�f]`_^b�efbj�f]hij|^�Hbj]h[^a�zH]hk8c^|�rtoq�Hbj]h[poqija�_^b�|^[�c^ef]hg^ijaHk`o��

�Koqija£c^ef]�zHaH�f�f]�c^efaH�f|^c^c�]h[^a�zt�^aøbji�c^ef]hg^ijaHk`o\bj[VaH�yoqk8aø�fbuoN��rtoxry] �hb }oXln]hefkV|^iuoxry]NzH]hefefa¡rfrtoqk8aH[sryaaH_�oq[poqijbj�H�Foxry]^��bj[^_^b���bj_^|poq[^_^]Vzt�^buoqeyoqk8aH[sryaBija�bj[�ln]hefk`oq�Hbj]h[^b�oV_^bj�fc�]h�fbj�Hbj]h[^a�c�aHe�]qrfryaH[^aHef[^aBiuo�f]hij|^�Hbj]h[^aq�

�do�l�oq�fa �`zH]h[^�fbj��rya8[^aHiCcpoq�f�yo���hbj]�_poqi<c^ef]hg^ijaHk`o�efaFoqija ; �p�fbjzH]^��zt�^bjk8bjzH]^�dbj[ �ha �h[^aHefbj��rybjzH]^�aHzH]h[^]hk8bjzH]^��ÓFÓFÓR?2oq_ |^[ c^ef]hg^ijaHk`o�k`oxryaHk`oxrybjzH]^� k8aH_^buoq[srya�i�� bj[^_^b���bj_^|poq�Hbj]h[^a�_^b\efaHiuoq�Hbj]h[^bij]�hbjzH]x�®k`oxryaHk`oxrybjzt�^a ; aH~�|poq�Hbj]h[^b�oqi*�haHg^efbjzt�^aq��aH~�|poq�Hbj]h[^b»_^b5AmaHefaH[^�Hbuoqijb��h_^bj�faH~�|poq�Hbj]h[^b��pÓFÓFÓ ?<zt�^azH]hijija �h�^bj[^]�ryeyo6ij]hef]�ija2bj[�ln]hefk`oq�Hbj]h[^bK[^]qryaq�KefaHiuoxryb��ha8oqiCc^ef]hg^ijaHk`o6bj[aH�yoqk8aq�Ka8iuo6�f]hij|^�Hbj]h[^aq�Ô b�[^]qrybmzt�^a�bj[:~�|^aH��rto�l�oq�faB_^a¡�h]h[^]2aH�f�faHefa�bj[^_^b���bj_^|poxryaNija�~�|poq[sryb�rx}oVzt�^a��fb�efb�rybjaH[^aN_^aH�fzHefb��qoq[^]aH�yoq|^��ryb��qoqk8aH[srya bjihc^ef]hg^ijaHk`o ; b�Æs§q­�Ãq_^aHi�c^ef]hg^ijaHk`o@?�a ~�|^aHijijaCzt�^a �f]h[^]Noq�f�f|^[srya£zH]hk8aø¨3¦xª¬«hÉHÃ�¦xÄm¢_^aHi�c^ef]hg^ijaHk`oV��ryaH�f�f]^�

ÖP×KØfÙFÚ�Û^Ü�Ý2Þqß��sß#àFá�â ã¡äxå�áçæ3è�é�á»ý©äxú�å �xú¥é�é úBâ á ÷ á äxæ3é�áçâ�ú�á ä2ùqãHå�á %�á ã¡ä©èXã¡é�á % %�ã¡äHû ú¥÷ è�#�â ã¡ä�ý©ä+) èfånû�é�è�üøá ûE'ú �qå�å�ú�è÷*) ú¥÷ û�é úK÷ á �qè�é ú©õpàFá -Fý©ã¡÷ èdæ3è û�è�é�üøá äxú¥é�è�÷çú�þÿã¡é�ü�ú#ú¡å�å�ý©äHû ú#æ©ú¥÷ ÷çú#å �xú¥é�é ú )Hýxú¥äxæ3ãCú¥÷ ÷*) èfånû�é�è�üøá ûE'úK÷ á �qè�é úU'è�ú¥ù©ù©÷ áçâ�útû úý©äxúøþÿã¡é %fú N�æ3è��xèfå�å�á ã¡ä©è<è�÷çú¡ånû�áçâ�ú�æ3è�÷ ÷çúXå �xú¥é�é ú�P�#xú -¡è�äxæ3ã�úXæ3áçå�ùqãHå�á %�á ã¡ä©è<÷ è å�è �¡ý©è�äHû�á�á ä3þÿã¡é�ü�ú&%�á ã¡ä©á :�÷ ý©ä���'©è % %fúè<å�è %�á ã¡ä©è<æ3è�÷ ÷çú�å �xú¥é�é ú�#Hü�útû�è�é�áçú¥÷ è<æ3á�â ý©áZ'è<â ãHånû�á û�ý©á û úø÷çúøå �xú¥é�é ú�#¡þÿã¡é %fúøú¥ù©ù©÷ áçâ�útû úøú¥÷ ÷*) èfånû�é�è�üøá ûE'ú ÷ á �qè�é ú©õàFá"-Fý©ã¡÷ èXâ ãHånû�é�ý©á é�è�ý©ä�üøã3æ3è�÷ ÷ ã�ü�útû�è�ü�útû�áçâ ãBùqè�éCû ú¥÷ èXù©é�ã��©÷ è�ü�ú©õBàFè\å�ámå�ý©ù©ùqã¡ä©è\â('©èX÷çúBå�è %�á ã¡ä©è\æ3è�÷ ÷çúå �xú¥é�é ú<å�áçú<ù©áçâ�â ã¡÷çú<é�áçå�ùqè û�û�ã ú¥÷ ÷çúCå�ýxú<÷ ý©ä���'©è % %fú�#Fâ áqå�áqé�áçâ ã¡äxæ3ýxâ è�ú¡æøý©äXù©é�ã��©÷ è�ü�ú �©áçæ3á üøè�äxå�á ã¡äxú¥÷ è¡õpà3â è�÷ û�ã ý©äå�áçånû�è�ü�ú<æ3áxé�á þÿè�é�á üøè�äHû�ãCâ�ú¥é�û�èfå�áçú¥ä©ã#ã¡é�û�ã��¡ã¡äxú¥÷ è F�ê��#Hâ ã¡ä�÷*) ã¡é�á �¡á ä©è�â ã¡á äxâ áçæ3è�äHû�è�â ã¡äø÷*) èfånû�é�è�üøá ûE'ú �qå�å�ú�#¡÷*) ú¡å�å�è�êû ú¥÷ è�â('©èd÷çúCå �xú¥é�é ú �¡áçú¡â�â áçú<å�ýXæ3áxèfå�å�ã<è�÷*) ú¡å�å�è��£ã¡é�á è�äHû útû�ã#ä©è�÷ ÷çúCæ3á é�è %�á ã¡ä©è�æ3è�÷ ÷çú#þÿã¡é %fú<ú¥ù©ù©÷ áçâ�útû ú<ú¥÷ ÷*) èfånû�é�è�üøá ûE'ú÷ á �qè�é ú N! �á �¡ý©é ú�ì¡õ ïQP�#�å�á 'xú�â('©èø÷çú�æ3è��xèfå�å�á ã¡ä©èøè�÷çú¡ånû�áçâ�úG'èø÷çú�å�ã¡÷ ý�%�á ã¡ä©èøæ3è�÷�å�è �¡ý©è�äHû�è£ù©é�ã��©÷ è�ü�ú�æ3á 1hè�é�è�ä�%�áçú¥÷ è

Page 6: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

��=

�� ���� ���� ��º

��

� �

� �

� b*�h|^eyo0�h�>= ����a�paH�f�fbj]h[^a£aHiuoq��rybjzFo\_^b�|^[poN�fgpoqefeyo���\oqc^c^efaH�faH[srtoq�Hbj]h[^a£_^aHijiuo\�fgpoqefeyo\c^efbjk`oX_^aHijiuo�paH�f�fbj]h[^aq�pbj[6|^[�efb�lnaHefbjk8aH[sry]VzFoqe�ryaH�fbuoq[^]^�

Nÿüøã3æ3è�÷ ÷ ã . P:��I I Nÿê�P

Nnì � � I Nÿê�P � P 2�� � ë�� N�� ! ê�P���� N�í�P»ë�í� I N�í�P»ë6í

æ3ãE-¡è � 'è�ý©äxúKâ ãHånû ú¥äHû�è�æ3á ùqè�äxæ3è�äHû�è�æ©ú¥÷Fü�útû�è�é�áçú¥÷ è Nÿüøã3æ3ý©÷ ã#æ3áHû�é ú&%�á ã¡ä©è�ãKæ3á��»ã¡ý©ä�� P�# � á ÷Füøã¡üøè�äHû�ã#æ3áFá ä©è�é %�áçúæ3è�÷ ÷çú�å�è %�á ã¡ä©èXû�é ú¡åP-¡è�é å�ú¥÷ è�æ3è�÷ ÷çúYå �xú¥é�é ú�#���÷çúB÷ ý©ä���'©è % %fúYæ3è�÷ ÷çúYå �xú¥é�é ú�# � ÷çú�þÿã¡é %fú�ú¥ù©ù©÷ áçâ�útû úYú¥÷ ÷*) èfånû�é�è�üøá ûE'ú÷ á �qè�é ú©õ^ö�èdâ ã¡äxæ3á %�á ã¡ä©á� N�í�P»ë�í#è ��I N�í�P»ë6í#á äxæ3áçâ�ú¥ä©ãCâ('©è�÷ ãCå�ùqãHånû ú¥üøè�äHû�ã#èd÷çú#ùqè�äxæ3è�ä�%fúCæ3è�÷ ÷*) èfånû�é�è�üøá ûE'ú �qå�å�úæ3è�÷ ÷çú\å �xú¥é�é úXå�ã¡ä©ãXäFý©÷ ÷ ájõ àFá^ä©ã¥û�ápâ('©è � # � #�� # � è£êVå�ã¡ä©ã �¡é ú¥äxæ3è % %�èCä©ã¥û�è èøå�ã¡ä©ãNæ3è û�û�è ����K*� � � �MI1P�; Kpæ3è�÷üøã3æ3è�÷ ÷ ã ü�útû�è�ü�útû�áçâ ã*VH÷çúU)Hýxú¥äHû�á ûE'ú �#3æ©ú æ3è û�è�é�üøá äxú¥é�è�# 'è�÷çú å�ã¡÷ ý�%�á ã¡ä©èKæ3è�÷qüøã3æ3è�÷ ÷ ã èfæ0'è�æ3è�ä©ã¡üøá äxútû ú ����KM>�� �>1; K P�; KjõàFá�ãHå�å�è�é -Fáhâ('©è#ä©è�÷ ÷çú�â ãHånû�é�ý�%�á ã¡ä©è#æ3áhû ú¥÷ è<üøã3æ3è�÷ ÷ ã�å�ã¡ä©ãøånû útû�á�á äHû�é�ã3æ3ã¥û�û�á�æ3è �¡÷ ásè�é�é�ã¡é�ásæ3ãE-Fý3û�á�ú¥÷hþ�útû�û�ã�â('©èå�ásû�é ú¡å�â ý©é ú£ä©ã¡ä�å�ã¡÷ ãø÷çúXå�è %�á ã¡ä©èCæ3è�÷ ÷çú�å �xú¥é�é ú�#3ü�úXú¥äxâ('©è<÷çú£û�ã¡é å�á ã¡ä©è�#©÷çú£þÿã¡é %fú�æ3á �¡é ú -Fá ûE'ú�#3è û â¥õ��dä+) ý©÷ û�è�é�á ã¡é�èå�è�üøù©÷ á �qâ�ú&%�á ã¡ä©èKæ3è�÷hù©é�ã��©÷ è�ü�ú£å�áhù©ýQ'ã£ã¥û�û�è�ä©è�é�è#å�ý©ù©ùqã¡ä©è�äxæ3ãøâ('©èKá ÷sùqèfå�ãøú¥ù©ù©÷ áçâ�útû�ã£å�áçú���ù©áçâ�â ã¡÷ ã��<èD)Hý©á äxæ3áh÷çúùqè�äxæ3è�ä�%fú ��I Nÿê�P�å�áçú�û ú¥÷ è£â('©è ��I Nÿê�P�� � �ì¡õ �Èä�û ú¥÷^â�ú¡å�ã\å�á$'xúXá ÷på�è �¡ý©è�äHû�è ù©é�ã��©÷ è�ü�úXæ3á 1hè�é�è�ä�%�áçú¥÷ è(Nÿüøã3æ3è�÷ ÷ ãc P:

� I I Nÿê�P�ë�� N�� ! ê�P���� N�í�P»ë�í� I N�í�P»ë6í£î

W�ý©èfånû ) ý©÷ û�á ü�ú�å�è�üøù©÷ á �qâ�ú&%�á ã¡ä©è^þ�ú¡â á ÷ á û ú�á ÷¡â�ú¥÷çâ ã¡÷ ã�æ3è�÷ ÷çú�å�ã¡÷ ý�%�á ã¡ä©è�#fü�ú�#fãE-&-Fáçú¥üøè�äHû�è�#�á äHû�é�ã3æ3ýxâ èpá ä#èfå�å�ú�ý©÷ û�è�é�á ã¡é�áè�é�é�ã¡é�ájõ!�Èä �á �¡ý©é ú�ì¡õ ôXå�ã¡ä©ã�é�á ùqã¡é�û útû�á�á$�¡é ú �qâ á�æ3è�÷ ÷ è å�ã¡÷ ý�%�á ã¡ä©á�é�è�÷çútû�á -¡èCú¥á�üøã3æ3è�÷ ÷ á . è c�ùqè�é�ý©äxúXå �xú¥é�é ú�æ3áú¡â�â áçú¥á ã N � ë>ï¥í¡í"$#&% T ��ë>ï¥í¡íKM8ìfí')(�*+% N T T-,. �0P,PDNXæ3á�÷ ý©ä���'©è % %fú/��ë ìfí¡íNüøü�èXå�è %�á ã¡ä©èXæ3á»é ú&���¡á ã0 ë ì¡î ó�üøü #dú¥÷ -tú¥é�áçú¥é�èBæ3è�÷ ÷çú�þÿã¡é %fú � N � ë ìfí 3 # < ó&,hìfí 3 # < ìfí&,hìfí 3 # < ìyó1,hìfí 3 # P ' àFá�ä©ã¥û�á�â('©è�ùqè�éN¥ì2"$#�ëDìfí�h3#�ë?ìfí�h)(�*4,/T5%6. �'fàFèX÷çúBå�è %�á ã¡ä©è£û�é ú¡åP-¡è�é å�ú¥÷ èøæ3è�÷ ÷çúBå �xú¥é�é úG'èXâ á é â ã¡÷çú¥é�èøæ3á�é ú&���¡á ã 0 #�á ÷mâ è�äHû�é�ãBæ3á�ü�ú¡å�å�ú 'èXú¥÷�â è�äHû�é�ãBæ3è�÷ ÷çúâ á é â ã¡ä3þÿè�é�è�ä�%fúCèfæ\á ÷süøã¡üøè�äHû�ã�æ3ásá ä©è�é %�áçú£÷ ý©ä��¡ãøâ áçú¡å�â ý©ä�ú¡å�å�è#â('©è �¡áçú¡â èKä©è�÷ ÷çú£å�è %�á ã¡ä©èKû�é ú¡åP-¡è�é å�ú¥÷ èdè#ùxú¡å�å�ú¥äHû�èùqè�é�á ÷�â è�äHû�é�ã #O'è � ë�7 3 0 3¡õ

Page 7: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# �=�

� ë�ìfí�3)#Ñá �¡é ú �qâ ámé�è�÷çútû�á -Fá�ú¥÷ ÷ è�æ3ý©è�å�ã¡÷ ý�%�á ã¡ä©á�å�ã¡ä©ãYù©é útû�áçâ�ú¥üøè�äHû�èNâ ã¡á äxâ áçæ3è�äHû�á*#»ùqè�é � ë�ó ,xìfí�32#Ñèfå�å�áæ3á 1hè�é�áçå�â ã¡ä©ã\÷ è ���¡è�é�üøè�äHû�è�#qùqè�é � ë�ìfí ,Fìfí�3)#�û ú¥÷ èøæ3á 1hè�é�è�ä�%fú\æ3á -¡è�äHû ú\å�è�äxå�á �©á ÷ è Nÿ÷ è£ùqãHå�á %�á ã¡ä©á^æ3è�÷ ÷*) èfånû�é�è�üøã÷ á �qè�é�ã�æ3è�÷ ÷çúNå �xú¥é�é ú\å�èfâ ã¡äxæ3ãNá ÷^üøã3æ3è�÷ ÷ ã . èfæ�á ÷püøã3æ3è�÷ ÷ ãQc�æ3áçånû ú¥ä©ã�â á é â�úYìfíXüøü P�#�ùqè�é � ë>ìyó$,3ìfí�3)#�÷ èæ3ý©èCå�ã¡÷ ý�%�á ã¡ä©á�å�á�æ3áçå�â ãHånû ú¥ä©ãøå�á �¡ä©á �qâ�útû�á -tú¥üøè�äHû�è¡õ Ö

®[ �haH[^aHeyoqijaq��oq[^zt�^aY�fa��fbdefbjaH�fzHa�o2_^a¡ryaHefk8bj[poqefa�|^[�� aH�fc^efaH�f�fbj]h[^aVoq[poqijb�rybjzFoV_^aHijiuo��f]hij|^�Hbj]h[^a_^b�|^[�c^ef]hg^ijaHk`oNk`oxryaHk`oxrybjzH]^�©�fc�aH�f�f]Y[^]h[�}aøc�]h�f�fbjg^bjija |�rybjijbj�H�FoqefiuoXc�aHe#zFoqijzH]hiuoqefa bji��qoqij]hefa£_^aHijiuo�f]hij|^�Hbj]h[^a�bj[|^[bj[^�fbjaHk8a2oq�f�fa �h[poxry]w_^b<c^|^[sryb��D�£b }]�oqzHzFoq_^aq�#oq_aH�faHk8c^bj]^�dc�aHe�bji<k8]�_^aHijij] 3_^aHijiuoø_^a�paH�f�fbj]h[^a aHiuoq��rybjzFo�_^bs|^[poX�fgpoqefeyo � �©]^�xc^by}|B�faHk8c^ijbjzHaHk8aH[sryaq�xc�aHediuoø�faHefbja������� ; ���� ��2?¥�©iuoln|^[^�Hbj]h[^a�bj[srya �heyoqija���� ; �fbj[�������?����¥��iuo�ln|^[^�Hbj]h[^a�ij]�<�d��a¡ryzq� �doYl�oq�fa =�zH]h[^�fbj��rya�[^aHimcpoq�f�yoqefaB_poqik8]�_^aHijij]øk`oxryaHk`oxrybjzH]XoqipÅ`¦FÆ�¢¡ªÈªj¦\Äp«�Å`¢¡¤¥Ã�Ët¦x�xzHbj]^}a oq_Y|^[Ycpoqe�rybjzH]hiuoqefa<k8]�_^aHijij]øk`oxryaHk`oxrybjzH]�zt�^azH]hbj[s�h]hi*�haX|^[`[�|^k8aHef] 6CÄpÃÈ­®¦�_^b»[�|^k8aHefb»efaFoqijbpaXefaHiuoq�Hbj]h[^b^k`oxryaHk`oxrybjzt�^a�ryeyo�~�|^aH��ryb���zFoqijzH]hiuoqg^bjijbzH]h[�|^[�[�|^k8aHef] 6CÄpÃÈ­®¦�_^b�]hc�aHeyoq�Hbj]h[^b�oqefb�ryk8a¡rybjzt�^aq�

ÖP×KØfÙFÚ�Û^Ü�Ý2Þqß���ß#àFá�â ã¡äxå�áçæ3è�é�ás÷ ã�åP-Fá ÷ ý©ù©ùqãøá äBå�è�é�á è<æ3á���ú¡â#ö�ú¥ý©é�á ä�æ3è�÷ ÷çú þÿý©ä�%�á ã¡ä©è �"!�:� ! ë$#%&(' d

ê &)+* ëDì ��ê � ê��ï * � ê%2ô * ��î�î�î�î Nnì¡õ ô�PàFèBå�ádå�ãHånû�á û�ý©áçå�â è�ú¥÷ ÷çú�å�è�é�á èN÷çú�å�ã¡üøü�ú�æ3è�á�ù©é�á üøá ) û�è�é�üøá ä©ádå�á�ã¥û�û�á è�ä©è�ý©ä+) èfå�ù©é�èfå�å�á ã¡ä©è -tú¥÷ ý3û ú&�©á ÷ èBâ ã¡ä8ý©ääFý©üøè�é�ã �xä©á û�ãXæ3á�ã¡ùqè�é ú&%�á ã¡ä©á�ú¥é�á û�üøè û�áçâ('©è¡õ-,�útû�ý©é ú¥÷ üøè�äHû�è#û ú¥÷ è<å�ãHånû�á û�ý�%�á ã¡ä©èCâ ã¡üøùqã¡é�û ú ÷*) á äHû�é�ã3æ3ý�%�á ã¡ä©èCæ3á�ý©äè�é�é�ã¡é�è�#Hæ3è û�û�ã " FF.>�F " � ��K F.>�I�C(�1< " I KM> ��I� �MK*�5CL> #xâ('©èKæ3á ùqè�äxæ3è<æ©ú¥÷ ÷çú å�â è�÷ û ú æ3á ) õ � áP'ýXù©é�èfâ áçå�ú¥üøè�äHû�è�#Fæ©útû�ãøâ('©è

� ! ëDì ��ê � ê��ï * ��î�î�î � ê &)+* � 0 & Nÿê�P < Nnì¡õ ñ Pæ3ãE-¡è 0 & Nÿê�P4'è á ÷^é�èfånû�ãNæ3á^á äxæ3áçâ è ) æ3è�÷ ÷çúNå�è�é�á è£â ã¡äxå�áçæ3è�é útû ú�#hå�á$'xúNâ('©è 0 & Nÿê�P�é ú¥ù©ù©é�èfå�è�äHû ú�÷*) è�é�é�ã¡é�è æ3á�û�é�ã¡ä�Yâ�ú¥üøè�äHû�ãBú¥äxú¥÷ á û�áçâ ãxõ . )Hý©èfånû�ã�ù©ý©äHû�ã �©áçå�ã��¡äxúNúE1hé�ã¡äHû ú¥é�è£á ÷»ù©é�ã��©÷ è�ü�úNæ3á�ånû�á ü�ú¥é�èø÷*) è�é�é�ã¡é�èøæ3ápû�é�ã¡äxâ�ú¥üøè�äHû�ãú¥äxú¥÷ á û�áçâ ãxõ��Èä )Hý©èfånû�ã�â�ú¡å�ãøå�á+'xú�:

0 & Nÿê�P�ë.��/ ê & ; �N ) ��ìRP * <æ3ãE-¡è10 ú¥ù©ùxú¥é�û�á è�ä©èKú¡æ�ý©ä\ã¡ù©ùqã¡é�û�ý©ä©ã á äHû�è�é -tú¥÷ ÷ ã£æ3è�÷qû�á ùqã�2 !13 < 354 VFæ3ý©ä*)Hý©è�#3ý©äxú ånû�á ü�ú£â�ú¥÷çâ ã¡÷çú&�©á ÷ è�æ3è�÷ ÷*) è�é�é�ã¡é�èæ3ásû�é�ã¡äxâ�ú¥üøè�äHû�ãøú¥äxú¥÷ á û�áçâ ã 'è#÷çú�å�è �¡ý©è�äHû�èQ:

� 0 & Nÿê�P/� E �76 � ê � & ; �N ) ��ìRP * î Nnì¡õ óQP8 �Èä )Hý©èfånû�ã�â�ú¡å�ãø÷çúøå�ã¡÷ ý�%�á ã¡ä©è�Nÿè�#*)Hý©á äxæ3á*#©÷çú£þÿã¡é�ü�ú£æ3è�÷ ÷çú�å �xú¥é�é ú�P 'xú£÷*) èfå�ù©é�èfå�å�á ã¡ä©èKùxú¥é ú¥üøè û�é�áçâ�ú�:

êIë 9 ï � ��

N;: â ãHå�< d ! : â ãHå�< d ! â ãHå�<�P�-ë 9 � �ï �>= 7 � �d

â ãHå�<5?5<@ â ãHå�< d ! â ãHå�<æ3ãE-¡èA<£ëB< NDC P 'èK÷*) ú¥ä��¡ã¡÷ ãCþÿã¡é�ü�útû�ã£æ©ú¥÷ ÷çúCû ú¥ä��¡è�äHû�è#ú¥÷ ÷çú£å �xú¥é�é ú<ä©è�÷ �¡è�ä©è�é�áçâ ã ù©ý©äHû�ãøú£æ3áçånû ú¥ä�%fú�C�æ©ú¥÷ ÷*) èfånû�é�è�üøã�qå�å�útû�ã #3èCæ©ú¥÷ ÷*) ú¡å�å�è �hõ . ÷ ÷*) èfånû�é�è�üøã��qå�å�útû�ã NDC�ë6í�P�#�< N�í�P�ë 7 � #©üøè�äHû�é�èCú¥÷ ÷*) èfånû�é�è�üøãø÷ á �qè�é�ã NDC�ë � #©÷ ý©ä���'©è % %fúæ3è�÷ ÷çúøå �xú¥é�é ú�P�å�á�ùqã¡ä©èE< N���P�ëB< d õ

Page 8: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

� "

0 20 40 60 80 100−5

−4

−3

−2

−1

0

x (mm)

y (

mm

)

F=104

modello Amodello B

0 20 40 60 80 100−25

−20

−15

−10

−5

0

x (mm)

y (

mm

)

F=5*104

modello Amodello B

0 20 40 60 80 100−50

−40

−30

−20

−10

0

x (mm)

y (

mm

)

F=10*104

modello Amodello B

0 20 40 60 80 100−120

−100

−80

−60

−40

−20

0

x (mm)

y (

mm

)

F=15*104

modello Amodello B

� b*�h|^eyo �h� �$�C��a�paH�f�fbj]h[^a�aHiuoq��rybjzFoV_^b�|^[po2�fgpoqefeyo�� Ô ]hij|^�Hbj]h[^b�efaHiuoxryb��ha�oqb�k8]�_^aHijijbX3�aB�N�

Page 9: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# ���

àFá�ãHå�å�è�é -Fá�â ã¡ü ý©ä*)Hý©è<â('©è<÷çúøü�ú&���¡á ã¡é ú&%�á ã¡ä©è Nnì¡õ óQPmé�áçå�ã¡÷ -¡è<å�ã¡÷ ã�ú¥ù©ùxú¥é�è�äHû�è�üøè�äHû�è#á ÷�ù©é�ã��©÷ è�ü�úøæ3è�÷ ÷çú�ånû�á ü�ú�æ3á0 & Nÿê�P V3á ä3þ�útû�û�áqá ÷qù©é�ã��©÷ è�ü�úCå�áqå�ùqãHånû úCú¥÷ ÷çúCæ3è û�è�é�üøá äxú&%�á ã¡ä©è�æ3á 3 #Hã üøè �¡÷ á ã æ3áxý©ä -tú¥÷ ã¡é�è�æ3á �Cùqè�é�â ý©á�� 6�� ! � ����� & ; �� �ä©ã¡ä�å�áçú ��èfâ�â èfå�å�á -tú¥üøè�äHû�è �¡é ú¥äxæ3è2�©õ Ö

ÖP×KØfÙFÚ�Û^Ü�Ý2Þqß��sß#àFáçú���Nÿê�P�ý©äxú�þÿý©ä�%�á ã¡ä©è�é�èfú¥÷ èmæ3á&-tú¥é�áçú&�©á ÷ è�é�èfú¥÷ è�#tæ3è�é�á -tú&�©á ÷ è¡õ^àFè�å�áFå�ãHånû�á û�ý©áçå�â è�ú¥÷ ÷çú�æ3è�é�á -tútû úæ3á���Nÿê�Pmá ÷�é ú¥ù©ùqã¡é�û�ã£á äxâ é�è�üøè�äHû ú¥÷ èQ: ?��?¡ê Nÿê�P J ��Nÿê ���%P ! ��Nÿê�P�

<

æ3ãE-¡è�� 'è�å�â è�÷ û�ãKã¡ù©ùqã¡é�û�ý©äxú¥üøè�äHû�è��&#¡å�á©ã¥û�û�á è�ä©è�ý©ä+) èfå�ù©é�èfå�å�á ã¡ä©è"-tú¥÷ ý3û ú&�©á ÷ è�â ã¡ä£ý©äøäFý©üøè�é�ã �xä©á û�ã<æ3á©ã¡ùqè�é ú&%�á ã¡ä©áú¥é�á û�üøè û�áçâ('©è�#^å�ý©ù©ùqãHånû�ã #^ãE-&-Fáçú¥üøè�äHû�è�#�â('©èXá ÷�â�ú¥÷çâ ã¡÷ ã�æ3á���Nÿê�P<á ä2ý©ä�ù©ý©äHû�ã�ùqãHå�å�úBèfå�å�è�é�è�è1hè û�û�ýxútû�ãYâ ã¡ä�ý©ääFý©üøè�é�ã �xä©á û�ã�æ3á�ã¡ùqè�é ú&%�á ã¡ä©á�ú¥é�á û�üøè û�áçâ('©è¡õRC '©áçú¥é ú¥üøè�ä3û�èN÷*) è�é�é�ã¡é�èNæ3á�û�é�ã¡äxâ�ú¥üøè�äHû�ã�ú¥äxú¥÷ á û�áçâ ã #»ã " FF.>�F "�� �� �>G*C F " K*� ILI �I��*>�I" h&#xæ3á ùqè�äxæ3èCæ©ú���è�#3ùqã¡áçâ(',+è

÷ á ü��� d

��Nÿê ���%P ! ��Nÿê�P� ë ?��?¡ê Nÿê�P <

÷*) è�é�é�ã¡é�è#æ3áhû�é�ã¡äxâ�ú¥üøè�äHû�ã�ú¥äxú¥÷ á û�áçâ ã û�è�äxæ3èCú�íøú¥÷hû�è�äxæ3è�é�è<æ3á��Búøí©õ Ö

�dol�oq�fa ��zH]h[^�fbj��ryaD[^aHiji�� bj[^_^b���bj_^|poqefa�|^[�§qª 4�¦x¤¥ÃÈ­�Å`¦�efbj�f]hij|�ryb��h]c�aHe`bjiXk8]�_^aHijij]�[�|^k8aHefbÿ�zH]^� zHbj]^}a:|^[po��faH~�|^aH[^�Fo �p[^b�rto?_^bø]hc�aHeyoq�Hbj]h[^bøoqefb�ryk8a¡rybjzt�^a:zt�^aq�£�fa6aH�fa �h|^b�ryaq� zH]h[^�faH[sry]h[^]�_^bzFoqijzH]hiuoqefaq�^o�cpoqe�rybjefaB_poqb�_poxryb�_^b�bj[^c^|�rF�^b�_poxryb�_^b�]h|�ryc^|�rF�

®iøzH]h[^zHa¡rfry]�_^b�oqi*�h]hefb�ryk8]�}a:bj[^_^bj�f�f]hij|^g^bjijk8aH[srya8ija �soxry]�oqi£�f|^]�aH�faHzH|�ry]hefaq�£[^aHiø�faH[^�f]zt�^aija�]hc�aHeyoq�Hbj]h[^bøzt�^a���b�zH]hk8cpoqbj]h[^]?_^a¡�h]h[^]�aH�f�faHefa�zt�^buoqeyoqk8aH[srya��fc�aHzHb��pzFoxrya�aH_PaH�fa �h|^bjg^bjijb£bj[efaHiuoq�Hbj]h[^a6oqijija�c�]h�f�fbjg^bjijb�rx}ow_^aHiji�� aH�faHzH|�ry]hefaq� {\aHiX[^]h��ryef]zFoq�f]�i�� aH�faHzH|�ry]hefa�_^b\oqi*�h]hefb�ryk8b\}a�|^[zFoqijzH]hiuoxry]hefaBzH]h[w|^[�¨¡Ãj¨¡­®¢¡Å8§6§q¤¥ÃÈ­�Å`¢¡­�Ã�Ët¦:§YÇ�¤y¢tË¡Ãj¨¡Ã�¦xÄm¢ 6CÄpÃÈ­ §q�»zHbj]^}a�zt�^a�]hc�aHeyo8�f]hij]2�f|D[�|^k8aHefbeyoqc^c^efaH�faH[srtoxrybdk8aH_^buoq[sryaB|^[�[�|^k8aHef]��p[^b�ry]2_^b�zHb�lnefaB�fb*�h[^b��pzFoxryb��haq�

� |^aH��rtoBl�oq�fa�_^aHipc^ef]�zHaH�f�f]Yefbj�f]hij|�ryb��h]�zH]hk8c�]he�rto�i�� bj[sryef]�_^|^�Hbj]h[^a£_^b^|^[8aHefef]hefaq�s_^a¡rfry]:¢¡¤¥¤y¦x¤y¢ÆqÃK¤y¦x«�Ä�Æ�¦��X��_^]3��|�ry]2c^ef]hc^efbj]Voqiji�� |�rybjijbj�H�H]�_^bm|^[��fbj��ryaHk`oVoqefb�ryk8a¡rybjzH]�oVc^efaHzHbj�fbj]h[^aK�p[^b�rto��_���^o~�|^aH��ry]8rybjc�]�_^b�aHefef]hefa�zt�^a��yoqe©}o2_^aH_^bjzFoxry]8oqk8c^bj]��fcpoq�Hbj]2bj[6�fa �h|^b�ry]^��do�l�oq�fa " zH]h[^�fbj��rya2[^aHiji�� bjk8c^ijaHk8aH[srtoq�Hbj]h[^a�_^aHiji�� oqi*�h]hefb�ryk8]8bj[�|^[^]6�fc�aHzHb��pzH]woqkVg^bjaH[srya2_^b

zFoqijzH]hij] ; ¨3¦�¥­�# §q¤y¢ø]VÇ�¤y¦R4q¤f§qÅ�Å8§�?¥�<_poxryb�_^bm]h|�ryc^|�rø_^aHimc^ef]�heyoqk8k`oB_^bjc�aH[^_^]h[^]V~�|^bj[^_^b�[^]h[��f]hij]�_poqiji�� oqi*�h]hefb�ryk8]^�xk`o�oq[^zt�^a_poqijijaXzFoqeyoxrfryaHefbj��rybjzt�^a\�fc�aHzHb��pzt�^a�_^aHiji�� oqkVg^bjaH[srya\_^b»zFoqijzH]hij]^��zH]hk8aq��oq_`aH�faHk8c^bj]^��b�zH]hk8c^bjiuoxry]hefb��bjipcpoqe�rybjzH]hiuoqefaø�fbj��ryaHk`oYoqefb�ryk8a¡rybjzH]^�ha¡ryzq� #_poxryb»_^bp]h|�ryc^|�r£_^aHi»c^ef]�heyoqk8k`oB�f]h[^]�oq�f�f|^[sryb�zH]hk8aiuo����f]hij|^�Hbj]h[^aH�2_^aHi�c^ef]hg^ijaHk`oVefaFoqijaB[^aHiji�� oqc^c^ef]�zHzHbj]�zH]hk8c^|�rtoq�Hbj]h[poqijaq�

� . û ú¥÷ è<ù©é�ã¡ùqãHå�á û�ãøå�á�-¡èfæ©ú ÷*) èfå�è�é â á %�á ã�ì�L�#3æ3è�÷! fì¡õ ìfí©õ ïh�,dè�÷ ÷çú�ä©ãHånû�é úVú¡â�â è %�á ã¡ä©èBý©ä�ù©é�ã��©÷ è�ü�úVü�útû�è�ü�útû�áçâ ã 'è�â�ú¥é útû�û�è�é�á % %fútû�ã�æ©ú¥÷�â ã¡äxâ è û�û�ã2æ3á�á ä��xä©á û�ã2èfæ:ý©äù©é�ã��©÷ è�ü�ú£äFý©üøè�é�áçâ ãXæ©ú¥÷ ÷çú �xä©á û�è % %fú Vxý©äBù©é�ã��©÷ è�ü�ú£ü�útû�è�ü�útû�áçâ ã�ù©ýQ'ã�èfå�å�è�é�è#á ä©ã¡÷ û�é�èCâ ã¡äHû�á äFý©ãøã�äFý©üøè�é ú&�©á ÷ è¡õ

" ú¥÷ -¡ã¡÷ û úNá ÷»ùxú¡å�å�ú&���¡á ã�æ©ú¥÷»üøã3æ3è�÷ ÷ ãBü�útû�è�ü�útû�áçâ ã�ú¥÷»üøã3æ3è�÷ ÷ ãBäFý©üøè�é�áçâ ã�é�áçâ('©á èfæ3è�ù©é�á ü�ú�á ÷»ùxú¡å�å�ú&���¡á ã�æ©ú�ý©äù©é�ã3â èfå�å�ã�â ã¡äHû�á äFý©ã�ú¡æNý©ä©ãXäFý©üøè�é ú&�©á ÷ è#èCùqã¡á�æ©ú()Hý©èfånû�ãXú¡æNý©äBù©é�ã3â èfå�å�ã �xä©á û�ã*Vxá ä�û ú¥÷�â�ú¡å�ã #xå�ùqèfå�å�ãø÷*) è�é�é�ã¡é�èú¡å�å�ã3â áçútû�ã£ú¥÷sù©é�á üøãøùxú¡å�å�ú&���¡á ã 'è<æ3è û�û�ãXæ3á � �>G*C F " K*� ILI �I��*>�I"dèD)Hý©è�÷ ÷ ã�ú¡å�å�ã3â áçútû�ã£ú¥÷�å�èfâ ã¡äxæ3ã 'è<æ3è û�û�ã�æ3á"K F.>�I�C(�$#< " I KM>���I� �MK*�5CL>tõ,W�ý©áså�ásýxå�è�éE'ú ÷çú£æ3è�ä©ã¡üøá äxú&%�á ã¡ä©è K F.>�I�C(�1< " I KM> ��I� �MK*�5CL>#ùqè�é�á äxæ3áçâ�ú¥é�èK÷*) è�é�é�ã¡é�èKâ ã¡üøù©÷ èfå�å�á -¡ãæ3ãE-Fý3û�ã�ú¥÷ ÷çú£å�ãHånû�á û�ý�%�á ã¡ä©è<æ3è�÷�üøã3æ3è�÷ ÷ ã�ü�útû�è�ü�útû�áçâ ã�â ã¡÷süøã3æ3è�÷ ÷ ãøäFý©üøè�é�áçâ ãxõ

Page 10: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

�=�

021�� ��J#;FJ�9#@BT ;PGVAC;¡9<TU@N9<;FW\; GIQ?A<@BW\;FJ#;H587D@��7?;¡9#G®ihryaHefk8bj[^aø¨¡Ãj¨¡­®¢¡Å8§�§q¤¥ÃÈ­�Å`¢¡­�Ã�Ët¦�§ Ç�¤y¢tË¡Ãj¨¡Ã�¦xÄm¢"6CÄpÃÈ­ § �fb�efb�lnaHefbj�fzHaCbj[ �haH[^aHeyoqija oqb�zHefb�ryaHefbs_^b�eyoqc^c^efa¥��faH[srtoq�Hbj]h[^aXbj[2k8aHk8]hefbuoN_^aHbp_poxrybp_^b�rybjc�]�[�|^k8aHefbjzH]�aH_`oqijijaø]hc�aHeyoq�Hbj]h[^b^aHijaHk8aH[srtoqefb ; _^aHiji�� |^[^b�rx}ooqefb�ryk8a¡rybjzFo@?C_^a2�p[^b�ryaB�f|w_^b�aH�f�fb��¬� Brybjc^b£_^bø_poxrybn�®�Y[�|^k8aHefbjzHb£aHijaHk8aH[srtoqefb£~�|^b�zH]h[^�fbj_^aHeyoxryb£�f]h[^]�bji rybjc�]�ÃÈÄp­®¢¡¤y¦`aH_>bji rybjc�]�¤y¢y§qªj¢� ¦F§q­�ÃÈÄ�4$�jÇp¦xÃÈÄp­ ; efaFoqija:l � c��B?y� � � c�]hbjzt���aq� zH]hk8a6�fbø�haH_^e©}obj[>�fa �h|^b�ry]^� �f]h[^]~�|^aHijijbøzt�^a6c�]h�f�f]h[^]�haH[^aHeyoqefaC�fb�ry|poq�Hbj]h[^bqzH]hk8c^|�rtoq�Hbj]h[poqijbqzt�^aCefbj_^|^zH]h[^]X]Noq[^[�|^ijiuoq[^]�i�� ox�8_poqg^bjijb�rx}o _^bh|^[�oqi*�h]hefb�ryk8][�|^k8aHefbjzH]^�

��� �� ������������������������� ���!� ��"$#%��&����'� #®iBzHefb�ryaHefbj]*_^b�eyoqc^c^efaH�faH[srtoq�Hbj]h[^a_^aHb�_poxrybB_^b�rybjc�]>bj[sryaHef]Õ}a�~�|^aHijij]>c�]h�fbj�Hbj]h[poqijaDbj[�gpoq�fa)(; (+*-, �!(+. � ?B� � ��Z\�f|poqijk8aH[srya/(PÏ =���zHbj]^}aV�fb#|�rybjijbj�H�Fo8iuo`eyoqc^c^efaH�faH[srtoq�Hbj]h[^a2g^bj[poqefbuo+B�oq_aH�faHk8c^bj] �

� � Ï � � ?� � Ï �O� � ?�=� � Ï ���O�O�O� � ?, � " � Ï ���Q�O���Q�O�O�O��� � ?

a�iuo�k8aHk8]hefbj�H�Foq�Hbj]h[^a�oF����bjaH[^aBc�]h[^aH[^_^]2bj[:|^[po�ij]�zFoq�Hbj]h[^a\iuo���ryefbj[ �soV_^bmg^b�rF��_po�_^aH��ryeyo��haHef�f]�fbj[^bj��ryeyo�� ®i�g^b�rXc^by}|6o��fbj[^bj��ryeyo�}aBefbj�faHe��qoxry]8oqiji�� bj[�ln]hefk`oq�Hbj]h[^a ; g^bj[poqefbuo@? _^aHi��fa �h[^]^��do\ij|^[ �h�^aH�H�FoK�p[^b�rto10p_^b�|^[poNij]�zFoq�Hbj]h[^aC_^b�k8aHk8]hefbuo�zH]hk8c�]he�rto\i�� aH�fbj��ryaH[^�FoN_^b�|^[Vk`oq�f�fbjk8]; aX_^bp|^[`k8bj[^bjk8]�?dbj[sryaHef]�eyoqc^c^efaH�faH[srtoqg^bjijaq� Ô bpzH]h[^�fbj_^aHefb���oq_`aH�faHk8c^bj]^�s|^[`�fbj��ryaHk`o�oqefb�ryk8a¡rybjzH]bj[sryaHef]�zH]h[20�Ï � � � B^�fb��po2oqijij]heyoYzt�^a�bji�[�|^k8aHef] � " � Ï ���O�O�O�O�O� � }aBeyoqc^c^efaH�faH[srtoqg^bjijaBa���bjaH[^ak8aHk8]hefbj�H�Foxry]V[^aHi�k8]�_^]��fa �h|^aH[srya��¬�43

5 � � � � � � �

®[ rtoqijaw�fbj��ryaHk`o�bji\c^by}| �heyoq[^_^aD[�|^k8aHef]�eyoqc^c^efaH�faH[srtoqg^bjijaq�\zt�^a?�yoqe©}o�_^a¡rfry]76 98h�d��}a ��= , � �c�]hbjzt��}aBiuoV�f|po2k8aHk8]hefbj�H�Foq�Hbj]h[^aV}a

�Md àFá�ù©é�è�äxæ3è�éE'úBá ä`â ã¡äxå�áçæ3è�é ú&%�á ã¡ä©èXå�ã¡÷ ã�÷çú�é ú¥ù©ù©é�èfå�è�äHû ú&%�á ã¡ä©èXæ3è�á�äFý©üøè�é�ámá ä8üøè�üøã¡é�áçú�Nÿé ú¥ù©ù©é�èfå�è�äHû ú&%�á ã¡ä©èá äHû�è�é�äxú�P®õ�ö�úVé ú¥ù©ù©é�èfå�è�äHû ú&%�á ã¡ä©è�æ3è�á�äFý©üøè�é�á�å�ý©á�æ3áçå�ùqãHå�á û�á -FáKæ3á �.f2F N -Fáçæ3è�ã #�ånû ú¥üøùxú¥äHû�è�#�è û â¥õ P�'èB÷çúVä©ã¡éPYü�ú¥÷ è£é ú¥ù©ù©é�èfå�è�äHû ú&%�á ã¡ä©è£æ3èfâ á ü�ú¥÷ è2V�ú¥÷ ÷ ã�ånû�èfå�å�ãNüøã3æ3ãBå�ápù©é�è�äxæ3è�é ú¥ä©ä©ã\á äVâ ã¡äxå�áçæ3è�é ú&%�á ã¡ä©è£å�ã¡÷ ãN÷ èøã¡ùqè�é ú&%�á ã¡ä©áú¥é�á û�üøè û�áçâ('©è#èfå�è �¡ý©á �©á ÷ á�æ©ú¥÷ ÷*) ý©ä©á ûE'úøú¥é�á û�üøè û�áçâ�ú©õ� � àFá�æ3è��xä©áçå�â è�K*� P5> � � ����KM>#÷*) á äxå�á è�üøèCæ3è�á@-tú¥÷ ã¡é�á�â('©è#ý©äBæ©útû�ãøù©ýQ'ãXú¡å�å�ý©üøè�é�è¡õ� �3�È÷hû�á ùqãXæ3á�æ©útû�ã CL>1< P "G�G`> EJU P U 'èKèfå�å�è�ä�%�áçú¥÷ üøè�äHû�è#é�áçâ ã¡äxæ3ýxâ á �©á ÷ è#ú )Hý©è�÷ ÷ ãøé�èfú¥÷ è�þnõ ù�õ #xúøâ ý©á�ú¥ù©ùxú¥é�û�è�ä��¡ã¡ä©ã÷çú£ùxú¥é�û�è#é�èfú¥÷ èKèfæNá ÷�â ãFè�:Xâ á è�äHû�è<æ3è�÷ ÷*) á üøü�ú&�¡á äxú¥é�á ãxõ� 24;�á äxæ3áçâ�ú£÷*) á äxå�á è�üøèCæ3è�á�äFý©üøè�é�ásäxútû�ý©é ú¥÷ ájõ� 3�,dè�÷ ÷çú£ü�ú&���¡á ã¡é�ùxú¥é�û�è<æ3è �¡÷ á�â�ú¥÷çâ ã¡÷çútû�ã¡é�á C�ë6ôHï < L¥ñ < ìyï�<©õ� N � è�émü�ú&���¡á ã¡é�èKâ('©áçú¥é�è % %fú�#Hùqè�é�é ú¥ù©ù©é�èfå�è�äHû ú¥é�èdá ÷ �©á ûdæ3è�÷�å�è �¡ä©ãøå�ã¡ä©ã ýxå�útû�ásá�å�á ü �qã¡÷ á �wè ! ú¥÷sùqãHånû�ãøæ3è�÷ ÷ èâ á þÿé�è �©á äxú¥é�á è<í£è�ì¡õ

Page 11: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# � ,

5 � � � � � � �

k8aH[sryefaøiuo\eyoqc^c^efaH�faH[srtoq�Hbj]h[^aøg^bj[poqefbuo\_^bX��= � � Ï ���O�O�O�O�O�O� � }a£zH]h��ryb�ry|^b�rto\_po � zHb�lnefa£aø[^]h[Vc^|�}]aH�f�faHefa£k8aHk8]hefbj�H�Foxrto�� Ô bsc^|�}]NzH]h[^zHij|^_^aHefaCzt�^a£|^[Ybj[sryaHef] *�� ����}a |^[Ybj[sryaHef]Y¤f§¥ÇsÇ�¤y¢t¨3¢¡Äp­ § �HÃȪj¢�fa�a��f]hij]2�fa

�� 6 98h�a�zt�^aYi�� bj[^�fbjaHk8a��`_^a �hijb�bj[sryaHefb�eyoqc^c^efaH�faH[srtoqg^bjijb<}aBi�� bj[^�fbjaHk8a �p[^b�ry]

Ò 6 98h�D?FÒ 6 98h� 5 ��?FÓ�Ó�Ó5?FÒG��?E�+?&��?FÓ�Ó�Ó5? 6 98h�8Ò ��? 6 98h��

_^]3�ha6 98h��Ï = ��� � Ò ��Ó

Ô aB *��:�#k`o <*��»�<oqijij]heyo [^]h[ }a:eyoqc^c^efaH�faH[srtoqg^bjija`aH_*|^[�a¡�haH[sry|poqija:ryaH[srtoxryb��h]D_^bk8aHk8]hefbj�H�Foq�Hbj]h[^a c^ef]3�h]�zFoBiuo��fb�ry|poq�Hbj]h[^a£aHzHzHaH�Hbj]h[poqija�_^a¡rfrto8¦*5q¢¡¤ � ¦5#6_^a �hijb�bj[sryaHefb�� ®[ �haH[^aHeyoqijai�� bj[^�fbjaHk8a��D}a�bj[^_^b���bj_^|poxry]V|^[^b��h]�zFoqk8aH[srya�_poqijiuoV�f]hiuoVij|^[ �h�^aH�H�Fo�0®�ø� �Ô |^bC_poxrybC_^b<rybjc�]�bj[sryaHef]��f]h[^]w_^a2�p[^b�rya8ija2]hc�aHeyoq�Hbj]h[^b �`§sÆsÆqÃ�ÉHÃ�¦xÄm¢tÊX¨3¦x­�­�¤f§3ÉHÃ�¦xÄm¢tÊNÅ`¦xª¬­�à Ç�ª¬ÃM�

Ëy§3ÉHÃ�¦xÄm¢�a?ÆqÃ653Ãj¨¡Ã�¦xÄm¢t���H�#_^aH[^]qrtoxrya`~�|^b£bj[�k8]�_^]w_^b��haHef�f]D_poqijija2zH]hefefbj�fc�]h[^_^aH[sryb£]hc�aHeyoq�Hbj]h[^bCbj[��� 5 � Ò � � � � � ®i#efbj�f|^i�rtoxry]��:_^b#|^[�� ]hc�aHeyoq�Hbj]h[^a � ; _^]3�ha��'��rto�c�aHe 5 ?FÒ ?�� ?(�O?N�fb_^bjzHaDÆ�¢ 6CÄpÃÈ­®¦2�fa�� *��»�#oqi�ryefbjk8aH[sryb#�fb _^bjzHa`zt�^a`aH�f�f]*}a8bj[^_^a2�p[^b�ry]wa8zt�^a:�fb\}a2�haHefb��pzFoxrtow|^[po�fb�ry|poq�Hbj]h[^aBaHzHzHaH�Hbj]h[poqijaB_^b�]3�haHe)�p]�� _^a �hijb�bj[sryaHefb��

ÖP×KØfÙFÚ�Û^Ü�Ý2Þqß���ß#àFáçú¥ä©ã�� T 3 êNë?ìfí¡í�# ) ë�ó¥í�#&T ë L¡í©õ»àFá+'xú�:� ) ��T��¡ë?ì¡ìfí���� T 3 ê

èD)Hý©á äxæ3á�á ÷�é�áçå�ý©÷ û útû�ã�æ3è�÷ ÷*) ã¡ùqè�é ú&%�á ã¡ä©è<ó¥í � L¡í 'è#á äxæ3è��xä©á û�ãxõ Ö

®[��haH[^aHeyoqijaq�p�fa �a0 "�f]h[^]8bj[sryaHefb�rtoqijbmzt�^aÐ Ð�� 6 98h�D?<Ð �Ð�� 6 98h� ?

� ' ��á äxæ3áçâ�ú£÷*) á äxå�á è�üøèCæ3è �¡÷ ásá äHû�è�é�á�é�è�÷çútû�á -Fájõ� 8 F�÷ û�é�è�ú¥÷ ÷çú<é ú¥ù©ù©é�èfå�è�äHû ú&%�á ã¡ä©è�ùqè�émå�è �¡ä©ãCèdüøã3æ3ý©÷ ã #�)Hý©áxý3û�á ÷ á % %fútû ú�#Hèfå�áçånû�ã¡ä©ã ú¥÷ û�é�èdé ú¥ù©ù©é�èfå�è�äHû ú&%�á ã¡ä©áxæ3è �¡÷ áá äHû�è�é�ápé�è�÷çútû�á -Fájõ �ÈäVùxú¥é�û�áçâ ã¡÷çú¥é�è�#�å�á»é�áçâ ã¡é æ©ú )Hý©è�÷ ÷çú�æ3è�÷ CL>1< P "J< " I KM> ���1; "�2 % 4 :<á ÷»ù©é�á üøã �©á û æ3á�å�á ä©áçånû�é ú 'xúú¥äxâ ã¡é úNá ÷�é�ý©ã¡÷ ã�æ3á �©á û£æ3ámå�è �¡ä©ã #�èfæVá�äFý©üøè�é�á�ä©ã¡ä�ä©è �Hútû�á -Fá 'xú¥ä©ä©ãB÷çú�ånû�èfå�å�ú\þÿã¡é�ü�úNþÿã¡é�ä©á û ú�æ©ú¥÷ ÷çú�é ú¥ù©ù©é�èYå�è�äHû ú&%�á ã¡ä©èXá ä`å�è �¡ä©ãYè\üøã3æ3ý©÷ ã #»üøè�äHû�é�è\á ÷,-tú¥÷ ã¡é�è�#^á ä8üøã3æ3ý©÷ ã #�æ3á�ý©ä8äFý©üøè�é�ã�ä©è �Hútû�á -¡ãYé ú¥ù©ù©é�èfå�è�äHû útû�ãBá äâ ã¡üøù©÷ è�üøè�äHû�ã\úXæ3ý©è£å�á�ã¥û�û�á è�ä©è á ä&-¡è�é�û�è�äxæ3ãXá ÷�-tú¥÷ ã¡é�è æ3á�ã��¡ä©á�å�á ä��¡ã¡÷ ã �©á û�N�â ã¡üøù©÷ è�üøè�äHû�ã\ú�ý©ä©ã P�#O-tú¥÷ ý3û ú¥äxæ3ã÷çúøé ú¥ù©ù©é�èfå�è�äHû ú&%�á ã¡ä©è#â ã¡üøè<á äHû�è�é�ãXå�è�ä�%fú�å�è �¡ä©ã #3èfæBú&���¡á ý©ä��¡è�äxæ3ã£ý©ä©ã\ú¥÷�é�áçå�ý©÷ û útû�ã*V©á äBèfå�å�ú£ùqè�é�û ú¥äHû�ã #©á ÷�û�á ùqãá äHû�è�é�ãøä©ã¡ä$'è<å�á üøüøè û�é�áçâ ãøé�áçå�ùqè û�û�ãXú¥÷ ÷ ã�%�è�é�ã #3èfå�å�è�äxæ3ã�� T�� ) ë ! � T 3 ê ! ì N!� T�� ) ë?üøá ä©á üøã�á äHû�è�é�ãøé ú¥ù©ù©é�èYå�è�äHû ú&�©á ÷ èRP®õ . æ�èfå�è�üøù©á ã�ä©è�÷ ÷çúNé ú¥ù©ù©é�èfånû ú&%�á ã¡ä©è£á ä CL>1< P "J< " I KM> � �1; " á ÷päFý©üøè�é�ãVìfí©ìfí � ë ! L �Md #�á ä3þ�útû�û�á :ìfí©ìfí<ë ! N�í©ìfí©ì ��ìRP»ë ! Njó ��ìRP»ë ! L©õ� �3�È÷�é�áçå�ý©÷ û útû�ã�æ3è�÷ ÷çúøæ3á -Fáçå�á ã¡ä©è#á äHû�è�é ú 'è#÷çú£ùxú¥é�û�è#á äHû�è�é úøæ3è�÷�)Hý©ã�%�á è�äHû�è¡õ

Page 12: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

� �

oqijij]heyo � ��� ��$ �fa ��$ ��+6 98h�

bj[^_^a2�p[^b�ry]��fa Ð ��$ �Ð . 6 98h��Ó ; �h� �@?

ÖP×KØfÙFÚ�Û^Ü�Ý2Þqß���ß �Èä�ý©ä�å�áçånû�è�ü�ú�ú¥é�á û�üøè û�áçâ ãøá äHû�è�é�ãXâ ã¡äQA�ëwï£è C�ë�ôHï£å�á�-Fý©ã¡÷ èCâ�ú¥÷çâ ã¡÷çú¥é�è )+* õ � è�é ) �Dìfôû ú¥÷ è<â�ú¥÷çâ ã¡÷ ã£ù©é�ãE-¡ã3â�úCãE-¡è�é��xã���á ä )Hýxú¥äHû�ãìfô * ��ï 2 � ! ìKë�� T 3 ê

N�å�ásé�áçâ ã¡é æ3á�â('©è#ý©ä©ã�æ3è�á�ôHï �©á û�úøæ3áçå�ùqãHå�á %�á ã¡ä©è#ùqè�éd÷çú£é ú¥ù©ù©é�èfå�è�äHû ú&%�á ã¡ä©èKæ3è�÷�äFý©üøè�é�ã 'è#é�áçå�è�é -tútû�ãøú¥÷�å�è �¡ä©ã P®õ Ö

ÖP×KØfÙFÚ�Û^Ü�Ý2Þqß�Þ���ß#àFáçú¥ä©ã�� T 3 ê\ë?ìfí¡í�# ) ë L¡í�#�T ë6ó¥í£è�Në ! ñHí èCå�á@-¡ã��¡÷ áçú£â�ú¥÷çâ ã¡÷çú¥é�è) ��T ���î�È÷sé�áçå�ý©÷ û útû�ã�æ3è�÷ ÷*) ã¡ùqè�é ú&%�á ã¡ä©è

N ) � T P � �Në N�L¡í � ó¥í�P � N ! ñHí�P'è#á äxæ3è��xä©á û�ã #©èfå�å�è�äxæ3ã

� L¡í �8ó¥í � �wìfí¡í<ë � T 3 ê <üøè�äHû�é�è<å�á 'xú�â('©è

) � N T � ��P»ëRL¡í � Njó¥í � N ! ñHí�P,Ppë L¡í � ìfíCë %tí£î,dã¡ä -tú¥÷ è<æ3ý©ä*)Hý©è<÷çúøù©é�ã¡ù©é�á è ûE'ú£ú¡å�å�ã3â áçútû�á -tú£æ3è�÷ ÷*) ú¡æ©æ3á %�á ã¡ä©è¡õ Ö

®[��haH[^aHeyoqijaq�©iuo ; �h� �@?�zH]hk8c�]he�rtoøzt�^a oqijzH|^[^a#c^ef]hc^efbja¡rx}oø_^aHiji�� oqefb�ryk8a¡rybjzFo<ryeyoq_^bj�Hbj]h[poqija ; c^ef]hc^efbja¡rx}ooq�f�f]�zHbuoxryb��qoB_^aHiji�� oq_^_^bj�Hbj]h[^a�aNc^ef]hc^efbja¡rx}o�_^bj��ryefbjg^|�ryb��qo�_^aHijiuoBk8]hi�rybjc^ijbjzFoq�Hbj]h[^a£efbj�fc�a¡rfry]Voqiji�� oq_^_^bÿ��Hbj]h[^a�?£[^]h[6�fbuoq[^]2c^by}|:�qoqijbj_^aq�

®[�zH]h[^zHij|^�fbj]h[^aq��efbjzFoqc^b�ry]hiuoq[^_^]`~�|poq[sry]�_^a¡rfry]6c^efaHzHaH_^aH[sryaHk8aH[sryaq�K|^[��fbj��ryaHk`o�oqefb�ryk8a¡rybjzH]bj[sryaHef]6}aBzH]h��ryb�ry|^b�ry]2_po$�

� |^[�bj[^�fbjaHk8a �p[^b�ry] � ���:��b�zH|^bmaHijaHk8aH[sryb ; �hijb»aHijaHk8aH[srybm_^aHi�rybjc�]�bj[sryaHef]�?¥���f]h[^]�_^a¡rfryb �hijbÃÈÄp­®¢¡¤¥Ã<¤f§¥ÇsÇ�¤y¢t¨3¢¡Äp­ § �HÃȪ¬Ã�B� |^[DzHefb�ryaHefbj]8_^bdeyoqc^c^efaH�faH[srtoq�Hbj]h[^a�_^a �hijbdaHijaHk8aH[srybd_^b �»���f|DzH|^b }aYgpoq�yoxrto8iuo2k8aHk8]hefbj�H�Fo©��Hbj]h[^a�_^a �hijb�aHijaHk8aH[sryb�_^aHi�rybjc�]�B� |^[:bj[^�fbjaHk8aX_^b»]hc�aHeyoq�Hbj]h[^b��s_^a¡rfryaN]hc�aHeyoq�Hbj]h[^b»oqefb�ryk8a¡rybjzt�^aX�f|^iprybjc�]Ybj[sryaHef]^��bj[8zH|^b»�fbuo �hijb]hc�aHeyoq[^_^b��fbuo�b�efbj�f|^i�rtoxrybm�f]h[^]2aHijaHk8aH[sryb�_^aHi�rybjc�]�B� |^[`bj[^�fbjaHk8aø_^b»oqi*�h]hefb�ryk8b ; zFoqg^iuoxryb^[^aHiji�� |^[^b�rx}oBoqefb�ryk8a¡rybjzFo@?¥�qzt�^a\zH]h[^�faH[sry]h[^]�i�� aH�faHzH|^�Hbj]h[^a_^aHijijaB]hc�aHeyoq�Hbj]h[^b�oqefb�ryk8a¡rybjzt�^a��f|^i�rybjc�]�bj[sryaHef]^�

Page 13: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# � 1

��� �� ��� ������������� ������� ���!� ��"$#�� #� � ��&������ #���&��®ipzHefb�ryaHefbj]B_^bpeyoqc^c^efaH�faH[srtoq�Hbj]h[^a\_^a �hijb^aHijaHk8aH[srybp_^b^rybjc�]�efaFoqijaB}aX~�|^aHijij]�_^aHijiuo�eyoqc^c^efaH�faH[srtoq�Hbj]h[^a� ¦F§q­�ÃÈÄ�4$�jÇp¦xÃÈÄp­»Äm¦x¤¥Å8§qª¬Ã�ÉtÉ3§q­ § ; l � c�� [��B?�bj[�gpoq�fa�(6��3� �Koqija#zHefb�ryaHefbj]��fbhgpoq�yo��f|^iji�� bj_^aFoø_^bh|�rybjijbj�H�Foqefa�f]hij]�ija\zHb�lnefa\�fb*�h[^b��pzFoxryb��ha\_^bm|^[:[�|^k8aHef]VefaFoqijaq��bj[`k8]�_^]Yzt�^a�~�|^aH��ry]��fbuoYeyoqc^c^efaH�faH[srtoxry]�zH]hk8ac^ef]�_^]qrfry]8_^bd|^[w[�|^k8aHef]���»zH]h[-( � � �ÑÐ CÐQ� �h�»c�aHe\|^[po8c�]qryaH[^�Fo`bj[sryaHeyo2_^aHijiuo�gpoq�fa (C� 8�aHeaH�faHk8c^bj]^�^bji�[�|^k8aHef]��6Ï =�=�Ó ��� " �po2iuoVeyoqc^c^efaH�faH[srtoq�Hbj]h[^a�l � c�� [��Cbj[6gpoq�fa ���$�

�6Ï>��Ó>=�=$��� " � ��� � ? zH]h[�6Ï>��Ó>=�=$��� " ? ����� � � ��Ó>=�=$��� " ���CÓÔ b [^]qryb zt�^a`iuo�ijbjk8b�rtoq�Hbj]h[^a ( � � � Ð CÐ � � ; Äm¦x¤¥Å8§qª¬Ã�ÉtÉ3§3ÉHÃ�¦xÄm¢ ?Nbjk8c^ijbjzFo�zt�^a:iuo6c^efbjk`o6zHb�lneyo_^aHijiuoYcpoqe�ryaNlneyoq�Hbj]h[poqefbuo�_^b ; [^aHijiuoYgpoq�fa ( ?C�fbuo�_^b��haHef�yoV_poV�HaHef] � a�efaH[^_^aB|^[^bjzFo�iuoYeyoqc^c^efa¥��faH[srtoq�Hbj]h[^aq� � |poqij�fbuoq�fbd[�|^k8aHef]:efaFoqijaY� <Ï �`c^|�}]�aH�f�faHefa�eyoqc^c^efaH�faH[srtoxry]�bj[wln]hefk`o2l � c�� [���aV�fb�po

�6Ï���2(��Q?�� *�� ? (�� � �(Ð CÐ����CÓ��a¡rfry] �bjiC[�|^k8aHef]wbj[sryaHef]�ln]hefk`oxry]�_poqijija2zHb�lnefa2_^aHijiuo6cpoqe�rya8lneyoq�Hbj]h[poqefbuo�_^b�>aH_*oF�haH[srya:ij]��ryaH�f�f]?�fa �h[^]D_^b���KbjiC[�|^k8aHef]�� Ï� � ( � }a`bj[^_^b���bj_^|poxry]6_poqijiuo6zH]hc^c^buo ; ?�� ?�BD -}a8_^a¡rfry]k`oq[srybj�f�yo�aH_��8}aB_^a¡rfry]8aH�fc�]h[^aH[sryaY_^aHijiuo�eyoqc^c^efaH�faH[srtoq�Hbj]h[^aBl � c�� [��

ÖP×KØfÙFÚ�Û^Ü�Ý2Þqß�ÞqÞqßDC�ã¡äxå�áçæ3è�é útû�ã AYë?ìfí�#©å�á 'xú�:êNë6ôHó3î ì�L¡ð¡í¥ñ ��� ��� & �ë�� ê�ë�í©î ôHó3ì�L¡ð¡í¥ñ M�ìfí�� ë�� N T < ��P»ë N �KôHó3ì�L¡ð¡í¥ñ < �#ïQPêNë6í©î í¡í¡í©ìyï ��� ��� & �ë�� ê�ë�í©î ìyï M�ìfí ��2 ë�� N T < ��P»ë N � ìyï < ! ô�PêNë ! ô¡í¡í¡í¡í¡í¡í¡í¡í���� ��� & �ë�� ê�ë ! í©î ôUM�ìfí�h ë�� N T < ��P»ë N ! ô < �Kð�PêNë ! í©î í¡í¡í¡í¡í¥ñ�%tð���� ��� & �ë�� ê�ë ! í©î ñ�%tð M�ìfí��ON ë�� N T < ��P»ë N ! ñ�%tð < ! óQP

Ö

�do�k8aHk8]hefbj�H�Foq�Hbj]h[^a#_^bs|^[�[�|^k8aHef]\efaFoqija<eyoqc^c^efaH�faH[srtoxry]\k8aH_^buoq[srya<bjihzHefb�ryaHefbj]øl � c�� [���efbjzt�^bja¥�_^aY~�|^bj[^_^bdiuo2k8aHk8]hefbj�H�Foq�Hbj]h[^aB_^aHijija�bj[�ln]hefk`oq�Hbj]h[^bm�fa �h|^aH[sryb �øbji��fa �h[^]`_^b ?��ija�zHb�lnefa�_^b ?��bji�fa �h[^]8_^b �Ya�ijaBzHb�lnefa�_^b��h� � ��3 zFoq|^�yo2_^aHijiuoVij|^[ �h�^aH�H�Fo��p[^b�rtoV_^b�|^[po�ij]�zFoq�Hbj]h[^a�_^b�k8aHk8]hefbuo�fb��poV|^[�[�|^k8aHef]$�p[^b�ry]�_^bmzHb�lnefa�c�aHeøiuo�eyoqc^c^efaH�faH[srtoq�Hbj]h[^a�_^b" 'aH_6|^[6[�|^k8aHef]$�p[^b�ry]V_^b�zHb�lnefac�aHeXiuoVeyoqc^c^efaH�faH[srtoq�Hbj]h[^aY_^b��h�

®i�zHefb�ryaHefbj] _^b�eyoqc^c^efaH�faH[srtoq�Hbj]h[^a�l � c�� [�� }a�~�|^bj[^_^b�|^[^b��h]�zFoqk8aH[srya�zFoqeyoxrfryaHefbj�H�Foxry] _poqijija~�|poq[sryb�rx}o$�� h�ö�úVå�â è�÷ û úVæ3á A 'èNþ�útû�û úVå�ý©÷ ÷çú �xú¡å�è�æ3á�â ã¡äxå�áçæ3è�é ú&%�á ã¡ä©á�æ3á�è�:Xâ á è�ä�%fú�èBå�è�üøù©÷ áçâ á ûE'úVæ3ádá üøù©÷ è�üøè�äHû ú&%�á ã¡ä©è

'xú¥é æ���ú¥é�è¡õ^ö ) ý©ä©áçâ ãøé�è�)Hý©áçå�á û�ã£û�è�ã¡é�áçâ ã 'è<â('©è�A�å�áçú£ý©ä�äFý©üøè�é�ãøá äHû�è�é�ã£ü�ú&���¡á ã¡é�è#æ3è�÷ ÷*) ý©ä©á ûE'ú©õ� d �Èä3þ�útû�û�á�å�á 'xú�:

� �Z�¥ë6í©î ? � ? � î�î�î < ì E ? � E A ! ì <í E ? � E A ! ì < ( �wì»î� � àFá�ä©ã¥û�á�â('©èBå�èQA�ë�ïY÷çú�ù©é�á ü�úVâ á þÿé úVæ3á T 'è�ä©èfâ èfå�å�ú¥é�áçú¥üøè�äHû�èYì�è0)Hý©á äxæ3áD'è�á äFý3û�á ÷ è�üøè�üøã¡é�á % %fú¥é�÷çúèfå�ù©÷ áçâ á û ú¥üøè�äHû�è Nÿüøè�üøã¡é�á % %fú&%�á ã¡ä©è#â ã¡ä�á ÷sù©é�á üøã ���MK � < P �5C��MKM>*P®õ

Page 14: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

= �

�h��iuo�gpoq�fa ( B=���bji�k`oq�f�fbjk8]�[�|^k8aHef] �ø_^b�zHb�lnefa ; [^aHijiuoVgpoq�fa ( ?£_^aHijiuoVk`oq[srybj�f�yo� B

����bji�k`oq�f�fbjk8]�[�|^k8aHef]8_^b�zHb�lnefa ; [^aHijiuoVgpoq�fa ( ?£c�aHeXi�� aH�fc�]h[^aH[srya �h�»]3���haHef]`|^[wk8bj[^bjk8]�aH_|^[�k`oq�f�fbjk8]�aH�fc�]h[^aH[sryaYeyoqc^c^efaH�faH[srtoqg^bjijaq�

®id[�|^k8aHef] ��}a�_^a¡rfry]2Ç�¤y¢tË¡Ãj¨¡Ã�¦xÄm¢�_^aHi��fbj��ryaHk`o`oqefb�ryk8a¡rybjzH]�l � c�� [���� Ô b�_^aH[^]qryaHeyoq[^[^]:bj[^]hi�ryefaBzH]h[�& 6 ÕaH_ �& 98h��efbj�fc�a¡rfryb��qoqk8aH[srya6bji�k8bj[^bjk8]DaH_ bjiøk`oq�f�fbjk8]�aH�fc�]h[^aH[sryaDeyoqc^c^efaH�faH[srtoqg^bjijaq� � �®[?ryaHefk8bj[^b�_^b#k`oq[srybj�f�yo:aH_�aH�fc�]h[^aH[sryaq��i�� |^[^bjzHb�rx}o`_^aHijiuo`eyoqc^c^efaH�faH[srtoq�Hbj]h[^a � � ( � ��efbj�f|^i�rtoija �soxrto\oqiji�� aH�f�faHefa��& 6 � � � �& 98h�8aH_�oqijiuo�zH]h[^_^bj�Hbj]h[^a _^b�[^]hefk`oqijbj�H�Foq�Hbj]h[^a#c�aHeKiuoXk`oq[srybj�f�yo��(�� � � � � (���Ò>�h��{\a �hijb�aH�faHk8c^b<�fa �h|^aH[sryb (P�yoqe©}o�eyoqc^c^efaH�faH[srtoxrto�bj[?gpoq�faR������k8aH[sryefa$ aH_����yoqeyoq[^[^]8bj[ �haH[^aHefa�eyoqc^c^efaH�faH[srtoxrybdbj[6gpoq�fa (C�

ÖP×KØfÙFÚ�Û^Ü�Ý2Þqß�Þ �sß#àFádâ ã¡äxå�áçæ3è�é�á�á ÷dâ é�á û�è�é�á ãYþnõ ù�õ ä�õ�ú�ù©é�èfâ áçå�á ã¡ä©è �xä©á û úVâ�ú¥é útû�û�è�é�á % %fútû�ãVæ©ú A�ë�ï�#��\ë�% #��T�� ) ë ! %Cèfæ ��T 3 êBë % #qú�â ý©á�â ã¡é�é�áçå�ùqã¡äxæ3è#ý©äxúø÷ ã3â�ú&%�á ã¡ä©èCæ3á�üøè�üøã¡é�áçú N!�qå�å�útû ú�æ3á�÷ ý©ä���'©è % %fú�ìyïQP�å�ýxæ©æ3á Y-Fáçå�ú£â ã¡üøè<å�è �¡ý©èQ:

� �

� �èfå�ùqã¡ä©è�äHû�è � �ü�ú¥äHû�áçå�å�úàFá�â ã¡äxå�áçæ3è�é�á ä©ãøásäFý©üøè�é�á*#©ùqè�édå�è�üøù©÷ áçâ á ûE'ú��¡á6'ú á äNþÿã¡é�ü�ú �©á äxú¥é�áçú�#

3 ëDìfí©ì¡ì¡î ì¡ì < ë ! ìfí©ì¡î ìfí¡í©ì¡ì¡ì < edëDìfí©ì¡ì¡ìfí©ìfí©ì¡î ì <â('©è#á ä�ä©ã¥û ú&%�á ã¡ä©è�þnõ ù�õ ä�õ�å�ã¡ä©ãøé ú¥ù©ù©é�èfå�è�äHû útû�ásâ ã¡üøè<å�è �¡ý©èQ:

3 ë6í©î ìfí©ì¡ì¡ì¡ì MBï �Md.d â á ã*'è T ë � ìfí©ì¡ì¡ì¡ì < �#ë � ìfí¡í < ë ! í©î ìfí©ì¡ìfí¡í©ì¡ì¡ì MBï � � â á ã*'è T ë ! ìfí©ì¡ìfí¡í©ì¡ì¡ì < �#ë � ì¡ì <edë6í©î ìfí©ì¡ì¡ìfí©ìfí©ì¡ì MBï �Md.d�� â á ã*'è T ë � ìfí©ì¡ì¡ìfí©ìfí©ì¡ì < �#ë � ìfí¡í©ì¡î

àFã¡÷ ã 3 ù©ýQ'ã<èfå�å�è�é�è�é ú¥ù©ù©é�èfå�è�äHû útû�ã#å�èfâ ã¡äxæ3ã#á ÷qâ é�á û�è�é�á ãKþnõ ù�õ ä�õpú#ù©é�èfâ áçå�á ã¡ä©è �xä©á û ú �qå�å�útû�ã<è )Hý©á äxæ3á©üøè�üøã¡é�á % %fútû�ãä©è�÷ ÷çú�â ã¡é�é�áçå�ùqã¡äxæ3è�äHû�è�÷ ã3â�ú&%�á ã¡ä©è�æ3á�üøè�üøã¡é�áçú©õ �Èä3þ�útû�û�á*# 'xú�÷*) èfå�ùqã¡ä©è�äHû�è�é ú¥ù©ù©é�èfå�è�äHû ú&�©á ÷ è ND��T�� ) E ô E��T 3 ê�P�#tü�ú�á ÷FäFý©üøè�é�ã�æ3á3â á þÿé�èmæ3è�÷ ÷çú�å�ýxú�ü�ú¥äHû�áçå�å�úD'èmð�#yüøè�äHû�é�è�ùqè�é�epä©ã¡ä å�ã¡ä©ãdé ú¥ù©ù©é�èfå�è�äHû ú&�©á ÷ á¡ä�'è»÷*) èfå�ùqã¡ä©è�äHû�èN�ð �B��T 3 ê�P�#©ä�'è#÷çú£ü�ú¥äHû�áçå�å�ú(Nnìfí����,P®õ � 2 Ö� �fàFáhä©ã¥û�ásâ('©è�#Fùqã¡áçâ(',+è�÷ ã %�è�é�ã 'xú ü�ú¥äHû�áçå�å�ú<äFý©÷ ÷çú�#F÷çú£å�ýxúCé ú¥ù©ù©é�èfå�è�äHû ú&%�á ã¡ä©è�þnõ ù�õ ä�õ 'èKú¥ä©ã¡ü�ú¥÷çú VFå�ású¡å�å�ý©üøè�éE'ú)Hý©á äxæ3á

í ��� ��� & �ë�� í©î í¡í�î�î�î�M A � S�� &#î� 2�ö�úVé ú¥ù©ù©é�èfå�è�äHû ú&%�á ã¡ä©èBæ3è�÷ ÷*) èfå�ùqã¡ä©è�äHû�è � # )Hý©áKâ ã¡äxå�áçæ3è�é útû ú�# 'èYæ3è û�û ú F �.P�P�F "G " I K �I��*>�I" G " H�I�>��$< > �1;� >tõàFùqèfå�å�ã #pùqè�éE'ã #»÷*) èfå�ùqã¡ä©è�äHû�è ��ä©ã¡ä 'èNé ú¥ù©ù©é�èfå�è�äHû útû�ãBá ä2û ú¥÷ è\þÿã¡é�ü�ú�#pü�úYä©è�÷ ÷çú�þÿã¡é�ü�ú ��� ����#�æ3ãE-¡è ��� 'èNý©ääFý©üøè�é�ã<á äHû�è�é�ã N�æ3è û�û�ã ��� �RG,P^û ú¥÷ è�â('©èA� � �����:í©õ . æ�èfå�è�üøù©á ã #Hå�è A�ë6ï#èdá ÷xäFý©üøè�é�ã æ3áqâ á þÿé�èdùqè�é�÷*) èfå�ùqã¡ä©è�äHû�è

N�å�è �¡ä©ãNá äxâ ÷ ýxå�ã P 'è <�#�å�á�â ã¡äxå�áçæ3è�é úNå�ã¡÷ á û ú¥üøè�äHû�è ���CëPìyï %Xè�í E � �B��� E ï¡ó¡ó N�å�á»ä©ã¥û�á»â('©è�÷*) á äxå�á è�üøè�æ3è �¡÷ áèfå�ùqã¡ä©è�äHû�áxé ú¥ù©ù©é�èfå�è�äHû ú&�©á ÷ á©ä©ã¡ä 'è�å�á üøüøè û�é�áçâ ãCé�áçå�ùqè û�û�ã£ú¥÷ ÷ ã %�è�é�ã #¡á äK)Hýxú¥äHû�ã ��T�� ) ë ! ìyï %�èfæ���T 3 êNë?ìyï�<�P®õ" ú¥÷ è#é ú¥ù©ù©é�èfå�è�äHû ú&%�á ã¡ä©è 'è#æ3è û�û ú�"�C�C("G�G`>gZ1\���õ

Page 15: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# =��

í �� �� ì ì¡î %¥ó

a T�� ) a T 3 ê

� b*�h|^eyo �h� " � \oqc^c^efaH�faH[srtoq�Hbj]h[^a �heyo*�pzFo._^aHiji�� bj[^�fbjaHk8a�� ; ( ?���?��& 6 .?��& 98h�"?¥�Oa _^aHizH]hefefbj�fc�]h[^_^aH[srya�bj[^�fbjaHk8aB_^b�eyoqc^c^efaH�faH[srtoqg^bjijb�rx}o ; z¡lneH�<aH�faHk8c^bj] �h���=�@?

®idl�oxrfry]8zt�^a�ija�ij]�zFoq�Hbj]h[^b�_^bdk8aHk8]hefbuo2oqg^g^buoq[^]8ij|^[ �h�^aH�H�Fo �p[^b�rto2zH]hk8c�]he�rto2zH]h[^�fa �h|^aH[^�Ha[^]qrya¡�h]hijb ��h�XaH�fbj��ry]h[^]D|^[*k`oq�f�fbjk8]�[�|^k8aHef]DefaFoqija`eyoqc^c^efaH�faH[srtoqg^bjija��* 98h� ; zH]hk8a`c�aHeVbji<rybjc�]wbj[srya¥�ef]�?øaH_D|^[wk8bj[^bjk8]V[�|^k8aHef]`efaFoqija�c�]h�fb�ryb��h]8eyoqc^c^efaH�faH[srtoqg^bjija �* 6 ; zHb }]8_^bj�fzHaH[^_^a�_poqijiuo[^aHzHaH�f�yoqefbuo��p[^b�ryaH�H�Fo2_^aHijiuoVeyoqc^c^efaH�faH[srtoq�Hbj]h[^aY_^b � ?�B=��X[^]h[ry|�rfryb<bC[�|^k8aHefbCefaFoqijbCoqc^cpoqe�ryaH[^aH[sryb£oqiji�� bj[sryaHe��qoqijij] � �* 6 .? �* 98h� � �f]h[^]weyoqc^c^efaH�faH[��rtoqg^bjijb£�faHzH]h[^_^]�bjiøzHefb�ryaHefbj]Dl � c�� [�� o?c^efaHzHbj�fbj]h[^aR�p[^b�rto ; zHb }]�_^bj�fzHaH[^_^a�_poqijiuow[^aHzHaH�f�yoqefbuo�p[^b�ryaH�H�Fo2_^aHijiuoVeyoqc^c^efaH�faH[srtoq�Hbj]h[^a�_^b ?¥�

� bj�f�yoxry]�|^[2cpoqe�rybjzH]hiuoqefa£zHefb�ryaHefbj]�l � c�� [��<oBc^efaHzHbj�fbj]h[^a �p[^b�rto��hzH]h[^�fa �h|^a\zt�^aXefbj�f|^i�rto�|^[^b��h]�zFo©�k8aH[sryaV_^a¡ryaHefk8bj[poxry]:i�� bj[^�fbjaHk8a�� Ï�� ; ( ?���?��& 6 .?��& 98h�"?N_^a �hijb�aHijaHk8aH[srybK_^aHi�rybjc�]:efaFoqija�zt�^a�h]�_^aB_^aHijijaB�fa �h|^aH[srybdc^ef]hc^efbja¡rx}o$�

� � �po�|^[6aHijaHk8aH[sry]2_^b�k`oq�f�fbjk8]Vk8]�_^|^ij]^� �* 98h�d�paH_6|^[�aHijaHk8aH[sry]2_^b�k8bj[^bjk8]�k8]�_^|^ij]^��* 6 .B

� � zH]h[srybjaH[^a�ij]��HaHef]�B� ��}a�zH]h��ryb�ry|^b�ry]�_po�|^[�[�|^k8aHef] �p[^b�ry]2_^b�aHijaHk8aH[sryb B� �hijbmaHijaHk8aH[sryb�_^b�� �f]h[^]8b�[�|^k8aHefb�efaFoqijb�zt�^a�oqc^cpoqe�ryaH[ �h]h[^]8oqiji�� bj[^�fbjaHk8a

�¬Ò �* 98h�D?FÒ �* 6 ��� � �� � �* 6 .? �* 98h� � ; �h� , ?aH_Dbj[w[^]qrtoq�Hbj]h[^a�l � c�� [��N�poq[^[^]`iuo8k`oq[srybj�f�yo8zH]h[D|^[D[�|^k8aHef]:_^b�zHb�lnefaYk8bj[^]hefa�]:| �h|poqijao �¥�

� ijb aHijaHk8aH[srybø_^b��"�fb�_^bjzH]h[^]*Äp«�Å`¢¡¤¥ÃN¤f§¥ÇsÇ�¤y¢t¨3¢¡Äp­ § �HÃȪ¬ÃB¢t¨F§q­�­ §qÅ`¢¡Äp­®¢�]>Äp«�Å`¢¡¤¥ÃNÅ8§�ËtËE��ÃÈÄ�§`ai�� bj[^�fbjaHk8a ; �h� , ?ø�fb�_^bjzHa�ÃÈÄ�¨¡Ã�¢¡Å`¢�ÆqÃ#¤f§¥ÇsÇ�¤y¢t¨3¢¡Äp­ § �HÃȪ¬ÃÈ­�ͧ ;�� b*�h|^eyo �h� " ?

Page 16: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

=�=

ÖP×KØfÙFÚ�Û^Ü�Ý2Þqß�Þ�A�ß#àFá�â ã¡äxå�áçæ3è�é�á � Njï < ô < ! ì < ìRP®õ � 'è#á ÷�å�ã¥û�û�ã¡á äxå�á è�üøè �xä©á û�ã�æ3á��.:� ë��yí < � í©î ìfí¡íUMBï � � < � í©î ìfí©ì MBï � � < î�î�î < � í©î ì¡ìfíUMBï � < � í©î ì¡ì¡ì M�ï ��� ëë��yí < � í©î ï¡ó �Md < � í©î ô©ìyï¡ó �Md < î�î�î < � ì¡î ó �Md < � ì¡î %¥ó �Md � <

â ãHånû�á û�ý©á û�ãNæ©ú\ï¡ó�è�÷ è�üøè�äHû�á N! �á �¡ý©é úBì¡õ ñ P®õ �Èä )Hý©èfånû�ã\â�ú¡å�ã a T 3 êVëDí©î ì¡ì¡ì4M�ï � ë�ì¡î %¥ó �Md è a T�� ) ë?í©î ìfí¡í Mï � � ë6í©î ï¡ó �Md õ Ö

®[��haH[^aHeyoqijaq�^_poxry] � ; ( ?���?��& 6 .?��& 98h�"?¥�»bjim[�|^k8aHef]8_^b�aHijaHk8aH[sryb�_^b�� }a � �= ; (wÒ � ?�( � � � ; �& 98h��Ò �& 6 5 � ? 5 �

aBefbj�f|^i�rto�* 98h��Ï>��Ó � �\ÓFÓFÓ����-( �P��� � Ï ( �P��� � ; �XÒ7( � �P? ; �VÏ (wÒ � ?�B�* 6 DÏ>��Ó�����ÓFÓFÓ�� �-( �P�� � Ï ( �P�� � � �

� ijb�aHijaHk8aH[sryb�_^b �Õ[^]h[��f]h[^]B|^[^b�ln]hefk8aHk8aH[sryaø�fcpoq�Hbuoxryb�lneyo�ij]hef]^�qk`oN�f]hij]Nlneyo�c�]qryaH[^�Ha�_^aHijiuogpoq�fa��f|^zHzHaH�f�fb��haq�X��|^[^~�|^aYiuo��fcpoq�Hbuoxry|^eyo2oq�f�f]hij|�rto�[^]h[�}a�|^[^b�ln]hefk8aq�^k`o�~�|^aHijiuo�efaHiuoxryb��qoVij]6}a%Bbj[�l�oxrfryb�]�h[^b�aHijaHk8aH[sry]:_^b � _^b5AmaHefbj�fzHa�_po��hijb�oq_^buoqzHaH[sryb�c�aHe�|^[po`�f]hiuo8|^[^b�rx}o8�f|^iji�� |^i�rybjk`o�zHb�lneyo_^aHijiuoVk`oq[srybj�f�yo��

8�aHeX]�h[^b�[�|^k8aHef]2efaFoqija��6Ï���2( � �fb��haHefb��pzFo2|^[po2aH_�|^[po2�f]hiuo�_^aHijijaNryefa��fb�ry|poq�Hbj]h[^b ��h�ø�-* � aH_}a�~�|^bj[^_^b�aH�yoxrfrtoqk8aH[srya�eyoqc^c^efaH�faH[srtoqg^bjija%B=��ø�D[^]h[}a�eyoqc^c^efaH�faH[srtoqg^bjija�bj[ ��B

���ø�*}aBeyoqc^c^efaH�faH[srtoqg^bjijaq�pk`o�[^]h[6aH�yoxrfrtoqk8aH[sryaq�»bj[ �2�®i�zFoq�f] =V�fb�c^efaH�faH[srto8�fa6Ð �<Ð�� �* 6 K�paH_woqijij]heyo��fb�_^bjzHa�zt�^a��fbC}a��haHefb��pzFoxry]8|^[«�Ä�Æ�¢¡¤ � ¦5#<�

]hc^c^|^efa`�faÐ �<Ð . �* 98h�d�#aH_*oqijij]heyo��fb _^bjzHa`zt�^a`�fbN}a2�haHefb��pzFoxry]D|^[Õ¦*5q¢¡¤ � ¦5# ; �* 6 a��* 98h��f]h[^]�zt�^buoqk`oxrybKefbj�fc�a¡rfryb��qoqk8aH[srya2¨3¦R4qª¬Ã�§wÆqã«�Ä�Æ�¢¡¤ � ¦5#>a`¨3¦R4qª¬Ã�§wÆqÃX¦*5q¢¡¤ � ¦5# ?�B�rtoqijaV�fb�ry|poq�Hbj]h[^abj[c^eyoxrybjzFo6�fbC�po�~�|poq[^_^] � � �& 6 ]hc^c^|^efa �2. �& 98h�d� 8�aHe�iuo �haH��rybj]h[^a8_^aHijija2zH]h�fbj_^_^a¡rfrya¨¡ÃÈ­�«^§3ÉHÃ�¦xÄpÃ#¢tËtËt¢yÉHÃ�¦xÄ�§qª¬Ã^�fb�efbjk`oq[^_poYoqi�cpoqeyo��heyoxln] �h� , �f|^im�fbj��ryaHk`oVoqefb�ryk8a¡rybjzH]���rtoq[^_poqef_�J�������ÖP×KØfÙFÚ�Û^Ü�Ý2Þqß�Þ 9�ß�Kútû�ãB÷*) á äxå�á è�üøè � Njï < ô < ! ì < ìRP N�â®þÿéfõ�èfå�è�üøù©á ã�ì¡õ ìfô�P�#�å�á 'xú�ý©äxæ3è�é��xã���ùqè�é ã��¡ä©á»ê�� �û ú¥÷ è<â('©è#ê��`í©î ï¡ó èfæ\ãE-¡è�é��xã���ùqè�édã��¡ä©ásê �wì¡î %¥ó3õ Ö

®iXzFoq�f] ���fb��haHefb��pzFo~�|poq[^_^] �& 6 � � � �& 98h��a�bji�[�|^k8aHef]_^bXzHb�lnefa6g^bj[poqefbja6_^b( }awk`o���hbj]hefa6_^b �¥� Ô b\c�]h[^aw_^|^[^~�|^a?bjiXc^ef]hg^ijaHk`o_^b\zH]hk8aweyoqc^c^efaH�faH[srtoqefaD��k8aH_^buoq[srya�|^[aHijaHk8aH[sry]2_^b �2��,3 �áçå�å�útû�ãNý©ä$-tú¥÷ ã¡é�èøæ3è�÷ ÷*) èfå�ùqã¡ä©è�äHû�è�#�÷çúNù©é�á ü�ú�â á þÿé ú�æ3è�÷ ÷çúNü�ú¥äHû�áçå�å�úNæ3ápý©äVäFý©üøè�é�ã�æ3á -¡è�é å�ã�æ©ú %�è�é�ãNù©ýQ'ãú¡å�å�ý©üøè�é�è�A ! ìD-tú¥÷ ã¡é�á�æ3áçånû�á äHû�á�è â áçú¡å�â ý©äxúøæ3è�÷ ÷ èCé�á ü�ú¥ä©è�äHû�á � ! ìCâ á þÿé�èCù©ýQ'ãXú¡å�å�ý©üøè�é�è_A�-tú¥÷ ã¡é�á�æ3áçånû�á äHû�á*#qùqè�éý©ä�û�ã¥û ú¥÷ è�æ3á�A�� � � N�A ! ìRPpùqãHå�å�á �©á ÷ áqâ ã¡ü �©á äxú&%�á ã¡ä©áqæ3è�÷ ÷ èKâ á þÿé�èKæ3è�÷ ÷çúCü�ú¥äHû�áçå�å�ú©õpàFèKå�áqãHå�å�è�é -tú<â('©è �¡÷ áqèfå�ùqã¡ä©è�äHû�áé ú¥ù©ù©é�èfå�è�äHû ú&�©á ÷ áhå�ã¡ä©ã���T 3 ê ! ��T�� ) �:ì#èCâ('©è#ùqè�é�ã��¡ä©áhäFý©üøè�é�ãøé ú¥ù©ù©é�èfå�è�äHû ú&�©á ÷ è�ùqãHå�á û�á -¡ãøá ÷�å�ý©ã£ã¡ù©ùqãHånû�ã 'èé ú¥ù©ù©é�èfå�è�äHû ú&�©á ÷ è�#tè�å�áqâ ã¡äxå�áçæ3è�é ú<ú¥äxâ('©è�÷ ã %�è�é�ã #Hå�á©û�é�ãE-tú<â('©è�á ÷xäFý©üøè�é�ã æ3áxè�÷ è�üøè�äHû�áqæ3á � 'è )Hý©è�÷ ÷ ãCå�ùqèfâ á �qâ�útû�ãxõ

Page 17: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# =$�

ÖP×KØfÙFÚ�Û^Ü�Ý2Þqß�Þ�sß#àFá�â ã¡äxå�áçæ3è�é�á � Nnìfí < ï < ! ð < ð�P®õ��dä�äFý©üøè�é�ãVòê � � 'è<æ3è�÷hû�á ùqãòê�ë � í©î ? � ? � M�ìfí � < �#ë ! ð < ! < < î�î�î < < < ð�î�È÷�äFý©üøè�é�ã�êwë ì¡î ï LYä©ã¡ä`ú¥ù©ùxú¥é�û�á è�ä©è�ú � #�ü�úYù©ýQ'ãVèfå�å�è�é�èNé ú¥ù©ù©é�èfå�è�äHû útû�ã�üøèfæ3áçú¥äHû�è�ý©ä`è�÷ è�üøè�äHû�ãVæ3á � õ�Èä3þ�útû�û�á�ê�'è<ånû�é�è û�û ú¥üøè�äHû�è<â ã¡üøù©é�èfå�ã û�é úøæ3ý©è<äFý©üøè�é�á�æ3á � #xâ á ã*'èì¡î ï �8ê���ì¡î ôèfæNý©ä©ã�æ3è�á�æ3ý©èCå�ásù©ýQ'ãXå�â è �¡÷ á è�é�è#â ã¡üøè<ú¥ù©ù©é�ãHå�å�á ü�ú&%�á ã¡ä©èKæ3á�êBüøèfæ3áçú¥äHû�è<ý©äNè�÷ è�üøè�äHû�ãXæ3á � õ Ö

8�aHe�oqc^c^ef]h�f�fbjk`oqefa�|^[w[�|^k8aHef]`efaFoqijaB�DÏ���Ó � � � � ÓFÓFÓ"� � � � 7 � ÓFÓFÓ � ( � oqc^cpoqe�ryaH[^aH[sryaVoqiji�� bj[���fbjaHk8a`_^b eyoqc^c^efaH�faH[srtoqg^bjijb�rx}o�_^b �2�#k`o�[^]h[�eyoqc^c^efaH�faH[srtoqg^bjija:aH�yoxrfrtoqk8aH[sryaq�#zH]h[*|^[�[�|^k8aHef] 0 ; �"? * �2�^�fb�c^|�}]8|�rybjijbj�H�Foqefa�|^[po�_^aHijijaB_^|^aBryaHzH[^bjzt�^aY�fa �h|^aH[sryb �� ­�¤y¦xÄmËy§qÅ`¢¡Äp­®¦$� 0 ; �"?��fb<]qrfrybjaH[^a�ryef]h[^zFoq[^_^]�ija�zHb�lnefa�_^aHijiuo:k`oq[srybj�f�yo��fa �h|^aH[sryb iuo � �®k`o��zHbj]^}a

0 ; �"?CÏ���Ó � � � � ÓFÓFÓ7� � �-(���B

� §q¤¥¤y¦x­®¦xÄ�Æs§qÅ`¢¡Äp­®¦$� 0 ; �"?£�fb�]qrfrybjaH[^aB_poV�w�f]hk8k`oq[^_^] �� (�oqijiuo ; � 5 � ? �®aH�fbjk`o�zHb�lneyoV_^aHijiuo�f|po�k`oq[srybj�f�yo�aBc�]hbmryef]h[^zFoq[^_^]8ija�zHb�lnefaB�fa �h|^aH[srybdiuo � �®k`o���]3���haHef]`bj[^zHefaHk8aH[srtoq[^_^]2_^b��� � �faV}a � � 7 � � �� (�a�c�]hb�ryef]h[^zFoq[^_^]VijaNzHb�lnefaN_^aHijiuoYk`oq[srybj�f�yo��f|^zHzHaH�f�fb��haYoqijiuo � �®k`o���zHbj]^}a�� 0 ; �"?CÏ � ��Ó � � � � ÓFÓFÓ"� � �2( � � � 7 � � �� (��Ó � � � � ÓFÓFÓ ; � � 5 � ?��2( � � � 7 � � �� (

®[�efaHiuoq�Hbj]h[^aBoqiji�� aH�faHk8c^bj] �h�����V�fb��po$�ryef]h[^zFoqk8aH[sry] � 0 ; �hÓ>=$�@?<Ï �hÓ>=�Boqefef]qry]h[^_poqk8aH[sry] � 0 ; �hÓ>=$�@?<Ï �hÓ ��Ó

Ô a�bj[s�haHzHaY_^b��6Ï �hÓ>=$���fb�zH]h[^�fbj_^aHeyo��6Ï �hÓ>= " �^�fb��po$�ryef]h[^zFoqk8aH[sry] � 0 ; �hÓ>= " ?<Ï �hÓ>=�Boqefef]qry]h[^_poqk8aH[sry] � 0 ; �hÓ>= " ?<Ï �hÓ>=øÓ

®[|^[�fbj��ryaHk`o�oqefb�ryk8a¡rybjzH]:efaFoqijaVl � c�����ija�zFoqeyoxrfryaHefbj��rybjzt�^a2ija �soxrya2oqijiuo`eyoqc^c^efaH�faH[srtoq�Hbj]h[^aq�KzHbj]^}ai�� bj[^�fbjaHk8a ��a�ija�ln|^[^�Hbj]h[^b�_^b�ryef]h[^zFoqk8aH[sry]aH_ oqefef]qry]h[^_poqk8aH[sry]^� �f]h[^]�_^a¡rfrya�ija�Ëy§q¤f§q­�­®¢¡¤¥ÃM�¨¡­�Ã�ËE�p¢�¨¡­ §q­�Ã�ËE�p¢`Æ�¢¡ªd¨¡Ãj¨¡­®¢¡Å8§q� Ô b#_^a2�p[^bj�fzH]h[^]�Ëy§q¤f§q­�­®¢¡¤¥Ãj¨¡­�Ã�ËE�p¢8ÆqÃÈÄ�§qÅ�Ã�ËE�p¢\ijaVc^ef]hc^efbja¡rx}o:ija �soxryaoqijija`]hc�aHeyoq�Hbj]h[^b l � c���_^a2�p[^b�rya6bj[ �"aH_ o��hijb£oqi*�h]hefb�ryk8bC|�rybjijbj�H�FoxrybC_poqiji�� |^[^b�rx}oDoqefb�ryk8a¡rybjzFo�c�aHebjk8c^ijaHk8aH[srtoqefijaq���aB]hc�aHeyoq�Hbj]h[^b»l � c��<�f]h[^] �Coq_^_^bj�Hbj]h[^a\l � c������^�f]qrfryeyoq�Hbj]h[^aNl � c������^k8]hi�rybjc^ijbjzFoq�Hbj]h[^aøl � c������

_^b���bj�fbj]h[^aNl � c������»~�|^b�_^aH[^]qrtoxrya�bj[6k8]�_^]2_^b��haHef�f]8_poqijija�zH]hefefbj�fc�]h[^_^aH[sryb�|^�f|poqijb�]hc�aHeyoq�Hbj]h[^b�bj[� � � 3� N�,dã¡äYâ á�å�á�ã3â�â ý©ùqè�éE'úXæ3è�÷ ÷*) ã¡ùqè�é ú&%�á ã¡ä©èCæ3á^â ã¡ä&-¡è�é å�á ã¡ä©è<æ©ú¥÷ ÷çúøé ú¥ù©ù©é�èfå�è�äHû ú&%�á ã¡ä©è#á ä��xú¡å�è�A2ú¥÷ ÷çú�é ú¥ù©ù©é�èfå�è�ä�Yû ú&%�á ã¡ä©è£æ3èfâ á ü�ú¥÷ è£èøæ©úK)Hý©è�÷ ÷çú\æ3èfâ á ü�ú¥÷ èøúK)Hý©è�÷ ÷çú\á ä �xú¡å�è A #�â á ã*'è£æ©ú¥÷ ÷çúXé ú¥ù©ù©é�èfå�è�äHû ú&%�á ã¡ä©è<á äHû�è�é�äxú\ú )Hý©è�÷ ÷çúèfånû�è�é�äxú£è�-Fáçâ è/-¡è�é å�ú©õ

Page 18: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

= "

®i#efbj�f|^i�rtoxry]���`Ï ��� ��� ; _^]3�ha��'��rto�c�aHe 5 ?FÒ ?�� ?(�O?¥��zH]h[?�D?�� * �2���fb#_^bjzHa�Æ�¢ 6CÄpÃÈ­®¦��fa�`Ï � ���Doqc^cpoqe�rybjaH[^a2oqiji�� bj[^�fbjaHk8aY_^b#eyoqc^c^efaH�faH[srtoqg^bjijb�rx}o���oqi�ryefbjk8aH[sryb�bjiKefbj�f|^i�rtoxry]D}a�bj[^_^a2�p[^b�ry]a8�fbC_^bjzHa8zt�^a`�fb\}a��haHefb��pzFoxrto�|^[po��fb�ry|poq�Hbj]h[^a2aHzHzHaH�Hbj]h[poqija8_^b�¦*5q¢¡¤ � ¦5# �fa�Ð �»Ð�. �* 98h�d�K]w_^b«�Ä�Æ�¢¡¤ � ¦5# �fa�Ð �»Ð�� �* 6 K� ���K]hc^c�]he�ry|^[^]w�f]qrfry]hijbj[^aFoqefa8zt�^a�oq[^zt�^a�[^aHi zFoq�f]?_^b efbj�f|^i�rtoxry] �eyoqc^c^efaH�faH[srtoqg^bjija��fbC�po6bj[ �haH[^aHeyoqija?��=<Ï ���dbj[�~�|poq[sry]P���oqc^cpoqe�rybjaH[^a2�faHk8c^efa8o �2�dk8aH[sryefa �c^|�}]2[^]h[woqc^cpoqe�ryaH[^aHefaYo �2�

ÖP×KØfÙFÚ�Û^Ü�Ý2Þqß�Þ���ß#àFámâ ã¡äxå�áçæ3è�é�á ä©ã � Nnìfí < ï < ! ï < ïQP�è\÷çúBüøã¡÷ û�á ù©÷ áçâ�ú&%�á ã¡ä©èøþnõ ù�õ�û�é ú�ê�ë ï3î <Bè �8ë %Fî %Fõ�� )á üøüøèfæ3áçútû�ã -¡è�é�á �qâ�ú¥é�è�â('©èmê < � � � VKá ä3þ�útû�û�á3á ä£é ú¥ù©ù©é�èfå�è�äHû ú&%�á ã¡ä©èpþnõ ù�õ ä�õpå�á 'xú�êNë6í©î ï�<XMCìfí � è �øë6í©î %2%ZMCìfí �è )Hý©á äxæ3á»êVè ��å�ã¡ä©ãNé ú¥ù©ù©é�èfå�è�äHû ú&�©á ÷ á�èfå�útû�û ú¥üøè�äHû�è¡õ4�È÷pé�áçå�ý©÷ û útû�ã aNëê$M �Yë�ï3ì¡î ó L\ë�í©î ï3ìyó L(M2ìfí��£æ3è�÷ ÷çúüøã¡÷ û�á ù©÷ áçâ�ú&%�á ã¡ä©èCýxå�ýxú¥÷ è á ä �Dú¥ù©ùxú¥é�û�á è�ä©è£ú¥÷ ÷*) á äxå�á è�üøè£æ3á^é ú¥ù©ù©é�èfå�è�äHû ú&�©á ÷ á ûE'ú�#xü�ú�#sú -¡è�äxæ3ãXý©äxúXü�ú¥äHû�áçå�å�ú\æ3á�ñâ á þÿé�è�#3ä©ã¡ä 'è#é ú¥ù©ù©é�èfå�è�äHû ú&�©á ÷ èKèfå�útû�û ú¥üøè�äHû�è�#©â á ã*'èKä©ã¡ä�ú¥ù©ùxú¥é�û�á è�ä©è<ú � õ Ö

®i�efbj�f|^i�rtoxry] ��D_^bX|^[po�]hc�aHeyoq�Hbj]h[^a:l � c��>_^bjc�aH[^_^a�_poqiji�� oqi*�h]hefb�ryk8]6|�rybjijbj�H�Foxry]?_poqiø�fbj��ryaHk`ooqefb�ryk8a¡rybjzH]�l � c�����zHbj]^}aq��bj[`_^a2�p[^b�ryb��qo��s_poqiji�� ]he �soq[^bj�H�Foq�Hbj]h[^aøa\_poqi»c^ef]�ha¡rfry] ; oYijb��haHijij]B�poqef_ �øoqefa�?_^aHiji�� |^[^b�rx}oNoqefb�ryk8a¡rybjzFo�� Ô b�zH]h[^�fbj_^aHefaHe©}oB]qrfrybjk`oqijaq�x_po�|^[�c^|^[sry]�_^b���bj��rto�_^bj[poqk8bjzH]^�x|^[��fbj��ryaHk`ooqefb�ryk8a¡rybjzH]�l � c��<c�aHeXbji�~�|poqijaB�fbdoqg^g^buo

���Ï���� ��2Ï 0 ; �O?CÏ 0 ; � ��� ?mÓ ; �h� � ?�do ; �h� � ?�bjk8c^ijbjzFoVzt�^a�bji�efbj�f|^i�rtoxry]?���_^aHiji�� ]hc�aHeyoq�Hbj]h[^aBl � c����»zt�^a:}aY|^[w[�|^k8aHef]`k`oqzHzt�^bj[po��d}aaH�yoxrfrtoqk8aH[sryaNiuoYeyoqc^c^efaH�faH[srtoq�Hbj]h[^a�l � c�� [��Co�c^efaHzHbj�fbj]h[^aK�p[^b�rtoY_^aHimefbj�f|^i�rtoxry] �Y_^aHijiuoYzH]hefefbj�fc�]h[��

_^aH[sryaY]hc�aHeyoq�Hbj]h[^a�bj[ � �øZ\[6rtoqija��fbj��ryaHk`o2oqefb�ryk8a¡rybjzH] �soqeyoq[srybj�fzHa�_^|^[^~�|^a�zt�^a�bji�efbj�f|^i�rtoxry]2_^b~�|poqij�fbuoq�fbK]hc�aHeyoq�Hbj]h[^aVl � c��V_^b5AmaHefbj�fzFo�_poqi#efbj�f|^i�rtoxry]�_^aHiji�� ]hc�aHeyoq�Hbj]h[^a�bj[ � zH]hefefbj�fc�]h[^_^aH[srya ; bjiefbj�f|^i�rtoxry]�aH�yoxrfry]�?¥��_^b<|^[po6~�|poq[sryb�rx}o6zt�^aw}a2bji<�f]hij]6aHefef]hefa2_^bCeyoqc^c^efaH�faH[srtoq�Hbj]h[^a8_^b ����a���bjaH[^ac�aHefzHb }]�_^a¡rfry]�¨¡Ãj¨¡­®¢¡Å8§:§`Å8§©¨t¨¡ÃÈÅ8§:§�ËtË¡«�¤f§q­®¢yÉtÉ3§:ÆqÃÈÄ�§qÅ�Ã�Ëy§q�Ô b»zH]h[^�fbj_^aHefb���oq_8aH�faHk8c^bj]^��i�� oq_^_^bj�Hbj]h[^aøl � c����<�do��f|poYaH�faHzH|^�Hbj]h[^aNzH]h[^��rtoYaH�f�faH[^�Hbuoqijk8aH[sryaN_^b

~�|poxrfryef]�l�oq�fb ��h��zH]h[�lnef]h[sry]8ryeyo �hijb�aH�fc�]h[^aH[srybK_^aHijiuo�eyoqc^c^efaH�faH[srtoq�Hbj]h[^a�l � c�� [���_^a �hijb�oq_^_^aH[^_^b�c�aHe\bj[^_^b���bÿ�_^|poqefa�i�� oq_^_^aH[^_^]8zH]h[�aH�fc�]h[^aH[sryaYc^by}|�c^bjzHzH]hij]�B

=��8���f�^b�lÈry��_^aHijija\zHb�lnefaN_^aHijiuo�k`oq[srybj�f�yoY_^b�rtoqijaNoq_^_^aH[^_^]Vbj[`k8]�_^]�zt�^a�bji»efaHiuoxryb��h]�aH�fc�]h[^aH[sryaefbj�f|^i�ryb�| �h|poqija�o�~�|^aHijij]�_^aHiji�� oqi�ryef]�oq_^_^aH[^_^]�B

���Xoq_^_^bj�Hbj]h[^aB_^aHijijaBk`oq[srybj�f�faB_^a �hijb�oq_^_^aH[^_^b B" �Xoqefef]qry]h[^_poqk8aH[sry]2a�[^]hefk`oqijbj�H�Foq�Hbj]h[^aN_^aHi�efbj�f|^i�rtoxry]^�

Z\�f|poqijk8aH[srya�i�� |^[^b�rx}o?oqefb�ryk8a¡rybjzFowk8aHk8]hefbj�H�Fo �hijbø]hc�aHeyoq[^_^b£l � c����£aH_ oq[^zt�^a6b�_poxrybøbj[sryaHefk8aH_^b�haH[^aHeyoxrybm_^|^eyoq[srya�i�� aH�faHzH|^�Hbj]h[^a�_^aHiji�� ]hc�aHeyoq�Hbj]h[^aXl � c�����bj[:ij]�zFoq�Hbj]h[^b»_^a¡rfrya�efa �hbj��ryefb ; �p�fbjzFoqk8aH[sryaoqiji�� bj[sryaHef[^]2_^aHiji�� |^[^b�rx}o@?£zH]h[Dc^efaHzHbj�fbj]h[^a �� � � . �¥� � |^bj[^_^bdb�cpoq�f�fb�=2a �8�f]h[^]:aH�fa �h|^b�ryb�oF�haH[^_^]o2_^bj�fc�]h�fbj�Hbj]h[^aBc�aHe\iuo�k`oq[srybj�f�yo2|^[w[�|^k8aHef]`_^b�zHb�lnefaYk`o���hbj]hefa�_^aHijiuo�c^efaHzHbj�fbj]h[^a�_^aHid�fbj��ryaHk`ooqefb�ryk8a¡rybjzH]VzH]h[^�fbj_^aHeyoxry]^�

Page 19: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# = �

Ô a��fbX�f|^c^c�]h[^a�zt�^a ((Ï ����a.�:Ï " a�zt�^a�bXcpoq�f�fb(=?a����fbuoq[^]aH�fa �h|^b�rybøl�oqzHaH[^_^]�|^�f]_^bN|^[ efa �hbj��ryef]*o�_^]hc^c^buo�c^efaHzHbj�fbj]h[^aq�Xbji\zFoqijzH]hij]�_^b��� Ï 8 ���3�XzH]h[ 8�Ï ��Ó 1�1 � � �>��� ��a��Ï���Ó " � ��= � ��� � �¥�»oF����bjaH[^a�[^aHidk8]�_^]��fa �h|^aH[srya���h�XzH]h[�lnef]h[sry]2_^a �hijb�aH�fc�]h[^aH[sryb �C]3����buoqk8aH[srya�=/.�ÒG��B=��X�f�^b�lÈrX_^aHijiuoVk`oq[srybj�f�yo�_^b���� ��Ó " � ��=VÒ�� ��Ó �O�O� " � ��= �+B���X�f]hk8k`oV_^aHijijaBk`oq[srybj�f�fa��4��Ó 1�1 � �O�O�O�O� 5 ��Ó �O�O� " � ��= ��Ï>��Ó 1�1 � , � ��= �+B" �\oqefef]qry]h[^_poqk8aH[sry]�a�[^]hefk`oqijbj�H�Foq�Hbj]h[^aN_^aHi�efbj�f|^i�rtoxry] � � Ï>��Ó 1�1 �O� � ��� �F�

Ô bB[^]qrybBzt�^a_^|^eyoq[sryabjiBcpoq�f�f] " �f]h[^]P��rtoxryac�aHef�fa�ryefa�zHb�lnefa��fb*�h[^b��pzFoxryb��ha?_^aHi�efbj�f|^i�rtoxry]*a~�|^bj[^_^b � [^]h[�}a6bji£efbj�f|^i�rtoxry]?aH�yoxrfry]�_^aHiji�� oq_^_^bj�Hbj]h[^a`_^b 8a��3�£oq[^zt�^a��fa 8a��8�f]h[^][�|^k8aHefbk`oqzHzt�^bj[po+B�ry|�rfrtoF��buo � Ï 0 ; ��Ó 1�1 � , � ��= � � ��� � ? aN_^|^[^~�|^aq�pbj[:~�|^aH��ry]�zFoq�f]^���fbm�poViuoYk`oq�f�fbjk`ooqzHzH|^eyoxryaH�H�Fo � �©�3Ni�zH]h[sryeyoqefbj]^���faYb�efa �hbj��ryefb��poq[^[^] �� � � Ï �ø�fb�]qrfrybjaH[^a���h�XzH]h[�lnef]h[sry]2_^a �hijb�aH�fc�]h[^aH[sryb � =/.�ÒG��B=��X�f�^b�lÈrX_^aHijiuoVk`oq[srybj�f�yo�_^b���� ��Ó " � ��=VÒ�� ��Ó �O�O� " B���X�f]hk8k`oV_^aHijijaBk`oq[srybj�f�fa��4��Ó 1�1 � � 5 ��Ó �O�O� " Ï>��Ó 1�1 � , B" �\oqefef]qry]h[^_poqk8aH[sry]�a�[^]hefk`oqijbj�H�Foq�Hbj]h[^aN_^aHi�efbj�f|^i�rtoxry] � � Ï>��Ó 1�1 � , � ��� � B

~�|^bj[^_^b�[^]h[6�fb��po2iuoVk`oq�f�fbjk`o�oqzHzH|^eyoxryaH�H�Fo��

021�� ��@BACA<58A<@�=?; A<58L?7?=D5 ��<� aHefef]hefaøzt�^aX�fb^zH]hk8k8a¡rfryaøoqc^c^ef]h�f�fbjk`oq[^_^]�|^[2[�|^k8aHef]BefaFoqija£�8k8aH_^buoq[srya 0 ; �"?��fb^_^bjzHaY¢¡¤¥¤y¦x¤y¢ÆqÃ#¤y¦x«�Ä�Æ�¦�� Bpc^by}|�c^efaHzHbj�yoqk8aH[sryaq�p�fb��po$�¯�°s±#²�³n´�³nµ�²d°�¶p·��d·`¸�¹�º©º©µ»º©° ��³ º©µ�¿�² �dµ��VÁ� ¦xÄ�¨¡Ã�Æ�¢¡¤f§q­®¦�«�Äw¨¡Ãj¨¡­®¢¡Å8§:§q¤¥ÃÈ­�Å`¢¡­�Ã�Ët¦ ��� Ç���§BÇ�¤y¢tË¡Ãj¨¡Ã�¦xÄm¢�6CÄpÃÈ­ §©ÊøËt¦xÄ � ; ( ?���?��& 6 .?��& 98h�"?yÊC¨¡Ã�§�P«�Ä?Äp«�Å`¢¡¤y¦8¤y¢y§qªj¢V§¥ÇsÇ^§q¤¥­®¢¡Äm¢¡Äp­®¢�§qªÈª�� ÃÈÄ�¨¡Ã�¢¡Å`¢�ÆqÃ<¤f§¥ÇsÇ�¤y¢t¨3¢¡Äp­ § �HÃȪ¬ÃÈ­�ͧ:ÆqÃ���¢B¨¡Ã�§ 0 ; �"?�ª�§2¨¡«^§¤f§¥ÇsÇ�¤y¢t¨3¢¡Äp­ §3ÉHÃ�¦xÄm¢BÃÈÄ ��� à ÆqÃ�Ët¢V¢¡¤¥¤y¦x¤y¢V§©¨t¨3¦xª¬«�­®¦`ÆqÃ<¤y¦x«�Ä�Æ�¦�� ÃȪ�Äp«�Å`¢¡¤y¦

Ð 0 ; �"?KÒ��<Ð�?¢B¨¡ÃCÆqÃ�Ët¢V¢¡¤¥¤y¦x¤y¢Y¤y¢¡ª�§q­�Ã65q¦:ÆqÃ#¤y¦x«�Ä�Æ�¦�� ÃȪ�Äp«�Å`¢¡¤y¦

Ð 0 ; �"?KÒ��<ÐÐ �<Ð Ó� ' � )©ùqãHå�å�á �©á ÷ èCæ3á üøãHånû�é ú¥é�è<â('©è ��� ��� ë � �8ô�#©á äxå�á è�üøèCâ ã¡äBú¡â�â ã¡é �¡á üøè�äHû�á�æ3è û�û�á"GK*�5C���� ���MK�è F.>1; I� �MI H ���MK*#�'èû ú¥÷ è<æ©ú��Hú¥é ú¥äHû�á é�è�÷çú£ü�ú¡å�å�á ü�úøú¡â�â ý©é útû�è % %fúCá ä )Hýxú¥÷çå�áçú¡å�ásã¡ùqè�é ú&%�á ã¡ä©è�þnõ ù�õ12 ìfô 4 õ

Page 20: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

=$�

�6Ï�� �2( � 0 � ; �"?CÏ> � �-( � 0�� ; �"?CÏ �� �-( �Ò ��Ó��O����=$� , � ��� � � Ò ��Ó��O����=$� � ��� � � Ò ��Ó��O����= " � ��� � ���Ó��O�O�O��= " � � ��� � ��Ó��O�O�O��= � ��� � ��Ó��O�O�O��= � ��� ���Ó � , =$��� , � � ��� � ��Ó � , =$��� � ��� � ��Ó � , =$� " � ��� �Ò ��Ó " �Q� ,, � � ����� Ò ��Ó " �Q� ,, � ����� Ò ��Ó " �Q� , � � �����

� ¼ �<°�¾�¾n¼ ¶p·�¶��:¼ � ��º©°�½H°�²�À3¼^´�³nµ�²d°��Kµ»¼�À©³�² �� �<µ�³�²�À6²dµ»º���¼p¾�³n´s´s¼�À3¼��VÀ3º©µ�²��h¼ ��°�²�À3µ*°��¼^º©º©µpÀ3µ�² �d¼ ��°�²�À3µ �d°�¾�¾n¼ ��¼p²�À©³�½F½H¼

�<� aHefef]hefaB_^b�ef]h|^[^_^]%A?�fb �haH[^aHeyo��fbuo�~�|poq[^_^]2|^[6[�|^k8aHef]2efaFoqijaq�^_poxry]2[^aHijiuo�|^�f|poqijaBeyoqc^c^efaH�faH[��rtoq�Hbj]h[^a`_^aHzHbjk`oqijaq�K��bjaH[^a:eyoqc^c^efaH�faH[srtoxry]bj[ �2�<zHbj]^}a:[^aHi£cpoq�f�yo���hbj]?_poqijiuo�eyoqc^c^efaH�faH[srtoq�Hbj]h[^aaH��ryaHef[po�o:~�|^aHijiuo:bj[sryaHef[po ; ¢¡¤¥¤y¦x¤y¢`Æqã¤y¦x«�Ä�Æ�¦���Æqã¤f§¥ÇsÇ�¤y¢t¨3¢¡Äp­ §3ÉHÃ�¦xÄm¢ ?¥���fbuo:[^aHiji�� aH�faHzH|^�Hbj]h[^a2_^b]�h[^b�]hc�aHeyoq�Hbj]h[^a�l � c�� ; ¢¡¤¥¤y¦x¤y¢VÆqÃ<¤y¦x«�Ä�Æ�¦��/Æ�¢¡ªÈªj¢V¦tÇp¢¡¤f§3ÉHÃ�¦xÄpÃQ��� Ç���?¥�

��� �� ��� �'� � #� ��� � � #�� &�� #���� � � � � � � �'� ��&������'��#�& �Ô buoq[^] ( Ï ����a �`Ï �?a6�fb�bj[^_^bjzt�^bj[^]�zH]h[ 0 � ; �"?`Ï � �7( � a 0�� ; �"?:Ï �� � ( � ijaeyoqc^c^efaH�faH[srtoq�Hbj]h[^b^l � c�� [��#_^bp|^[8[�|^k8aHef]�efaFoqijaø�:]qrfryaH[�|�ryaq��efbj�fc�a¡rfryb��qoqk8aH[sryaq�hc�aHe<ryef]h[^zFoqk8aH[sry]a8c�aHeVoqefef]qry]h[^_poqk8aH[sry]^��a`zH]h[�� iuo6k`oq[srybj�f�yo�[^]hefk`oqijbj�H�Foxrto�_^b<�>bj[�oqefb�ryk8a¡rybjzFo�]hef_^bj[poqefbuo; zHbj]^}a�oq_6bj[Q�p[^b�ryaBzHb�lnefa�?¥� � � Ô b�zH]h[^�fbj_^aHefbj[^]2b�[�|^k8aHefb�efbjc�]he�rtoxryb�bj[ �Koqg�aHijiuo �h���h�

���#bjk8k8aH_^buoxry]�]h�f�faHe��qoqefa�zt�^a � $aR � _^b5AmaHefbj�fzH]h[^]D_^bø|^[poD~�|poq[sryb�rx}oDbj[�lnaHefbj]hefa:o ��� � 3F�k8aH[sryefa!� 'a ��w_^b5AmaHefbj�fzH]h[^]�_^bm|^[po�~�|poq[sryb�rx}o�[^]h[��f|^c�aHefbj]hefa�o$��Ó � � ��� � 3 B�i�� aHefef]hefaNefaHiuoxryb��h]_^b�ef]h|^[^_^]%A>}a�k8bj[^]hefa�_^bD��� � � [^aHi�c^efbjk8]�zFoq�f]8a�_^b ��Ó � � ��� � � [^aHid�faHzH]h[^_^] ; �Koqg�aHijiuo �h�>=O?¥�

Ô b^��|^]hijaX]heyoB��rtoqg^bjijbjefaøbj[ �haH[^aHeyoqijaXbji^k`oq�f�fbjk8]�aHefef]hefaXefaHiuoxryb��h]Bzt�^aN�fbpzH]hk8k8a¡rfryaXoqc^c^ef]h�f�fbÿ�k`oq[^_^]ø|^[�[�|^k8aHef]XefaFoqijaK� <Ï>��zH]h[ 0 ; �"?�bj[B|^[��fbj��ryaHk`oXoqefb�ryk8a¡rybjzH]£zH]h[ � ; ( ?���?��& 6 .?��& 98h�"?¥�Ô b�zH]h[^�fbj_^aHefb��poq_6aH�faHk8c^bj]^�^bji�zFoq�f]8_^aHiji�� oqefef]qry]h[^_poqk8aH[sry]�B��fbuoq[^]

� Ï � �2( � ? 0 ; �"? Ï �� �-( �

efbj�fc�a¡rfryb��qoqk8aH[srya�iuoVeyoqc^c^efaH�faH[srtoq�Hbj]h[^a�l � c�� [��<_^b�� ; bj[6oqefb�ryk8a¡rybjzFo�]hef_^bj[poqefbuo@?Ca�iuoVeyoqc^c^efaH�faH[��rtoq�Hbj]h[^a�l � c�� [��Co2c^efaHzHbj�fbj]h[^a �p[^b�rto ; bj[ � ?øzH]hefefbj�fc�]h[^_^aH[sryaq� ��|^[^~�|^aÐ"� �Ð � (�� �

� 8 Kú6ã¡é ú:á ä�ùqã¡á£á ÷ û�è�é�üøá ä©è2ü�ú¥äHû�áçå�å�ú 'è2ý3û�á ÷ á % %fútû�ã�å�áçú�ùqè�é�á äxæ3áçâ�ú¥é�è2÷çú�ü�ú¥äHû�áçå�å�ú6æ3á£ý©ä�äFý©üøè�é�ã6á äé ú¥ù©ù©é�èfå�è�äHû ú&%�á ã¡ä©èdþnõ ù�õ ä�õ #xå�áçú£ùqè�édá äxæ3áçâ�ú¥é�è#÷çú£ùxú¥é�û�èKþÿé ú&%�á ã¡äxú¥é�áçú æ3è�÷�äFý©üøè�é�ãøá ä\û ú¥÷ è<é ú¥ù©ù©é�èfå�è�äHû ú&%�á ã¡ä©è¡õ

Page 21: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# = ,

� Ð"� �Ò= � Ð Ð"� �Ò �� Ð Ð 0 � ; �"?KÒ��<Ð �»Ð �<Ð Ð 0�� ; �"?#Ò��<Ð �»Ð �<ÐÒ ��Ó��O����=$� , � ��� � � ��Ó , � � ��� � 3 ��Ó �O� � ��� � 3 ��Ó �Q� � ��� � � ��Ó>=$� � ��� � ���Ó��O�O�O��= " � � ��� � ��Ó " � � ��� � 3 ��Ó " � � ��� � 3 ��Ó " � � ��� � � ��Ó " � � ��� � ���Ó � , =$��� , � � ��� � ��Ó , � � ��� � 3 ��Ó>=�= � ��� � 3 ��Ó�� " � ��� � � ��Ó � � � ��� � 3Ò ��Ó " �Q� ,, � � ����� ��Ó � � � ��� � 3 ��Ó � � � ��� � 3 ��Ó��O� � ��� � � ��Ó��O� � ��� � �

� ¼ �<°�¾�¾n¼P¶p·87�� :¼ � ��º©°�½H°�²�À3¼^´�³nµ�²d°��Kµ»¼�À©³�² �� �<µ�³�²�À`²dµ»º���¼p¾�³n´s´s¼�À3¼��B°�º©º©µ»º©° ��³ º©µ�¿�² �� µ���¼p½F½Hµ�¾�¿�À3µ�°Dº©°�¾n¼�À©³69�µ

a � )a0 ��D_^b5AmaHefbj�fzH]h[^]2c�aHeX|^[po�~�|poq[sryb�rx}o2[^]h[6�f|^c�aHefbj]hefaYo ��Ó � �-( � �®�^zHbj]^}aÐ � �Ò �� Ð�� �= ( � � Ó

Ô b��po$�Ð 0 ; �"?KÒ��<ÐÐ �<Ð Ï Ð ��:Ò � *Ð

Ð"� �Ð ��� ( � �( � � Ï

�= ( � � � ?

_po`zH|^bKefbj�f|^i�rto2zt�^a �� (#� � �X}a�bjiCÅ8§©¨t¨¡ÃÈÅ`¦6¢¡¤¥¤y¦x¤y¢V¤y¢¡ª�§q­�Ã65q¦Yzt�^aV�fb�zH]hk8k8a¡rfryaVoqc^c^ef]h�f�fbjk`oq[^_^]8�zH]h[ 0 ; �"?¥�D3N[poqij]�h]2efbj�f|^i�rtoxry]V�f|^�f�fbj��ryaYc�aHeXbjimryef]h[^zFoqk8aH[sry] �Ð 0 ; �"?KÒ��<ÐÐ �<Ð �+( � � � Ó

¯�°s±#²�³n´�³nµ�²d°�¶p· � ·`¸�� ¼p½F½F³���¼�¼ ���s¿dº©¼�À3°�´s´s¼º©°�¾n¼�À©³69^¼�Á ÃdÆqÃ�Ët¢�Å8§©¨t¨¡ÃÈÅ8§�§�ËtË¡«�¤f§q­®¢yÉtÉ3§Y¤y¢¡ª�§q­�Ã65x§�ÆqÃ�«�Ä8¨¡Ãj¨¡­®¢¡Å8§V§q¤¥ÃÈ­�Å`¢¡­�Ã�Ët¦D��� Ç��\§�Ç�¤y¢tË¡Ãj¨¡Ã�¦xÄm¢ 6CÄpÃÈ­ §�Ët¦xÄ� ; ( ?���?��& 6 .?��& 98h�"?VÃȪ�Å8§©¨t¨¡ÃÈÅ`¦6¢¡¤¥¤y¦x¤y¢8ËE�p¢�¨¡Ã�Ët¦xÅ�Å`¢¡­�­®¢VÄm¢¡ªd¤f§¥ÇsÇ�¤y¢t¨3¢¡Äp­ §q¤y¢V«�ÄÄp«�Å`¢¡¤y¦��Äm¢¡ª�¨¡Ãj¨¡­®¢¡Å8§ �

� Ï> 98h� Ð 0 ; �"?#Ò��<ÐÐ �<Ð Ï �= ( � � � ? � 0 � 8����>��� 0 0 : 8�� ������ ��8@ � ��£Ó� :^Ï 98h� Ð 0 ; �"?#Ò��<ÐÐ �<Ð Ï ( � � � ? � 0 � 8����>��� 0 � ��� � 8@ � ��£Ó

�£]hk8a8�fbC�haH_^e©}owbj[��fa �h|^b�ry]^�Kiuo6k`oq�f�fbjk`o6oqzHzH|^eyoxryaH�H�FoDefaHiuoxryb��qo�}a8|^[po�_^aHijija8zH]h��rtoq[srybCln]h[^_po©�k8aH[srtoqijbm_^b�|^[��fbj��ryaHk`o2oqefb�ryk8a¡rybjzH]�l � c�� o�c^efaHzHbj�fbj]h[^a �p[^b�rto��ÖP×KØfÙFÚ�Û^Ü�Ý2Þqß�Þ��sß#àFáçú¥ä©ã,AYë?ìfí£è �»ë�ó3õpö�ú£ü�ú¡å�å�á ü�úøú¡â�â ý©é útû�è % %fú é�è�÷çútû�á -túK'è

� ë í©î ó M�ìfí��%3 N�ú¥é�é�ã¥û�ã¡äxæ©ú¥üøè�äHû�ã P <� I ë ìfí��%3 N¬û�é�ã¡äxâ�ú¥üøè�äHû�ã Pqî

Ö

Page 22: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

= �

Ô b�]h�f�faHe���bdzt�^aq�»c�]h��ry]��BÏ ; 0 ; �"?KÒ��"?��©�d�p�f|^�f�fbj��ryaYiuoVefaHiuoq�Hbj]h[^a�� 0 ; �"?CÏ�� ; � 5 � ?�? Ð �pÐ�� � ; Ð �pÐ�� � : ?mÓ ; �h� 1 ?

�Bo`]heyo:bj[?c�]hb#�fb#c^efaH[^_^aHe©}o�bj[?zH]h[^�fbj_^aHeyoq�Hbj]h[^a��f]hij]�i�� oqefef]qry]h[^_poqk8aH[sry]^�mc�aHefzt���a6}a�ij]:�fzt�^aHk`o|^�yoxry]2[^aHijiuoVk`o���hbj]he£cpoqe�rya�_^a �hijb�zFoqijzH]hiuoxry]hefb��

��� �� ��� �'� � #� ��� � � #�� &�� #���� ������� #�� �'� ���'��#�& � � #� � ��&������ #���&��®[|^[�fbj��ryaHk`o6oqefb�ryk8a¡rybjzH]`l � c��8o�c^efaHzHbj�fbj]h[^a��p[^b�rto:�fbCbj[sryef]�_^|^zHa�|^[aHefef]hefa2[^aHiCefbj�f|^i�rtoxry]�_^b]�h[^b ]hc�aHeyoq�Hbj]h[^a:oqefb�ryk8a¡rybjzFo�� �£]hk8a`�yoqe©}owzt�^buoqefb�ry]wbj[��fa �h|^b�ry]^�<oq[^zt�^a�iuo�efbj�f]hij|^�Hbj]h[^a8_^b |^[c^ef]hg^ijaHk`o�oqc^cpoqefaH[sryaHk8aH[srya\�faHk8c^ijbjzHaq�h~�|poqija\oq_2aH�faHk8c^bj]B|^[�� aH~�|poq�Hbj]h[^aNoqi*�haHg^efbjzFoN_^bp�faHzH]h[^_^]�heyoq_^]^��c^|�}]*efbj�f|^i�rtoqefa�ry|�rfrF� oqi�ryef]�zt�^a?gpoq[poqijaq�Xbj[ ~�|poq[sry]*[^aHi�c^ef]�ha¡rfry]�_^bN|^[�oqi*�h]hefb�ryk8]P}aaH�f�faH[^�HbuoqijaCryaH[^aHefa£zH]h[sry]�_^aHijiuoXc^efaH�faH[^�FoN_^aHiji�� aHefef]hefa _^b�ef]h|^[^_^]%AK�qzt�^aøoqi�ryefbjk8aH[srybsc^|�}]NzH]h[^_^|^efefaoq_6|^[po2�f]hij|^�Hbj]h[^aBzH]hk8c^ija¡rtoqk8aH[srya�aHefeyoxrto��Ô bK�f|^c^c�]h[ �so�zt�^a�bji��fbj��ryaHk`o:oqefb�ryk8a¡rybjzH] �soqeyoq[srybj�fzFo:iuo8k`oq�f�fbjk`o:oqzHzH|^eyoxryaH�H�Fo�_^bj[poqk8bjzFo+B

oqijij]heyo���bj[^_^bjzFoxrto�zH]h[ �$|^[po*~�|poqij�fbuoq�fb�_^aHijijaD~�|poxrfryef]>]hc�aHeyoq�Hbj]h[^bNaHijaHk8aH[srtoqefb�a?zH]h[ � �$iuozH]hefefbj�fc�]h[^_^aH[srya�]hc�aHeyoq�Hbj]h[^a�l � c����p�fa 8`a �N�f]h[^]8_^|^a�[�|^k8aHefb�k`oqzHzt�^bj[po��fb��po$�

8 � � ��Ï� 0 ; 8 � �2?CÏ ; 8 � �2? ; � 5 � ?�? ; �h�����@?zH]h[*Ð �pÐ�� � �

ÖP×KØfÙFÚ�Û^Ü�Ý2Þqß�Þ���ß#àFá¡â ã¡äxå�áçæ3è�é�á ä©ãbA�ë?ìfímè �»ë:ñ�è�å�á¡å�ã¡üøüøá ä©ã�á¡äFý©üøè�é�á 3 ë6í©î ótñHð2L�M�ìfí�� # ë6í©î <&%Fì�ñ�M�ìfí �è_edë6í©î ì�ñHð¡ô M�ìfí�� � õ��å�è �¡ý©è�äxæ3ã£÷ è<ú¡æ©æ3á %�á ã¡ä©á�å�èfâ ã¡äxæ3ãø÷*) ã¡é æ3á ä©è N 3�� P � e #xâ ã¡äNý©ä�é�è �¡áçånû�é�ã£ú�æ3ã¡ù©ù©áçú£ù©é�èfâ áçå�á ã¡ä©è�#3å�á+'xú�:

N�í©î ótñHð2LUM�ìfí � � í©î <&%Fì�ñUM�ìfí � P � í©î ì�ñHð¡ô M�ìfí � � ëë í©î L¡ô2L&%DM�ìfí � � í©î ì�ñHð¡ô M�ìfí � � ë�í©î L¡ô2L < M�ìfí � <â�ú¥÷çâ ã¡÷çú¥äxæ3ã á ä&-¡èfâ è 3�� N � e*P�å�á 'xú�:

í©î ótñHð2L M�ìfí � � N�í©î <&%Fì�ñ M�ìfí � � í©î ì�ñHð¡ô M�ìfí � � P»ëë í©î ótñHð2L M�ìfí � � í©î <&%¥ï¥ð M�ìfí � ë�í©î L¡ô2L¡ð M�ìfí � î�»é�áçå�ý©÷ û útû�áhã¥û�û�è�äFý3û�ásä©è�á�æ3ý©è<â�ú¡å�ásæ3á 1hè�é�áçå�â ã¡ä©ã£æ3ápìKý©ä©á ûE'úøå�ý©÷ ÷*) ý©÷ û�á ü�úøâ á þÿé ú£å�á �¡ä©á �qâ�útû�á -tú èD)Hý©á äxæ3áhä©ã¡ä -tú¥÷ èK÷çúù©é�ã¡ù©é�á è ûE'ú£ú¡å�å�ã3â áçútû�á -tú£æ3è�÷ ÷*) ú¡æ©æ3á %�á ã¡ä©è¡õ Ö

ÖP×KØfÙFÚ�Û^Ü�Ý2Þqß�Þ���ß#àFáçú¥ä©ã AYë?ìfí è �»ë:ñøè<å�á�â ã¡äxå�áçæ3è�é�á ä©ã ásäFý©üøè�é�á 3 ë�í©î ï¡ïtñHí4MYìfí�� # ë�í©î %tðHó¥ô4MYìfí � èedë�í©î ô¡ôHï¥ð M�ìfí � õ��û�á ÷ á % %fú¥äxæ3ãøý©ä�é�è �¡áçånû�é�ãøúøæ3ã¡ù©ù©áçú£ù©é�èfâ áçå�á ã¡ä©è�#©å�á 'xú�:

N 3�� P�� edëë N�í©î ï¡ïtñHíUM�ìfí � � í©î %tðHó¥ôUM�ìfí � P��2í©î ô¡ôHï¥ð M�ìfí � ëë í©î ô¡í¡ôHó4M�ìfí � �2í©î ô¡ôHï¥ð M�ìfí � ë�í©î ìfí©ìfí M�ìfí 3

Page 23: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# = 1

èN 3 � e*P � N � e*P»ëë N�í©î ï¡ïtñHí M�ìfí � �2í©î ô¡ôHï¥ð M�ìfí � P � N�í©î %tðHó¥ô M�ìfí � �2í©î ô¡ôHï¥ð M�ìfí � P�ëë í©î %yñFó %DM�ìfí 2 � í©î ï L¥ñ < M�ìfí 2 ë�í©î ìfí©ì¡ì MVìfí 3 î

ö�èXæ3ý©èXå�è�)Hý©è�ä�%�è�æ3á»ã¡ùqè�é ú&%�á ã¡ä©ápä©ã¡ä�â ã¡äxæ3ýxâ ã¡ä©ãBú¥÷ ÷ ãBånû�èfå�å�ã�é�áçå�ý©÷ û útû�ã�è )Hý©á äxæ3á»ä©ã¡ä 'è(-tú¥÷ áçæ©úN÷çú�ù©é�ã¡ù©é�á è ûE'úæ3áçånû�é�á �©ý3û�á -tú�æ3è�÷ ÷çú£üøã¡÷ û�á ù©÷ áçâ�ú&%�á ã¡ä©è#é�áçå�ùqè û�û�ã�ú¥÷ ÷*) ú¡æ©æ3á %�á ã¡ä©è¡õ Ö

021�� ��@BQ?J#;FEH587 TUGVW\W��?;F7DGÔ |^c^c�]h[^buoqk8]6_^b aH�fa �h|^bjefaq�Kbj[�|^[��fbj��ryaHk`owoqefb�ryk8a¡rybjzH]:l � c��6zH]h[ (ÕÏ ���6a �YÏ ����_^]qrtoxry]w_^bk`oq�f�fbjk`o�oqzHzH|^eyoxryaH�H�Fo2_^bj[poqk8bjzFo���i�� oq_^_^bj�Hbj]h[^a��

��Ó 1�1 " � ��� � ��Ó�� ,, � ����� � Ï���Ó 1�1 " � ��� � ��Ó �O�O�Q� ,, � ��� Ï>��Ó 1�1 " � ��� Ó®[ �haH[^aHeyoqijaq�m~�|poq[^_^]��fbKaQAma¡rfry|po�|^[�� oq_^_^bj�Hbj]h[^aYl � c��Nryeyo:[�|^k8aHefbKoF�haH[sryb<]hef_^bj[^b�_^b��heyoq[^_^aH�H�Fo_^b5AmaHefaH[sryb�c^|�}]�oqzHzFoq_^aHefaVzt�^aVbjidefbj�f|^i�rtoxry]8�fbuo`| �h|poqija�oqiji�� oq_^_^aH[^_^]`k`o���hbj]hefa ; bj[��qoqij]hefa�oq�f�f]x�ij|�ry]�?¥� � |^aH��ry]8lnaH[^]hk8aH[^]w}a�_^]3��|�ry]:oqijiuoVc^efaHzHbj�fbj]h[^a �p[^b�rto�_^aHid�fbj��ryaHk`o8oqefb�ryk8a¡rybjzH]�|�rybjijbj�H�Foxry]aH_�aH�fbj��ryaYoq[^zt�^aY�fa�bji��fbj��ryaHk`o �soqeyoq[srybj�fzHaBiuo�k`oq�f�fbjk`oVoqzHzH|^eyoxryaH�H�Fo8_^bj[poqk8bjzFo��®[�cpoqe�rybjzH]hiuoqefaq�^}aøbj[sryaHefaH�f�yoq[sryaX��ry|^_^buoqefaøbji�zFoq�f]�bj[VzH|^b^i�� oq_^_^aH[^_^]Bk`o���hbj]hefa bj[V�qoqij]hefaøoq�f�f]hij|�ry]}a�bji�[�|^k8aHef] �h�¯�°s±#²�³n´�³nµ�²d°�¶p·���·`¸�¹ �K½F³�¾nµ�² ��¼ ������³�²d¼�Á�¥Ä�«�Ä�¨¡Ãj¨¡­®¢¡Å8§D§q¤¥ÃÈ­�Å`¢¡­�Ã�Ët¦0��� Ç��?Ët¦xÄ � ; ( ?���?��& 6 .?��& 98h�"?yÊ\¨¡ÃXÆqÃ�Ët¢�¢®Çp¨¡ÃȪj¦xÄ�Å8§�ËtËE��ÃÈÄ�§�ÃȪ�Ç�Ã�Í«Ç�Ã�ËtËt¦xªj¦2Äp«�Å`¢¡¤y¦ * �'­ §qªj¢�ËE�p¢

����£Ï� 0 ; � 5 E? .��CÓ ; �h���O� ?

Ô b#�f|^c^c�]h[ �so���oq_�aH�faHk8c^bj]^�$( Ï ����a ��Ï ��a2�fb<zFoqijzH]hijb�bji#c^by}| �heyoq[^_^a2[�|^k8aHef] �)* ��rtoqijazt�^a � �

�����2Ï �CÓ ; �h����=O?�£�^buoqeyoqk8aH[srya � ���?Ï ��Ó � � � � � � � �����s��zH]h[ � � ��� Ô b<�f|^c^c�]h[ �so6bj[^]hi�ryefa�zt�^a8i�� oq_^_^bj�Hbj]h[^al � c��<�fbuo�aH�fa �h|^b�rto2zH]h[�|^[�efa �hbj��ryef]2o2_^]hc^c^buoVc^efaHzHbj�fbj]h[^aq�� � � )^þ�ú¡â á ÷ è�â ã¡ä&-Fá äxâ è�é å�á�â('©è�÷çúY÷ á üøá û útû�è % %fúVæ3è�÷�å�áçånû�è�ü�úYþnõ ù�õ`ú�ù©é�èfâ áçå�á ã¡ä©è��xä©á û ú�á üøù©÷ áçâ�ú�÷*) èfå�áçånû�è�ä�%fú�æ3áè�÷ è�üøè�äHû�á�æ3á -¡è�é å�á�æ©ú¥÷ ÷ ã %�è�é�ãXâ('©è å�á�â ã¡üøùqã¡é�û ú¥ä©ãXâ ã¡üøè<÷ ã�%�è�é�ã�ä©è�÷ ÷çú�å�ã¡üøü�ú£þnõ ù�õdâ ã¡äVì¡õ��ÈäBé�èfú¥÷ ûE'ú£èfå�áçånû�èCý©äá äxå�á è�üøèCæ3ásäFý©üøè�é�á�ü�ú¡â�â('©á äxú£â('©è �¡ã3æ3ã¡ä©ãøæ3á�)Hý©èfånû ú£ù©é�ã¡ù©é�á è ûE'ú©õ

Page 24: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

�O�

3\�8[^zt���aY�fbuoV�qoqijbj_po ; �h����=O?¥��i�� ]hc�aHeyoq�Hbj]h[^a���� �:_^a¡�ha�aH�f�faHefaYaQAma¡rfry|poxrto2zH]hk8aB�fa �h|^a��

��Ó � � � � � � � ����� 5��Ó � � � � � � � � � � ����� Ï��Ó � � � � � � � � � � ����� ; �h���=�@?

aViuo8ryaHef�Fo:zHb�lneyo:�fb*�h[^b��pzFoxryb��qo2_^aHi#efbj�f|^i�rtoxry]:[^]h[?_^a¡�ha2aH�f�faHefa2k8]�_^b��pzFoxrto8~�|poq[^_^]6�fbKaQAma¡rfry|poi�� oqefef]qry]h[^_poqk8aH[sry]�o ��zHb�lnefaq� � |^aH��rF� |^i�rybjk`oDzH]h[^_^bj�Hbj]h[^a6aH~�|^b��qoqija�o�bjk8c�]hefefa:zt�^a � � � ���k8aH[sryefa�_po ; �h���=�@?¥��oF�haH[^_^] aQAma¡rfry|poxry] |^[^]P�f�^b�lÈr�_^aHijiuo*k`oq[srybj�f�yo>_^b � _^bBryefazHb�lnefaD�haHef�f]_^aH��ryeyo��p�fb�efbjzFoF�qo

� 5 ��Ï ��? _po2zH|^b ��Ï �XÒ �YÏ/Ò =�Ó8�aHe�rtoq[sry] �2Ï���Ó " 1�1 � ��� � � a�~�|^bj[^_^b £Ï>��Ó � �O� � ��� � � �®[ �haH[^aHeyoqijaq�pbj[6|^[��fbj��ryaHk`o2oqefb�ryk8a¡rybjzH]�l � c��<zH]h[ � ; ( ?���?��& 6 .?��& 98h�"?¥�p�fb��po$�

£Ï �= ( � � � ?[^aHi�zFoq�f]8_^aHiji�� oqefef]qry]h[^_poqk8aH[sry]^��k8aH[sryefaq�

£Ï ( � � �[^aHi�zFoq�f]8_^aHi�ryef]h[^zFoqk8aH[sry]^��<� aHc^�fbjij]h[�k`oqzHzt�^bj[poDzH]hbj[^zHbj_^a`zH]h[>iuowk`oq�f�fbjk`owoqzHzH|^eyoxryaH�H�Fo�efaHiuoxryb��qo � � 8�aHe�rtoq[sry]^�C[^aHi

�fa �h|^b�ry]^�»bdryaHefk8bj[^bdaHc^�fbjij]h[wk`oqzHzt�^bj[po8a�k`oq�f�fbjk`o2oqzHzH|^eyoxryaH�H�Fo:efaHiuoxryb��qo2�yoqeyoq[^[^]:zH]h[^�fbj_^aHeyoxryb�fbj[^]h[^bjk8b��{\]qrto�iuo�gpoq�fa`_^b [�|^k8aHeyoq�Hbj]h[^aq�KzH]h[^]h�fzHaHefa:i�� aHc^�fbjij]h[k`oqzHzt�^bj[po�aH~�|^b��qoqija`owzH]h[^]h�fzHaHefa:iuo

c^efaHzHbj�fbj]h[^a�_^aHim�fbj��ryaHk`o�oqefb�ryk8a¡rybjzH]�B�_�� oqi�ryeyo�cpoqe�ryaBiuo�_^a2�p[^bj�Hbj]h[^a���ryaH�f�yo2_^bmaHc^�fbjij]h[:k`oqzHzt�^bj[po�f| ��haHefbj�fzHaB|^[�k8]�_^]Vc�aHe�zFoqijzH]hiuoqefaXrtoqija�~�|poq[sryb�rx}oV[^aHi��fbj��ryaHk`oVoqefb�ryk8a¡rybjzH]�_^b�|^[�zFoqijzH]hiuoxry]hefaq�~�|poq[^_^]�[^]h[:}a£[^]qrto\iuo\c^efaHzHbj�fbj]h[^aq�K�<� oqi*�h]hefb�ryk8]Xc�aHe�bji�zFoqijzH]hij]\_^aHiji�� aHc^�fbjij]h[Yk`oqzHzt�^bj[poY}a£bj[�l�oxrfrybgpoq�yoxry]2�f|^ijiuoV_^a2�p[^bj�Hbj]h[^a��h� �+Bpbji��qoqij]hefa �p[poqijaB_^b � }aq�»oqc^c^|^[sry]^�pi�� aHc^�fbjij]h[�k`oqzHzt�^bj[po��

Page 25: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# �Q�

��º©µ��h°���¿dº©°:c^efaHzHefaHi ; µ�¿�À � � ?��� �������� Ëy§qªjËt¦xªj¦`Æ�¢¡ªÈª�� ¢®Çp¨¡ÃȪj¦xÄwÅ8§�ËtËE��ÃÈÄ�§��� ��� � �� � ��������������������� � �9^¼^º � � �Cº©°�¼p¾n° ¢®Çp¨¡ÃȪj¦xÄwÅ8§�ËtËE��ÃÈÄ�§

����� ��� � ��� ��� ��� � � � � � �9^¼^º � � � �Cº©°�¼p¾n° 5x§q¤¥Ã�§ �HÃȪj¢YÆqà §¥ÇsÇp¦R4 4qÃ�¦�9^¼^º � � = �Cº©°�¼p¾n° 5x§q¤¥Ã�§ �HÃȪj¢YÆqà §¥ÇsÇp¦R4 4qÃ�¦�

��� ��! �#" � � � $���&% " � � ! � �� ���çÏ �hÓ5Bº©°��<°�¼�À

� �çÏ � ��B� ���çÏ � ��� ( B� = �çÏ � � 5 �hÓ5B

¿�²�À©³�¾ ; � =�Ï �hÓB?°�² ��c^efaHzHefaHi

' (*),+.-./01(3254167418:9<; =>)1(@? A#B�)DC>-E($?F;,+G21; +.)H; )I/1-E; ;FJ -ECLK#?F; )HMNB�2L+.+EO>?FME2

®k8c^ijaHk8aH[srtoq[^_^]�i�� oqi*�h]hefb�ryk8]Bbj[ � ]he�ryeyoq[`�f|:|^[�zFoqijzH]hiuoxry]hefa�J��v � Ô � Ô ���ryaHk5P$�O�O�O����bj[�fbj[ �h]hiuoVaH_�bj[�_^]hc^c^buo�c^efaHzHbj�fbj]h[^aq�p�fb��po�efbj�fc�a¡rfryb��qoqk8aH[srya�� � �

� Ï = � � �RQ ��Ó��O� 1 = � 1 � � ��� � �D?� Ï = � 3 �RQ ��Ó>=�=�= � "�" � � ��� � �43dÓ ; �h��� " ?

�£]h[`|^[`eyo��hbj]h[poqk8aH[sry]�oq[poqij]�h]�oqipc^efaHzHaH_^aH[srya��fbp�haHefb��pzFoYzt�^aq���faX�D?�� * ��aX�6Ï � ( � �efbj�f|^i�rto$�

� � � <Ï��DSNTUÐ �dÐ � � Ï �= ( � � � ?�.h � )sá äHû�è�é�èfå�å�ú¥äHû�è£ä©ã¥û ú¥é�è�â('©èø÷*) á üøù©÷ è�üøè�äHû ú&%�á ã¡ä©è�æ3á é�è û�û ú�æ3è�÷ ÷*) ú¥÷ �¡ã¡é�á û�üøã�æ3èfå�â é�á û�û�ã�ä©è�÷ ÷çú � é�ã3â èfæ3ý©é ú�ì¡õ ì�#ù©ýQ'ã<ùqã¡é�û ú¥é�è�úKé�áçå�ý©÷ û útû�áxæ3á -¡è�é å�á©å�ý©÷ ÷ ã<ånû�èfå�å�ã<â�ú¥÷çâ ã¡÷çútû�ã¡é�èmå�èdå�á3ýxå�ú¥ä©ã<â ã¡üøù©á ÷çútû�ã¡é�á©æ3á -¡è�é å�ájõ . ÷çâ ý©ä©áxâ ã¡üøù©á ÷çútû�ã¡é�á

N�ú¡æ�èfå�õ�Báçâ é�ãHå�ã¥þ¬û ©ã¡é�û�é ú¥ä(L©õ í�P^ã¥û�û�á üøá % %fú¥ä©ãCá ä3þ�útû�û�áhá ÷qù©é�ã��¡é ú¥üøü�úKèfå�è �¡ý©á �©á ÷ èdü�ú¥äHû�è�ä©è�äxæ3ãCá -tú¥÷ ã¡é�áqæ3á � ìdèfæ� ïøå�è�üøù©é�è<ä©è�á�é�è �¡áçånû�é�ájõ �Èä�û ú¥÷�â�ú¡å�ãø÷*) ú¥÷ �¡ã¡é�á û�üøã�æ3è/-¡èCèfå�å�è�é�è#á üøù©÷ è�üøè�äHû útû�ã ��â ãHånû�é�á ä��¡è�äxæ3ã��£á ÷�â ã¡üøù©á ÷çútû�ã¡é�èú�â ã¡äxå�è�é -tú¥é�è � ï£á ä�üøè�üøã¡é�áçúøâ è�äHû�é ú¥÷ è2V©å�á�-Fý©ã¡÷ èCâ�ú¥÷çâ ã¡÷çú¥é�èK÷*) è�ùxå�á ÷ ã¡äNü�ú¡â�â('©á äxú£é�è�÷çútû�á -¡ã�ú¥÷ ÷çú£ù©é�èfâ áçå�á ã¡ä©è<æ3è�÷ ÷ è÷ ã3â�ú&%�á ã¡ä©á�æ3ásüøè�üøã¡é�áçú�#3ä©ã¡ä�ú()Hý©è�÷ ÷çúøæ3è�á�é�è �¡áçånû�é�á *

Page 26: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

� =

zHbj]^}aq���fawÐ �dÐZ� � � �6[^]h[?_�}o`zH]h[sryefbjg^|�ry]�oqijiuo2�f]hk8k`o�� � |^bj[^_^b���_poxry]:|^[?~�|poqij|^[^~�|^aV[�|^k8aHef]k`oqzHzt�^bj[poV�d��aH�fbj��ryaY|^[wbj[^�fbjaHk8aB_^bd[�|^k8aHefb�k`oqzHzt�^bj[po2zt�^a��fbdzH]hk8c�]he�rtoq[^]2zH]hk8a�ij]8�HaHef]`[^aHiÿ�i�� oq_^_^bj�Hbj]h[^a�l � c��\zH]h[w�d� � |^aH��rto`]h�f�faHe��qoq�Hbj]h[^a`}a�_^b�bjk8c�]he�rtoq[^�FoVln]h[^_poqk8aH[srtoqijaY[^aHi�c^ef]�ha¡rfry]_^b�oqi*�h]hefb�ryk8b�[�|^k8aHefbjzHb��Ô b�[^]qryb�zt�^a

� Ï �= ( � � � Ï ( � � � ��= ( � � � Ï ( � � � � � �(Ð �<Ð � � B

bj[ �haH[^aHeyoqijaq�p~�|^bj[^_^b��^�fbdoq�f�f|^k8a � ÏÑÐ �<Ð � � ?

bj[�~�|poq[sry]2rtoqijaB[�|^k8aHef]8�fb�zFoqijzH]hiuoVbjk8k8aH_^buoxrtoqk8aH[sryaNo2cpoqe�rybjefa�_po��Da�_po � �Ô b��f|^c^c�]h[ �so:_^bd�h]hijaHeNzFoqijzH]hiuoqefa � � �mzH]h[D� . ��� � |^aH��ry]�c^ef]hg^ijaHk`o8k`oxryaHk`oxrybjzH]2c^|�}]�aH�f�faHefaoqc^c^ef]h�f�fbjk`oxry]�_poqi�c^ef]hg^ijaHk`oV[�|^k8aHefbjzH]���zFoqijzH]hij]�_^b�|^[po��f]hk8k`oq��_^aHi�rybjc�] �

��� �� ���

� � �� Ï � 5 � 5 � �= 5 � ���� 5 ÓFÓFÓ 5 ����� Ó ; �h�����O?

��� }a\|^[�� oqc^c^ef]h�f�fbjk`oq�Hbj]h[^a\_^b � � ��iuo�zH|^bø��g�]h[srx}oq�V_^bjc�aH[^_^aN_po�"aH_?}a\rtoq[sry]�k8b*�hijbj]hefaø~�|poq[sry]k`o���hbj]hefa�}a� � ��Bo�|^[�c^|^[sry]8_^bm��bj��rto8oqi*�h]hefb�ryk8bjzH]Ybji�zFoqijzH]hij]�_^b ��� }a�gpoq�yoxry]2�f|^ijij]2�fzt�^aHk`o$�

� Ï �� � Ï � � � � 5 � � �� �� ? wÏ ��?�=�?FÓFÓFÓ ?��(Ó ; �h���=�@?

Z\[po6ln]hefkV|^iuo6_^aHiCrybjc�]�_^aHijiuo ; �h���=�@?B�yoqe©}o�_^a¡rfrto �F¦x¤¥Å�«�ª�§w¤¥Ã�Ët¦x¤¥¤y¢¡Äp­®¢�a`iuo�zH]hk8c^|�rtoq�Hbj]h[^agpoq�yoxrto��f|�_^b�aH�f�yo8�yoqe©}o�_^a¡rfrtoVÇ�¤y¦3Ët¢t¨t¨3¦2ÃÈ­®¢¡¤f§q­�Ã65q¦x��<� oq�fc�a¡rfry]�zHaH[sryeyoqija�_poqimc^|^[sry]2_^b���bj��rtoV[�|^k8aHefbjzH]�zH]h[^�fbj��rya�[^aHijiuo�_^a¡ryaHefk8bj[poq�Hbj]h[^aN_^aHi�[�|��

k8aHef]�� _^a �hijbKoq_^_^aH[^_^bK_po`�f]hk8k`oqefaq� ®[ �Koqg�aHijiuo �h� �8�f]h[^]�efbjc�]he�rtoxryb�b��qoqij]hefbd_^bd� � �� ���aV_^b� � c�aHe��6Ï ��a �Ï ��?�=�?FÓ�Ó�Ó5?�=$����zFoqijzH]hiuoxrybm|�rybjijbj�H�Foq[^_^] , zHb�lnefaB_^aHzHbjk`oqijbm�fb*�h[^b��pzFoxryb��haq� � �Ô bB[^]qryb�zt�^a�o*cpoqe�rybjefa?_po Ï =��h� � � [^]h[��qoqefbuo��\]3���haHef] bji\ryaHefk8bj[^aD� � �� ��ø[^]h[Õ_�}ozH]h[sryefbjg^|�ry]�oqijiuo��f]hk8k`o � � � � 5 � � �� ��5B�bpzFoqijzH]hijbpaQAma¡rfry|poxryb�o�cpoqe�rybjefaX_po wÏ =��\�f]h[^]�_^|^[^~�|^a

bj[�|�rybjijb��Ô |^ijiuo8gpoq�fa�_^aHiji�� ]h�f�faHe��qoq�Hbj]h[^aVc^efaHzHaH_^aH[sryaq�d�fb#c^|�}]6o AmaHefk`oqefaVzt�^a�c^bj|�rfry]h��ry]:zt�^a��p�f�yoqefa2o

c^efbj]hefb ���}aXc^by}|2zH]h[s�haH[^bjaH[sryaNzt�^aN�fbuoBi�� oqi*�h]hefb�ryk8]N��ryaH�f�f]�o�_^a¡ryaHefk8bj[poqefaX_^bj[poqk8bjzFoqk8aH[sryaq�hzHbj]^}a[^aHi�zH]hef�f]B_^aHijiuoX�f|po�aH�faHzH|^�Hbj]h[^aq�sbjis�qoqij]hefa c^by}|�]hc^c�]he�ry|^[^]�_^b���<{\aHi�zFoq�f]B[^]h[V�fbuoN[^]qrtoNoqijzH|^[pobj[�ln]hefk`oq�Hbj]h[^a �f|^iji�� aHefef]hefa£_^b�ryef]h[^zFoqk8aH[sry]�oq[poqijb�rybjzH]^�xiuo��fzHaHi�rto�c^by}|V[poxry|^eyoqija�}a�zt�^a�i�� oqi*�h]hefb�ryk8]zH]h[srybj[�|^b�bji�c^ef]�zHaH�f�f]`b�ryaHeyoxryb��h] ; �h���=�@?øoqefefaH��rtoq[^_^]h�fbdoqi�c^by}|�c^bjzHzH]hij]V�qoqij]hefa �c�aHeXzH|^b��fbdoqg^g^buo2 d �Èä )Hý©èfånû�ã�â�ú¡å�ãøå�á+'xú£÷ á ü�� � #�� �wë.�"!2 � �Èüøù©÷ è�üøè�äHû ú&%�á ã¡ä©è<å�ý/�[c ��� ��à*C�à��3ånû�è�ügfEL¡í¡í¡í�#Há ä ©ã¡é�û�é ú¥ä+#3ý3û�á ÷ á % %fú¥äxæ3ãø÷çúøå�á ä��¡ã¡÷çú£ù©é�èfâ áçå�á ã¡ä©è¡õ

Page 27: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# ���

� � �� �� � �� ��Ó����O�O�O�O�O� � ����� ��Ó����O�O�O�O�O� � ������ ��Ó � �O�O�O�O�O� � ����� ��Ó �O�O�O�O�O�O� � �����= ��Ó���= � �O�O�O� � ��� � ��Ó�� � � �O�O�O� � ��� �� ��Ó>= � � ������� � ��� � ��Ó � 1 ������� " � ��� �" ��Ó>=$�O� " �=� , � ��� � ��Ó � �$� , � �O� � ��� �� ��Ó>=$�O� " �=��� � ��� � ��Ó 1 � " �=����� � ��� �� ��Ó>=�� , �Q�=� 1 � ��� � ��Ó��O�=�Q�O� � � � ��� �, ��Ó���� � �O� 1�1 � ��� � ��Ó���= � �Q� 1 � � ��� �� ��Ó 1 � �O� �O� 1 � ����� ��Ó��=� � �O� , = � ��� �1 ��Ó �$� � =�= �O� � ����� ��Ó�� " ��� � 1 � � ��� ���� ��� ���� � ��Ó � 1 � � = � � � ��� � � ��Ó�� " � " ��= 1 � ��� �� 1 ��Ó����$� , 1 � = � ��� � � ��Ó�� " � " �=�Q� � ��� �= � ��Ó � 1 � 1�1 � � � ��� � � ��Ó�� " � " �=� = � ��� �=�� ��Ó 1 �����Q��� , � ��� � 3 ��Ó�� " � " �=� = � ��� �=�= ��Ó>=���=��O�=�Q� � ��� � 3 ��Ó�� " � " �=� = � ��� �=$� ��Ó " �Q�O��=�� 1 � ��� � � ��Ó�� " � " �=� = � ��� �

� ¼ �<°�¾�¾n¼�¶p·������Y¼p¾nµ»ºx³K��³\� � �� ��<° ��³ � � µpÀFÀ3°�²�¿�À©³ �hµ�²�¿�²���º©µ��»º©¼ � ��¼>³�² ��µ»º3À3º©¼p²��¿�À©³�¾�³n´s´s¼p² �dµ¿�²d¼ �Vµ»º���½¡À3¼�À©³nµ�²�� � ��� ��� ��d½¡À3°���� �������

Ð �<Ð � � ���� �����<ÏÑÐ � � � � Ð�� � ? ; �h��� , ?

zHbj]^}aC~�|poq[^_^]Ni�� oq_^_^aH[^_^]X� � � ��©�fbszH]hk8c�]he�rtoXzH]hk8aCij]X�HaHef]\[^aHijiuo��f]hk8k`oøl � c��KzH]h[ � � � � � ®[^]hi�ryefaiuo ; �h��� , ?#�qoqija\c�aHe� . waN~�|^bj[^_^b��fa . wbji»c^ef]�zHaH�f�f]Vb�ryaHeyoxryb��h]�[^]h[`k8]�_^b��pzFoYoqijzH|^[poYzHb�lneyo�fb*�h[^b��pzFoxryb��qoY_^aHijiuo��f]hij|^�Hbj]h[^aNzFoqijzH]hiuoxrtoY_poqiji�� oqi*�h]hefb�ryk8]^�d�<� oqi*�h]hefb�ryk8]Y�poV_^|^[^~�|^a�_^a¡ryaHefk8bj[po©�ry]8_^bj[poqk8bjzFoqk8aH[srya�bji�k8b*�hijbj]he �qoqij]hefaB_^b � ; �)Ï ?¥�£�do ; �h��� , ?£eyoqc^c^efaH�faH[srto`|^[�aH�faHk8c^bj]8_^bzH]h[^_^bj�Hbj]h[^a£zt�^aq���fa£�haHefb��pzFoxrto��szH]hk8c�]he�rto\iuoK�p[^a�_^aHi�c^ef]�zHaH�f�f]�b�ryaHeyoxryb��h] ; z¡lneH��8<ef]�zHaH_^|^eyo��h�>=O?¥�Ô b�_^bje©}o�Ë¡¤¥ÃÈ­®¢¡¤¥Ã�¦2ÆqÃ#§q¤¥¤y¢t¨¡­®¦\_^b�|^[�oqi*�h]hefb�ryk8]Bgpoq�yoxry]V�f|:|^[:c^ef]�zHaH�f�f]�b�ryaHeyoxryb��h]Yi�� bj[^�fbjaHk8aX_^aHijijazH]h[^_^bj�Hbj]h[^b#zt�^aq�d�fa2�haHefb��pzFoxryaq�d_^a¡ryaHefk8bj[poq[^]6i�� oqefefaH��ry]6_^aHi<c^ef]�zHaH�f�f]�b�ryaHeyoxryb��h]^���<� bj[^_^b���bj_^|po©��Hbj]h[^aw_^bN|^[ zHefb�ryaHefbj]�_^bNoqefefaH��ry]*a¡�8zHbjaH[srya ; zHbj]^}awzt�^a?[^]h[ zH]hk8c�]he�ryb\b�ryaHeyoq�Hbj]h[^bXbj[�|�rybjijb6?VaH_ox�8_poqg^bjija ; zHbj]^}a�zt�^aY[^]h[�oqefefaH��ryb�c^efaHk`oxry|^eyoqk8aH[sryaBijaBb�ryaHeyoq�Hbj]h[^b6?�}a�|^[^]2_^a �hijb�oq�fc�a¡rfryb�c^efbj[^zHbÿ�cpoqijb�_^aHi�c^ef]�ha¡rfry]2_^a �hijb�oqi*�h]hefb�ryk8b»gpoq�yoxryb��f|�c^ef]�zHaH�f�fbdb�ryaHeyoxryb���b���<�Koqi*�h]hefb�ryk8]6_^aH�fzHefb�rfry]?[^aHijiuo@8<ef]�zHaH_^|^eyo �h�>=�|^�yowbji zHefb�ryaHefbj]w_^bøoqefefaH��ry]Dc^efaHzHaH_^aH[sryaq� zt�^a��fb_^bje©}o�Ë¡¤¥ÃÈ­®¢¡¤¥Ã�¦:Æqà §q¤¥¤y¢t¨¡­®¦2Ä�§q­�«�¤f§qªj¢��

Page 28: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

� "

��º©µ��h°���¿dº©°�aH�fc�] ; ³�²��£�D? � Bpµ�¿�À � � � ?��� �������� Ëy§qªjËt¦xªj¦`Æ�¢¡ª,5x§qªj¦x¤y¢�Æ�¢¡ªÈª�§ �¥«�Ä�ÉHÃ�¦xÄm¢�¢t¨®Çp¦xÄm¢¡Ä�ÉHÃ�§qªj¢��� ��� � �� � ����� ��������������� � ���� ��������������� �� � ��! � %�� �9^¼^º �ø� �Cº©°�¼p¾n° §q¤ 4�¦xÅ`¢¡Äp­®¦`Æ�¢¡ªÈª�§ �¥«�Ä�ÉHÃ�¦xÄm¢

¢t¨®Çp¦xÄm¢¡Ä�ÉHÃ�§qªj¢ 9^¼^º � � �Cº©°�¼p¾n° ¢®Çp¨¡ÃȪj¦xÄDÅ8§�ËtËE��ÃÈÄ�§�

��� ��������������� �� � � %���� %�� �9^¼^º � � � �Cº©°�¼p¾n° 5x§qªj¦x¤y¢�§©¨t¨¡«�Äp­®¦:Æs§qªÈª�§ �¥«�Ä�ÉHÃ�¦xÄm¢

¢t¨®Çp¦xÄm¢¡Ä�ÉHÃ�§qªj¢BÃÈÄ � ��� � ��� � ��� ��� ��� � � � � � �9^¼^º �� � ³�²�À3°�º©µ Äp«�Å`¢¡¤y¦`ÆqÃ<­®¢¡¤¥Å�ÃÈÄpÃCÆ�¢¡ªÈª�§2¨3¢¡¤¥Ã�¢ 9^¼^º � 8 � � �Cº©°�¼p¾n° ­®¢¡¤¥Å�ÃÈÄm¢(4�¢¡Äm¢¡¤f§qªj¢�Æ�¢¡ªÈª�§�¨3¢¡¤¥Ã�¢

��� ��! �#" � � � $���&% " � � ! � �� � �çÏ���Ó5B �çÏ>�+B8 � � �çÏ �hÓ5Bº©°��<°�¼�À

� � �çÏ � � 5 8 � � B �çÏ 5 ��B8 � � �çÏ 8 � � � ; ���� ?�B Ëy§qªjËt¦xªj¦`ÆqÃm� � �� ��!

¿�²�À©³�¾ ; Ð 8 � ��ÐO�/Ð � � �Ð�� � ? Ë¡¤¥ÃÈ­®¢¡¤¥Ã�¦:ÆqÃC§q¤¥¤y¢t¨¡­®¦8Ä�§q­�«�¤f§qªj¢ °�² ��aH�fc�]

' (*),+.-./01(32N416 �8:9<; =>)1(@? A#B�)DC>-E($?F;,+G21; +.)H; )I/H? °����aNzH]h[^�fbj_^aHeyoq�Hbj]h[^b»c^efaHzHaH_^aH[sryb��fbmoqc^c^ijbjzFoq[^]Ybj[ �haH[^aHeyoqija�oq_`|^[:c^ef]�zHaH�f�f]�b�ryaHeyoxryb��h]Ygpoq�yoxry]

�f|�|^[poVln]hefkV|^iuoVefbjzH]hefefaH[srya�_^aHi�rybjc�]� � 7 � Ï � � 5 8 � Ó ; �h��� � ?

®i�zHefb�ryaHefbj]�_^bdoqefefaH��ry]8[poxry|^eyoqija�}aÐ 8 � Ð�� �� � ?

Page 29: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# � �

_^]3�ha�� � ÏÑÐ � � Ð � � Ó

����]hc^c�]he�ry|^[^]P[^]qrtoqefa?zt�^a�oq[^zt�^abji�zFoqijzH]hij]*_^b �� � }a��f]qrfry]hc�]h��ry]PoqijijaDijbjk8b�rtoq�Hbj]h[^bX_^aHi�fbj��ryaHk`o2oqefb�ryk8a¡rybjzH]�l � c��Co2c^efaHzHbj�fbj]h[^a �p[^b�rto��

ÖP×KØfÙFÚ�Û^Ü�Ý2Þqß�� ��ß#àFá�â ã¡äxå�áçæ3è�é�ádý©ä�å�áçånû�è�ü�ú�ú¥é�á û�üøè û�áçâ ãVþnõ ù�õwâ ã¡ä � Nnìfí < % < ! ð < ð�PCè�å�á�â�ú¥÷çâ ã¡÷ á ä©ãVá -tú¥÷ ã¡é�áæ3á�����á äVé�è�÷çú&%�á ã¡ä©è�ú¥ápäFý©üøè�é�á �Vë ! í©î ìyï¥ô <¥ñFó¡ïUM2ìfí��£è���ë�í©î ï¥ôHó L©ì¡ìyïUM8ìfí�� �tõNàFá»é�áçâ ã¡é æ3ápâ('©è�÷*) è�ùxå�á ÷ ã¡äü�ú¡â�â('©á äxú�èfæ£á ÷©üøá ä©á üøã#äFý©üøè�é�ã#é�èfú¥÷ èmùqãHå�á û�á -¡ã#é ú¥ù©ù©é�èfå�è�äHû ú&�©á ÷ èmá ä � å�ã¡ä©ã #té�áçå�ùqè û�û�á -tú¥üøè�äHû�è�# � ë6í©î ó"M£ìfí�� 'è�a T�� ) ë�í©î ìDM�ìfí���htõ�àFá+'xú�:

���<ë N�í©î ìyï¥ô <¥ñFó¡ïDM�ìfí��*P,M N�í©î óUM�ìfí�� '*P�ë�í©î L©ìfðHï¡ï L¡í4M�ìfí��ON <���Kë N�í©î ï¥ôHó L©ì¡ìyïDM�ìfí�� �*P"M N�í©î óUM�ìfí�� '*P�ë�í©î ì¡ìE% <¡íHó L4M�ìfí�� � 3 �Ya T�� )

èD)Hý©á äxæ3á�á ÷�â�ú¥÷çâ ã¡÷ ãøæ3á���C,N���0Pmù©é�ãE-¡ã3â�ú ý©äxæ3è�é��xã��<õ Ö

�<� oqi*�h]hefb�ryk8]�_^aH�fzHefb�rfry]�[^aHijiuo 8<ef]�zHaH_^|^eyo �h� �DzFoqijzH]hiuo � c�aHe�|^[*�qoqij]hefa��haH[^aHefbjzH]?� * �)bj[k8]�_^]2oqzHzH|^eyoxry]^�®[^]hi�ryefaq�pi�� oqi*�h]hefb�ryk8]�efbjzt�^bjaH_^aYzt�^aY�fbuo2[^]qry]8bji��qoqij]hefa�_^b��* 6 K��zt�^aYc^|�}]`aH�f�faHefa�zFoqijzH]hiuoxry]2zH]h[i�� oqi*�h]hefb�ryk8]Y_^aH�fzHefb�rfry]8[^aHijiuo 8<ef]�zHaH_^|^eyo �h� " � � �Ô bK�f|^c^c�]h[ �so`_^b��h]hijaHeN|�rybjijbj�H�Foqefa�iuo ; �h���=�@?Xc�aHeNzFoqijzH]hiuoqefaY|^[�� oqc^c^ef]h�f�fbjk`oq�Hbj]h[^aY_^b�� � zH]h[��zHb�lnefaB_^aHzHbjk`oqijbmzH]hefefa¡rfryaq� �£bj�f]�h[po�_^|^[^~�|^a�oqefefaH��rtoqefa�bji�c^ef]�zHaH_^bjk8aH[sry]2b�ryaHeyoxryb��h]�~�|poq[^_^]

Î/Ï Ð � � Ò � � Ð�������� � Ï �� 0�Ó®i��qoqij]hefaø_^b»Î�[^]h[6}a�[^]qry]�a�g^bj�f]�h[po�~�|^bj[^_^b^zFoqijzH]hiuoqef[^aø|^[poB��rybjk`o�� � � 3 rtoqiQ�p[^a��fbpefbjzH]hef_^b

zt�^a� � Ò � � Ï� � ; �"?�?

a�zt�^a�oqi�_^b��haHe �haHefa�_^b �bjimefaH��ry]8_^aHijiuo�ln]hefkV|^iuoV_^b �Ko��ij]heB}a�bj[Q�p[^b�ryaH�fbjk8]ijbjk� � � ; �"?� � �� B

2 � " ú¥÷ è�ú¥÷ �¡ã¡é�á û�üøã�å�á �xú¡å�úNå�ý©÷ ÷*) á ùqã¥û�èfå�á*# �¡è�ä©è�é ú¥÷ üøè�äHû�èU-¡è�é�á �qâ�útû ú�#�â('©è�#�á äVâ�ú¡å�ã�æ3á»ý©äxæ3è�é��xã�� #�ú¡æBaK-¡è�ä��Húú¡å�å�è �¡äxútû�ã á ÷�-tú¥÷ ã¡é�è<í©õ2 2)�däxú�ånû�á ü�úøæ3á � 'èKþÿã¡é�ä©á û úøæ©ú¥÷ ÷çú Nnì¡õ óQP.N�â®þÿéfõpèfå�è�üøù©á ã�ì¡õ L�P:

��ë � � ! ! � & �¡ë � 0 & Nÿê�P/� E �76 � ê � & ; �N ) ��ìRP * <â ã¡ä 3 ã¡ù©ùqã¡é�û�ý©äxú¥üøè�äHû�èNå�â è�÷ û�ãxõ�C�ã¡üøè �¡á6'ú�ãHå�å�è�é -tútû�ãBä©è�÷ ÷*) èfå�è�üøù©á ã8ì¡õ L�#pá ÷�â�ú¥÷çâ ã¡÷ ã�è1hè û�û�á -¡ãVæ3á�û ú¥÷ èNånû�á ü�úé�áçâ('©á èfæ3èBùqè�éE'ã�÷çú2æ3è û�è�é�üøá äxú&%�á ã¡ä©èBæ3á 3 #�ã�üøè �¡÷ á ã2æ3á�ý©äR-tú¥÷ ã¡é�è�æ3á 3 ùqè�é\â ý©á�� 6 � ê � & ; � % N ) ��ìRP * ä©ã¡ä6å�áçú��èfâ�â èfå�å�á -tú¥üøè�äHû�è �¡é ú¥äxæ3è2�©õ " ú¥÷ è ù©é�ã��©÷ è�ü�ú�ä©ã¡ä 'è æ3á�þ�ú¡â á ÷ è é�áçå�ã¡÷ ý�%�á ã¡ä©èCèU)Hý©á äxæ3á�á ä �¡è�ä©è�é ú¥÷ è<ä©ã¡äYå�á�é�áçâ ã¡é�é�èú¥÷ ÷çúøånû�á ü�úøå�ýxæ©æ3è û�û ú©õ

Page 30: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

���

��º©µ��h°���¿dº©°�aHc^�fefaHi ; ³�²��£�D? �* 6 .? � Bpµ�¿�À � � ?��� �������� Ëy§qªjËt¦xªj¦`Æ�¢¡ªÈª�� ¢®Çp¨¡ÃȪj¦xÄwÅ8§�ËtËE��ÃÈÄ�§2¤y¢¡ª�§q­�Ã65q¦:§@���� ��� � �� � ����� ��������������� � ���� ��������������� �� � ��! � %�� �9^¼^º �ø� �<º©°�¼p¾n° Äp«�Å`¢¡¤y¦2Å8§�ËtËE��ÃÈÄ�§`Æqãˡ«�ãËy§qªjËt¦xª�§q¤y¢

ª�� ¢®Çp¨¡ÃȪj¦xÄwÅ8§�ËtËE��ÃÈÄ�§2¤y¢¡ª�§q­�Ã65q¦ 9^¼^º � �* 6 �<º©°�¼p¾n° Å�ÃÈÄpÃÈÅ`¦�Äp«�Å`¢¡¤y¦8¤y¢y§qªj¢øÇp¦3¨¡ÃÈ­�Ã65q¦

¤f§¥ÇsÇ�¤y¢t¨3¢¡Äp­ § �HÃȪj¢ 9^¼^º � � �<º©°�¼p¾n° ¢®Çp¨¡ÃȪj¦xÄwÅ8§�ËtËE��ÃÈÄ�§

��� ��������������� �� � � %���� %�� �9^¼^º � � �<º©°�¼p¾n° ¢®Çp¨¡ÃȪj¦xÄwÅ8§�ËtËE��ÃÈÄ�§8¤y¢¡ª�§q­�Ã65q¦`§2�

��� ��! �#" � � � $���&% " � � ! � �³��Ð �<Ð�. �* 6 � � À �d°�² � �çÏ Ð �<Ð � � B

°�¾�½H° � �çÏ��* 6 .B

°�² ��³�°�² ��aHc^�fefaHi

' (*),+.-./01(325416 �8:9<; =>)1(@? A#B�)DC>-E($?F;,+G21; +.)H; )I/H? �

zH]h[^zHij|^_^aH[^_^]^�xo�k8aH[^]�_^aHihl�oxrfry]hefa#zH]h��rtoq[srya � � ; ] ���dzH]h[�� * �¬Ò18 ? 8��6?¥�©�qoqija#i�� oqc^c^ef]h�f�fbjk`oq�Hbj]h[^a��Î/Ï Ð � � Ò � � Ð Q Ð �<Ð � 7 �

; 5 � ? � Ï Ð � � 7 � Ò � � Ð¥Ó�do�~�|poq[sryb�rx}o

Ð � � 7 � Ò � � Ð�fb#oq�f�f|^k8a�c�aHe�rtoq[sry]�zH]hk8a���rybjk`o`_^aHiji�� aHefef]hefaVoq�f�f]hij|�ry]6Î oqi�cpoq�f�f]. �a��fbK|�rybjijbj�H�Fo8bji�zHefb�ryaHefbj]_�� oqefefaH��ry]

Ð � � 7 � Ò � � Ð�� �� 0�Ó ; �h��� 1 ?

Page 31: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# � ,

��º©µ��h°���¿dº©°:efk8bj[ ; ³�²��( Bpµ�¿�À ���* 6 ?��� �������� Ëy§qªjËt¦xªj¦`Æ�¢¡ª�Å�ÃÈÄpÃÈÅ`¦�Äp«�Å`¢¡¤y¦8¤y¢y§qªj¢øÇp¦3¨¡ÃÈ­�Ã65q¦

¤f§¥ÇsÇ�¤y¢t¨3¢¡Äp­ § �HÃȪj¢��� ��� � �� � ��������������������� � ���� ��������������� �� � ��! � %�� �9^¼^º � ( �C³�²�À3°�º©µ �t§©¨3¢�Æ�¢¡ª�¨¡Ãj¨¡­®¢¡Å8§:§q¤¥ÃÈ­�Å`¢¡­�Ã�Ët¦

��� ��������������� �� � � %���� %�� �9^¼^º ���* 6 �<º©°�¼p¾n° Å�ÃÈÄpÃÈÅ`¦�Äp«�Å �N¤y¢y§qªj¢øÇp¦3¨¡ÃÈ­�Ã65q¦2¤f§¥ÇsÇ�¤��

����� ��� � ��� ��� ��� � � � � � �9^¼^º ��� �<º©°�¼p¾n° 5x§q¤¥Ã�§ �HÃȪj¢YÆqà §¥ÇsÇp¦R4 4qÃ�¦

��� ��! �#" � � � $���&% " � � ! � �� �çÏ �hÓ5Bº©°��<°�¼�À

�* 6 �çÏ ��B� �çÏ���� ( B

¿�²�À©³�¾ ; ��Ï���ÓB?°�² ��efk8bj[

' (*),+.-./01(325416 � 8:9<; =>)1(@? A#B�)DC>-E($?F; +G21; +.)H; )&/H? �* 6 8�aHeX~�|poq[sry]`_^a¡rfry]2�f|^idzHefb�ryaHefbj]�_�� oqefefaH��ry]8[poxry|^eyoqijaq��[^]h[��po2�faH[^�f]8efbjzt�^bjaH_^aHefa�zt�^a

�� 0 ����� � ÏÑÐ � � Ð�� � Ó ; �h�>= �@?3N[poqij]�soqk8aH[sryaq�s�faN�fbp��|^]hijaXzFoqijzH]hiuoqefa�|^[�� oqc^c^ef]h�f�fbjk`oq�Hbj]h[^a�_^b � � zH]h[�`zHb�lnefaX�fb*�h[^b��pzFoxryb��hazH]hefefa¡rfryaq�p�fb�|�rybjijbj�H�FoVbjimzHefb�ryaHefbj]�_�� oqefefaH��ry]

Ð � � 7 � Ò � � ÐÐ � � Ð ������� � Ï �� 0 ; �h�>=�� ?a�bj[6~�|^aH��ry]8zFoq�f]8[^]h[��po��faH[^�f]`efbjzt�^bjaH_^aHefa�zt�^a

�� 0 � � Ó ; �h�>=�=O?

Page 32: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

� �

� 3 �� 0 Î :��Ó�� " � �O� 1 � � ��� � ��� � � ��Ó , � � ��� � ���Ó�� " � " � = 1 � ��� � ��� � � ��Ó � 1 � ��� � ���Ó�� " � " ��= " � ��� � ��� � 3 ��Ó ��� � ��� � 3��Ó�� " � " �=�Q� � ��� � ��� � � ��Ó " � � ��� � ���Ó�� " � " �=� = � ��� � ��� � � ��Ó>=$� � ��� � ���Ó�� " � " �=� = � ��� � ��� � � ��Ó>=$� � ��� � ���Ó�� " � " �=� = � ��� � ��� � � ��Ó>=$� � ��� � ���Ó�� " � " �=� = � ��� � ��� � � ��Ó>=$� � ��� � ���Ó�� " � " �=� = � ��� � ��� � � ��Ó>=$� � ��� � ���Ó�� " � " �=� = � ��� � ��� � � ��Ó>=$� � ��� � �

� ¼ �<°�¾�¾n¼P¶p·�� � �Y¼p¾nµ»ºx³ ��³ � 3�µpÀFÀ3°�²�¿�À©³ �hµ�²�¿�² ��º©µ��»º©¼ � ��¼�³�²-��µ»º3À3º©¼p²��C¿�À©³�¾�³n´s´s¼p² �dµ¿�²d¼ �Vµ»º���½¡À3¼�À©³nµ�²�� � ��� ��� ��d½¡À3°���� ��������° ��³69�°�ºx½F³49^¼p¾nµ»ºx³U��³ À3µ�¾�¾n°�º©¼p²d´s¼

®[ �Koqg�aHijiuo �h� " �f]h[^]:efbjc�]he�rtoxryb�bd�qoqij]hefb�_^b � 3 ��zFoqijzH]hiuoxryb�|�rybjijbj�H�Foq[^_^]8iuo 8<ef]�zHaH_^|^eyo �h�>=2zH]h[DbjizHefb�ryaHefbj]8_�� oqefefaH��ry] ; �h�>=�� ? � � �mc�aHe\�qoqij]hefb�_^b1�� 0C_^b5AmaHefaH[sryb��»aV�f]h[^]:bj[^_^bjzFoxryb �hijbdaHefef]hefb�efaHiuoxryb���b��Î : ��zH]hefefbj�fc�]h[^_^aH[sryb�� Ô bm]h�f�faHe���b�zt�^aBc�aHe ��� � � � �� 0 � ��� � � i�� aHefef]hefaNefaHiuoxryb��h]:}a�_^aHiji�� ]hef_^bj[^a\_^b�� 0®��k8aH[sryefaVc�aHe �� 0�� ��� � � rtoqija�aHefef]hefa:}aV�faHk8c^efa ��Ó>=$��� ��� � �F��bj[?oqzHzH]hef_^]�zH]hi��qoqij]hefa�_^b �efbjc�]he�rtoxry]Vbj[ ; �h��� " ?¥� � 3

8�aHe�rtoq[sry]^��bdzHefb�ryaHefb�_�� oqefefaH��ry] ; �h��� 1 ?øa ; �h�>=�� ?ø_^a¡�h]h[^]:aH�f�faHefaVk8]�_^b��pzFoxryb��^c�aHe\a¡��b�rtoqefaYzt�^a�fbuoVefbjzt�^bjaH��rto�|^[�� oqzHzH|^eyoxryaH�H�Fo`zt�^a�[^]h[�c^|�}]8aH�f�faHefaY]qrfryaH[�|�rto$�Ð � � 7 � Ò � � Ð�� �� 0 5 Ð � � Ð�� � ?

Ð � � 7 � Ò � � ÐÐ � � Ð � �� 0 5 � Ó

� |poq[sry]Y_^a¡rfry]Y�f|^ipzHefb�ryaHefbj]B_^b»oqefefaH��ry]Y[poxry|^eyoqija�c�aHeCbji^zFoqijzH]hij]B_^b � � �fbpaH��ryaH[^_^aNbj[ �haH[^aHeyoqijaoqihc^ef]�ha¡rfry]�_^aHbszHefb�ryaHefbh_^bsoqefefaH��ry]\_^bsoqi*�h]hefb�ryk8b©gpoq�yoxrybh�f|Yc^ef]�zHaH�f�fb�b�ryaHeyoxryb���b��Fzt�^a oqc^c^ef]h�f�fbjk`oq[^]|^[��qoqij]hefa �8zH]hk8a£ijbjk8b�rya<_^b�|^[poN�f|^zHzHaH�f�fbj]h[^a � � ����� � ®[Vrtoqi�zFoq�f]^�h�fb^oq�f�f|^k8]h[^]�zH]hk8aø��rybjk8a_^aHiji�� aHefef]hefaBoq�f�f]hij|�ry]2a�_^aHiji�� aHefef]hefaBefaHiuoxryb��h]�efbj�fc�a¡rfryb��qoqk8aH[sryaBija�~�|poq[sryb�rx}o

Ð � � 7 � Ò�� � Ða

Ð � � 7 � Ò�� � ÐÐ � � Ð2,3)�Èüøù©÷ è�üøè�äHû ú&%�á ã¡ä©è<å�ý�ý©ä+) �[c ��� ��à*C�à��3ånû�è�ügfEL¡í¡í¡í�#Fá ä ©ã¡é�û�é ú¥ä+#Fý3û�á ÷ á % %fú¥äxæ3ãø÷çú�å�á ä��¡ã¡÷çú ù©é�èfâ áçå�á ã¡ä©è¡õ2 N��K÷ á�è�é�é�ã¡é�ásé�è�÷çútû�á -Fá�å�ã¡ä©ãXånû útû�á�â�ú¥÷çâ ã¡÷çútû�á�ý3û�á ÷ á % %fú¥äxæ3ã�ý©ä+) ú¥ù©ù©é�ãHå�å�á ü�ú&%�á ã¡ä©è<æ3á � NCâ ã¡ä�ð�â á þÿé�è å�á �¡ä©á �qâ�útû�á -¡èâ ã¡é�é�è û�û�è�ND� N J í©î ì�ñ <¥ñxìfô©ìyó¥ðDM�ìfíQ2*P®õ

Page 33: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# � 1

a��fb�|�rybjijbj�H�Foq[^]Vb�zHefb�ryaHefb�_�� oqefefaH��ry] � �� � 7 � �� � �� �� 0 5 � � � � ; �h�>=$�@?

aÐ � 7 � Ò�� � ÐÐ � � Ð � �� 0 5 � Ó ; �h�>= " ?

021�� �dE WX587?=?;HMN;H587DGVTU@B7:9#5 =?; L?7RQ?A<58S?EH@BTUG TUGY9#@BTUGY9<;FWX5{\aHijiuoVefbj�f]hij|^�Hbj]h[^aNzH]hk8c^|�rtoq�Hbj]h[poqija�_^b�|^[�c^ef]hg^ijaHk`o���b�_poxryb��f]h[^]`o Ama¡rfryb�_poqiji�� aHefef]hefa�_^b�ef]h|^[��_^]%AK� ®[ �haH[^aHeyoqijaq��aH�f�fb<�f]h[^]6o Ama¡rfryb<oq[^zt�^a�_po:aHefef]hefb#�fc�aHefbjk8aH[srtoqijb���zHbj]^}aV_^]3��|�ryb<oqbKc^ef]�zHaH_^bÿ�k8aH[sryb�_^b�k8bj�f|^eyoq�Hbj]h[^a£|�rybjijbj�H�Foxryb�c�aHe#iuoNij]hef]Nefbjija¡�qoq�Hbj]h[^a%Bqrtoqijb�aHefef]hefb��qbj[^a¡��b�rtoqg^bjijb��©[^]h[��yoqeyoq[^[^]aH�fc^ijbjzHb�rtoqk8aH[srya�zH]h[^�fbj_^aHeyoxryb��^bj[6~�|poq[sry]`oq�f�fbjk8bjiuoqg^bjijb»o��hijbmaHefef]hefb�_^b�ef]h|^[^_^]%AK�®[�gpoq�fa�o�~�|poq[sry]2_^a¡rfry]^��zHbm�fb�oq| �h|^eyo�zt�^aq�pbj[:|^[�c^ef]hg^ijaHk`oYk`oxryaHk`oxrybjzH]^��|^[�aHefef]hefa�efaHiuoxryb��h]; oq�f�f]hij|�ry]�?:[^aHb�_poxrybB_^b�|^[�zHaHe�ry] ]hef_^bj[^a�_^b��heyoq[^_^aH�H�FoP�fbBryeyoq_^|^zFoPbj[Õ|^[�aHefef]hefa�efaHiuoxryb��h]; oq�f�f]hij|�ry]�?£_^aHijij]���ryaH�f�f]`]hef_^bj[^aB[^aHijiuoV�f]hij|^�Hbj]h[^aq�¯�°s±#²�³n´�³nµ�²d°�¶p·��m·`¸ �¿dµ�² �hµ�² ��³n´�³nµ�²d¼ ��°�²�À3µ ��³£¿�² ��º©µ��K¾n°���¼�Á% Ä�Ç�¤y¦��Hªj¢¡Å8§�¨¡Ã#ÆqÃ�Ët¢@�¥¢¡ÄDËt¦xÄ�ÆqÃ�ÉHÃ�¦xÄ�§q­®¦B¨3¢Nª�� ¢¡¤¥¤y¦x¤y¢N¤y¢¡ª�§q­�Ã65q¦D� §©¨t¨3¦xª¬«�­®¦1 �Äm¢¡ªÈª�§Y¨3¦xª¬«hÉHÃ�¦xÄm¢(�^§2§qªÇ�Ã�Í«�ªj¦V¨¡­®¢t¨t¨3¦�¦x¤fÆqÃÈÄm¢�Æqà 4q¤f§qÄ�Æ�¢yÉtÉ3§`Æ�¢¡ªÈª�� ¢¡¤¥¤y¦x¤y¢�¤y¢¡ª�§q­�Ã65q¦ � §©¨t¨3¦xª¬«�­®¦1 2Äm¢¡ÃCÆs§q­�à �

®[6oqzHzH]hef_^]�zH]h[�iuoY_^a2�p[^bj�Hbj]h[^a�c^efaHzHaH_^aH[sryaq�p|^[6c^ef]hg^ijaHk`oYc�aHeøbji�~�|poqijaNi�� aHefef]hefaNefaHiuoxryb��h]V[^aHijiuo�f]hij|^�Hbj]h[^a��po�]hef_^bj[^a�_^b �heyoq[^_^aH�H�Fo�k`o���hbj]hefa£efbj�fc�a¡rfry]�oqiji�� aHefef]hefaøefaHiuoxryb��h]B[^aHbp_poxrybp�fbp_^bjzHa�Å8§qªËt¦xÄ�ÆqÃ�ÉHÃ�¦xÄ�§q­®¦x�

ÖP×KØfÙFÚ�Û^Ü�Ý2Þqß��sÞqß �È÷�å�áçånû�è�ü�úøæ3á�è�)Hýxú&%�á ã¡ä©áh÷ á ä©èfú¥é�á :� ï3î ì ê � ô©î ó ��ë <ñxî ìfð ê � % ��ë ìyó Nnì¡õ ï¡óQP

ú¥üøüøè û�û�è\â ã¡üøèXå�ã¡÷ ý�%�á ã¡ä©è Nnìfí¡í < ! ó %Fî %Fì�ñdî�î�î P®õ�ö�úBé�áçå�ã¡÷ ý�%�á ã¡ä©è�æ3á.Nnì¡õ ï¡óQPU'è�ý©ä�ù©é�ã��©÷ è�ü�ú�ámâ ý©ámæ©útû�á�å�ã¡ä©ãBáâ ãFè�:Xâ á è�äHû�á�æ3è�÷ ÷çúøü�útû�é�áçâ è<æ3è�÷�å�áçånû�è�ü�ú£èfæ\á ÷�-¡è û�û�ã¡é�è<æ3è�áhû�è�é�üøá ä©á�ä©ã¥û�ájõ2 'fàFáFãHå�å�è�é -FáHâ('©è�á3â é�á û�è�é�áFæ ) ú¥é�é�èfånû�ã Nnì¡õ ï¥ô�Psè.Nnì¡õ ïtñ P �Hú¥é ú¥äHû�áçå�â ã¡ä©ãdâ('©è�á ÷Fù©é�ã3â èfå�å�ãdá û�è�é útû�á -¡ã�å�á3ú¥é�é�èfånû�á )Hýxú¥äxæ3ãæ3ý©è#ú¥ù©ù©é�ãHå�å�á ü�ú&%�á ã¡ä©áqå�ýxâ�â èfå�å�á -¡èKå�ã¡ä©ã ý��¡ýxú¥÷ ású üøè�ä©ãøæ3áhý©äxúCû�ã¡÷ ÷ è�é ú¥ä�%fúCù©é�è��qå�å�útû ú�#Hü�ú ä©ã¡äNú¡å�å�áçâ ý©é ú¥ä©ã â('©è÷*) è�é�é�ã¡é�è�å�áçúdüøá ä©ã¡é�èmæ3áHû ú¥÷ è�û�ã¡÷ ÷ è�é ú¥ä�%fú©õ � è�é�û ú¥÷ è�üøã¥û�á -¡ã�èfå�å�á �¡è�ä©è�é ú¥÷ üøè�äHû�èmå�áFý3û�á ÷ á % %fú¥ä©ã�á ä£â ã¡ü �©á äxú&%�á ã¡ä©èmâ ã¡äú¥÷ û�é�á©â é�á û�è�é�á*#¥â('©èmû�è�ä��¡ã¡ä©ã<â ã¡äHû�ã #¥ú¡æ èfå�è�üøù©á ã #¡æ3á©â�ú¥é útû�û�è�é�áçånû�áçâ('©èmùxú¥é�û�áçâ ã¡÷çú¥é�áFæ3è�÷©ù©é�ã3â èfå�å�ã�á û�è�é útû�á -¡ãKá äøèfå�ú¥üøè¡õàFá»ãHå�å�è�é -Fápá ä©ã¡÷ û�é�è�â('©èXå�èøê & ë*íNá�â é�á û�è�é�á�å�ýxæ©æ3è û�û�á»é�áçå�ý©÷ û ú¥ä©ã�á äxú¡æ3è �¡ýxútû�á Nÿá ÷�â é�á û�è�é�á ã Nnì¡õ ïtñ P 'è�ú¡æ©æ3á é�á û�û�ý©é úá äFý3û�á ÷ á % %fú&�©á ÷ è * P®õ �däøüøã3æ3ã<ùqè�émå�ý©ùqè�é ú¥é�èmû ú¥÷ è�ù©é�ã��©÷ è�ü�úKù©ýQ'ã<èfå�å�è�é�è )Hý©è�÷ ÷ ãCæ3áxý3û�á ÷ á % %fú¥é�è�ý©äXâ é�á û�è�é�á ã<æ ) ú¥é�é�èfånû�ãæ3è�÷sû�á ùqã

� ê & ; �"! ê & �� ê & �2��ì � ���7C+� � î

Page 34: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

" �

àFá�â ã¡äxå�áçæ3è�é�ásã¡é ú á ÷�å�áçånû�è�ü�ú�:� ï3î ì ê � ô©î ó �-ë <ñxî ìfðHï ê � % �-ë ìyó Nnì¡õ ï L�P

â('©èmú¥üøüøè û�û�èmâ ã¡üøèmå�ã¡÷ ý�%�á ã¡ä©è.Nnìyï¡ó < ! %¥ï3î %Fì�ñdî�î�î P®õ �È÷Få�áçånû�è�ü�ú Nnì¡õ ï L�Psù©ýQ'ã�èfå�å�è�é�è�-Fáçånû�ãKâ ã¡üøè�ý©äxúdùqè�é�û�ý©é �xú&%�á ã Yä©è#æ3è�÷så�áçånû�è�ü�ú(Nnì¡õ ï¡óQPDNÿä©è�÷�â ãFè�:Xâ á è�äHû�è#æ3áhê�ä©è�÷ ÷çú£å�èfâ ã¡äxæ©úCè�)Hýxú&%�á ã¡ä©è4'è#ånû útû�ã á äHû�é�ã3æ3ã¥û�û�ã£ý©ä\è�é�é�ã¡é�èdé�è�÷çútû�á -¡ãæ3è�÷ ÷*) ã¡é æ3á ä©è#æ3á^ìfí ��2 P�èK÷çú é�áçå�ã¡÷ ý�%�á ã¡ä©èKæ3á"Nnì¡õ ï L�P�â ã¡üøèK÷çú£é�áçå�ã¡÷ ý�%�á ã¡ä©èKæ3ásý©ä\ù©é�ã��©÷ è�ü�ú ùqè�é�û�ý©é �xútû�ã£é�áçå�ùqè û�û�ãøú¥÷ù©é�ã��©÷ è�ü�ú�æ3á�é�á þÿè�é�á üøè�äHû�ã Nÿ÷çú�é�áçå�ã¡÷ ý�%�á ã¡ä©èCæ3á Nnì¡õ ï¡óQP,P®õdö�ú�ùqè�é�û�ý©é �xú&%�á ã¡ä©èCá äHû�é�ã3æ3ã¥û�û ú�#qâ ã¡é�é�áçå�ùqã¡äxæ3è�äHû�è�#xæ©úXý©äù©ý©äHû�ã£æ3á -Fáçånû ú �¡è�ã¡üøè û�é�áçâ ã #Hú¡æ�ý©äxúCüøã3æ3á �qâ�úCä©è�÷ ÷*) á äxâ ÷ á äxú&%�á ã¡ä©èKæ3è�÷ ÷çúCå�èfâ ã¡äxæ©úCæ3è�÷ ÷ èKæ3ý©èKé�è û�û�è�é ú¥ù©ù©é�èfå�è�äHû útû�èæ©ú¥÷ ÷ èCè�)Hýxú&%�á ã¡ä©á*#hâ ã¡üøùqã¡é�û ú�ý©ä+) ú¥üøù©÷ á �qâ�ú&%�á ã¡ä©è æ3è�÷ ÷*) è�é�é�ã¡é�èCä©ã¥û�è/-¡ã¡÷ è�#qæ3ãE-Fý3û úXú¥÷�þ�útû�û�ãNâ('©è ÷ è£æ3ý©è é�è û�û�è£å�ã¡ä©ã� )Hýxú¡å�áxùxú¥é ú¥÷ ÷ è�÷ è2�<è )Hý©á äxæ3ásý©äxúC÷ á è/-¡è�ùqè�é�û�ý��xú&%�á ã¡ä©èKæ3è�÷sâ ãFè�:Xâ á è�äHû�èKú¥ä��¡ã¡÷çú¥é�è�æ3áhý©äxú æ3è�÷ ÷ è#æ3ý©èKù©é�ãE-¡ã3â�ú#ý©ä©ãå�ùqãHånû ú¥üøè�äHû�ãøæ3è�÷�ù©ý©äHû�ã�æ3á�á äHû�è�é å�è %�á ã¡ä©è¡õ " ú¥÷ è<å�á û�ýxú&%�á ã¡ä©è 'è#á ÷ ÷ ýxånû�é útû ú á ä �á �¡ý©é úNì¡õ ó3õ Ö

2.1x+3.5y=8 r

4.19x+7y=15 s

4.192x+7y=15 t

� b*�h|^eyo �h� � � ®[sryaHefc^efa¡rtoq�Hbj]h[^a��haH]hk8a¡ryefbjzFo�_^aHi zH]h[^_^bj�Hbj]h[poqk8aH[sry] ��k`oqiCzH]h[^_^bj�Hbj]h[poqk8aH[sry]w_^b|^[6�fbj��ryaHk`o�ijbj[^aFoqefaq�

ÖP×KØfÙFÚ�Û^Ü�Ý2Þqß�� �sßDC�ã¡äxå�áçæ3è�é�áçú¥üøã ã¡é ú£á ÷�å�áçånû�è�ü�ú�æ3ásè�)Hýxú&%�á ã¡ä©ás÷ á ä©èfú¥é�á :� ! ô©î ó ê � ï3î ì ��ë ìyóï3î ì ê � ô©î ó ��ë < Nnì¡õ ï %�P

â('©è<ú¥üøüøè û�û�èCâ ã¡üøè<å�ã¡÷ ý�%�á ã¡ä©è N ! í©î L&%¥ï¥ô < ï3î L <¡ð©ìRP�#¥èfæ\á ÷�å�áçånû�è�ü�ú�:� ! ô©î ó¥íHï ê � ï3î ì �-ë ìyóï3î ì ê � ô©î ó �-ë < Nnì¡õ ï�<�P

â('©èXú¥üøüøè û�û�è\â ã¡üøè\å�ã¡÷ ý�%�á ã¡ä©è N ! í©î L&%¥ï¥í < ï3î L < <¡ð�P®õKC�ã¡üøè�ä©è�÷ ÷*) èfå�è�üøù©á ã�ù©é�èfâ èfæ3è�äHû�è�#�á ÷må�áçånû�è�ü�ú Nnì¡õ ï�<�PKù©ýQ'ãèfå�å�è�é�è -Fáçånû�ã�â ã¡üøè\ý©äxú�ùqè�é�û�ý©é �xú&%�á ã¡ä©èNæ3è�÷�å�áçånû�è�ü�ú Nnì¡õ ï %�P N�ú¥äxâ('©è\á ä )Hý©èfånû�ãYèfå�è�üøù©á ã #pä©è�÷�â ãFè�:Xâ á è�äHû�è�æ3áêYä©è�÷ ÷çúNå�èfâ ã¡äxæ©ú�è�)Hýxú&%�á ã¡ä©è 'è ånû útû�ã\á äHû�é�ã3æ3ã¥û�û�ã\ý©äYè�é�é�ã¡é�è<é�è�÷çútû�á -¡ã\æ3è�÷ ÷*) ã¡é æ3á ä©è£æ3ámìfí���20P�è£÷çúXé�áçå�ã¡÷ ý�%�á ã¡ä©è æ3áNnì¡õ ï�<�P�â ã¡üøè<÷çúøé�áçå�ã¡÷ ý�%�á ã¡ä©èCæ3á�ý©äBù©é�ã��©÷ è�ü�úøùqè�é�û�ý©é �xútû�ã�é�áçå�ùqè û�û�ã\ú¥÷�ù©é�ã��©÷ è�ü�ú�æ3á�é�á þÿè�é�á üøè�äHû�ã Nÿ÷çú�é�áçå�ã¡÷ ý�%�á ã Yä©èNæ3á Nnì¡õ ï %�P,P®õ�ö�ú�ùqè�é�û�ý©é �xú&%�á ã¡ä©è\á äHû�é�ã3æ3ã¥û�û ú�ä©ã¡ä`â ã¡üøùqã¡é�û ú0)Hý©èfånû úG-¡ã¡÷ û úBý©ä+) ú¥üøù©÷ á �qâ�ú&%�á ã¡ä©è�æ3è�÷ ÷*) è�é�é�ã¡é�è�#

Page 35: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# " �

æ3ãE-Fý3û ú\ú¥÷�þ�útû�û�ãNâ('©è£á ä )Hý©èfånû�ãNâ�ú¡å�ãX÷ èøæ3ý©è é�è û�û�èøå�ã¡ä©ã � )Hýxú¡å�á�ã¡é�û�ã��¡ã¡äxú¥÷ á � 2 8 èU)Hý©á äxæ3á^ý©äxú\÷ á è/-¡è ùqè�é�û�ý��xúEY%�á ã¡ä©è#æ3è�÷�â ãFè�:Xâ á è�äHû�è<ú¥ä��¡ã¡÷çú¥é�è�æ3á�ý©äxú£æ3è�÷ ÷ è<æ3ý©è#ä©ã¡äNù©é�ãE-¡ã3â�ú<ý©ä©ã�å�ùqãHånû ú¥üøè�äHû�ãøå�á �¡ä©á �qâ�útû�á -¡ã£æ3è�÷sù©ý©äHû�ã�æ3áá äHû�è�é å�è %�á ã¡ä©è¡õ " ú¥÷ è<å�á û�ýxú&%�á ã¡ä©èU'è#á ÷ ÷ ýxånû�é útû ú£á ä �á �¡ý©é úNì¡õ L©õ Ö

2.1x+3.5y =8

−3.5x+2.1y =15

−3.502x+2.1y =15

r

s

t

� b*�h|^eyo �h� �$� ®[sryaHefc^efa¡rtoq�Hbj]h[^a �haH]hk8a¡ryefbjzFo�_^aHi zH]h[^_^bj�Hbj]h[poqk8aH[sry] ��g�aH[*zH]h[^_^bj�Hbj]h[poqk8aH[sry]w_^b|^[��fbj��ryaHk`o�ijbj[^aFoqefaq�

ÖP×KØfÙFÚ�Û^Ü�Ý2Þqß��*A�ß#àFá�â ã¡äxå�áçæ3è�é�áh÷*) è�)Hýxú&%�á ã¡ä©èQ:Nÿê ! ïQP 3 ë�í < Nnì¡õ ï¥ð�P

â('©è 'xú()Hýxútû�û�é�ã£é ú¡æ3áçâ á�â ã¡á äxâ áçæ3è�äHû�á�ý��¡ýxú¥÷ á�ú�ï3õ»àFá�â ã¡äxå�áçæ3è�é�ásùqã¡ás÷*) è�)Hýxú&%�á ã¡ä©èQ:Nÿê ! ïQP 3 ë?ìfí � � < Nnì¡õ ô¡í�P

â('©èKù©ýQ'ã£èfå�å�è�é�è -Fáçånû ú£â ã¡üøè�ý©äxú ùqè�é�û�ý©é �xú&%�á ã¡ä©èKæ3è�÷ ÷çú ù©é�èfâ èfæ3è�äHû�è�#Fá ä )Hýxú¥äHû�ãK'èKã¥û�û�è�äFý3û ú å�ã¡üøü�ú¥äxæ3ã\ìfí�� �Kú¥÷å�èfâ ã¡äxæ3ã�üøè�ü �©é�ã\æ3á Nnì¡õ ï¥ð�P V�'èCá üøüøèfæ3áçútû�ã\ãHå�å�è�é -tú¥é�è<â('©èONnì¡õ ô¡í�P 'xúXæ3ý©è£å�ã¡÷ ý�%�á ã¡ä©á�é�èfú¥÷ á�Nnì¡õ ð¡ð�è£ï3õ í©ìRP�è æ3ý©èå�ã¡÷ ý�%�á ã¡ä©ásâ ã¡üøù©÷ èfå�å�è Njï ! í©î í©ì �pèCï ��í©î í©ì � P®õpö�ú ùqè�é�û�ý©é �xú&%�á ã¡ä©è#á äHû�é�ã3æ3ã¥û�û ú 'xú )Hý©á äxæ3ásüøã3æ3á �qâ�útû�ãøá ÷�â�ú¥üøùqãæ3á�ú¥ù©ùxú¥é�û�è�ä©è�ä�%fúBæ3è�÷ ÷ è\å�ã¡÷ ý�%�á ã¡ä©á�èfæ 'xúBá äHû�é�ã3æ3ã¥û�û�ã�ä©è�÷ ÷ èXé ú¡æ3áçâ á�é�èfú¥÷ á�ý©äxúBùqè�é�û�ý©é �xú&%�á ã¡ä©è\æ3á�ý©ä2ã¡é æ3á ä©è\æ3á�¡é ú¥äxæ3è % %fú ñ -¡ã¡÷ û�è<å�ý©ùqè�é�á ã¡é�è¡õ Ö

ÖP×KØfÙFÚ�Û^Ü�Ý2Þqß���9�ß#àFá�â ã¡äxå�áçæ3è�é�áhá ÷�ù©é�ã��©÷ è�ü�úøæ3è�÷�â�ú¥÷çâ ã¡÷ ãøæ3è�÷ ÷çúøæ3á 1hè�é�è�ä�%fú û�é úøæ3ý©è<äFý©üøè�é�ájõàFáçú¥ä©ãøêNë?ìyï¥ô¥ñFó L&% <©î í#è ��ëDìyï¥ô¥ñFó L&%2%Fî í<è<å�áçú �ø÷çúøæ3á 1hè�é�è�ä�%fú� ë�ê ! ��ëDì¡î í¡í¡í¡í¡í¡í¡í¡í V

2 8 � è�é\æ3ý©è�é�è û�û�è�ã¡é�û�ã��¡ã¡äxú¥÷ ádæ3áKâ ãFè�:Xâ á è�äHû�èYú¥ä��¡ã¡÷çú¥é�è�#�é�áçå�ùqè û�û�á -tú¥üøè�äHû�è�# T�è0T IKå�ýxå�å�áçånû�èB÷çú�é�è�÷çú&%�á ã¡ä©èQ:T�,/T Iqë ! ì¡õ

Page 36: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

" =

å�á�â ã¡äxå�áçæ3è�é�ásã¡é ú á ÷�ù©é�ã��©÷ è�ü�ú�#3â('©èCå�ásã¥û�û�á è�ä©è<ùqè�é�û�ý©é �xú¥äxæ3ãøá ÷sù©é�èfâ èfæ3è�äHû�è�#� I ë�ê I ! � I < ê I ëDìyï¥ô¥ñFó L&% <©î ì < � I ëDìyï¥ô¥ñFó L&%EL©î ð <

÷çúøâ ý©á�å�ã¡÷ ý�%�á ã¡ä©èU'è� I ë?ì¡î ï¥í¡í¡í¡í¡í¡í¡í�î

àFásãHå�å�è�é -Fásâ('©è#ý©äxúøùqè�é�û�ý©é �xú&%�á ã¡ä©èKä©è�÷ ÷çúøä©ã¡äxú£â á þÿé úøå�á �¡ä©á �qâ�útû�á -túøæ3è�á�æ©útû�á 'xúøù©é�ãE-¡ã3â�útû�ã ý©äxú£ùqè�é�û�ý©é �xú&%�á ã¡ä©èä©è�÷ ÷çúøå�èfâ ã¡äxæ©ú£â á þÿé úøå�á �¡ä©á �qâ�útû�á -túøæ3è�÷ ÷çúøå�ã¡÷ ý�%�á ã¡ä©è�#©æ3ý©ä*)Hý©è<÷*) è�é�é�ã¡é�èKé�è�÷çútû�á -¡ã£ä©è�á�æ©útû�á� ê ! ê I6�� ê � ë6í©î <©ì�M�ìfí � � < � � ! ��I �� ��� ë6í©î <©ì�M�ìfí � �

å�áZ'è<ú¥üøù©÷ á �qâ�útû�ãøá ä� � ! � I8�� �O� ë6í©î ï3î

àFèBå�á�â ã¡äxå�áçæ3è�é úYá ÷däFý©üøè�é�ã�æ3è�÷ ÷ è�â á þÿé�è�å�á �¡ä©á �qâ�útû�á -¡è�â ã¡é�é�è û�û�èBæ3è �¡÷ ádú¡æ©æ3è�äxæ3ádè�æ3è�÷dé�áçå�ý©÷ û útû�ã #må�á 'xú�â('©èBå�áùxú¡å�å�ú£æ©ú£ã¥û�û�ãXâ á þÿé�è�#3á ä�ê I�è ��I #xú¡æNý©äxúøâ á þÿé ú�#3á ä � I #xâ á ã*'è#å�ã¡ä©ã ��â�ú¥äxâ è�÷ ÷çútû�è2�Q2 �Kå�è û�û�èCâ á þÿé�è<å�á �¡ä©á �qâ�útû�á -¡è¡õ " ú¥÷ èþÿè�ä©ã¡üøè�ä©ã 'è#æ3è û�û�ã #xú¥ù©ù©ý©äHû�ã #�C(��I�C("� ! �I��*>�I"dèfæNý©äxú �¡á ýxånû�á �qâ�ú&%�á ã¡ä©è<æ3á�èfå�å�ã 'è<æ©útû ú£ä©è�÷ ÷*) èfå�è�üøù©á ã�ì¡õ ï %Fõ Ö

�Bo��hijb»aH�faHk8c^b�c^efaHzHaH_^aH[sryb��fb�_^aH_^|^zHa�zt�^a�[^aHb�c^ef]hg^ijaHk8b�k`oqimzH]h[^_^bj�Hbj]h[poxrybø��c^bjzHzH]hijb�aHefef]hefbj�[^aHb�_poxryb�c�]h�f�f]h[^]8efbj�f|^i�rtoqefaBoqk8c^ijb��pzFoxrybm[^aHijiuoV�f]hij|^�Hbj]h[^aq�^efaH[^_^aH[^_^]hiuoVrtoqi��h]hi�rtoVbj[poqzHzHa¡rfrtoqg^bjijaq���a¡rfry] ��i�� aHefef]hefaB[^aHb�_poxryb�a���i�� aHefef]hefaBzH]hefefbj�fc�]h[^_^aH[sryaY[^aHijiuoV�f]hij|^�Hbj]h[^aq�^a�c�]h��ry]

�DÏ���� � ?�>}a�_^a¡rfry]6ÃÈÄ�ÆqÃ�Ët¢�ÆqãËt¦xÄ�ÆqÃ�ÉHÃ�¦xÄ�§qÅ`¢¡Äp­®¦\_^aHi�c^ef]hg^ijaHk`o+B^efbj�f|^i�rto���~�|^bj[^_^b��^zt�^a��

�fa�� � �Ybjimc^ef]hg^ijaHk`o�}a �¥¢¡Ä�Ët¦xÄ�ÆqÃ�ÉHÃ�¦xÄ�§q­®¦ B�fa�� .���bji�c^ef]hg^ijaHk`o:}a2Å8§qª�Ët¦xÄ�ÆqÃ�ÉHÃ�¦xÄ�§q­®¦ � �3�

Ô bKefbjzH]hef_^bKzt�^aq��aH�f�faH[^_^]�bj[sryaHefaH�f�yoxryb<oqijiuo2efbj�f]hij|^�Hbj]h[^a�_^bKc^ef]hg^ijaHk8b�k8aH_^buoq[srya�zFoqijzH]hiuoxry]hefaq�ª�� §qÄ�§qª¬Ãj¨¡ÃøÆ�¢¡ªCËt¦xÄ�ÆqÃ�ÉHÃ�¦xÄ�§qÅ`¢¡Äp­®¦�Æ�¢25q¢8¢t¨t¨3¢¡¤y¢ �H§q­�­ §�¨3¢¡ÅNÇ�¤y¢¡�mc�aHefzt���a2b#_poxrybK�f]h[^]6o Ama¡rfryb<oqijk8a¥�[^]w_poqiji�� aHefef]hefa2_^b ef]h|^[^_^]%AK� 3N_�aH�faHk8c^bj]^�KbjiCc^ef]hg^ijaHk`o ; �h� �O�@?Bc^|�}]DaH�f�faHefa`��bj��ry]wzH]hk8a`|^[po�haHef�fbj]h[^a�_^b ; �h�>= 1 ?£c�aHe�ry|^efgpoxrto�_poqi��f]hij]�aHefef]hefa�_^b�ef]h|^[^_^]%A?[^aHid�faHzH]h[^_^]8k8aHkVg^ef]^�}��~�|^bj[^_^b»bjk8c�]he�rtoq[srya�efbj|^�fzHbjefa\oY_poqefaX|^[poY��rybjk`o�~�|poq[sryb�rtoxryb��qoB_^aHiji�� bj[^_^bjzHa�_^bpzH]h[^_^bj�Hbj]h[po©�k8aH[sry]^�

2 ��ö�úNâ�ú¥äxâ è�÷ ÷çú&%�á ã¡ä©è£æ3è�÷ ÷ è�â á þÿé�è�å�á �¡ä©á �qâ�útû�á -¡èK'èøæ3ãE-Fý3û úNú¥÷�þ�útû�û�ãBâ('©è£ä©è�÷ ÷*) è1hè û�û�ýxú¥é�èø÷çúNå�ã¥û�û�é ú&%�á ã¡ä©è á äVâ ã¡éPYé�áçå�ùqã¡äxæ3è�ä�%fú\æ3è�÷ ÷ è�â á þÿé�èøý��¡ýxú¥÷ á Nÿé�áçå�ùqè û�û�á -tú¥üøè�äHû�èøá ä�êVèøá ä ��PKå�ápã¥û�û�á è�ä©è£ý©äxú�â á þÿé ú\äFý©÷ ÷çú©õ . å�è �¡ý©á û�ã�æ3è�÷ ÷çúä©ã¡é�ü�ú¥÷ á % %fú&%�á ã¡ä©è<÷ è£â á þÿé�è äFý©÷ ÷ èU-¡è�ä��¡ã¡ä©ã�û�é ú¡å�÷çútû�è4-¡è�é å�ã\å�á ä©áçånû�é ú�è ùqè�é�û ú¥äHû�ãNâ ã¡é�é�áçå�ùqã¡äxæ3ã¡ä©ãXúXùqãHå�á %�á ã¡ä©á^â('©è-¡è�ä��¡ã¡ä©ã ùqè�é å�è¡õ2.h�ö�ú�ü�ú&���¡á ã¡é#ùxú¥é�û�èXæ3è�á�ù©é�ã��©÷ è�üøá�æ3è�÷ ÷*) . äxú¥÷ áçå�á ,dý©üøè�é�áçâ�ú�é�áçå�ý©÷ û ú�#�ù©áP'ý�ãBüøè�ä©ã #�ü�ú¥÷�â ã¡äxæ3á %�á ã¡äxútû úNèfæ 'èå�è�üøù©é�è����`í©õ

Page 37: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# " �

Ô b<�f|^c^c�]h[ �so�_^bK�qoqij|�rtoqefaV|^[po:ln|^[^�Hbj]h[^a ���efaFoqija�_^bK�qoqefbuoqg^bjija�efaFoqijaq��bj[�|^[�c^|^[sry]��d� ®[�haH[^aHeyoqijaq��[^aHiji�� oqc^c^ef]�zHzHbj]BzH]hk8c^|�rtoq�Hbj]h[poqijaq���[^]h[`�yoqe©}oB�qoqij|�rtoxrto�bj[2�d��k`o�bj[�� 5�� ��o�zFoq|^�yo_^aHiji�� aHefef]hefa�_^b�ef]h|^[^_^]%AK�C�<� aHefef]hefa � �wbj[^_^|^zHa�~�|^bj[^_^b�|^[6aHefef]hefaB[^aHi�efbj�f|^i�rtoxry]^� Ô b�[^]qryb�zt�^a�iuoln|^[^�Hbj]h[^a 8eyoqc^c^efaH�faH[srtoBbji�c^ef]hg^ijaHk`o\k`oxryaHk`oxrybjzH]^�©�8b��f|^]hb�_poxryb ; � �w}a£iuoNij]hef]Nc�aHe�ry|^efgpoq�Hbj]h[^aoq�f�f]hij|�rto@?£a ; �"?£iuo��f]hij|^�Hbj]h[^aq��8�]h��ry]

� �Ï ; � 5�� �"?#Ò ; �"?�? ; �h� �Q� ?�fa *}aB_^aHefb��qoqg^bjija�bj[��d�p�fb��po$�

� 6Ï��: ; �"? � � 5 ; � �"?�? ijbjk� � � ; � �"?� � Ï>�

a�~�|^bj[^_^bР� CРQ Р�: ; �"? � �<�B

Ð : ; �"?3Ðm}aq�moqijij]heyo��^bji�l�oxrfry]hefa�_^bKoqk8c^ijb��pzFoq�Hbj]h[^aB[^aHijiuo��f]hij|^�Hbj]h[^aY_^aHiji�� aHefef]hefaYoq�f�f]hij|�ry]`c^efaH�faH[srya[^aHbs_poxrybsaH_8}aq�x~�|^bj[^_^b��©i���ÃÈÄ�ÆqÃ�Ët¢£ÆqÃ�Ët¦xÄ�ÆqÃ�ÉHÃ�¦xÄ�§qÅ`¢¡Äp­®¦\§©¨t¨3¦xª¬«�­®¦x��®[Yk8]hi�rybqzFoq�fb»}a<c^by}|�bj[sryaHefaH�f�yoq[sryazH]h[^�fbj_^aHeyoqefa�i�� aHefef]hefaBefaHiuoxryb��h]^��c�aHeXbji�~�|poqija��qoqijaBiuoVefaHiuoq�Hbj]h[^a

����

� ����Q ����

: ; �"?®� ; �"?���������

� ������Ó ; �h� � =O?

�do�~�|poq[sryb�rx}o� ; "?f�"?CÏ Ð �: ; �"?®�<Ð �»Ð ; �"?3Ð ; �h� ���@?

}abjiNl�oxrfry]hefa�_^b�oqk8c^ijb��pzFoq�Hbj]h[^aD[^aHijiuo*�f]hij|^�Hbj]h[^a�_^aHiji�� aHefef]hefa?efaHiuoxryb��h]>[^aHb�_poxrybBaH_ }aq�B~�|^bj[��_^b�� i���ÃÈÄ�ÆqÃ�Ët¢�ÆqÃ�Ët¦xÄ�ÆqÃ�ÉHÃ�¦xÄ�§qÅ`¢¡Äp­®¦?¤y¢¡ª�§q­�Ã65q¦x� � ; "?f�"?�~�|poq[sryb��pzFooqc^c^|^[sry]�iuo?�faH[^�fbjg^bjijb�rx}o?_^aHic^ef]hg^ijaHk`o�o��hijbmaHefef]hefb�[^aHb�_poxryb��

����|�rybjijaXefbjzH]hef_poqefaXiuo��fa �h|^aH[srya�efa �h]hiuo�]hc�aHeyoxryb��qo�l�oqzHbjijk8aH[sryaX_^aH_^|^zHbjg^bjijaX_poqijiuo ; �h� � =O?&�K�faNb_poxryb»_^aHi�c^ef]hg^ijaHk`o��poq[^[^] �:zHb�lnefa\�fb*�h[^b��pzFoxryb��haXzH]hefefa¡rfrya ; � � �¥��zH]h[ �<c^efaHzHbj�fbj]h[^a\_^aHi��fbj��ryaHk`ooqefb�rybjk8a¡rybjzH]wl � c��B?¥��oqijij]heyo?|^[ bj[^_^bjzHa6_^bXzH]h[^_^bj�Hbj]h[poqk8aH[sry]efaHiuoxryb��h] � Ï �����8bj[^_^bjzFo?zt�^awiuo�f]hij|^�Hbj]h[^a8[^]h[�c^|�}]DaH�f�faHefa�]qrfryaH[�|�rto�zH]h[�c^by}|_^b ��Ò��zHb�lnefa8�fb*�h[^b��pzFoxryb��ha2zH]hefefa¡rfryaq� Ô b c^|�}]zH]h[^zHij|^_^aHefaDoqijij]heyoDzt�^aw|^[ c^ef]hg^ijaHk`o�k`oqi�zH]h[^_^bj�Hbj]h[poxry]P}a�zH]hk8c^ija¡rtoqk8aH[srya6bj[sryeyoxrfrtoqg^bjija:�fa� Q � B�oqi�ryefbjk8aH[sryb»aH�f�f]�}a\ryeyoxrfrtoqg^bjija\o�cpoxrfry]�zt�^aB�fb��fbuo�_^bj�fc�]h��rybmoY�f]hc^c�]he�rtoqefa�|^[po�c�aHef_^b�rtoY_^b�VzHb�lnefaB�fb*�h[^b��pzFoxryb��haB[^aHijiuoV�f]hij|^�Hbj]h[^aq�

ÖP×KØfÙFÚ�Û^Ü�Ý2Þqß�� �sß#àFáçú���Nÿê�P�ë�ê��¥õ ��áçå�ý©÷ û ú� N � < ê�Pmë6ï�hê � �Yî

� áP'ýøá ä��¡è�ä©è�é ú¥÷ è�#� ) � ; # � Nÿê & < ê�P�ë ) è )Hý©á äxæ3á©á ÷qâ ã¡äxæ3á %�á ã¡äxú¥üøè�äHû�ã<æ3è�÷©ù©é�ã��©÷ è�ü�ú#æ3è�÷ ÷çú�-tú¥÷ ý3û ú&%�á ã¡ä©èdæ3è�÷ ÷çúþÿý©ä�%�á ã¡ä©è<ê & ä©ã¡ä�æ3á ùqè�äxæ3è æ©ú£ê�V©á ä©ã¡÷ û�é�è#á ÷�ü�ú¥÷�â ã¡äxæ3á %�á ã¡äxú¥üøè�äHû�ãøæ3è�÷ ÷çú þÿý©ä�%�á ã¡ä©èCâ é�èfå�â è#ú¥÷�â é�èfå�â è�é�è#æ3á ) õ Ö

Page 38: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

"�"

ÖP×KØfÙFÚ�Û^Ü�Ý2Þqß�����ß#àFáçú���Nÿê�P�ë @ ì ! ê$#©ê E ì¡õ��Èä )Hý©èfånû�ã�â�ú¡å�ã� N � < ê�P�ë � ê �

ï�Nnì ! ê�P Vùqè�é�ê�ë�í©î ð¡ð¡ð¡ð�# � N � < ê�P�ë:ñHð¡ð¡ð©î óCèD)Hý©á äxæ3á�á ÷�ù©é�ã��©÷ è�ü�úK'è#ü�ú¥÷�â ã¡äxæ3á %�á ã¡äxútû�ã£ùqè�é�ê J ì¡õ Ö

8�aHe�c^ef]hg^ijaHk8b\c^by}| zH]hk8c^ijaH�f�fb�_^aHijiuo�faHk8c^ijbjzHa��qoqij|�rtoq�Hbj]h[^aw_^bN|^[po�ln|^[^�Hbj]h[^aD�fzFoqiuoqefaD_^b|^[po��f]hiuo��qoqefbuoqg^bjija � �fb�c^|�}]>zHaHefzFoqefa?_^bN�qoqij|�rtoqefaDbji\l�oxrfry]hefaD_^bBoqk8c^ijb��pzFoq�Hbj]h[^a�_^aHiji�� aHefef]hefa�haH[^aHeyoqijbj�H�Foq[^_^]Yi�� oqc^c^ef]�zHzHbj]Y~�|^b�c^efaH�faH[srtoxry]^�CZ\[:zFoq�f]V�faHk8c^ijbjzHaX_po�oq[poqijbj�H�FoqefaY}a\~�|^aHijij]Y_^aHijiuo�f]qrfryeyoq�Hbj]h[^a�ryeyo�_^|^a�[�|^k8aHefb��3 d . å�å�è �¡äxútû ú ý©äxú£þÿý©ä�%�á ã¡ä©è�-¡è û�û�ã¡é�áçú¥÷ è<ú¡æKTÕâ ã¡üøùqã¡ä©è�äHû�á*#©æ3á ) -tú¥é�áçú&�©á ÷ á :

� ë N � � < î�î�î < � S P� � ë � � Nÿê P �pë?ì < î�î�î < Tê ë Nÿê � < î�î�î < ê & P

æ©ú¥÷ ÷ ãøåP-Fá ÷ ý©ù©ùqã�á äBå�è�é�á è<æ3á " ú �F÷ ã¡émèfæ�ú¡æ3ã¥û�û ú¥äxæ3ã #3á ä©ã¡÷ û�é�è�#3ùqè�édæ3ý©èD-¡è û�û�ã¡é�á�ú¡å�å�è �¡äxútû�á� ë N � � < î�î�î < � ��P� ë N � � < î�î�î < � ��P÷çú£ä©ã¥û ú&%�á ã¡ä©è � � ë � � �� � < î�î�î < � �� ���å�á 'xú �

�� �ë � � � �

�yê�� , ê��� � ��êê Nnì¡õ ô¥ñ P

æ3ãE-¡è#â ã¡ä�ê è

�� å�á�á äxæ3áçâ�ú¥ä©ã #3é�áçå�ùqè û�û�á -tú¥üøè�äHû�èQ:�

ê ë N�ê � < î�î�î <

�ê & P�

� ë N�� � < î�î�î <

�� S P

æ©úøâ ý©á�å�è �¡ý©è �� � ë � � Nÿê �

�ê P ! � � Nÿê P �pë?ì < î�î�î < T�î

� ú¡å�å�ú¥äxæ3ã£ú¥÷ ÷ è#ä©ã¡é�üøè#á ä��xä©á û�ã�ä©è�÷ ÷çú Nnì¡õ ô¥ñ P�#©å�ásã¥û�û�á è�ä©è��� # E � � � ��yê�� , ê��� � � # ,

�êê #ö�ú )Hýxú¥äHû�á ûE'ú

� N � < ê P�ë � � � ��yê�� , ê��� � � #'è#æ3è û�û ú �MI� �5C("�� �&CL>�I� � I��*>�I�1< " I KM>OF "� ��K*�ML1><æ3è�÷ ÷çú þÿý©ä�%�á ã¡ä©è � ä©è�÷�ù©ý©äHû�ãøê õ

Page 39: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# " �

ÖP×KØfÙFÚ�Û^Ü�Ý2Þqß�� �sß#àFáçú���Nÿê < ��P�ë:ê ! �hõ � ãHånû�ã��Në ��Nÿê �

�ê < � �

���P ! ��Nÿê < ��P�#xå�á+'xú�:�

��ë�ê ��ê ! � !

�� ! ê � ��ë

�ê !

�� <

èD)Hý©á äxæ3á���Z�� �Z� E �

�ê �2� �

����

� ê ! ��� ë � ê �2� � ���� ê ! ��� ,��ê �2� �

����

� ê �2� � ��� ëë � ê � ! �2� � ��� ���� ��Nÿê < ��P/� , � � N

�ê <���P/� �

� � Nÿê < ��P/� � <

æ3ãE-¡è(� � , � �Hë � � , � � � á ä � �&# � ! ë�� �� ! è � �<ë�� �� � õ �Èä )Hý©èfånû�ã�â�ú¡å�ãø÷*) á äxæ3áçâ èCæ3á�â ã¡äxæ3á %�á ã¡äxú¥üøè�äHû�ãøé�è�÷çútû�á -¡ãK'è� N � < ê < ��P»ë N � ê �2� � ��� P % � ê ! ���®î

� )Fè/-Fáçæ3è�äHû�è#â('©è � N � < ê < ��Pmâ é�èfå�â èKú¥÷sæ3á üøá äFý©á é�è<æ3è�÷ ÷çú£æ3áçånû ú¥ä�%fú<û�é ú ê�è �OV©æ3ý©ä*)Hý©èKá ÷�â�ú¥÷çâ ã¡÷ ã£æ3è�÷ ÷çú£æ3á 1hè�é�è�ä�%fúû�é úVæ3ý©è�äFý©üøè�é�áD'èBý©ä:ù©é�ã��©÷ è�ü�ú�ü�ú¥÷�â ã¡äxæ3á %�á ã¡äxútû�ã�)Hýxú¥äxæ3ãVáKæ3ý©èBäFý©üøè�é�á�å�ã¡ä©ã �P-Fáçâ á ä©á �©õ C�á6'ã2å�ù©á è �Hú�á ÷þÿè�ä©ã¡üøè�ä©ã�æ3è�÷ ÷çú�â�ú¥äxâ è�÷ ÷çú&%�á ã¡ä©èKá ÷ ÷ ýxånû�é útû�ãøä©è�÷ ÷*) èfå�è�üøù©á ã�ì¡õ ïtñxõ Ö

ÖP×KØfÙFÚ�Û^Ü�Ý2Þqß�� ��ß���áçå�ý©÷ û ú�#3å�ùqèfå�å�ã #Fý3û�á ÷ èKä©è�÷ ÷ è#ú¥ù©ù©÷ áçâ�ú&%�á ã¡ä©á*#3æ3è û�è�é�üøá äxú¥é�è�ý©äxú ånû�á ü�ú£æ3è�÷ ÷*) á äxæ3áçâ è<æ3ásâ ã¡äxæ3á Y%�á ã¡äxú¥üøè�äHû�ã #Fùqè�édá ÷�ù©é�ã��©÷ è�ü�ú

��ë��wê <á äVâ ý©á»å�ã¡ä©ã\ä©ã¥û�áp÷çú\ü�útû�é�áçâ è�� èfæYá ÷ -¡è û�û�ã¡é�è£êVè�å�á»æ3èfå�áçæ3è�é ú\æ3è û�è�é�üøá äxú¥é�è£á ÷Z-¡è û�û�ã¡é�è �#�útû�û�é ú -¡è�é å�ãXá ÷p÷ ã¡é�ãù©é�ã3æ3ã¥û�û�ãxõ»àFè�#xæ3ý©ä*)Hý©è�#xå�ás÷ è ���¡è#á ÷�ù©é�ã��©÷ è�ü�ú£â ã¡üøèQ:

�øë��wêNë ��Nÿê�P <á ä �xú¡å�è<ú )Hýxú¥äHû�ã�æ3èfå�â é�á û�û�ãø÷*) á äxæ3áçâ èCæ3á�â ã¡äxæ3á %�á ã¡äxú¥üøè�äHû�ã�ú¡å�å�ã¡÷ ý3û�ã 'è#æ©útû�ãXæ©ú

� � I Nÿê�P/�Hë���� <üøè�äHû�é�è#÷*) á äxæ3áçâ èCæ3á�â ã¡äxæ3á %�á ã¡äxú¥üøè�äHû�ã£é�è�÷çútû�á -¡ã£é�áçå�ý©÷ û ú

�����OI Nÿê�PÈê��Nÿê�P

����ë ��wê����wê�� ëDì¡î

Ö

021� *GIJ�9#GVS?;FEF;¡9��G =?; L?7RGVE��258AC;¡9<TU53N[^zt�^a8�fa8bji#c^ef]hg^ijaHk`o:_po6efbj�f]hi��haHefa6}a2g�aH[zH]h[^_^bj�Hbj]h[poxry]^��iuo�g�]h[srx}o6_^aHi<efbj�f|^i�rtoxry]�zH]hk8c^|�rto©��Hbj]h[poqija�_^bjc�aH[^_^a�_poqiX��zH]hk8c�]he�rtoqk8aH[sry]h��_^aHiji�� oqi*�h]hefb�ryk8]Y|�rybjijbj�H�Foxry]^�

Page 40: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

" �

ÖP×KØfÙFÚ�Û^Ü�Ý2Þqß�� ��ß� í©î í¡í¡í©ì)ê � ��ë ìê � ��ë ï

àFá�ù©ýQ'ãNæ3á üøãHånû�é ú¥é�èCâ('©èU)Hý©èfånû�ã\ù©é�ã��©÷ è�ü�ú 'è �qè�äYâ ã¡äxæ3á %�á ã¡äxútû�ã*Vq÷çú\å�ýxú\å�ã¡÷ ý�%�á ã¡ä©è('è á ÷Q-¡è û�û�ã¡é�èONnì¡õ í¡í¡í©ìfí¡í¡í©õ õ õ #í©õ ð¡ð¡ð <¡ð¡ð¡ð¡ð©õ õ õ P®õàFãHånû�á û�ý©è�äxæ3ã ú¥÷ ÷çúCå�èfâ ã¡äxæ©ú<è�)Hýxú&%�á ã¡ä©è�÷çú<ù©é�á ü�úCè�)Hýxú&%�á ã¡ä©è�üøã¡÷ û�á ù©÷ áçâ�útû ú<ùqè�éKìfí�3dè�å�ã¥û�û�é útû�û úCæ©ú¥÷ ÷çúCå�èfâ ã¡ä�Yæ©ú�#©å�á 'xúøá ÷�å�áçånû�è�ü�úøè�)Hý©á -tú¥÷ è�äHû�è� í©î í¡í¡í©ì"ê � �-ë ì

! ð¡ð¡ð¡ð �-ë ! ð¡ð¡ð <æ©úøâ ý©á

� �-ë ��h.h.h ���h.h.h.h ë ì¡î í¡í¡í©ìfí¡í¡í�î�î�îêUë �d � d.d.d�� Nnì ! h.h.h �h.h.h.h P'ë í©î ð¡ð¡ð <¡ð¡ð¡ð <�î�î�î��å�è �¡ý©è�äxæ3ã )Hý©èfånû�ã�ú¥÷ �¡ã¡é�á û�üøã£á äNý©äBå�áçånû�è�ü�úøú¥é�á û�üøè û�áçâ ã£þnõ ù�õ�â ã¡äNù©é�èfâ áçå�á ã¡ä©è �»ë�ôøå�á�ã¥û�û�á è�ä©èQ:

� �-ë � d � �Md.d�� �Md��� d � �Md.d�� �Md � ë(í©î ìfí¡í M�ìfí � ë ìêUë d � �Md.d�� �Md �d � �Md.d�� �Md���� N�í©î ìfí¡íUM�ìfí � ! í©î ìfí¡í M�ìfí � P"ë í

â á ã*'è�á ÷O-¡è û�û�ã¡é�è#å�ã¡÷ ý�%�á ã¡ä©è#â�ú¥÷çâ ã¡÷çútû�ã£æ©ú¥÷ ÷*) ú¥÷ �¡ã¡é�á û�üøã 'xú ÷çú£å�èfâ ã¡äxæ©ú£â ã¡üøùqã¡ä©è�äHû�è#å�è�ä�%fú£ú¥÷çâ ý©äxú£â á þÿé ú£å�á �¡ä©á �qâ�úEYû�á -túøâ ã¡é�é�è û�û ú é�áçå�ùqè û�û�ãXú¥÷ ÷çúøå�ã¡÷ ý�%�á ã¡ä©èKèfå�útû�û úøæ3è�÷�ù©é�ã��©÷ è�ü�ú©õ�Èä&-¡è�é�û�è�äxæ3ãN÷*) ã¡é æ3á ä©è�æ3è�÷ ÷ è�è�)Hýxú&%�á ã¡ä©ápèøüøã¡÷ û�á ù©÷ áçâ�ú¥äxæ3ã�ùqè�éXìfí��%3ø÷çúNù©é�á ü�úNè�)Hýxú&%�á ã¡ä©è�#�å�ápã¥û�û�á è�ä©è�#�ä©è�÷ ÷ ãånû�èfå�å�ãøå�áçånû�è�ü�ú þnõ ù�õ #x÷çúøå�ã¡÷ ý�%�á ã¡ä©èKê�ë?ì�# ��ëDì�#©â('©è 'è#ý©ä+) ú¥ù©ù©é�ãHå�å�á ü�ú&%�á ã¡ä©è#ú¡â�â è û�û ú&�©á ÷ è<æ3è�÷ ÷çúøå�ã¡÷ ý�%�á ã¡ä©è¡õW�ý©èfånû�ã2þÿè�ä©ã¡üøè�ä©ã 'èYæ3ãE-Fý3û�ã`ú¥÷Küøã3æ3ã8á äwâ ý©á<å�áU'èYù©é�ã¡ùxú&�Hútû�ã�÷*) è�é�é�ã¡é�èYæ3á#é�ã¡ý©äxæ3ã 1?æ3ý©é ú¥äHû�è�÷*) èfå�èfâ ý�Y

%�á ã¡ä©è\æ3è�÷ ÷*) ú¥÷ �¡ã¡é�á û�üøãxõ5�Èä3þ�útû�û�á*#»ä©è�÷mù©é�á üøã�â�ú¡å�ã�÷çú�æ3á -Fáçå�á ã¡ä©è\ùqè�é£á ÷�â ãFè�:Xâ á è�äHû�èNí©î í¡í¡í©ì 'xú�ù©é�ã3æ3ã¥û�û�ãYâ ã¡üøèþ�útû�û�ã¡é�èCüøã¡÷ û�á ù©÷ áçâ�útû�á -¡ã�ìfí�3 ú¥üøù©÷ á �qâ�ú¥äxæ3ãX÷*) è�é�é�ã¡é�èCæ3á�é�ã¡ý©äxæ3ã 1^õ �Èä&-¡è�é�û�è�äxæ3ãX÷*) ã¡é æ3á ä©è æ3è�÷ ÷ è è�)Hýxú&%�á ã¡ä©á*#qá ÷�þ�útûPYû�ã¡é�è#üøã¡÷ û�á ù©÷ áçâ�útû�á -¡ã�æ3á -¡è�äHû úNìfí��%3<â('©è�#xú¥÷�â ã¡äHû�é ú¥é�á ãøæ3è�÷�ù©é�á üøã�â�ú¡å�ã #3é�áçæ3ýxâ è#ä©ã¥û�è/-¡ã¡÷ üøè�äHû�è#÷çúøù©é�ã¡ùxú&�Hú&%�á ã¡ä©èæ3è�÷ ÷*) è�é�é�ã¡é�è¡õ Ö

®[ �haH[^aHeyoqijaq�s_^|^aXoqi*�h]hefb�ryk8bhc�aHe<iuoNefbj�f]hij|^�Hbj]h[^a _^b^|^[^]B��ryaH�f�f]�c^ef]hg^ijaHk`oNc�]h�f�f]h[^]�c^ef]�_^|^efefaefbj�f|^i�rtoxrybs_^b5AmaHefaH[sryb��qbj[V~�|poq[sry] �hijb�aHefef]hefb�zH]hk8c^ijaH�f�fb���b�_^b�ef]h|^[^_^]%A`�f]h[^]�_^b��haHef�fb�� ®i�zH]h[^zHa¡rfry]B_^b¨¡­ § �HÃȪ¬ÃÈ­�ͧ ; aø~�|^bj[^_^b^_^b^bj[^��rtoqg^bjijb�rx}o@?��fb^efb�lnaHefbj�fzHa�oqijiuo\�faH[^�fbjg^bjijb�rx}o\_^b^|^[8oqi*�h]hefb�ryk8]Noqijiuo\c^efaHzHbj�fbj]h[^a�p[^b�rto���zHbj]^}a�o��hijb�aHefef]hefb�_^b�ef]h|^[^_^]%A�_^aHb�_poxryb�a�_^aHijijaB]hc�aHeyoq�Hbj]h[^b�l � c����¯�°s±#²�³n´�³nµ�²d°�¶p·�·8¸ �H²�½¡À3¼ �K³�¾�³ÈÀ��¼ ��³ø¿�² ¼p¾�»µ»ºx³ÈÀ��µ�Á% Ä�§qª 4�¦x¤¥ÃÈ­�Å`¦8¨¡ÃøÆqÃ�Ët¢�ÃÈÄ�¨¡­ § �HÃȪj¢tÊ ¨3¢ 4qª¬Ã�¢¡¤¥¤y¦x¤¥Ã�Æqà ¤y¦x«�Ä�Æ�¦��/ÃÈÄp­�¤y¦FÆ�¦x­�­�ãÄm¢¡ÃøÆs§q­�Ã<¨¡Ã�Ç�¤y¦tÇ^§ 4s§qÄm¦§qÅNÇ�ª¬Ã 6øËy§qÄ�Æ�¦3¨¡Ã�ÃÈÄ?Å8§qÄpÃ�¢¡¤f§8­ §qªj¢VËE�p¢YÃ#¤¥Ãj¨¡«�ª¬­ §q­�Ã#¨¡Ã�§qÄm¦2ÃÈÄ�§�ËtËt¢¡­�­ § �HÃȪ¬Ã �

®[*oqi�ryefbCryaHefk8bj[^b���|^[>oqi*�h]hefb�ryk8]�}a:bj[^��rtoqg^bjija8�faq�CowzFoq|^�yoD_^aHijiuo�c^ef]hcpo��soq�Hbj]h[^a`_^aHiji�� aHefef]hefa`_^bef]h|^[^_^]%AK��bji�efbj�f|^i�rtoxry]�_^aHiji�� oqi*�h]hefb�ryk8]�_^b5AmaHefbj�fzHa��f]h��rtoq[^�Hbuoqijk8aH[sryaN_poqijiuo��f]hij|^�Hbj]h[^aN_^aHimc^ef]hg^ija¥�k`oV[�|^k8aHefbjzH]^�Z\[woqi�ryef]�aH�faHk8c^bj]2_^bdoqi*�h]hefb�ryk8]�bj[^��rtoqg^bjijaV}aBbjijij|^��ryeyoxry]V~�|^b�_^b��fa �h|^b�ry]^�

Page 41: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# "+,

� �� 5 ��Ó ��� =���= ��� � ��� � 5 ��Ó ��� , � , 1 � � ��� = 5 ��Ó>=$� " = " �O� � ��� � 5 ��Ó>= � , = , � , � ��� " 5 ��Ó�� , � � 1 � = � ��� � 5 ��Ó�� " � �$� " � � ��� � 5 ��Ó���=$� , 1 � � � ��� , 5 ��Ó��O��= " = 1 � � ��� � 5 ��Ó����O� �$���O� � ��� 1 5 ��Ó 1 " 1 � = �$� � ��� � ���� 5 ��Ó � ��� ,5"�"�" � ��� � ��O� 5 ��Ó "�" = � � ��= � ��� ��= Ò ��Ó " �Q��� 1 ,5" � ��� ��=� 5 ��Ó � , � " =$��� � ��� �� " Ò ��Ó , 1 , � 1 , � � ��� �

� ¼ �<°�¾�¾n¼�¶p· � � ��¼p¾��hµ�¾nµ ��³ ¿�²I³�²�À3° �»º©¼p¾n° �<°�º'ºx³��hµ»º©º©°�²d´s¼��.ºx³�½F¿�¾ÈÀ3¼�À©³ µpÀFÀ3°�²�¿�À©³³�� �K¾n°���°�²�À3¼p² �dµ?¾n¼ � µ»º���¿�¾n¼*¸H¶p·�� ��Á �hµ�² wÏ � " ½F¿>¿�²d¼ �� � ��� �� ��d½¡À3°���� �������

ÖP×KØfÙFÚ�Û^Ü�Ý2ÞqßBA ��ß#àFá�â ã¡äxå�áçæ3è�é�áh÷*) á äHû�è �¡é ú¥÷ è� & ë = �

dê & � !&� � ?¡ê < ) �`í�î

�ÈäHû�è �¡é ú¥äxæ3ã£ùqè�é�ùxú¥é�û�á�å�á 'xú�:

= �

dê &�� !&� � ?¡êNë�� ê &�� !&� ��� �d ! ) = �

dê & � � � !&� � ?¡ê�ëDì ! ) = �

dê & � � � !&� � ?¡ê <

â á ã*'èQ:� & ë?ì ! ) � & � � < ) �wì»î Nnì¡õ ôHóQP

Kútû�ã�â('©è� d ë = �

d� !&� � ?¡ê�ë?ì ! ì� <

å�ásù©ýQ'ãXâ�ú¥÷çâ ã¡÷çú¥é�è � & üøèfæ3áçú¥äHû�è#ý©äBú¥÷ �¡ã¡é�á û�üøã �xú¡å�útû�ãøå�ý©÷ ÷çú þÿã¡é�ü ý©÷çú£é�áçâ ã¡é�é�è�äHû�è� d ë?ì ! ì %7� <� & ëDì ! ) � & � � < ) �wì»î Nnì¡õ ô2L�P

àFáhâ ã¡äxå�áçæ3è�é�á*#Fú¡æ�èfå�è�üøù©á ã #Há ÷sâ�ú¥÷çâ ã¡÷ ã æ3á � � 3 õ��Èüøù©÷ è�üøè�äHû ú¥äxæ3ã á ÷hù©é�ã3â èfæ3á üøè�äHû�ãCá ä ©ã¡é�û�é ú¥ä+#Hå�ý\ý©ä+) �[c � � ��à*Cà��3ånû�è�ügfEL¡í¡í¡í�#Hý3û�á ÷ á % %fú¥äxæ3ã�÷çúøå�á ä��¡ã¡÷çú ù©é�èfâ áçå�á ã¡ä©è�#©å�ásã¥û�û�è�ä��¡ã¡ä©ãøásé�áçå�ý©÷ û útû�á�é�á ùqã¡é�û útû�áhá ä " ú&�qè�÷ ÷çú�ì¡õ ó3õ

Page 42: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

" �

�È÷�-tú¥÷ ã¡é�è â�ú¥÷çâ ã¡÷çútû�ã�æ3á � � 3 'è â ã¡üøù©÷ è û ú¥üøè�äHû�èCè�é�é útû�ã*Vxá ä3þ�útû�û�á � � 3 æ3è/-¡èCèfå�å�è�é�èCùqãHå�á û�á -¡ãxõ � áP'ý�á ä �¡è�ä©è�é ú¥÷ è�#á@-tú¥÷ ã¡é�ásã¥û�û�è�äFý3û�á�ùqè�é ) �:ðøå�ã¡ä©ã�â ã¡üøù©÷ è û ú¥üøè�äHû�è#è�é�é útû�ájõ 3 � � è�é â(',+èKå�á@-¡è�é�á �qâ�úU)Hý©èfånû ú�å�á û�ýxú&%�á ã¡ä©è�� � è�édæ©ú¥é�èý©äxú�é�áçå�ùqãHånû ú�ú¥÷ )Hý©èfå�á û�ãKã3â�â ã¡é�é�è�ú¥äxú¥÷ á % %fú¥é�è�÷çú�ù©é�ã¡ùxú&�Hú&%�á ã¡ä©è�æ3è�÷ ÷*) è�é�é�ã¡é�èmæ3á3é�ã¡ý©äxæ3ã 1Xæ3è�÷ ÷*) ý©ä©áçâ ã#æ©útû�ã#æ3á3á ä©ù©ý3ûæ3è�÷ ÷*) ú¥÷ �¡ã¡é�á û�üøã # � d #3å�ý©ù©ùqã¡ä©è�äxæ3ã #Fùqè�é�å�è�üøù©÷ áçâ á ûE'ú�#3æ3áhè1hè û�û�ýxú¥é�èK÷ è�ã¡ùqè�é ú&%�á ã¡ä©áqá äNú¥é�á û�üøè û�áçâ�ú<èfå�útû�û ú©õpö ) è�é�é�ã¡é�è� d á äHû�é�ã3æ3ã¥û�û�ãøá äVì %7� #Fè4)Hý©á äxæ3á�á ä � d #�'èKû ú¥÷ è<â('©èQ:

� d ë � d � � d < � � d � J ìfí � ' <æ3ãE-¡è � d æ3è�ä©ã¥û ú£á ÷�-tú¥÷ ã¡é�è#â�ú¥÷çâ ã¡÷çútû�ãøæ3á � d õ"W�ý©á äxæ3á*#3ùqè�édásû�è�é�üøá ä©á�å�ýxâ�â èfå�å�á -Fáså�á+'xú�:

� � ë?ì ! ìDM � d ë Nnì ! � d P ! � d ë � �"! � d <� � ë?ì ! ïUM � � ë Nnì ! ïUM � � P �8ïUM � d ë � � �8ïUM � dõõõ� & ë?ì ! ) M � & � � ë � & � N ! ìRP & )+* M � d îõõõ

�Èäxæ3áçâ�útû�ãøâ ã¡ä � & ÷*) è�é�é�ã¡é�èKú¡å�å�ã¡÷ ý3û�ã£ú¥÷sùxú¡å�å�ã ) #3ùqè�é ) ëDìfí é�áçå�ý©÷ û ú � & J ìKè�#©ú ùxú¥é�û�á é�è#æ©ú ) ëDìfí�# � & â é�èfå�â è��è�ä©ã¡é�üøè�üøè�äHû�è2��# -tú¥ä©á �qâ�ú¥äxæ3ãXá^â�ú¥÷çâ ã¡÷ á�å�ýxâ�â èfå�å�á -Fájõ �Èä�ùxú¥é�û�áçâ ã¡÷çú¥é�è�#xùqè�é ) ë�ì�ñ�� & J ì�ñ * MVìfí�� ' J <&%tí¡í¡í©õö ) ú¥÷ �¡ã¡é�á û�üøã �xú¡å�útû�ã å�ý©÷ ÷çú#þÿã¡é�ü ý©÷çúONnì¡õ ô2L�P,'è�æ3ý©ä*)Hý©è�á äxånû ú&�©á ÷ è�èKä©ã¡ä�ù©ýQ'ã èfå�å�è�é�èdý3û�á ÷ á % %fútû�ãxõ 3(�BàFáhæ3è/-¡è )Hý©á äxæ3áù©é�ã��¡è û�û ú¥é�è�ý©äBú¥÷ �¡ã¡é�á û�üøãøæ3á -¡è�é å�ã �xú¡å�útû�ãøå�ý�ý©äxú þÿã¡é�ü ý©÷çúøâ('©è#ä©ã¡äBú¥üøù©÷ á �qâ('©ás÷*) è�é�é�ã¡é�è#æ3á�é�ã¡ý©äxæ3ã 1^õàFá�ãHå�å�è�é -Fásâ('©è#÷çú£é�è�÷çú&%�á ã¡ä©è#æ3á�é�áçâ ã¡é�é�è�ä�%fúONnì¡õ ôHóQP�ù©ýQ'ã�èfå�å�è�é�èKé�áçå�â é�á û�û ú£ä©è�÷süøã3æ3ãXå�è �¡ý©è�äHû�èQ:� & � � ë Nnì ! � & P % ) ) ��ì»î Nnì¡õ ô&%�P

Kútû�ãdâ('©èp÷çú�å�ýxâ�â èfå�å�á ã¡ä©è � � & � 'è»æ3èfâ é�èfå�â è�äHû�è^èpû�è�äxæ3è»ú�í�#yå�á¡ù©ýQ'ã�ùqã¡é�é�è � � d ë6í�è»â�ú¥÷çâ ã¡÷çú¥é�è^ùqã¡á � � h < � � � < î�î�î < � � 3ý3û�á ÷ á % %fú¥äxæ3ã�÷çú Nnì¡õ ô&%�P®õmö ) á üøù©÷ è�üøè�äHû ú&%�á ã¡ä©èCæ3áQ)Hý©èfånû�ãXú¥÷ �¡ã¡é�á û�üøã�å�ý©÷ ÷*) �[c � � ��à*Cwà��3ånû�è�ügfEL¡í¡í¡í£â�ú¥÷çâ ã¡÷çú£ý©äxú�©ý©ã¡äxú#ú¥ù©ù©é�ãHå�å�á ü�ú&%�á ã¡ä©è�æ3á � � 3 è�#Hù©áP'ýøá ä �¡è�ä©è�é ú¥÷ è�#tþÿã¡é�ä©áçå�â è�é�áçå�ý©÷ û útû�áxú¡â�â è û�û ú&�©á ÷ á©ùqè�é ) E ìE% N " ú&�qè�÷ ÷çúøì¡õ L�P®õ 3�23 � C�á6'ã�'è�è/-Fáçæ3è�äHû�èXùqè�é ) � ìfí�#�á ä )Hýxú¥äHû�ã #�ùqè�é ã��¡ä©á ) # � & æ3è/-¡è�èfå�å�è�é�è�ùqãHå�á û�á -¡ãBè�û ú¥÷ è\â('©è � & ; � � � & õ,dè�÷ ÷çú£å�è �¡ý©è�äHû�è�û ú&�qè�÷ ÷çúøå�ã¡ä©ã£é�á ùqã¡é�û útû�áhá@-tú¥÷ ã¡é�ásâ ã¡é�é�è û�û�ásæ3á � & #3ùqè�é ) ë�í < ì < ï < î�î�î < ì�ñ #3ú¥é�é�ã¥û�ã¡äxæ©útû�áhú %Câ á þÿé�èå�á �¡á �qâ�útû�á -¡èQ:

) � &í õ L¡ôHï3ìyï¥í2L M�ìfí dì õ ô2L&% <&%tð¥ñ0M�ìfí dï õ ï L¥ñFïtñxì¡ì M�ìfí dô õ ï¥í&%¥ï %EL2L M�ìfí dñ õ ìE%tí <¡ð¡ô¥ñ0M�ìfí dó õ ì�ñFó¡ó¥ôHï¥ð M�ìfí dL õ ìyï L <¡íHïtñ0M�ìfí d% õ ì¡ìyï¥ô <¡ôHó M�ìfí d< õ ìfí¡í¡ð¡ôHï¥í M�ìfí dð õ ð©ì�L©ìyï¡ï¥ð M�ìfí�� �ìfí õ <¡ô <&%2%tí&% M�ìfí�� �ì¡ì õ %2%tôHó¡ï¡ï¥ô M�ìfí�� �ìyï õ %FìE%2%tôHï¡ó M�ìfí�� �ìfô õ L2L¡ð¥ñ�%2%tí M�ìfí � �ì�ñ õ LHï %tôHï3ì�L M�ìfí�� �

3(�3�È÷2-tú¥÷ ã¡é�èpæ3á ) ú�ùxú¥é�û�á é�èpæ©ú¥÷2)Hýxú¥÷ è»á¥é�áçå�ý©÷ û útû�á¡ä©ã¡ä<å�ã¡ä©ã�ù©áP'ýCútû�û�è�äxæ3á �©á ÷ áHæ3á ùqè�äxæ3è�æ©ú¥÷¡å�áçånû�è�ü�ú�þnõ ù�õ�ý3û�á ÷ á % %fútû�ãxõ

Page 43: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# " 1

� �= � ��Ó �O�O�O�O�O�O�� 1 ��Ó � �O�O�O�O�O� � ��� � �� � ��Ó � �O�O�O�O�O� � ��� � �� , ��Ó ��= ,,,, � � ��� � ��=� ��Ó � � , � � 1 � � ��� � ���� ��Ó � 1 �Q� , � , � ��� � �� " ��Ó � = , � =��=� � ��� � �

� ¼ �<°�¾�¾n¼�¶p·���� ��¼p¾��hµ�¾nµ ��³ ¿�²I³�²�À3° �»º©¼p¾n° �<°�º'ºx³��hµ»º©º©°�²d´s¼��.ºx³�½F¿�¾ÈÀ3¼�À©³ µpÀFÀ3°�²�¿�À©³³�� �K¾n°���°�²�À3¼p² �dµ?¾n¼ � µ»º���¿�¾n¼*¸H¶p·�� �»Á �hµ�² wÏ � " ½F¿>¿�²d¼ �� � ��� �� ��d½¡À3°���� �������

ö ) ú¥÷ �¡ã¡é�á û�üøã 'è#á ä3þ�útû�û�á�ånû ú&�©á ÷ è2V3÷*) è�é�é�ã¡é�è � � d á äHû�é�ã3æ3ã¥û�û�ãøá ä � � d å�á�ù©é�ã¡ùxú&�HúCä©è�÷�üøã3æ3ã�å�è �¡ý©è�äHû�èQ:� � d ë � � d � � � d <� � h ë Nnì ! � � d P %¥ï¥íCë � � h ! � � d %¥ï¥í <� � � ë Nnì ! � � h P %FìfðCë � � � � � � d % Njï¥íUM�ìfð�P <õõõ� & � � ë Nnì ! � & P % ) ë � & � � � N ! ìRP & � � � � d % Njï¥íUM�ìfð M ,),), ) P <õõõ

ö ) è�é�é�ã¡é�è�á ä©á %�áçú¥÷ è � � d ú¥÷ �¡è�ä©è�é�áçâ ã£ùxú¡å�å�ã ) 'è<æ3á -Fáçå�ãøùqè�é�ï¥íUM�ìfðUM ,),), M N ) ��ìRP�èD)Hý©á äxæ3á�ä©ã¡ä 'è<ú¥üøù©÷ á �qâ�útû�ã #ú¥ä�%�áZ'è#é�áçæ3ã¥û�û�ãxõ Ö

�Èä3þ�útû�û�á*#»á ä2ý©ä8å�áçånû�è�ü�úYú¥é�á û�üøè û�áçâ ãYú�ù©é�èfâ áçå�á ã¡ä©èXù©áP'ý2è�÷ è/-tútû ú�á ÷"-tú¥÷ ã¡é�è\æ3á ) ú¥ý©üøè�äHû ú Vmå�ámãHå�å�è�é -Fá»ùqè�éE'ã�â('©è)Hýxú¥÷ ý©ä*)Hý©è<å�áçú£÷çú£ù©é�èfâ áçå�á ã¡ä©è �xä©á û úøèfå�áçånû�è<å�è�üøù©é�è#ý©ä -tú¥÷ ã¡é�è#æ3á ) ùqè�édâ ý©á�á ÷�é�áçå�ý©÷ û útû�ã 'è#á äxú¡â�â è û�û ú&�©á ÷ è¡õ3�23��-tú¥÷ ã¡é�áhâ ã¡é�é�è û�û�áhæ3á � & #3ùqè�é ) ë6ï¥í < ìfð < î�î�î < ì�ñ #Fú¥é�é�ã¥û�ã¡äxæ©útû�áqú(%<â á þÿé�è#å�á �¡ä©á �qâ�útû�á -¡è�#Få�ã¡ä©ã é�á ùqã¡é�û útû�áhä©è�÷ ÷çúû ú&�qè�÷ ÷çúøå�è �¡ý©è�äHû�èQ:

) � &ï¥í õ ñFó¡ótñ¡ñ < <0M�ìfí � �ìfð õ ñ�%2%¥ï¡ï %EL0M�ìfí�� �ì < õ ó¥í©ì¡ìfð <Hó M�ìfí�� �ìE% õ ó¡ï %2%Fì¡ìyï M�ìfí�� �ì�L õ ó¡ó %Fìfð¡ôHó M�ìfí�� �ìyó õ ó¥ð¡í©ìE%¥ótñGM�ìfí�� �ì�ñ õ LHï %tôHï3ì�L0M�ìfí�� �àFáFä©ã¥û�á3â('©è�á&-tú¥÷ ã¡é�áHâ�ú¥÷çâ ã¡÷çútû�áFâ ã¡äC÷çú Nnì¡õ ô&%�Psùqè�é ) ë�ï¥í < ìfðdå�ã¡ä©ãdá äxútû�û�è�äxæ3á �©á ÷ á©å�ã¡÷ ãdùqè�é â(',+è�á ÷ -tú¥÷ ã¡é�è�á ä©á %�áçú¥÷ è � � d'èmú¥é �©á û�é ú¥é�á ã*Vtú�æ3á 1hè�é�è�ä�%fúKæ3è�÷ ÷*) ú¥÷ �¡ã¡é�á û�üøãdù©é�èfâ èfæ3è�äHû�è�#¥å�áFù©ýQ'ã#â�ú¥÷çâ ã¡÷çú¥é�è»â ã¡ä ÷çú�ü�ú¡å�å�á ü�ú�ú¡â�â ý©é útû�è % %fú�ùqãHå�å�á �©á ÷ èá ÷ -tú¥÷ ã¡é�èBæ3á )Hýxú¥÷ ý©ä*)Hý©è � & #�ú�ùxútû�û�ã2æ3á�â ã¡äxå�áçæ3è�é ú¥é�è�â ã¡üøèBæ©útû�ãVá ä©á %�áçú¥÷ è � & ë�íVâ ã¡ä ) å�ý :Xâ á è�äHû�è�üøè�äHû�èü�ú&���¡á ã¡é�èKæ3á ) õ

Page 44: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

� � � � �� �� � �� 5 ��Ó����O�O�O�O�O� � ����� 5 ��Ó����O�O�O�O�O� � ������ Ò ��Ó����O�O�O�O�O� � ��� � Ò ��Ó 1 �O�O�O�O�O� � �����= 5 ��Ó � �O�O�O�O�O� � ��� � 5 ��Ó " ���O�O�O�O� � ��� �� Ò ��Ó��=��������� , � ��� � Ò ��Ó���= �$����� , � ��� �" 5 ��Ó " �=��������� � ��� � 5 ��Ó>= 1 ���O�O�O� � ��� ���� ��� ���� = 5 ��Ó � � �O��� 1 = � ��� � � 5 ��Ó ��� ���=� " � � ��� � ���� Ò ��Ó��O�����=��� " � ��� � � Ò ��Ó , � � " 1�1 , � ��� � �� " 5 ��Ó ��� � , ��� 1 � ��� � � Ò ��Ó "�"+,, � � � � ��� � �� � Ò ��Ó 1 � ,, � 1 � � ��� � 3 Ò ��Ó � "�" � � 1 , � ��� � ���� ��� ���" � Ò ��Ó>= 1 � 1 �Q�=� � ��� � � Ò ��Ó ��=$� " = � � � ��� � �" = 5 ��Ó , �O� ,5" ��= � ��� � � Ò ��Ó ��=$� " =���� � ��� � �" � Ò ��Ó��=� � ��=���= � ��� � � Ò ��Ó ��=$� " =$�Q� � ��� � �"�" 5 ��Ó � , �Q� � " � � ��� � � Ò ��Ó ��=$� " =�= � � ��� � �

� ¼ �<°�¾�¾n¼�¶p·�� ���Y¼p¾nµ»ºx³K��³\� � �� ��#°=��³ � � µpÀFÀ3°�²�¿�À©³ �hµ�²�¿�²���º©µ��»º©¼ � ��¼P³�² ��µ»º3À3º©¼p²��¿�À©³�¾�³n´s´s¼p² �dµ¿�²d¼ �Vµ»º���½¡À3¼�À©³nµ�² �� � ��� ��� ��d½¡À3°���� �������

Z\[ oqi�ryef]aH�faHk8c^bj]�_^bNoqi*�h]hefb�ryk8]?bj[^��rtoqg^bjija}awi�� oqi*�h]hefb�ryk8]Dc�aHe`bjiXzFoqijzH]hij]_^b\|^[�� oqc^c^ef]h� ��fbjk`oq�Hbj]h[^aN_^b�� � �pzH]h[6� � ���pgpoq�yoxry]2�f|^ijij]�����bjij|^c^c�]2bj[6�faHefbja�_^b�vwoqzY�doq|^efbj[��ÖP×KØfÙFÚ�Û^Ü�Ý2ÞqßBA�Þqß#àFá�å�ý©ù©ùqã¡ä��Hú �»ë %Cè<êNë ! ìfí èCå�á�ú¥ù©ù©é�ãHå�å�á üøá �"!øâ ã¡ä�÷çúøå�ã¡üøü�ú

�� Nÿê�P�ë

�%&(' dê &)+* ë

�%&(' dN ! ìRP & � ê � &)+* î Nnì¡õ ô <�P

�däVú¥÷ �¡ã¡é�á û�üøã �xú¡å�útû�ã�å�ý©÷�â�ú¥÷çâ ã¡÷ ã�æ3á é�è û�û�ãBæ3è�÷ ÷çú Nnì¡õ ô <�PKâ ã¡ä�á ÷�â é�á û�è�é�á ã�æ ) ú¥é�é�èfånû�ãNäxútû�ý©é ú¥÷ è N�â®þÿéfõ � é�ã3â èfæ3ý©é úì¡õ ïQPKþÿã¡é�ä©áçå�â è\á ÷mé�áçå�ý©÷ û útû�ã ! í©î ó¡ï¥ô¥ñFï¡ï�<(M`ìfí �%3 #püøè�äHû�é�è\á ÷,-tú¥÷ ã¡é�è\â ã¡é�é�è û�û�ã�æ3á � � �Md #»ú¥é�é�ã¥û�ã¡äxæ©útû�ã�ú$%Bâ á þÿé�èå�á �¡ä©á �qâ�útû�á -¡è�#�'è�í©î ñFó¥ô¡ð¡ð¡ð¡ô M8ìfí��%3¥õ\ö ) è�é�é�ã¡é�è£é�è�÷çútû�á -¡ã 'è�æ3ý©ä*)Hý©è\â á é â�ú�í©î ï3ìyó¡ï¥ð©ìyóUM2ìfí�èfæ�á ÷»é�áçå�ý©÷ û útû�ãBæ3è�÷ Y÷*) ú¥÷ �¡ã¡é�á û�üøã�ä©ã¡ä 'è å�áçâ ý©é ú¥üøè�äHû�è ý©äxú��©ý©ã¡äxú\ú¥ù©ù©é�ãHå�å�á ü�ú&%�á ã¡ä©èCæ3á+��� �Md õ��È÷^ù©é�ã��©÷ è�ü�ú�é�áçå�á èfæ3è ä©è�÷�þ�útû�û�ã�â('©è áû�è�é�üøá ä©áqæ3è�÷ ÷ ã£åP-Fá ÷ ý©ù©ùqã á ä\å�è�é�á è�æ3á���� �Md 'xú¥ä©ä©ã å�è �¡ä©áhú¥÷ û�è�é�ä©áxè�÷çúCå�è�)Hý©è�ä�%fúCæ3áhå�ã¡üøüøèdè1hè û�û�ýxútû�èKä©ã¡ä 'è�ú¥÷ û�é�ãâ('©è#ý©äxúøå�è�)Hý©è�ä�%fúøæ3á�å�ã¥û�û�é ú&%�á ã¡ä©áqû�é ú£äFý©üøè�é�á ��ú&���xú¡ånû ú¥ä�%fú -Fáçâ á ä©á � N " ú&�qè�÷ ÷çúNì¡õ %�P®õö�úøâ�ú¥ýxå�ú�æ3è�÷ ÷*) á äxånû ú&�©á ÷ á ûE'ú�æ3áQ)Hý©èfånû�ãXú¥÷ �¡ã¡é�á û�üøã�ånû úøä©è�÷ ÷çúøù©é�ã��¡é�èfå�å�á -tú ùqè�é æ3á û úXæ3á�â á þÿé�è å�á �¡ä©á �qâ�útû�á -¡è#è )Hý©á äxæ3áä©è�÷�â ã¡äxå�è �¡ý©è�äHû�èøú¥ý©üøè�äHû�ã�æ3è�÷ ÷*) è�é�é�ã¡é�è£é�è�÷çútû�á -¡ã*V�å�á�-¡è�é�á �qâ�ú¥ä©ã�â á ã*'è�å�ýxâ�â èfå�å�á -tú¥üøè�äHû�è -tú¥é�á^þÿè�ä©ã¡üøè�ä©á�æ3ámâ�ú¥ä�Yâ è�÷ ÷çú&%�á ã¡ä©èNä©è�÷ ÷ è�å�ã¥û�û�é ú&%�á ã¡ä©ájõ6àFádé�áçâ ã¡é æ3á�â('©èB÷çúVâ�ú¥äxâ è�÷ ÷çú&%�á ã¡ä©è 'è�÷çú�ü�ú¥ä©á þÿèfånû ú&%�á ã¡ä©èBæ3ádý©äxú�â�ú¥é útû�û�è�é�áçånû�áçâ�úá äHû�é�á äxå�èfâ�úNæ3è�÷ ÷*) ã¡ùqè�é ú&%�á ã¡ä©èøú¥é�á û�üøè û�áçâ�úNæ3á�å�ã¥û�û�é ú&%�á ã¡ä©è�#�â('©è 'èøü�ú¥÷»â ã¡äxæ3á %�á ã¡äxútû ú )Hýxú¥äxæ3ã �¡÷ ápã¡ùqè�é ú¥äxæ3á»å�ã¡ä©ã�P-Fáçâ á ä©á � N�â®þÿéfõCèfå�è�üøù©á�ì¡õ ïtñ\èNì¡õ ï %�P®õ$�È÷^þ�útû�û�ã�â('©èøå�á^ýxå�ápý©äVå�áçånû�è�ü�ú\ú¥é�á û�üøè û�áçâ ã�ú\ù©é�èfâ áçå�á ã¡ä©è �xä©á û úNá üøù©÷ áçâ�úå�ã¡÷ ã�â('©è(-Fá 'è�â è�é�û ú¥üøè�äHû�èøý©äVè�é�é�ã¡é�è(N�)Hý©è�÷ ÷ ã�æ3á»é�ã¡ý©äxæ3ã 1&PKä©è �¡÷ á»ú¡æ©æ3è�äxæ3á»è )Hý©á äxæ3á»á ÷»ü�ú¥÷»â ã¡äxæ3á %�á ã¡äxú¥üøè�ä�Yû�ã�æ3è�÷ ÷çúYå�ã¥û�û�é ú&%�á ã¡ä©è\å�á�ü�ú¥ä©á þÿèfånû ú�è1hè û�û�á -tú¥üøè�äHû�è¡õ8àFã¡ä©ãYù©é�ã¡ù©é�á ã��¡÷ ámè�é�é�ã¡é�á�ä©è �¡÷ á�ú¡æ©æ3è�äxæ3á�÷ èNâ�ú¥ýxå�èNæ3è�÷ ÷çúâ�ú¥äxâ è�÷ ÷çú&%�á ã¡ä©è�#Füøè�äHû�é�è#÷*) ã¡ùqè�é ú&%�á ã¡ä©è#å�ã¥û�û�é ú&%�á ã¡ä©è�#Fä©è�÷ ÷*) ý©ä©á ûE'ú�ú¥é�á û�üøè û�áçâ�ú�#@'è<â ã¡üøù©÷ è û ú¥üøè�äHû�è#á äxâ ã¡÷ ùqè/-¡ã¡÷ è¡õ�dä£ú¥÷ �¡ã¡é�á û�üøãKånû ú&�©á ÷ èmùqè�épá ÷©â�ú¥÷çâ ã¡÷ ãKæ3áFý©ä+) ú¥ù©ù©é�ãHå�å�á ü�ú&%�á ã¡ä©èmæ3á��"!�ùqè�é^ê��`í�#¥ý3û�á ÷ á % %fú¥äxæ3ã#å�è�üøù©é�èm÷ ã#åP-Fá ÷ ý©ù©ùqãá äXå�è�é�á è�æ3á���ú¡âdö�ú¥ý©é�á ä+#*'è �xú¡å�útû�ã å�ý©÷ ÷*) ãHå�å�è�é -tú&%�á ã¡ä©è�â('©èA�"! ëDì %7����!#èfæ�á ÷qù©é�ã��©÷ è�ü�ú 'è�û�é ú¡ånþÿã¡é�ü�útû�ã<á ä )Hý©è�÷ ÷ ã

Page 45: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# ���

æ3è�÷^â�ú¥÷çâ ã¡÷ ã\æ3á+�"!�ùqè�é#ê �wí©õ#ö ) ú¥÷ �¡ã¡é�á û�üøãXæ3èfå�â é�á û�û�ãNæ©ú¥÷ ÷çú � é�ã3â èfæ3ý©é ú�ì¡õ ï�æ3è�÷ fì¡õ ñ #hâ('©è 'è£ånû ú&�©á ÷ è£å�è ê �wíN �¡÷ áKú¡æ©æ3è�äxæ3á#å�ã¡ä©ãVû�ý3û�û�á#ùqãHå�á û�á -Fá�èG)Hý©á äxæ3áKä©ã¡ä6â&) 'èBþÿè�ä©ã¡üøè�ä©ã8æ3á#â�ú¥äxâ è�÷ ÷çú&%�á ã¡ä©èRP�#�'èBýxå�útû�ã2ùqè�é\á ÷#â�ú¥÷çâ ã¡÷ ãæ3è�÷ ÷*) èfå�ùqã¡ä©è�ä�%�áçú¥÷ è ùqãHå�á û�á -¡ãNèfæ�á ÷^é�èfâ á ù©é�ã3â ã\æ3è�÷pé�áçå�ý©÷ û útû�ã0'è£á ÷^é�áçå�ý©÷ û útû�ãNæ3è�÷^äFý©ãE-¡ãNú¥÷ �¡ã¡é�á û�üøã\ånû ú&�©á ÷ è£ùqè�é �"!â ã¡äNê��`í©õ Ö

����bjk8c�]he�rtoq[sryaN]h�f�faHe��qoqefaBzt�^aq�»oV_^b5AmaHefaH[^�FoV_^aHimzH]h[^_^bj�Hbj]h[poqk8aH[sry]^��i�� bj[^��rtoqg^bjijb�rx}o2}a�|^[poVzFoqeyoxr��ryaHefbj��rybjzFo8_^aHiji�� oqi*�h]hefb�ryk8]�aV[^]h[D_^aHiKc^ef]hg^ijaHk`o8aV~�|^bj[^_^b�i�� bj[^��rtoqg^bjijb�rx}oVc^|�}]�aH�f�faHefa�a¡��b�rtoxrto`zH]h[]hc^c�]he�ry|^[^a�k8]�_^b��pzt�^aB_^aHiji�� oqi*�h]hefb�ryk8]^�szH]hk8aq�poqc^c^|^[sry]^�p[^aHiji�� aH�faHk8c^bj]R�h� �Q�h�

021�� ��� AC;¡9<TU@N9<;FWXG��:9#GV7?=DGVAC= �������

{\aHi�� 1 � =�bji Cªj¦F§q­�ÃÈÄ�4 ��Çp¦xÃÈÄp­w¦x¤��xÃÈÄ�4� ø¤y¦x«3Ç�Æ�¢¡ªÈª�� � ����� �S� � ¦xÅNÇ�«�­®¢¡¤ ¦3Ë¡Ã�¢¡­ � � ¨ � Ã�Ë¡¤y¦tÇ�¤y¦5�Ët¢t¨t¨3¦x¤ ­ §qÄ�Æs§q¤fÆ � ¦xÅ�Å�ÃÈ­�­®¢t¢�c^ef]hc�]h�fa?|^[^]P��rtoq[^_poqef_Õ_^bB½F³�½¡À3°���¼ ¼^ºx³ÈÀ��°sÀ©³��hµ �Kµ»¼�À©³�² ��<µ�³�²�À �K³�²d¼^ºx³nµd��c�aHeXbji�c^ef]�ha¡rfry]2_^aHijijaB|^[^b�rx}o�oqefb�ryk8a¡rybjzt�^a�_^aHbdzFoqijzH]hiuoxry]hefb � 3¥��<� oqefb�ryk8a¡rybjzFo Ô rtoq[^_poqef_ J������_^aH�fzHefb��haX|^[2k8]�_^aHijij]N_^b^�fbj��ryaHk`oBoqefb�ryk8a¡rybjzH] �p]soxrybj[ ��c�]hbj[sr

; zHbj]^}a�_^a2�p[^bj�fzHaViuo8eyoqc^c^efaH�faH[srtoq�Hbj]h[^aV_^aHbK_poxryb�[�|^k8aHefbjzHb�aVijaY]hc�aHeyoq�Hbj]h[^bKoqefb�ryk8a¡rybjzt�^a�aH�fa �h|^bÿ�rya:�f|*rtoqijb _poxryb6?�ija:zH|^b£zFoqeyoxrfryaHefbj��rybj��rybjzt�^a`ln]h[^_poqk8aH[srtoqijbC�fbøc�]h�f�f]h[^]��fbj[srya¡rybj�H�Foqefa:[^aHi£k8]�_^]�fa �h|^aH[srya��

� k`oq�f�fbjk`oVoqzHzH|^eyoxryaH�H�Fo8��rtoxrybjzFo��^c�aHeXiuoVeyoqc^c^efaH�faH[srtoq�Hbj]h[^aY_^aHb�[�|^k8aHefb��p]soxrybj[ ��c�]hbj[sr B� k`oq�f�fbjk`oVoqzHzH|^eyoxryaH�H�Fo8_^bj[poqk8bjzFo���c�aHe�ijaB]hc�aHeyoq�Hbj]h[^b�oqefb�ryk8a¡rybjzt�^a%B� �haH��rybj]h[^aB_^aHijijaBaHzHzHaH�Hbj]h[^b B� �haH��rybj]h[^aB_^aHi 4q¤f§sÆq«^§qª�«�Ä�Æ�¢¡¤ � ¦5#�B� ~�|poxrfryef]2�fzt�^aHk8bdc�aHeXi�� oqefef]qry]h[^_poqk8aH[sry]^�

{\aHiC�fa �h|^b�ry]��fbC_^aH�fzHefb��haq�#g^efa¡�haHk8aH[sryaq�#ij]���rtoq[^_poqef_ J��������efbuoq�f�f|^k8aH[^_^]�ija2c^ef]hc^efbja¡rx}o6_^aHijiuoeyoqc^c^efaH�faH[srtoq�Hbj]h[^aD_^aHbN[�|^k8aHefb&�p]soxrybj[ ��c�]hbj[srF��iuo �haH��rybj]h[^aw_^aHijija��fb�ry|poq�Hbj]h[^bXaHzHzHaH�Hbj]h[poqijb��øbji�heyoq_^|poqi�|^[^_^aHe)�p]�� a��hijb��fzt�^aHk8b�_^aHiji�� oqefef]qry]h[^_poqk8aH[sry]^��h� :¼ � ��º©°�½H°�²�À3¼^´�³nµ�²d° �d°�³<²�¿���°�ºx³ �Kµ»¼�À©³�² � �<µ�³�²�À �Ki�� oqefb�ryk8a¡rybjzFo J�����>_^a2�p[^bj�fzHaNryefaln]hefk`oxryb �� �fbj[ �h]hiuoVc^efaHzHbj�fbj]h[^aq�� _^]hc^c^buo�c^efaHzHbj�fbj]h[^aq�� c^efaHzHbj�fbj]h[^a�aH��ryaH�yo��

3 3��®äxånû�á û�ý3û�èCã¥þd×�÷ èfâ®û�é�áçâ�ú¥÷�ú¥äxæN×�÷ èfâ®û�é�ã¡ä©áçâ#×�ä��¡á ä©è�è�é�á ä��3 N�ö ) ú¥é�á û�üøè û�áçâ�ú2ånû ú¥äxæ©ú¥é æ � � � � 'èYú¥äxâ('©è�ä©ã¥û ú2â ã¡üøèYú¥é�á û�üøè û�áçâ�ú����8V`æ©ú¥÷ ÷ è�á ä©á %�áçú¥÷ áKæ3á��?õ��<ú&'xú¥ä+#��xõC�ãFã¡ä©è�ä+#�ånû�ýxæ3è�äHû�è�æ3á��?õ��<ú&'xú¥äYù©é�èfå�å�ãN÷*) �dä©á -¡è�é å�á ûE'úNæ3è�÷ ÷çú0Cmú¥÷ á þÿã¡é�ä©áçú\æ3á cmè�é��¡è�÷ è ��#hè�æ3è�÷pù©é�ã¥þnõ��Cõ�àHû�ã¡ä©è¡õ

�?õ �<ú&'xú¥ä 'xúøé�áçâ è/-Fý3û�ãøùqè�é )Hý©èfånû�ãøüøã¥û�á -¡ãøá ÷�ù©é�è�üøá ã�!#"%$&�('*)�æ©ú¥÷ ÷*) . C�� #3ä©è�÷»ìfð <¡ð©õ

Page 46: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

��=

v�aH[sryefa6b c^efbjk8b _^|^a:�f]h[^]?��ryefa¡rfrtoqk8aH[srya:]hg^g^ijb*�soxry]hefb���bjiCryaHef�H]Dln]hefk`oxry]�}a`l�oqzH]hi�rtoxryb��h]bj[:~�|poq[sry]V��bjaH[^aN|�rybjijbj�H�Foxry]�oqi�c^by}|8[^aHbmefa �hbj��ryefbmoqiji�� bj[sryaHef[^]Y_^aHiji�� |^[^b�rx}oYoqefb�ryk8a¡rybjzH]x�®ij]�hbjzFoc�aHe`iuo�eyoqc^c^efaH�faH[srtoq�Hbj]h[^aD_^aHb\efbj�f|^i�rtoxryb�bj[sryaHefk8aH_^b\bj[P|^[po��f|^zHzHaH�f�fbj]h[^a?_^b\]hc�aHeyoq�Hbj]h[^b�p]soxrybj[ �2c�]hbj[srF�

{\aHijiuo\rtoqg�aHijiuoN�fa �h|^aH[srya\efbjc�]he�rybuoqk8]\ijaøzFoqeyoxrfryaHefbj��rybjzt�^aø_^aHi^ln]hefk`oxry]NaH��ryaH�f]Y_^aHijiuo��fbj[ �h]hiuoa�_^aHijiuoV_^]hc^c^buo�c^efaHzHbj�fbj]h[^a��

��bjc�] ��|^[ �h�^aH�H�Fo8cpoqef]hiuo c^efaHzHbj�fbj]h[^a aH�fc�]h[^aH[sryaYk8bj[ aH�fc�]h[^aH[srya�k`o7��fbj[ �h]hiuo � " � � � = ���E��� =�� � ��� = "_^]hc^c^buo � , 1 � � " � ÒG�=��� � � � �=��� � "

�do�ij]�zFoq�Hbj]h[^aV_^b#k8aHk8]hefbuo`c�aHeBbKln]hefk`oxrybKbj[?�fbj[ �h]hiuo:a�_^]hc^c^buo:c^efaHzHbj�fbj]h[^a6}a��f|^_^_^b���bj�yo[^aHidk8]�_^]��fa �h|^aH[srya��

��bjc�] Ô a �h[^] �<�fc�]h[^aH[srya vwoq[srybj�f�yo�fbj[ �h]hiuo � � =$�_^]hc^c^buo � �O� ��=

�������������� ������������������������� ��������������������������

� � �!���"��� �� ��� #$����%&� ����������������'��� ���)(����*%������ � ������%+����������#,%&������ ���.-,/0/0/

��aH[^aH[^_^]*zH]h[sry]�_^aHiji�� ]he �soq[^bj�H�Foq�Hbj]h[^a�_^aHijiuo?ij]�zFoq�Hbj]h[^a6_^bXk8aHk8]hefbuo��Cbji�[�|^k8aHef]�_^b\g^b�refbj�faHe��qoxryb�c�aHeXi�� aH�fc�]h[^aH[srya`}a � c�aHeXiuo��fbj[ �h]hiuoVc^efaHzHbj�fbj]h[^a�a �O��c�aHeXiuo�_^]hc^c^buo+Bpc�aHe�rtoq[sry]i�� bj[sryaHe��qoqijij]�_^b�eyoqc^c^efaH�faH[srtoqg^bjijb�rx}o:}aB_^aHijbjk8b�rtoxry]Y_po��hijbmaH�fc�]h[^aH[sryb<ÒG��= , a 5 ��= � ��bj[�gpoq�fa=��»c�aHe\iuo��fbj[ �h]hiuo�c^efaHzHbj�fbj]h[^a�aY_po6ÒG��� =$�8o 5 ��� = " c�aHe\iuo�_^]hc^c^buo2c^efaHzHbj�fbj]h[^aq� � |^aH��ry]zH]hefefbj�fc�]h[^_^aq�Kbj[gpoq�fa �����Koq_�|^[po6l�oq�fzHbuo6aH�fc�]h[^aH[^�Hbuoqija8zt�^a8�qoqefbuo�_poÒ � � o 5 � � c�aHeiuo*�fbj[ �h]hiuo*c^efaHzHbj�fbj]h[^a�a�_po�Ò �O� � o 5 �O� � c�aHe6iuo*_^]hc^c^buo>c^efaHzHbj�fbj]h[^aq� ®[^]hi�ryefaq�N|^[�f]hij]Pg^b�r6��bjaH[^aefbj�faHe��qoxry]>c�aHe�bjiB�fa �h[^] _^aHijiuo*k`oq[srybj�f�yo�� 8�aHewi�� aH�fc�]h[^aH[srya�fb�|^�yoPiuo[^]qrtoq�Hbj]h[^a8oq_ ¢tËtËt¢t¨t¨3¦21 354Y[^aHijiuo:�fbj[ �h]hiuo�c^efaHzHbj�fbj]h[^a�a`oq_ ¢tËtËt¢t¨t¨3¦21 6�3 7*[^aHijiuo�_^]hc^c^buoc^efaHzHbj�fbj]h[^a � � �W�£b }]6�fb*�h[^b��pzFo�zt�^a8bj[s�haHzHa`_^b<eyoqc^c^efaH�faH[srtoqefa`i�� aH�fc�]h[^aH[srya �8zH]h[PÒG��= , �� � ��= � ��bji��qoqij]hefaYzt�^a���bjaH[^a�k8aHk8]hefbj�H�Foxry]�}a�|^[Dbj[sryaHef]:�faH[^�Fo:�fa �h[^]���8�zH]h[ ��� ��8 �= � ���^c�aHeXiuoV�fbj[ �h]hiuo�c^efaHzHbj�fbj]h[^aq� 8�aHeX~�|poq[sry]8efb*�h|poqef_poViuoV_^]hc^c^buo�c^efaHzHbj�fbj]h[^aq�^bj[s�haHzHaY_^beyoqc^c^efaH�faH[srtoqefa�i�� aH�fc�]h[^aH[srya �YzH]h[ÒG��� =$� � � � ��� = " ��bjaH[^a�k8aHk8]hefbj�H�Foxry]V|^[�[�|^k8aHef]c�]h�fb�ryb��h]^� ��8���zH]h[ � ����8 � = � "+, � � |^aH��ry]:�fb*�h[^b��pzFo2zH]h[^�fbj_^aHeyoqefa�oq[^zt�^a�bji�g^b�r\_^aH_^bjzFoxry]oqi<�fa �h[^]wzH]hk8a�|^[�g^b�r�c�aHe�bji#zFoqk8c�]6_^aHiji�� aH�fc�]h[^aH[srya8a2zFoqijzH]hiuoqefaVbji#[�|^k8aHef]�_^aHzHbjk`oqija]qrfryaH[�|�ry]8_poqiji�� bj[sryaHeyoV�faH~�|^aH[^�Fo8_^b�g^b�rF�^zH]h[^�fbj_^aHeyoxry]2zH]hk8a�bj[sryaHef]2�faH[^�Fo2�fa �h[^]^�

3�' W�ý©èfånû�ã�â ã¡äxâ è û�û�ã 'è �¡á6'úøånû útû�ãøá äHû�é�ã3æ3ã¥û�û�ãøä©è�÷ ÷çú£ä©ã¥û ú Njï¥ô�Pmæ3è�÷�ùxú¥é ú&�¡é útþÿãXì¡õ ï3õ ï3õ

Page 47: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# �$�

ÖP×KØfÙFÚ�Û^Ü�Ý2ÞqßBAO�sßDC�ã¡äxå�áçæ3è�é útû�áhô �©á û #xæ3á�â ý©áhá ÷sù©é�á üøã£ùqè�é�á ÷så�è �¡ä©ã£èfæ\áhé�èfånû ú¥äHû�ásæ3ý©è#ùqè�é�á ÷säFý©üøè�é�ãånû�èfå�å�ã #3å�á�ã¥û�û�è�ä��¡ã¡ä©ã£á�å�è �¡ý©è�äHû�ásäFý©üøè�é�ásäxútû�ý©é ú¥÷ á :

å�è �¡ä©ã&�?é ú¥ù©ù�õ �©á ä é ú¥ù©ù�õpæ3èfâ¥õíPí¡í YÈííPí©ì Y ìí�ìfí Ynïí�ì¡ì YÈôì�í¡í �Kíì�í©ì � ììPìfí �#ïìPì¡ì �Kô

��û�á ÷ á % %fú¥äxæ3ã�á ä&-¡èfâ è\÷çúYä©ã¥û ú&%�á ã¡ä©èNå�è�ä�%fú�å�è �¡ä©ã #p÷çú�ånû�èfå�å�úYå�è�)Hý©è�ä�%fúYæ3á �©á û�é ú¥ù©ù©é�èfå�è�äHû úBý©ä:æ3á -¡è�é å�ãá äHû�è�é -tú¥÷ ÷ ãøæ3á�äFý©üøè�é�á�äxútû�ý©é ú¥÷ á NÿùqãHå�á û�á -FáMP:å�è �¡ä©ã �Dé ú¥ù©ù�õ �©á ä é ú¥ù©ù�õpæ3èfâ¥õ»å�è�ä�%fúøå�è �¡ä©ã

íPí¡í ííPí©ì ìí�ìfí ïí�ì¡ì ôì�í¡í ñì�í©ì óìPìfí LìPì¡ì %

�ÈäD)Hý©èfånû�ãdâ�ú¡å�ã�÷*) á äHû�è�é -tú¥÷ ÷ ãdæ3á¥é ú¥ù©ù©é�èfå�è�äHû ú&�©á ÷ á ûE'ú"-Fá è�ä©è»å�÷ á û�û útû�ãdæ©ú 2 ! ô < �Kô 4 ú 2 í < % 4 õ Ö

�<� oqefb�ryk8a¡rybjzFo J�����P�fbmoF���qoqijaN_^aHijiuoYeyoqc^c^efaH�faH[srtoq�Hbj]h[^a �p]soxrybj[ ��c�]hbj[sr£[^]hefk`oqijbj�H�Foxrto��sbj[zH|^b���c�aHe�iuo6eyoqc^c^efaH�faH[srtoq�Hbj]h[^a`_^aHijiuo6k`oq[srybj�f�yo����fbCl�o6|^�f]D_^aHi �K³ÈÀ?³�� �K¾�³��s³ÈÀ3µ � ��� bji<g^b�rbj[^bj�HbuoqijaC�poB�faHk8c^efa£�qoqij]hefa ��a�[^]h[�}a�aH�fc^ijbjzHb�rtoqk8aH[srya£eyoqc^c^efaH�faH[srtoxry]^� � |^aH��ry]BzH]hk8c�]he�rtoiuo2c�]h�f�fbjg^bjijb�rx}o�_^b�eyoqc^c^efaH�faH[srtoqefa�aQAma¡rfryb��qoqk8aH[srya = " g^b�rNc�aHeNiuo8�fbj[ �h]hiuo2c^efaHzHbj�fbj]h[^aYa �$�c�aHeXiuoV_^]hc^c^buo�c^efaHzHbj�fbj]h[^aq�

XaHiuoxryb��qoqk8aH[srya�oqiji�� aH�faHzH|^�Hbj]h[^a�_^aHijija�µ��<°�º©¼^´�³nµ�²�³ø¼^ºx³ÈÀ��°sÀ©³����d°��Kµ»¼�À©³�² � �<µ�³�²�Às�»i�� o©�efb�ryk8a¡rybjzFo J������|�rybjijbj�H�Fo���c�aHe�iuo�eyoqc^c^efaH�faH[srtoq�Hbj]h[^a?_^aHijijaDk`oq[srybj�f�fa?_^a �hijbN]hc�aHeyoq[^_^b��efa �hbj��ryefbCoqefb�ryk8a¡rybjzHb#zH]h[ =@4q«^§q¤fÆ?ÆqÃ54qÃȭȨBc^by}||^[^]�¨¡­�Ã�Ë � � �`�HÃÈ­Cc�aHe �soqeyoq[srybjefa�iuo6k`oq�f�fbjk`ooqzHzH|^eyoxryaH�H�Fo2_^bj[poqk8bjzFo�� _^|^aBg^b�rXo���hbj|^[sryb���b ; bD4q«^§q¤fÆ:ÆqÃ54qÃȭȨ ? �f]h[^]2|�rybjijbj�H�Foxryb�c�aHe�eyoqc��c^efaH�faH[srtoqefaNbpc^efbjk8b�_^|^a\g^b�rC�fb*�h[^b��pzFoxryb���b^_^aHijiuoBcpoqe�ryaX_^bpk`oq[srybj�f�yo ; _^a �hijbp]hc�aHeyoq[^_^bpa\_^aHi

3 8 W�ý©èfånû�ã�â ã¡äxâ è û�û�ã='è �¡á6'ú?ånû útû�ã?á äHû�é�ã3æ3ã¥û�û�ã�ä©è�÷ ÷çúDä©ã¥û ú Njï3ìRP�æ3è�÷\ùxú¥é ú&�¡é útþÿã�ì¡õ ï3õ ï�ä©è�÷ ÷*) ú¥ü �©á û�ãæ3è�÷ ÷çúæ3èfå�â é�á %�á ã¡ä©è#æ3á�ý©ä )Hýxú¥÷çå�áçú¡å�áså�áçånû�è�ü�ú�ú¥é�á û�üøè û�áçâ ã �xãHútû�á ä��øùqã¡á äHûfõ

Page 48: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

� "

efbj�f|^i�rtoxry]�?#zt�^aN��bjaH[^a\efbjk8]h�f�yo�oYzFoq|^�yo�_^aHiji�� oqefef]qry]h[^_poqk8aH[sry]^��®i»ryaHef�H]Vg^b�rF��_^a¡rfry]8¨¡­�Ã�ËE� ��HÃÈ­��N}a6]qrfryaH[�|�ry]zH]hk8a�� �ij]�hbjzH]?_^b£ry|�rfryb�b�g^b�r�efbjk`oq[^aH[sryb�� �£]hk8a�oq[^zt�^a �hb }o?_^a¡rfry]�fbCc^|�}]�_^bjk8]h��ryeyoqefaVzt�^a2ryefa2g^b�r��f]h[^]��f|��8zHbjaH[sryb o �soqeyoq[srybjefa�iuo�k`oq�f�fbjk`o�oqzHzH|^eyoxryaH�H�Fo_^bj[poqk8bjzFo��

=�����°�½¡À©³nµ�²d° �d°�¾�¾n°D½F³ÈÀ©¿d¼^´�³nµ�²�³C°����h°�´�³nµ�²d¼p¾�³���Koqijb��fb�ry|poq�Hbj]h[^b��f]h[^]2zHiuoq�f�fb��pzFoxrya�bj[ �� ]hc�aHeyoq�Hbj]h[^aB[^]h[6�qoqijbj_po ;  {�� 3��  �0?¥�� _^b���bj�fbj]h[^a�c�aHeX�HaHef]^�� ]3�haHe)�p]����� |^[^_^aHe)�p]����

3N_w]�h[^b�aHzHzHaH�Hbj]h[^a��f]h[^]`oq�f�f]�zHbuoxryb�|^[ � § 4�aH_w|^[­�¤f§¥Ç�� ®i � § 4�bj[^_^bjzFo�~�|poqija�aHzHzHaH�Hbj]h[^a�fbN}a2�haHefb��pzFoxrto�� ®iX­�¤f§¥Ç�oxrfryb��qo6iuo6�fzHaHi�rto6ryeyo�_^|^a`c�]h�f�fbjg^bjijb�rx}o$��zt�^buoqk`oxrto�oq_�|^[po�c^ef]x�zHaH_^|^eyo6_^b �haH��rybj]h[^a�_^aHiji�� aHzHzHaH�Hbj]h[^a�zH]h[�bj[sryaHefef|^�Hbj]h[^a�_^aHiji�� aH�faHzH|^�Hbj]h[^a�]hc^c^|^efa2[^aH�f�f|^[pobj[sryaHefef|^�Hbj]h[^a�_^aHiji�� aH�faHzH|^�Hbj]h[^aq�

Z\[�� ]hc�aHeyoq�Hbj]h[^a  {�� 3��  ��c^ef]�_^|^zHa8|^[���¼�� ; {\]qr�o�{\|^kVg�aHeE?¥��iuo�zH|^bCeyoqc^c^efaH�faH[srto©��Hbj]h[^a�bj[�k8aHk8]hefbuo:}aBiuo��fa �h|^aH[srya��

Ô a �h[^] �<�fc�]h[^aH[srya vwoq[srybj�f�yo� �hÓFÓFÓ � ~�|poqij�fbuoq�fb�zH]hkVg^bj[poq�Hbj]h[^a�_^b�g^b�r

� �� ��� #$����%&� ������������� �� �������� � ��� %�����%������"-,/0/0/

Z\[�efbj�f|^i�rtoxry]�{Noh{"}a8]qrfryaH[sry]waQAma¡ry|poq[^_^]D]hc�aHeyoq�Hbj]h[^b<[^]h[�_^a2�p[^b�rya8zH]hk8a`oq_�aH�faHk8c^bj]� �����Z� � ����� ����� 5 ��Ò�����do�_^b���bj�fbj]h[^a�c�aHeB�HaHef]�c^ef]�_^|^zHa�zH]hk8a�efbj�f|^i�rtoxry] �H² �y�diuo`zH|^b<eyoqc^c^efaH�faH[srtoq�Hbj]h[^a2bj[?k8a¥�k8]hefbuo:}a�iuoV�fa �h|^aH[srya��

Ô a �h[^] �<�fc�]h[^aH[srya vwoq[srybj�f�yo� �hÓFÓFÓ � � � � � � � � �

� �� ��� #$����%&� ������������ -$��� �������� � ��� %�����%�������-,/0/0/

®i£�fa �h[^]D_^b �H² �V}a`i�� |^�f|poqija8�fa �h[^]?_^b |^[�~�|^]h�HbjaH[srya:c�aHe�rtoq[sry]Diuo�_^b���bj�fbj]h[^a2c�aHe��HaHef]*}ai�� |^[^bjzFo�]hc�aHeyoq�Hbj]h[^a�oqi*�haHg^efbjzFoVzt�^a�efb��haHiuo�bji��fa �h[^]8_^b��HaHef]^���]B��rtoq[^_poqef_ J�����bj[sryef]�_^|^zHaXoqijzH|^[^a£ln|^[^�Hbj]h[^b�_^b^oqkVg^bjaH[sryaøc�aHe#iuo �haH��rybj]h[^a�_^aHijija£aHzHzHa¥��Hbj]h[^b BhryeyoB~�|^aH��rya\cpoqe�rybjzH]hiuoqefk8aH[sryaø|�rybjijb��f]h[^]��������������! �"Bzt�^a\efaH��ryb�ry|^bj�fzHa�bji��qoqij]hefa 5q¢¡¤y¦

Page 49: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# � �

�fa2i�� oqe �h]hk8aH[sry]?}a2eyoqc^c^efaH�faH[srtoqg^bjijaq�Z�H§qª ¨3¦Vbj[�zFoq�f]�zH]h[sryeyoqefbj]^��a2iuo:ln|^[^�Hbj]h[^a � � ����� �! �"zt�^aYefb�ry]hef[poVbji��qoqij]hefa�5q¢¡¤y¦��fa�i�� oqe �h]hk8aH[sry]�}a�|^[w{Noh{Y�O�H§qª ¨3¦�oqi�ryefbjk8aH[sryb��Ô buo ��¼���zt�^a �H² �\c�]h�f�f]h[^]�aH�f�faHefa`|�rybjijbj�H�Foxryb�[^aHiji�� aH�faHzH|^�Hbj]h[^a8_^aHijijaV|^�f|poqijb<]hc�aHeyoq�Hbj]h[^boqefb�ryk8a¡rybjzt�^aq� ��bjc^bjzFoqk8aH[srya<bjihefbj�f|^i�rtoxry]�_^bs|^[�� ]hc�aHeyoq�Hbj]h[^a<zt�^a zH]hbj[s�h]hi*�ha<|^[ ��¼��(zH]h[srybÿ�[�|poNoq_�aH�f�faHefa���¼���©ij]\��ryaH�f�f]N�qoqija c�aHe��H² �mzH]h[VijaC_^]3��|�ryaøaHzHzHaH�Hbj]h[^b�~�|poqijb��xoq_VaH�faHk8c^bj]^�[^aHijiuoY_^b���bj�fbj]h[^aq� ®[:rtoqimzFoq�f]��fa�bji�[�|^k8aHeyoxry]hefa�}a�_^b��haHef�f]�_poV�HaHef]�aH_�bji»_^aH[^]hk8bj[poxry]hefa�}a�H² �y�pbjim~�|^]h�HbjaH[srya8}a��HaHef]^�

��� ��º©¼Z��¿d¼p¾�¿�² �d°�º"�Kµ������]\��rtoq[^_poqef_ J�����?bj[sryef]�_^|^zHa bszH]h�fbj_^_^a¡rfryb»Äp«�Å`¢¡¤¥Ã�Æ�¢¡Äm¦x¤¥Å8§qª¬Ã�ÉtÉ3§q­�ÃÈ�xoF�haH[sryb��qzH]hk8aCaH�fc�]x�[^aH[sryaq��i�� aH�fc�]h[^aH[srya�k8bj[^bjk8]2zH]h[^�faH[sryb�ry]:aH_D|^[po`k`oq[srybj�f�yo8[^]h[D[^]hefk`oqijbj�H�Foxrto�� � |^aH��ry]�fb*�h[^b��pzFo:zt�^a8_^]hc�] �* 6 PÏ ��Ó�� �)( �P�� � �f]h[^]�oq[^zH]heyo6eyoqc^c^efaH�faH[srtoqg^bjijb#b<[�|^k8aHefb<_^aHirybjc�] �

��Ó �Q� � ( �P�� � Ï ��Ó�� �-( �P�� � � ���Ó �O�Q� � ( �P�� � Ï ��Ó�� �-( �P�� � � ���Ó �O�O�Q� � ( �P�� � Ï ��Ó�� �-( �P�� � � �ÓFÓFÓHÓFÓFÓ ÓFÓFÓHÓFÓFÓ ÓFÓFÓHÓFÓFÓHÓFÓFÓFÓFÓFÓ

��Ó �O��ÓFÓFÓ � � ( �P�� � Ï ��Ó�� �-( �P�� � � ��£]hk8aN�fb�c^|�}]V[^]qrtoqefaNbj[:~�|^aH��ry]�zFoq�f]ViuoY�f]�hijbuo�_^b�|^[^_^aHe)�p]��Õ��bjaH[^aNefbj_^]qrfrto$�Kbj[:cpoqe�rybjzH]x�iuoqefaq��c�aHeNiuo8�fbj[ �h]hiuo8c^efaHzHbj�fbj]h[^aq��aH�f�faH[^_^] �XÏ , ��aH�f�yo�efbj�f|^i�rto2cpoqefb#o ��� � � � � � Ï ��� � � 3F�k8aH[sryefa�c�aHe�iuo�_^]hc^c^buo�c^efaHzHbj�fbj]h[^a ; �CÏ �=�@?¥�^�fb��po2bjim�qoqij]hefa$��� � � � � ����Ï ��� � � � � �X[�|^k8aHefbK_^aH[^]hefk`oqijbj�H�Foxrybd�f]h[^]�cpoqe�rybjzH]hiuoqefk8aH[sryaY|�rybjijbd[^aHiji�� aH�faHzH|^�Hbj]h[^aV_^b#�f]hk8k`oxry]hefbjabj[8zH|^b»|^[^]�_^a �hijb»oq_^_^aH[^_^b»c^|�}]Voq[^_poqefa\bj[8|^[^_^aHe)�p]���� ®[8rtoqi»zFoq�f]�i�� |^[^_^aHe)�p]�� �heyoq_^|poqijazH]h[^�faH[sryaN_^bp[^]h[8c�aHef_^aHefa\zH]hk8c^ija¡rtoqk8aH[srya�bjipzH]h[sryefbjg^|�ry]�_^b»|^[2rtoqijaXoq_^_^aH[^_^]^��ln]hef[^aH[^_^]~�|^bj[^_^b�|^[6efbj�f|^i�rtoxry]�c^by}|6oqzHzH|^eyoxry]^�

" � ����d°���³U��³£¼^º©º©µpÀ3µ�² �d¼ ��°�²�À3µ{\aHijiuo:eyoqc^c^efaH�faH[srtoq�Hbj]h[^a8_^aHb<[�|^k8aHefb��p]soxrybj[ ��c�]hbj[srF��|^�f|poqijk8aH[sryaq�d�fb<|�rybjijbj�H�Fo:ij]��fzt�^a¥�k`o8_^aHiji�� oqefef]qry]h[^_poqk8aH[sry]^�Q�<�f�f]�}aV|�rybjijbj�H�Foxry]8c�aHe�_^a¡l�oq|^i�r�[^aHijij]`��rtoq[^_poqef_ J��������aH_*}abj[^_^bjzFoxry]�oq[^zt�^a2zH]hk8a � ¦x«�Ä�Æ�¢yÆ6­®¦6­6�p¢ !Y¢y§q¤y¢t¨¡­B� � ! � �3����]���rtoq[^_poqef_ J������c^efa¡�haH_^aiuo�c�]h�f�fbjg^bjijb�rx}oV_^bd|�rybjijbj�H�Foqefa�ryefaY|^i�ryaHefbj]hefbd½ ���d°���³ ��³ø¼^º©º©µpÀ3µ�² �d¼ ��°�²�À3µd��bj[^_^bjzFoxryb��^efbÿ��fc�a¡rfryb��qoqk8aH[sryaq�qzH]hk8a � ¦x«�Ä�Æ�¢yÆB­®¦5# §q¤fÆ©¨<"#¢¡¤y¦ � � "� x� � ¦x«�Ä�Æ�¢yÆ�­®¦5# §q¤fÆ©¨ � ª¬«s¨�� � �� Ca� ¦x«�Ä�Æ�¢yÆY­®¦5# §q¤fÆ©¨ � ÃÈÄp«s¨ � � �� x�®i^c^efbjk8]N_^aHb�ryefaq����K�^}aøi�� |^�f|poqija£À3º©µ�²��h¼ ��°�²�À3µ � �F�

3 ��� >1; I��"(� KM>� "(��F "GK,X���b]Xå�á �¡ä©á �qâ�ú � P�P�F.>RG�G�� < �I��*>�I" � �P�'G; L �5C��MI�> #mù©é�á äxâ á ù©á ã�å�ý©÷ )Hýxú¥÷ èNå�á �xú¡å�ú�÷çúû�èfâ ä©áçâ�ú�æ3áHú¥é�é�ã¥û�ã¡äxæ©ú¥üøè�äHû�ãmùqè�é�÷çú�é ú¥ù©ù©é�èfå�è�äHû ú&%�á ã¡ä©è^æ3á¡ý©ä4)Hýxú¥÷çå�áçú¡å�átäFý©üøè�é�ã�é�èfú¥÷ èDN�ú¥ù©ùxú¥é�û�è�ä©è�äHû�è»ú¥÷ ÷*) á äxå�á è�üøèæ3á�é ú¥ù©ù©é�èfå�è�äHû ú&�©á ÷ á ûE'ú�P»üøèfæ3áçú¥äHû�è#á ÷�äFý©üøè�é�ãøü�ú¡â�â('©á äxú£ú¡æNèfå�å�ã£ù©áP'ý -Fáçâ á ä©ãxõ3Lh�� >1; I��"(� KM>�� ��F �RG�� " F.>BX���7]#å�á �¡ä©á �qâ�ú�� P�P�F.>RG�G�� < �I��*>�I" L " F�G`>� > I " F.> #qù©é�á äxâ á ù©á ã\å�ý©÷Q)Hýxú¥÷ è å�á �xú¡å�úø÷çúû�èfâ ä©áçâ�ú�æ3è�÷�û�é�ã¡äxâ�ú¥üøè�äHû�ã #©á ä �xú¡å�èCú¥÷ ÷çú )Hýxú¥÷ èCý©äBäFý©üøè�é�ãøé�èfú¥÷ è<é ú¥ù©ù©é�èfå�è�äHû ú&�©á ÷ è�-Fá è�ä©èCú¥ù©ù©é�ãHå�å�á ü�útû�ã�â ã¡ä�á ÷äFý©üøè�é�ãøü�ú¡â�â('©á äxú ù©áP'ý -Fáçâ á ä©ã�â('©è�#©á ä -tú¥÷ ã¡é�è#ú¡å�å�ã¡÷ ý3û�ã #Fé�áçå�ý©÷ û ú£ù©áP'ý�ù©áçâ�â ã¡÷ ãxõ

Page 50: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

�$�

k8aH[sryefa �hijb�oqi�ryefb»_^|^a��fzHaHi*�h]h[^]VzH]hk8a�oqc^c^ef]h�f�fbjk`oq�Hbj]h[^a&�p]soxrybj[ ��c�]hbj[srøefbj�fc�a¡rfryb��qoqk8aH[sryabji�[�|^k8aHef]8c^by}| �heyoq[^_^a 3 aH_�bji�c^by}|�c^bjzHzH]hij] 3f� �Ö ×KØfÙFÚ�Û^Ü�Ý2ÞqßBA A�ßKàFáçú »ë Nnìfí�# L�# YÈð�# ð�Ppè<êNë?î ìyï¥ô¥ñFó L&% <¡ð�#¡å�á+'xú

��C,Nÿê�P»ë � õ ìyï¥ô¥ñFó % ùqè�é �1,Dè � �õ ìyï¥ô¥ñFó L ùqè�é ����è �1�àFèCâ ã¡äxå�áçæ3è�é�áçú¥üøã£êNë ! î ìyï¥ô¥ñFó L&% <¡ð�#¡å�á+'xú(:

��C,Nÿê�P»ë � Y�õ ìyï¥ô¥ñFó % ùqè�é �1,Dè �1�Y�õ ìyï¥ô¥ñFó L ùqè�é ����è � �Ö

{\aHijiuo6rtoqg�aHijiuo6�fa �h|^aH[srya�efbjc�]he�rybuoqk8]6ija`zFoqeyoxrfryaHefbj��rybjzt�^a`_^aHb£[�|^k8aHefb4�p]soxrybj[ �wc�]hbj[sr�[^aHijij]��rtoq[^_poqef_�J����� a�iuo�ij]hef]Vrybjc^bjzFoVeyoqc^c^efaH�faH[srtoq�Hbj]h[^a��

3Ne �h]hk8aH[sry] Ô bj[ �h]hiuoVc^efaHzHbj�fbj]h[^a ��]hc^c^buoVc^efaHzHbj�fbj]h[^a�£b�r�_^b��fa �h[^] � ��£b�r�c�aHeXi�� aH�fc�]h[^aH[srya � �O��£b�r�c�aHeXiuoVk`oq[srybj�f�yo =$� ��=�£b�røry]qrtoqijb � = � "\oqc^c^eH�<aH�fc�]h[^aH[srya �<zHzHaH�f�f] ��= , �<zHzHaH�f�f] ��� =$�� oq�fzHbuoVaH�fc�]h[^aH[^�Hbuoqija _po��E��= , o 5 ��= � _poV�E��� =$��o 5 ��� = "v�bj[^bjk8]V[^]hefk`oqijbj�H�Foxry] Ò = � � � � Ò = � � � �vwo7�6[^]hefk`oqijbj�H�Foxry] ; =�Ò = � � � ?0=s� � � ; =�Ò = � 3 � ?0=s� � �� oq�fzHbuoV_^aHzHbjk`oqija _po ��� � � �\o ��� 7 � � _po ��� � � �NoR��� 7 � �v�bj[^bjk8]V_^aH[^]hefk`oqijbj�H�Foxry] ��� � � 3 ��� � � � �

� ��#,%&������ ���.-,/0/0/

N d � >1; I��"(�.KM>�� ��F �RG�� ; G�� X����]�å�á �¡ä©á �qâ�ú � P�P�F.>RG�G�� < �I��*>�I" L " F�G`>4P�'G;��MI0J"I �MKM> #tè�â ã¡äxå�áçånû�èmä©è�÷ ÷*) ú¥ù©ù©é�ãHå�å�á Yü�ú¥é�èmý©ä£äFý©üøè�é�ãKé�èfú¥÷ èmé ú¥ù©ù©é�èfå�è�äHû ú&�©á ÷ è�â ã¡ä á ÷©ù©áP'ýU-Fáçâ á ä©ã#äFý©üøè�é�ãKü�ú¡â�â('©á äxúKå�á û�ýxútû�ã#ä©è�÷ ÷çú#æ3á é�è %�á ã¡ä©èmùqãHå�á û�á -túæ3è�÷ ÷*) ú¡å�å�èKé�èfú¥÷ è¡õN � � >1; I��"(� KM>�� ��F �RG� �MI5; G� X���,]�å�á �¡ä©á �qâ�ú � P�P�F.>RG�G�� < �I��*>�I"�L " F�G`> < " I�> �MI0J"I �MKM> #Hè�â ã¡äxå�áçånû�èdä©è�÷ ÷*) ú¥ù�Yù©é�ãHå�å�á ü�ú¥é�è#ý©ä�äFý©üøè�é�ã�é�èfú¥÷ è<é ú¥ù©ù©é�èfå�è�äHû ú&�©á ÷ èCâ ã¡äBá ÷�ù©áP'ý0-Fáçâ á ä©ãXäFý©üøè�é�ã�ü�ú¡â�â('©á äxúXå�á û�ýxútû�ãXä©è�÷ ÷çúXæ3á é�è %�á ã¡ä©èä©è �Hútû�á -tú£æ3è�÷ ÷*) ú¡å�å�è#é�èfú¥÷ è¡õ

Page 51: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# � ,

021�� � 7DGVEF;FJ#;)=D@BEFEHG Q?A<58QDG �2G�MN;H587D@ =D@BEFE �@BACA<58A<@ =?;'A<58L?7��=D5 �

�<� oq[poqijbj�fbd_^aHijiuo8c^ef]hcpo��soq�Hbj]h[^a�_^aHiji�� aHefef]hefa�_^bKef]h|^[^_^]%AK��zHbj]^}a�ij]:��ry|^_^bj]:_^aHiji�� aHefef]hefa �hij]hgpoqijaY_^bef]h|^[^_^]%Awbj[sryaH�f]VzH]hk8a\�f]hk8k`o ; oqi*�haHg^efbjzFo@?#_^b»ry|�rfryb �hijb»aHefef]hefb�_^b�ef]h|^[^_^]%A �haH[^aHeyoxryb�_poqiji�� oqi*�h]x�efb�ryk8]^��efb��haH��rya�|^[�ef|^]hij]�ln]h[^_poqk8aH[srtoqijaN[^aHijiuo�c^ef]�ha¡rfrtoq�Hbj]h[^aNaB����bjij|^c^c�]V_^b�|^[6oqi*�h]hefb�ryk8]^�F���_^|^[^~�|^aYbjk8c�]he�rtoq[sryaBoF�haHefa�o2_^bj�fc�]h�fbj�Hbj]h[^a���ryef|^k8aH[srybdzt�^a�zH]h[^�faH[srtoq[^]`_^b��qoqij|�rtoqefa �hijb�aQAma¡rfryb_^bprtoqijaXaHefef]hefa\�f|^ijiuoB�f]hij|^�Hbj]h[^aXzFoqijzH]hiuoxrtoB_poqiji�� oqi*�h]hefb�ryk8] ; aH�fa �h|^b�ry]�bj[`oqefb�ryk8a¡rybjzFoBo�c^efaHzHbj�fbj]h[^a�p[^b�rto@?¥�{\aHim�fa �h|^b�ry]��f]h[^]�c^efaH�faH[srtoxrybm_^|^a�k8a¡ry]�_^b�_^bmoq[poqijbj�fb»_^aHijiuoYc^ef]hcpo��soq�Hbj]h[^a\_^aHiji�� aHefef]hefa\_^bmef]h|^[��_^]%A �

� iuo �F¦x¤!# §q¤fÆ�¢¡¤¥¤y¦x¤�§qÄ�§qª � ¨¡Ãj¨ B� iuo��t§�Ë �5# §q¤fÆ�¢¡¤¥¤y¦x¤�§qÄ�§qª � ¨¡Ãj¨s�

ÖP×KØfÙFÚ�Û^Ü�Ý2ÞqßBA 9�ß#àFáçúøú¡å�å�è �¡äxútû�ã£ý©äBå�áçånû�è�ü�úøú¥é�á û�üøè û�áçâ ã£þnõpù�õ� :%N�AYë?ìfí < �pë6ô < ��� ��� ë�ï , �� ��� �S 6 ! � 6� � � � & �

ë L�P <

á äBâ ý©á � ë6í©î ó1A � � �^ë6í©î óUM�ìfí���tõ�àFè#ê�ë6ô©î ñFó¥ôCé�áçå�ý©÷ û úê� dë ��C,Nÿê�P»ë6í©î ô¥ñFó4M�ìfí � ë�ô©î ñFó <

àFá�æ3èfå�áçæ3è�é ú£â�ú¥÷çâ ã¡÷çú¥é�èKá ÷�)Hýxú¡æ3é útû�ãøæ3á�ê�õ �ÈäBú¥é�á û�üøè û�áçâ�ú "G ��K K �$.Kë:ê��Kë?ì¡ì¡î ðHï¥ôHï¥í¡ð©õ��Èä � #©á ä&-¡èfâ è�#.� dë�ê� ��ê� �ë�í©î ô¥ñFó M�ìfí �2í©î ô¥ñFó M�ìfíCë6í©î ì¡ìfð M�ìfí � ëDì¡ì¡î ð

ö ) è�é�é�ã¡é�è�é�è�÷çútû�á -¡ãøæ3á�é�ã¡ý©äxæ3ã 1Yå�ý5. 'è�#©æ3ý©ä*)Hý©è�#� I��� ë

� . ! . �. ë�í©î ìfðHó M�ìfí �� E ô M N�í©î óUM�ìfí �� P

� ��� ��

ë6ô � ë?ì¡î ó M�ìfí �� î

àFè �¡ý©è�#©ú¥÷ ÷ ã¡é ú�#3â('©è� I��� J ( � â ã¡ä (�ë6ô

â á ã*'è#÷çú�ånû�á ü�úXæ3è�÷ ÷*) è�é�é�ã¡é�èKé�è�÷çútû�á -¡ã 'èCæ3è�÷ ÷ ãXånû�èfå�å�ã�ã¡é æ3á ä©èCæ3á+�¡é ú¥äxæ3è % %fúøæ3è�÷ ÷*) è�é�é�ã¡é�è<æ3á�é�ã¡ý©äxæ3ã 1�å�ý©÷�æ©útû�ã #©ùqè�éâ ý©á�÷çúøå�ã¡÷ ý�%�á ã¡ä©è<å�ásé�á û�á è�ä©è �QC�C(" K K �����! " õ Ö

��Ama¡rfry|^buoqk8]wi�� ¼p²d¼p¾�³�½F³0�d°�¾�¾n¼ ��º©µ���¼ �»¼^´�³nµ�²d° �d°�¾�¾�ÿ°�º©º©µ»º©° ��³Yº©µ�¿�² �dµ��X�<]3���haHef]���ry|��_^buoqk8]�bj[�zt�^a�k8]�_^] �hijbNaHefef]hefbBbj[sryef]�_^]qrfrybN_poqi��fbj��ryaHk`o*oqefb�ryk8a¡rybjzH]*o*c^efaHzHbj�fbj]h[^a �p[^b�rto*�fbc^ef]hcpo��soq[^]2�f|^ijiuoV�f]hij|^�Hbj]h[^aq�C{\aHi�zFoqijzH]hij]�_^b

���£Ï ��� � �����fbuoq[^] �

Page 52: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

� �

�h� � � � � � �<aHefef]hefb�_^b<¤f§¥ÇsÇ�¤y¢t¨3¢¡Äp­ §3ÉHÃ�¦xÄm¢ _^aHbd_poxryb �����Ï 0 ; �"?CÏ�� ; � 5 � � ?IzH]h[ Ð � � Ð�� �

��� Ï 0 ; � ?<Ï � ; � 5 � � ?UzH]h[ Ð � � Ð�� �

=���� � �CaHefef]hefaB�f|^ijija`¦tÇp¢¡¤f§3ÉHÃ�¦xÄpà � ¦F§q­�ÃÈÄ�4$�jÇp¦xÃÈÄp­�bj[ � �����Ï 0 ; �"? * ���� Ï 0 ; � ?�* �

c�aHeXzH|^b��fb�efbjzFoF�qo$� 0 ; �"? � � 0 ; � ? Ï 0 ; 0 ; �"? � 0 ; � ?�?#Ï ; 0 ; �"? � 0 ; � ?�? ; � 5 � � ?zH]h[ Ð � � Ð�� �

�f|^c^c�]h[^aH[^_^]�_^b\aH�f�faHefa?bj[ c^efaH�faH[^�Fo�_^b\|^[ �fbj��ryaHk`ozt�^aDefaFoqijbj�H�Fo�iuok`oq�f�fbjk`ooqzHzH|^eyoxryaH�H�Fo_^bj[poqk8bjzFo�� ®[�cpoqe�rybjzH]hiuoqefaq���h]hijaH[^_^]8aH�fa �h|^bjefa�bji�zFoqijzH]hij]�_^aHi�~�|poq_^eyoxry]2_^b�|^[�_poxry]�� �

� � � 0 ; �"?C�� ; � 5 � � ?aH�fa �h|^aH[^_^]`iuoVk8]hi�rybjc^ijbjzFoq�Hbj]h[^a

���£Ï ��� �P���£Ï� 0 ; ��� �w���E?CÏ� 0 ; 0 ; �"?�� 0 ; �"?�?�fb�]qrfrybjaH[^a

0 ; 0 ; �"? � 0 ; �"?�? Ï ; � ; � 5 � � ?� ��� �� �� � � � � ��

�¡� ; � 5 � � ?� ��� �� ��� � � � � ��

? � ; � 5 � � ?� ��� �� ���� � � � ��� � � � � � �

Ï

Ï � � ; � 5 � � ? � ; � 5 � � ?zH]h[*Ð � � Ð>?qÐ � � Ð � � �^_po2zH|^b �

���øÏ ��� �����øÏ � � ; � 5 � � ? � ; � 5 � � ?¥Ó�<� aHefef]hefaBefaHiuoxryb��h]2efbj�f|^i�rto$�

Î :��� Ï Ð � � Ò ��ÐÐ ��Ð Ï Ð � � ; � 5 � � ? � ; � 5 � � ?KÒ�� � Ð� � Ï

Ï ÐB� 5 � � 5 � �� 5 � � � �� 5 = � � 5 = � � � � Ò ��Ð��� � � 5 � � � 5 � � Ï � � 5�� ; � � ? Q �p¿

]3���haHef]>Î :��� }aD_^aHijij]���ryaH�f�f]>]hef_^bj[^aw_^b �heyoq[^_^aH�H�Fo*_^aHijiuok`oq�f�fbjk`ooqzHzH|^eyoxryaH�H�Fo*efaHiuoxryb��qo+B\iuo¨3¦xª¬«hÉHÃ�¦xÄm¢£c^|�}]^�p_^|^[^~�|^aq�»efb�ryaH[^aHef�fb §�ËtËt¢¡­�­ § �HÃȪj¢¡�

Page 53: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# � 1

ÖP×KØfÙFÚ�Û^Ü�Ý2ÞqßBA �sß . å�å�è �¡äxútû�ã£ý©äBå�áçånû�è�ü�úøú¥é�á û�üøè û�áçâ ã£þnõ ù�õ� : N�A�ë?ìfí < �pë:ñ < ��� ��� ë�ï , �� ��� �

ë9<�P <

á äBâ ý©á � ë6í©î ó1A � � �^ë6í©î óUM�ìfí���2 #©èfæ\á�æ©útû�á :êNë %Fî ñxìfô¡ô¡í���ê ë ��C,Nÿê�P�ë�í©î %yñxìfô M�ìfí ��øë6ô©î ñFó¥ô&%yñHð�� � ë ��C,N ��P»ë�í©î ô¥ñFótñUM�ìfí �

â�ú¥÷çâ ã¡÷çú¥é�è�÷çú�æ3á 1hè�é�è�ä�%fú�� dë:ê� ! �

�ÈäBú¥é�á û�üøè û�áçâ�úøú£ù©é�èfâ áçå�á ã¡ä©èKá ä��xä©á û ú �£ë:ê ! ��ë6ô©î ðHó¥ðHó¡ó3ì2V¡ä©è�÷�å�áçånû�è�ü�ú þnõ ù�õ»â ã¡äxå�áçæ3è�é útû�ã :� ë ê ! � ëë í©î %yñxìfôUM�ìfí � ! í©î ô¥ñFótñUM�ìfí �ë í©î ô¡ðHó¥ðUM�ìfí �

ö ) è�é�é�ã¡é�è�é�è�÷çútû�á -¡ãøæ3á�é�ã¡ý©äxæ3ã 1Yå�ý � é�áçå�ý©÷ û ú�:� I� � ë � � ! �O�� �O� ë � í©î ô¡ðHó¥ð M�ìfí � ! ô©î ðHó¥ðHó¡ó3ì&�ô©î ðHó¥ðHó¡ó3ì ë

ë í©î ìfô¡ðHï4M�ìfí ��2 J í©î ôUM�í©î óUM�ìfí ��2� ��� ��ãE-&-¡è�é�ã

� I� � J ( � â ã¡ä (�ë6í©î ôö�úXånû�á ü�úXã¥û�û�è�äFý3û ú 'è�#hæ3ý©ä*)Hý©è�#sæ3è�÷ ÷ ãNånû�èfå�å�ãXã¡é æ3á ä©è æ3á$�¡é ú¥äxæ3è % %fú�æ3è�÷ ÷*) è�é�é�ã¡é�èCæ3á�é�ã¡ý©äxæ3ã 1Vå�ý©÷^æ©útû�ã #qùqè�é#â ý©á÷çúOG`> ;'I��*>�I"�ù©ýQ'ãøé�á û�è�ä©è�é å�á �QC�C(" K K �����! " õ Ö

ÖP×KØfÙFÚ�Û^Ü�Ý2ÞqßBA ��ß . å�å�è �¡äxútû�ã£ý©äBå�áçånû�è�ü�úøú¥é�á û�üøè û�áçâ ã£þnõ ù�õ� : N�A�ë?ìfí < �pë:ñ < ��� ��� ë�ï , �� ��� �

ë9<�P <

á äBâ ý©á � ë6í©î ó1A � � �^ë6í©î óUM�ìfí���2 #©èfæ\á�æ©útû�á :êNë6ô©î ñFó¥ô&%yñHð � ê ë ��C,Nÿê�P»ë6í©î ô¥ñFótñUM�ìfí ��øë6ô©î ñFó¥ô¥ñHôHï � � dë ��C,N ��P»ë�í©î ô¥ñFó¥ô M�ìfí �

â�ú¥÷çâ ã¡÷çú¥é�è�÷çú�æ3á 1hè�é�è�ä�%fú� ë:ê ! �

�ÈäBú¥é�á û�üøè û�áçâ�úøú£ù©é�èfâ áçå�á ã¡ä©èKá ä��xä©á û ú �£ë:ê ! ��ë6í©î í¡í¡í¡ô©ìE%&V¡ä©è�÷�å�áçånû�è�ü�ú þnõ ù�õ»â ã¡äxå�áçæ3è�é útû�ã :� ë ê ! � ëë í©î ô¥ñFótñ M�ìfí � ! í©î ô¥ñFó¥ô M�ìfí �ë í©î ìDM�ìfí ��

Page 54: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

�O�

ö ) è�é�é�ã¡é�è�é�è�÷çútû�á -¡ãøæ3á�é�ã¡ý©äxæ3ã 1Yå�ý � é�áçå�ý©÷ û ú�:� I� � ë � � ! �O�� �O� ë � í©î ìDM�ìfí��� ! í©î í¡í¡í¡ô©ìE% �í©î í¡í¡í¡ô©ìE% ë

ë í©î ï3ìyó¡ó4M�ìfí � ë:ñxî ô©ì�M�ìfí 2 M�í©î óUM�ìfí ��2� ��� �

�ãE-&-¡è�é�ã� I� � J ( � â ã¡ä (�ë:ñHô©ìfí

ö�úøånû�á ü�úøã¥û�û�è�äFý3û ú 'è�#©æ3ý©ä*)Hý©è�#xüøã¡÷ û�ãøù©áP'ý �¡é ú¥äxæ3è<æ3è�÷ ÷*) è�é�é�ã¡é�è#æ3á�é�ã¡ý©äxæ3ã 1Yå�ý©÷�æ©útû�ã N 'è<â á é â�úøæ3ásû�é�è#ã¡é æ3á ä©á�æ3á�¡é ú¥äxæ3è % %fúCù©áP'ý �¡é ú¥äxæ3èRP�#3ùqè�édâ ý©á�÷çúOG`> ;'I��*>�I"�ù©ýQ'ãøé�á û�è�ä©è�é å�á I�>�I �QC�C(" K K �����! " õ Ö

Xb*�h|poqef_poq[^_^]Pb�_^|^aaH�faHk8c^bYc^efaHzHaH_^aH[sryb��fb�c^|�}] _^aH_^|^efefa�zt�^aq��[^aHijij]>��ryaH�f�f] �fbj��ryaHk`ooqefb�ryk8a¡rybjzH]wo?c^efaHzHbj�fbj]h[^a �p[^b�rto��<i�� oqi*�h]hefb�ryk8]6c�aHe�bji zFoqijzH]hij]D_^b£�DÒ �c^ef]�_^|^zHa6|^[ ¤¥Ãj¨¡«�ª¬­ §q­®¦§�ËtËt¢¡­�­ § �HÃȪj¢£bj[6zH]hefefbj�fc�]h[^_^aH[^�Fo2_^b

�,Ï , Ó " �=���O�� Ï ��Ó " �$� ,5" 1

aH_�|^[�¤¥Ãj¨¡«�ª¬­ §q­®¦I! � !ѧ�ËtËt¢¡­�­ § �HÃȪj¢øc�aHe�,Ï ��Ó " �$� ,5" 1� Ï ��Ó " �$� " � =

�No��faH[^�f]^��oqijij]heyo��xzt�^bjaH_^aHef�fbp�faXi�� oqi*�h]hefb�ryk8]Xc�aHe<bji�zFoqijzH]hij]N_^aHijiuo�_^b5AmaHefaH[^�Fo��fbuo���rtoqg^bjijaø]B[^]h[2ij]�fbuo�� ��Ama¡rfry|^buoqk8]^�p~�|^bj[^_^b��»i�� oq[poqijbj�fb�_^aHijiuo�c^ef]hcpo��soq�Hbj]h[^aY_^aHiji�� aHefef]hefa�_^bdef]h|^[^_^]%AK��[^aHi�zFoqijzH]hij]_^aHijiuoV_^b5AmaHefaH[^�FoVryeyo�_^|^aY[�|^k8aHefb����Da����Ô buoq[^] �

� � Ï� 0 ; �"?CÏ � ; � 5 � � ?��� Ï� 0 ; � ?#Ï � ; � 5 � � ?

�fb��po$�� � Ï ��� � ��� Ï

Ï 0 ; ���CÒ ���E?<ÏÏ 0 ; 0 ; �"?#Ò 0 ; � ?�?<ÏÏ ; � ; � 5 � � ?KÒ � ; � 5 � � ?�? ; � 5 � � ?zH]h[ÕÐ � � Ð>?qÐ � � Ð>?qÐ � � Ð � � �6�Boqiji�� aH�fc^efaH�f�fbj]h[^a2_^aHiji�� aHefef]hefa8efaHiuoxryb��h]6�fb _^aH_^|^zHa:bj[zt�^a`k8]�_^]^�drtoqijbaHefef]hefb����fb�c^ef]hcpo��soq[^]2�f|^ijiuoV�f]hij|^�Hbj]h[^a � � �

Î0:�� Ï Ð � � Ò � Ð

Ð � Ð Ï Ð ; � ; � 5 � � ?#Ò � ; � 5 � � ?�? ; � 5 � � ?KÒ ; �:Ò � ?3ÐÐ �`Ò �dÐ ÏÏ Ð � ; � � 5 � � 5 � � � � ?KÒ � ; � � 5 � � 5 � � � � ?3ÐÐ �`Ò �dÐ �

�Р�<Р; = � 5 � � ? 5 Р�dР; = � 5 � � ?

Ð �`Ò �dÐ Ï ; = � 5�� ; � � ?�?Ð �<Ð 5 Ð �dÐÐ �`Ò �dÐ

Page 55: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# �Q�

]3���haHef]Π:

�� Ï ; = � 5�� ; � � ?�?

Ð �<Ð 5 Ð �dÐÐ �`Ò �dÐ

Ô b<]h�f�faHe���b��d_^|^[^~�|^aq��zt�^a8i�� aHefef]hefaVefaHiuoxryb��h]�[^aHi<zFoqijzH]hij]�_^aHijiuo`_^b5AmaHefaH[^�Fo�ryeyo�_^|^a2[�|^k8aHefb��_^bjc�aH[^_^a�_po���a ��� ®[6oqi�ryefaNcpoqef]hijaq��[^]h[��fbmc^|�}]�zH]h[^zHij|^_^aHefa��faBiuoY�f]hij|^�Hbj]h[^aN�fbuo�oqzHzHa¡rfrtoqg^bjija\][^]^�pbj[^_^bjc�aH[^_^aH[sryaHk8aH[srya�_poqbm�qoqij]hefb�_^bm�Da ��� 8�]h��ry]

� Ï Ð �<Ð 5 Ð �dÐÐ �:Ò �dÐ ?

bj[6zH]hefefbj�fc�]h[^_^aH[^�Fo2_^b�RÏ , Ó " �=���O�� Ï ��Ó " �$� ,5" 1

�fb^]qrfrybjaH[^a � Ï =�Ó ,5"�" �\aq�s~�|^bj[^_^b��hbji�efbj�f|^i�rtoxry]N_^aHijiuoN_^b5AmaHefaH[^�Fo���Ò �Vc^|�}]Befb�ryaH[^aHef�fbpoqzHzHa¡rfrtoqg^bjijaq�3Ni�zH]h[sryeyoqefbj]^��c�aHe

�RÏ ��Ó " �$� ,5" 1� Ï ��Ó " �$� " � =

�fb<�po � Ï =�Ó�� , � ��� � ��zH]h��rtoq[srya �heyoq[^_^a�c�aHe�zH|^b#bji#efbj�f|^i�rtoxry]:_^aHijiuo:_^b5AmaHefaH[^�Fo�c^|�}]�efb�ryaH[^aHef�fb{ �B{�oqzHzHa¡rfrtoqg^bjijaq��<� oq[poqijbj�fb�_^aHijiuo*c^ef]hcpo��soq�Hbj]h[^a�_^aHiji�� aHefef]hefa?bj[sryef]�_^]qrfry]>_poqijiuo�c^efaHzHbj�fbj]h[^a �p[^b�rto���Æq«�¤f§qÄp­®¢

ª�� ¢t¨3¢tË¡«hÉHÃ�¦xÄm¢2Æ�¢¡ªÈªj¢`¦tÇp¢¡¤f§3ÉHÃ�¦xÄpà ¤¥Ã�ËE��Ã�¢t¨¡­®¢8Æs§qªÈª�� §qª 4�¦x¤¥ÃÈ­�Å`¦x����bjaH[^a�_^a¡rfrto � µ»º �V¼^º*��°�º©º©µ»ºw¼p²d¼ ¾ �d½F³�½*¸ �y·ÿ°»·ÿ¼�·uÁ ; oq[poqijbj�fbCbj[�oF�qoq[sryb6?¥� 3N[poqijbj�H�Hbuoqk8]^�d]heyo��K~�|poq[^_^]?|^[*oqi*�h]hefb�ryk8]6�fb c^|�}]?_^bjefa��rtoqg^bjija�[^aHid�faH[^�f]8_^aHijiuo�ln]he �øoqef_�aHefef]he\oq[poqi*��fbj� �Ô bmbj[^_^bjzt�^bj[^]�zH]h[ � ; ��?<a ��� ; ��?¥��efbj�fc�a¡rfryb��qoqk8aH[sryaq��iuo��f]hij|^�Hbj]h[^a\aH�yoxrfrtoVa�~�|^aHijiuoYzFoqijzH]hiuoxrto���doVl � aq� o��C�fb�c^ef]hc�]h[^aB_^b�ln]hef[^bjefaB|^[po2��rybjk`oV_^aHiji�� aHefef]hefa�efaHiuoxryb��h] ; oq�f�f]hij|�ry]�?&�

Î :� Ï ÐjÐ � ; ��?�Ò ��� ; ��?3ÐjÐÐjÐ � ; ��?3ÐjÐ ; ÐjÐ � ; ��?�Ò ��� ; ��?3ÐjÐ ?�? ; �h� � 1 ?_^]3�ha�ÐjÐ �pÐjÐ�}aY|^[po8[^]hefk`o$�p�f�yoxrto����fa �h|^aH[^_^]^��cpoq�f�f]:_^]hc�]8cpoq�f�f]^��i�� bj[sryef]�_^|^�Hbj]h[^aB_^bdaHefef]hefbd[^aHic^ef]�zHaH�f�f]�efbj�f]hij|�ryb��h]^�»o8cpoqe�rybjefaY_poqiji�� aHefef]hefa�[^aHb�_poxryb�a�iuo2ij]hef]8c^ef]hcpo��soq�Hbj]h[^aq��®[Dcpoqe�rybjzH]hiuoqefaq��fa

Î0:� Ï ÐjÐ � ; ��?�Ò ��� ; ��?3ÐjÐÐjÐ � ; ��?3ÐjÐ � � � �

~�|poqij]heyo � �fbuoN|^[poNzH]h��rtoq[sryaø�h�f|��8zHbjaH[sryaHk8aH[srya�c^bjzHzH]hiuoq�^�xi�� oqi*�h]hefb�ryk8]��fb�_^bjzHa£��rtoqg^bjija [^aHi��faH[^�f]_^aHijiuoVl � aq� o��5B»�faq��bj[s�haHzHaq� � }a�|^[po8zH]h��rtoq[sryaY��heyoq[^_^aH�^�»i�� oqi*�h]hefb�ryk8]��fbd_^bjzHa�bj[^��rtoqg^bjija [^aHid�faH[^�f]_^aHijiuo?l � aq� o��H��3N_ aH�faHk8c^bj]^�ø�fb\]h�f�faHe��qo�zt�^a�ª�� §qª 4�¦x¤¥ÃÈ­�Å`¦�Çp¢¡¤�ÃȪ�Ëy§qªjËt¦xªj¦�Æ�¢¡ª\ÌH«^§sÆq¤f§q­®¦*ÆqÃ�«�ÄÄp«�Å`¢¡¤y¦ Í¢�¨¡­ § �HÃȪj¢¡�^aH�f�faH[^_^]

Î0:��� Ï � � Ó

Page 56: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

� =

{\aHi�zFoqijzH]hij]V_^aHijiuoV�f]qrfryeyoq�Hbj]h[^aq�pbj[s�haHzHaq��oqc^c^ijbjzFoq[^_^]�iuo�l � aq� o��C�fb<}a�]qrfryaH[�|�ry] �Î0:

�� Ï ; = � 5�� ; � � ?�?

Ð �<Ð 5 Ð �dÐÐ �`Ò �dÐ

� ��� ��

Ó

��]2��ryaH�f�f]:oqi*�h]hefb�ryk8]��po2c^ef]�_^]qrfry]�|^[�efbj�f|^i�rtoxry]2oqzHzHa¡rfrtoqg^bjijaB[^aHi�c^efbjk8]�zFoq�f]^�paH�f�faH[^_^]Î0:

�� Q � � zH]h[ � Ï���Ó �

]3���haHef]6i�� oqi*�h]hefb�ryk8]�}aV��rtoqg^bjijaV[^aHi#�faH[^�f]�_^aHijiuo8l � aq� o�����k8aH[sryefaq��[^aHi#�faHzH]h[^_^]�zFoq�f]^��bjiKefbj�f|^i�rtoxry]{ �B{(}aYoqzHzHa¡rfrtoqg^bjijaBaH�f�faH[^_^]

Î0:�� Q � � zH]h[ � Ï " �Q���

]3���haHef]6i�� oqi*�h]hefb�ryk8]6}a�bj[^��rtoqg^bjijaY[^aHi#�faH[^�f]6_^aHijiuo8l � aq� o��8�Y�do`l � aq� o���ln]hef[^bj�fzHaq��~�|^bj[^_^b���bj[DefaHiuo©��Hbj]h[^aYoqi�zFoqijzH]hij]2_^aHijiuo�_^b5AmaHefaH[^�Fo���|^[�l�oxrfry]hefa�_^b�oqk8c^ijb��pzFoq�Hbj]h[^a�_^aHiji�� aHefef]hefa�zt�^a�_^bjc�aH[^_^a��fbuo_poqiji�� oqi*�h]hefb�ryk8]�zt�^a�_poqb�_poxryb�_^aHi�c^ef]hg^ijaHk`o��{Noq�fzHaq�ø~�|^bj[^_^b�� ryeyoqaH[^_^]�fc^|^[sry]�_po~�|^aH��ry]�zFoq�f]^�øiuo�[^aHzHaH�f�fb�rx}o_^bXbj�f]hiuoqefa�i�� oqk8c^ijb��pzFoq�Hbj]h[^a_^aHiji�� aHefef]hefawbj[sryef]�_^]qrfry]�_poqijijaw]hc�aHeyoq�Hbj]h[^b\_^aHiji�� oqi*�h]hefb�ryk8]^� bj[^_^bjc�aH[^_^aH[sryaHk8aH[srya?_poqbN_poxrybN_^aHic^ef]hg^ijaHk`o�� 3�rtoqija`�fzH]hc�]D|^[�� bj_^aFo�}a:~�|^aHijiuo�_^b£efbjzH]h[^_^|^efefa8ry|�rfryb �hijbCaHefef]hefb£bj[sryef]�_^]qrfrybC_poqijija]hc�aHeyoq�Hbj]h[^b �p]soxrybj[ ��c�]hbj[srBo�c^efaHzHbj�fbj]h[^a �p[^b�rto���oq_�aHefef]hefbK�f|^b<_poxryb���oq�f�f|^k8aH[^_^]^�d~�|^bj[^_^b���zt�^aijaV]hc�aHeyoq�Hbj]h[^b#_^aHiji�� oqi*�h]hefb�ryk8]2�fbuoq[^]^��_^bKl�oxrfry]^��aH�fa �h|^b�rya�bj[oqefb�ryk8a¡rybjzFo`o�c^efaHzHbj�fbj]h[^a�ÃÈÄ 6CÄpÃÈ­ §q�®[rtoqi<k8]�_^]^�dbj[�l�oxrfryb��diuo�c�aHe�ry|^efgpoq�Hbj]h[^aq�K]heyoqk`oqb#efbjzH]h[^_^]qrfrto6�f|^iC_poxry]wbj[^bj�Hbuoqijaq���yoqe©}o�~�|^aHijiuobj[^_^]qrfrto�_poqiji�� oqi*�h]hefb�ryk8]YaH_�|^[po8�f|po ; a¡�haH[sry|poqija�?\oqk8c^ijb��pzFoq�Hbj]h[^aN�f|^ijiuo��f]hij|^�Hbj]h[^a��yoqe©}o2~�|^aHijiuobj[^_^]qrfrtoB_poqipc^ef]hg^ijaHk`o��s]3���haHef]�_poqi ; k`oqi6?�zH]h[^_^bj�Hbj]h[poqk8aH[sry]�_^aHi»c^ef]hg^ijaHk`o��K�<� oq[poqijbj�fb^_^aHiji�� aHe �ef]hefaVgpoq�yoxrto:�f|^iji�� bj_^aFo`_^b#zH]h[^�fbj_^aHeyoqefa�iuo`�f]hij|^�Hbj]h[^aq��zFoqijzH]hiuoxrto8_poqiji�� oqi*�h]hefb�ryk8]�bj[?|^[?�fbj��ryaHk`ooqefb�ryk8a¡rybjzH]Do?c^efaHzHbj�fbj]h[^a �p[^b�rto��CzH]hk8a:�f]hij|^�Hbj]h[^a�aH�yoxrfrto ; ]3���haHef]�]qrfryaH[�|�rto?bj[>oqefb�ryk8a¡rybjzFoDoc^efaHzHbj�fbj]h[^a:bj[Q�p[^b�rto@?�_^bø|^[*c^ef]hg^ijaHk`oD_^aHijij]D��ryaH�f�f]�rybjc�]^�#zH]h[>_poxrybøc�aHe�ry|^efgpoxryb��#��bjaH[^a�_^a¡rfrto��¼ ��� �V¼^º*� °�º©º©µ»º`¼p²d¼p¾ �d½F³�½�¸ � ·ÿ°»·ÿ¼�·uÁ ; oq[poqijbj�fb�oqiji�� bj[^_^bja¡ryef]�?¥�

ÖP×KØfÙFÚ�Û^Ü�Ý2ÞqßBA �sß . å�å�è �¡äxútû�ã£ý©äBå�áçånû�è�ü�úøú¥é�á û�üøè û�áçâ ã£þnõ ù�õ� : N�A�ë?ìfí < �^ë�ñ < ��� ��� ë�ï , �� ��� �S 6 ! � 6� � � � & �

ë <�P <

á ä�â ý©á � ë�í©î ó1A � � �pë6í©î ó M�ìfí���2 #©èfæNá�æ©útû�á :êNë6ô©î ñFó¥ô&%yñHð � ê ë ��C,Nÿê�P»ë�í©î ô¥ñFótñUMVìfí ���ë6ô©î ñFó¥ô¥ñHôHï � � ë ��C,N ��Ppë6í©î ô¥ñFó¥ô M�ìfí �

â�ú¥÷çâ ã¡÷çú¥é�è�÷çúøæ3á 1hè�é�è�ä�%fúê� �� �

�Èä�ú¥é�á û�üøè û�áçâ�ú£èfå�útû�û ú � ë�ê ! ��ë�í©î í¡í¡í¡ô©ìE%&VHä©è�÷�å�áçånû�è�ü�ú£þnõ ù�õ»â ã¡äxå�áçæ3è�é útû�ã :� ë ê � � ë

ë í©î ìDM�ìfí ��

Page 57: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# ���

� ã¡÷ è�äxæ3ãBú¥äxú¥÷ á % %fú¥é�è �¡÷ ápè�é�é�ã¡é�á^á äHû�é�ã3æ3ã¥û�û�ápä©è�÷�â�ú¥÷çâ ã¡÷ ãNæ3á�� ë�ê � � # �©áçå�ã��¡äxú\é�áçâ ã¡é æ©ú¥é�è£â('©èø÷*) è�é�é�ã¡é�è£å�ý©÷æ©útû�ãøêBé�áçå�ý©÷ û ú�:� � ë ����

ê ! ��C,Nÿê�Pê����ë

ë ����í©î ô¥ñFó¥ô&%yñHðDM�ìfí � ! í©î ô¥ñFótñUM�ìfí �í©î ô¥ñFó¥ô&%yñHðDM�ìfí �

����ë

ë í©î % M�ìfí �%3æ©úøâ ý©á*#©á ä�ùxú¥é�û�áçâ ã¡÷çú¥é�è�#

��C,Nÿê�P»ë:ê+Nnì�� � � P���í©î ô¥ñFótñUM�ìfí � ë6ô©î ñFó¥ô&%yñHð Nnì �2í©î %4M�ìfí �%3 P. äxú¥÷ ã��Hú¥üøè�äHû�èK÷*) è�é�é�ã¡é�è#å�ý©÷�æ©útû�ã �Xé�áçå�ý©÷ û ú�:

� � ë ����� ! ��C,N ��P

�����ë

ë ����í©î ô¥ñFó¥ô¥ñHôHï�M�ìfí � ! í©î ô¥ñFó¥ô M�ìfí �í©î ô¥ñFó¥ô¥ñHôHï�M�ìfí �

����ë

ë í©î ìDM�ìfí ��2æ©úøâ ý©á*#©á ä�ùxú¥é�û�áçâ ã¡÷çú¥é�è�#

��C,N ��P»ë � Nnì � � � P���í©î ô¥ñFó¥ô M�ìfí � ë6ô©î ñFó¥ô¥ñHôHï�Nnì �2í©î ì�M�ìfí ��2 P� ã¡áçâ(',+è

� ë ��C,Nÿê�P ����C,N ��Ppëë í©î ô¥ñFótñ M�ìfí � �2í©î ô¥ñFó¥ô M�ìfí �ë í©î ìDM�ìfí ��

å�á 'xú� � ë � � ! �O�� �O� ë ����

í©î ìDM�ìfí��� ! í©î í¡í¡í¡ô©ìE%í©î í¡í¡í¡ô©ìE%����ë

ë í©î ï M�ìfí �è�#*)Hý©á äxæ3á*#

� ë �%Nnì � � � P���í©î ìDM�ìfí �� ë6í©î í¡í¡í¡ô©ìE% Nnì �2í©î ï M�ìfí � PW�ý©á äxæ3á :

��C,Nÿê�P»ë�ê+Nnì � � � P � í©î ô¥ñFótñUM�ìfíCë N�í©î ô¥ñFó¥ô&%yñHð4M�ìfí�P Nnì!�2í©î % M�ìfí��%30P��C,N ��Ppë � Nnì!� � � P � í©î ô¥ñFó¥ô M�ìfíCë N�í©î ô¥ñFó¥ô¥ñHôHïDM�ìfí�P Nnì!�2í©î ìDM�ìfí���2*P

� ë ��C,N ��C,Nÿê�P ! ��C,N ��P,Ppë �%Nnì!� � � P � í©î ìDM�ìfí���dë�í©î í¡í¡í¡ô©ìE% Nnì �2í©î ïUM�ìfí�P Nnì¡õ ñHí�P

�Z1hè û�û�ýxú¥äxæ3ãø÷ è<ã¡ùqè�é ú&%�á ã¡ä©áhá äBú¥é�á û�üøè û�áçâ�úøú£ù©é�èfâ áçå�á ã¡ä©è#á ä��xä©á û ú�#xæ©ú¥÷ ÷ è Nnì¡õ ñHí�Pmå�è �¡ý©èQ:� ë N�í©î ô¥ñFó¥ô&%yñHðDM�ìfí�P Nnì �2í©î % M�ìfí �%3 P � N�í©î ô¥ñFó¥ô¥ñHôHïDM�ìfí�P Nnì �2í©î ì�MVìfí ��2 P»ë

ë 2 N�í©î ô¥ñFó¥ô&%yñHðDM�ìfí�P Nnì �2í©î % M�ìfí �%3 P ! N�í©î ô¥ñFó¥ô¥ñHôHï�M�ìfí�P Nnì �2í©î ìDM�ìfí ��2 P 4 M Nnì �2í©î ï M�ìfí�PJ í©î í¡í¡í¡ô©ìE% Nnì��2í©î ìDM�ìfí ��2 P Nnì��2í©î %4M�ìfí �%3 P

� ��� � !&� � � ;�� � � ;���� Nnì �2í©î ï M�ìfí�P� ��� � � ;� �

Page 58: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

� "

æ©úøâ ý©á*#�� ë ê� �� � dë Nÿê ! ��P Nnì � � � P Nnì�� � � P Nnì � � � P <â ã¡ä � � � � E � < � � � � E � < � � � � E �

F -&-¡è�é�ã :� ë 2 ê+Nnì�� � � P Nnì�� � � P Nnì � � � P 4� ��� �! ! 2 � Nnì!� � � P Nnì�� � � P Nnì�� � � P 4� ��� ��

�Èä )Hý©èfånû�ã\üøã3æ3ã �¡÷ á�è�é�é�ã¡é�á�á äHû�é�ã3æ3ã¥û�û�ápæ©ú¥÷ ÷çú\å�ã¥û�û�é ú&%�á ã¡ä©èCèfå�è �¡ý©á û ú\ú\ù©é�èfâ áçå�á ã¡ä©è �xä©á û úNå�ã¡ä©ãNånû útû�á�é�áçâ ã¡äxæ3ã¥û�û�áú¡æBè�é�é�ã¡é�á^å�ýYê�èøå�ý �hõ � è�éKånû�á ü�ú¥é�è )Hýxú¥äHû�ãG'è �¡é ú¥äxæ3è ÷çú\ùqè�é�û�ý©é �xú&%�á ã¡ä©è é�áçâ ã¡äxæ3ã¥û�û ú\å�ý©ápæ©útû�á^á ä©á %�áçú¥÷ á^êVè �â�ú¥÷çâ ã¡÷ áçú¥üøã ÷çú�æ3áçånû ú¥ä�%fú£é�è�÷çútû�á -túCû�é ú Nÿê < ��P�è Nÿê < � P®õ� I ! � � ë ��Nÿê < � P ! Nÿê < ��P�

��Nÿê < ��P� ëë � ê+N � � � � � � � � � � P ! � N � � � � � � � � � � P/�� ê �2� � ��� EE � ê+Njï � � � �0P/�2� � � Njï � � � �0P/�� ê �2� � ��� ë6ï � � � � ë6ï � ����N � � P J ï � Nnì¡õ ñxìRP

��å�å�è�äxæ3ã #3æ3ý©ä*)Hý©è�#� I ! � � J ( I � â ã¡ä ( I ë�ï

÷çú8ånû�á ü�ú8æ3è�÷ ÷*) è�é�é�ã¡é�è�é�è�÷çútû�á -¡ã2ã¥û�û�è�äFý3û ú 'è�æ3è�÷ ÷ ã`ånû�èfå�å�ã8ã¡é æ3á ä©è�æ3á �¡é ú¥äxæ3è % %fú2æ3è�÷ ÷çú2ü�ú¡å�å�á ü�ú8ú¡â�â ý©é útû�è % %fúé�è�÷çútû�á -túøè�#O)Hý©á äxæ3á*#�Nÿê < � P�'èCý©ä+) ú¥ù©ù©é�ãHå�å�á ü�ú&%�á ã¡ä©è<æ3á Nÿê < ��Pdâ('©è æ3áçånû úXæ©ú Nÿê < ��Pdæ3á�ý©äxú )Hýxú¥äHû�á ûE'úXæ3è�÷ ÷ ã\ånû�èfå�å�ãã¡é æ3á ä©èøæ3á �¡é ú¥äxæ3è % %fúNæ3è�÷ ÷çúNü�ú¡å�å�á ü�úNú¡â�â ý©é útû�è % %fúXé�è�÷çútû�á -tú©õ�àFãHånû ú¥ä�%�áçú¥÷ üøè�äHû�è Nÿê < � PU'è£ý©ä+) ú¥ù©ù©é�ãHå�å�á ü�ú&%�á ã¡ä©èæ3á+Nÿê < ��P��QC�C(" K K �����! " õ �ÈäK)Hý©èfånû�ãCâ�ú¡å�ã #¡÷çú<ùqè�é�û�ý©é �xú&%�á ã¡ä©è�á äHû�é�ã3æ3ã¥û�û úCæ©ú¥÷ ÷ èdã¡ùqè�é ú&%�á ã¡ä©á©èfå�è �¡ý©á û�è�æ©ú¥÷ ÷*) ú¥÷ �¡ã¡é�á û�üøãè\ùqã¡á�é�áçâ ã¡äxæ3ã¥û�û úYå�ý©÷�æ©útû�ã�á ä©á %�áçú¥÷ è�# 'èNæ3è�÷ ÷ ã�ånû�èfå�å�ãYã¡é æ3á ä©èNæ3á �¡é ú¥äxæ3è % %fú�æ3è�÷ ÷*) è�é�é�ã¡é�è\æ3á�é�ã¡ý©äxæ3ã 1�å�ý©÷�æ©útû�ãá ä©á %�áçú¥÷ è#ùqè�édâ ý©á���� 6��*3qÝ5/tÜ�fÚ2ÝB0�Ù���*6YØfÝ�� �`/167)¡Ü�Ý$(�Ù��Ù�Ø �6��^Ü���Ù@(�Ù�mØfÙ�(�ØfÝ 0�Ù���*6��mß Ùqß 6sß�õ Ö

®[ �haH[^aHeyoqijaq��zH]h[^�fbj_^aHeyoxry]�|^[oqi*�h]hefb�ryk8]^�pzt�^a�efbj�f]hi��haV|^[?c^ef]hg^ijaHk`o`_^bK_poxrybKbj[^bj�Hbuoqijb1�p���fa�fb�bj[^_^bjzFoq[^]�zH]h[ � � a �p�^efbj�fc�a¡rfryb��qoqk8aH[sryaBbji�_poxry]2c�aHe�ry|^efgpoxry]2aH_wbjim_poxry]8bj[^bj�Hbuoqijaq��iuo�g�� aq� o��C�fbc^ef]hc�]h[^aB_^b�ln]hef[^bjefaB|^[po���rybjk`oV_^aHiji�� aHefef]hefaBefaHiuoxryb��h] ; oq�f�f]hij|�ry]�?&�

� �YÒ � � �� � � ; � �YÒ � � � ?�?

_^]3�ha � � � }aB|^[po2[^]hefk`o$�p�f�yoxrto�� ®[�_^a¡rfrto��hijbj]^���faÎ0:� Ï � �YÒ � � �

� � � � � � �

~�|poqij]heyo � �fbuo\|^[po�zH]h��rtoq[sryaø�h�f|��8zHbjaH[sryaHk8aH[srya�c^bjzHzH]hiuoq�^�xi�� oqi*�h]hefb�ryk8]��fb�_^bjzHa£��rtoqg^bjija [^aHi��faH[^�f]_^aHijiuo�g�� aq� o��5B^�faq�pbj[s�haHzHaq� � }a�|^[po�zH]h��rtoq[srya���heyoq[^_^aH�^�^i�� oqi*�h]hefb�ryk8]��fb�_^bjzHa�bj[^��rtoqg^bjija [^aHi��faH[^�f]_^aHijiuoVg�� aq� o��H�{\aHiji�� aH�faHk8c^bj]VefaHiuoxryb��h]8oqi�zFoqijzH]hij]�_^aHijiuoV�f]qrfryeyoq�Hbj]h[^a��

Î0:�� Q = �

Ð �<Ð 5 Ð �dÐÐ �`Ò �dÐ

� ��� ��

Page 59: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# � �

XbjzH]hef_^buoqk8]Nzt�^aq��_poqiji�� oq[poqijbj�fb�_^aHipzH]h[^_^bj�Hbj]h[poqk8aH[sry]�_^aHipc^ef]hg^ijaHk`oN_^aHijiuo��f]qrfryeyoq�Hbj]h[^aq�si�� bj[^_^bjzHa_^b�zH]h[^_^bj�Hbj]h[poqk8aH[sry]�_^aHijiuoV�f]qrfryeyoq�Hbj]h[^a�}a

� Ï Ð �<Ð 5 Ð �dÐÐ �:Ò �dÐ ?

c�aHeXzH|^b��^bj[6zH]hefefbj�fc�]h[^_^aH[^�Fo2_^aHb�_poxryb�RÏ , Ó " �=���O�� Ï ��Ó " �$� ,5" 1

�fb�_^aH_^|^zHa � Ï =�Ó ,5"�" �VaH_�bji�c^ef]hg^ijaHk`oVefbj�f|^i�rto��¥¢¡ÄËt¦xÄ�ÆqÃ�ÉHÃ�¦xÄ�§q­®¦ B^oqi�zH]h[sryeyoqefbj]^��c�aHe�RÏ ��Ó " �$� ,5" 1� Ï ��Ó " �$� " � =

�fbm�po � Ï =�Ó�� , � ��� � ��zH]h��rtoq[sryaB��heyoq[^_^aH��c�aHe£zH|^b�bji»c^ef]hg^ijaHk`o`}aVÅ8§qª�Ët¦xÄ�ÆqÃ�ÉHÃ�¦xÄ�§q­®¦x� � |^bj[^_^b��c�aHeXi�� oqi*�h]hefb�ryk8]Y_^aHijiuoV�f]qrfryeyoq�Hbj]h[^aq�^_poqijiuo �y·ÿ°»·ÿ¼�·��fb�_^aH_^|^zHa��

Î0:� Q = Ð �<Ð 5 Ð �dÐÐ �`Ò �dÐ �]3���haHef]ª�� ¢¡¤¥¤y¦x¤y¢V¨¡Ã�§¥ÇsÇ�¤y¦3¨t¨¡ÃÈÅ8§wËt¦xÄ�«�Ä�§DËt¦3¨¡­ §qÄp­®¢:ËE�p¢8Æqà Çp¢¡Ä�Æ�¢8Æs§qªÈª�� §qª 4�¦x¤¥ÃÈ­�Å`¦�¢`Æs§qªCËt¦xÄ�ÆqÃM�ÉHÃ�¦xÄ�§qÅ`¢¡Äp­®¦8Æ�¢¡ª�Ç�¤y¦��Hªj¢¡Å8§%B�_poqijiuo � ·ÿ°»·ÿ¼�·��

Î0: Q = �

]3���haHef]Dª�� ¢¡¤¥¤y¦x¤y¢B¨¡Ã §¥ÇsÇ�¤y¦3¨t¨¡ÃÈÅ8§�Ët¦xÄ?«�Ä�§�Ët¦3¨¡­ §qÄp­®¢�ËE�p¢VÆqà Çp¢¡Ä�Æ�¢�¨3¦xªj¦:Æs§qªÈª�� §qª 4�¦x¤¥ÃÈ­�Å`¦x� ®[�zH]h[��zHij|^�fbj]h[^a�i�� oqi*�h]hefb�ryk8]�_^aHijiuo2�f]qrfryeyoq�Hbj]h[^a:}a���rtoqg^bjijaY[^aHiK�faH[^�f]6_^aHijiuo2g�� aq� o��Bk`o8c^|�}]`ln]hef[^bjefa�|^[efbj�f|^i�rtoxry]�bj[poqzHzHa¡rfrtoqg^bjija��fa�b�[�|^k8aHefb��f]h[^]���bjzHbj[^b���c�aHefzt���a8}a�k`oqi�zH]h[^_^bj�Hbj]h[poxry]^�®[��haH[^aHeyoqijaq�^iuoVl � aq� o��#ln]hef[^bj�fzHa��

Î0:� Q � �

]3���haHef]�ª�� ¢¡¤¥¤y¦x¤y¢Y¨¡Ã�§¥ÇsÇ�¤y¦3¨t¨¡ÃÈÅ8§�Ët¦xÄ�«�Ä�§wËt¦3¨¡­ §qÄp­®¢ � ËE�p¢2Æqà Çp¢¡Ä�Æ�¢2Æs§qªÈª�� §qª 4�¦x¤¥ÃÈ­�Å`¦6¢8Æs§qª<Ët¦xÄ �ÆqÃ�ÉHÃ�¦xÄ�§qÅ`¢¡Äp­®¦8Æ�¢¡ª�Ç�¤y¦��Hªj¢¡Å8§%B�c�aHeXiuo � ·ÿ°»·ÿ¼�·n�»bj[s�haHzHaq�

Î0: Q � : �]3���haHef]wª�� ¢¡¤¥¤y¦x¤y¢�¨¡Ã §¥ÇsÇ�¤y¦3¨t¨¡ÃÈÅ8§:Ët¦xÄ?«�Ä�§�Ët¦3¨¡­ §qÄp­®¢ � : ËE�p¢VÆqà Çp¢¡Ä�Æ�¢N¨3¦xªj¦`Æs§qªÈª�� §qª 4�¦x¤¥ÃÈ­�Å`¦x�Ô buoq[^]

� Ï _poxryb bj[^bj�Hbuoqijb� ; ��? Ï �f]hij|^�Hbj]h[^a�aH�yoxrfrto

�� Ï _poxryb c�aHe�ry|^efgpoxryb�� ; ��? Ï �f]hij|^�Hbj]h[^a�zFoqijzH]hiuoxrto

Page 60: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

���

� � �� ; ��?c^efaHzHbj�fbj]h[^a�bj[Q�p[^b�rto

� � � � � � � � � � � � � ���

c e a z b � b ] [ a l b [ b r o �� ; ��?

��°�º©º©µ»º©°D½¡À©³���¼�À3µ�d¼p¾�¾n¼ �y·ÿ°»·ÿ¼�·

�� �c^efaHzHbj�fbj]h[^a�bj[Q�p[^b�rto�

°�º©º©µ»º©°D½¡À©³���¼�À3µ�d¼p¾�¾n¼ � ·ÿ°»·ÿ¼�·

� b*�h|^eyo$�h� , �F�<efef]hefbp��rybjk`oxryb�zH]h[8iuoNln]he �øoqef_8aHefef]heCoq[poqi*��fbj�<aXzH]h[8iuoBgpoqz1���øoqef_`aHefef]heCoq[poqi*��fbj�H�

� oqijaq��_^|^[^~�|^aq��ij]��fzt�^aHk`o2�fa �h|^aH[srya��Î : ���� �� Ï Î :��� ��� T �øoqz1���øoqef_ aHefef]he oq[poqi*��fbj� �

oq[poqijbj�fb/_^aHiji���oqi*�h]hefb�ryk8]Î :��� ��� Ï ��Î : � T Ô rybjk`o�_^aHi(zH]h[^_^bj�Hbj]h[poqk8aH[sry] �

oq[poqijbj�fb/_^aHi(c^ef]hg^ijaHk`oÎ : � Ï � � T efbj�f|^i�rtoxry]�_^aHijiuoVg�� aq� o��

� ��� �Î : ��� �� Ï � � �

zH]h[�R_^bjc�aH[^_^aH[srya _poqi(c^ef]hg^ijaHk`o ; bj[^_^bjzHa _^b(zH]h[^_��8_^aHi(c^ef]hg^i��B?� _^bjc�aH[^_^aH[srya _poqiji���oqi*�h]hefb�ryk8] ; l�oxrfry]hefa�_^b(oqk8c^ijb�l � _^aHiji�� aHefef]hefa�?� _^bjc�aH[^_^aH[srya _poqijiuoOc^efaHzHbj�fbj]h[^a�_^aHi(�fbj��ryaHk`o ; k`o7� oqzHzq�$efaHi��B?

Ô a�bji£c^ef]hg^ijaHk`o�}a�g�aH[>zH]h[^_^bj�Hbj]h[poxry] ; � � � ?�i�� aHefef]hefa:�f|^b�_poxrybø[^]h[*��bjaH[^a6oqk8c^ijb��pzFoxry]waH_«�Ä>§qª 4�¦x¤¥ÃÈ­�Å`¦`¨¡­ § �HÃȪj¢2Äm¢¡ª�¨3¢¡Ä�¨3¦�Æ�¢¡ªÈª�§ � �n¢ �ȧ � � � � �� �F¦x¤¥ÄpÃj¨3Ët¢2«�Ä�¤¥Ãj¨¡«�ª¬­ §q­®¦D§�ËtËt¢¡­�­ § �HÃȪj¢QBd�fabjidc^ef]hg^ijaHk`ow}aVk`oqi�zH]h[^_^bj�Hbj]h[poxry] ; � . � ?Xi�� aHefef]hefaV�f|^bK_poxryb���bjaH[^a�oqk8c^ijb��pzFoxry]8aH_ §qÄmËE�p¢V«�ħqª 4�¦x¤¥ÃÈ­�Å`¦V¨¡­ § �HÃȪj¢�Äm¢¡ª�¨3¢¡Ä�¨3¦`Æ�¢¡ªÈª�§D� �n¢ �ȧ � � � ���� (�F¦x¤¥ÄpÃj¨3Ët¢�«�Ä?¤¥Ãj¨¡«�ª¬­ §q­®¦8ÃÈÄ�§�ËtËt¢¡­�­ § �HÃȪj¢¡�Z\[poDeyoqc^c^efaH�faH[srtoq�Hbj]h[^a �heyo*�pzFoD_^a �hijb£aHefef]hefbø��rybjk`oxryb _poqijiuo6l � aq� o?a�_poqijiuo�g�� aq� o?aq� c^by}|�bj[

�haH[^aHeyoqijaq��_^aHijija\efaHiuoq�Hbj]h[^b»zt�^aBbj[sryaHefzH]hefef]h[^]Yryeyo�b�_^|^a�k8a¡ry]�_^b�_^bmoq[poqijbj�fb»_^aHiji�� aHefef]hefaq��}a\ln]hef[^b�rtobj[ � b*�h|^eyo �h� , �

ÖP×KØfÙFÚ�Û^Ü�Ý2ÞqßBA ��ß#àFáçúøú¡å�å�è �¡äxútû�ã ý©äBå�áçånû�è�ü�úøú¥é�á û�üøè û�áçâ ã£þnõ»ù�õ� : N�A�ë?ìfí < �^ë6ô < ��� ��� ë�ï , �� ��� �S 6 ! � 6� � � � & �

ëRL�P <

Page 61: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# � ,

á äBâ ý©á � ë6í©î ó1A � � �^ë6í©î óUM�ìfí���tõ�àFè#ê�ë6ô©î ñFó¥ôCé�áçå�ý©÷ û úê ë ��C,Nÿê�P»ë6í©î ô¥ñFó4M�ìfí � ë�ô©î ñFó3î

Cmú¥÷çâ ã¡÷çú¥é�èKá ÷�)Hýxú¡æ3é útû�ãøæ3á�ê�õ �Èä�ú¥é�á û�üøè û�áçâ�ú "G ��K K �Kê��Kë6ô©î ñFó¥ô M�ô©î ñFó¥ô<ë?ì¡ì¡î ðHï¥ôHï¥í¡ð©õ��Èä � #3á ä&-¡èfâ è�#. ë�ê ��ê ë�í©î ô¥ñFó M�ìfí �2í©î ô¥ñFó M�ìfíCë6í©î ì¡ìfð M�ìfí � ëDì¡ì¡î ð

� è�é�ånû�ýxæ3áçú¥é�è �¡÷ ásè�é�é�ã¡é�áhá äHû�é�ã3æ3ã¥û�û�á�ä©è�÷�â�ú¥÷çâ ã¡÷ ãøæ3á�ê � á ä � #©å�á�é�áçâ ã¡é æ©ú£â('©è�#3÷*) è�é�é�ã¡é�è#å�ý©÷�æ©útû�ã 'è#á ÷�å�è �¡ý©è�äHû�èQ:� � ë ����

ê ! ��C,Nÿê�Pê����ë

ë ����í©î ô¥ñFó¥ô M�ìfí � ! í©î ô¥ñFó4M�ìfí �í©î ô¥ñFó¥ô M�ìfí �

����ë

ë í©î < M�ìfí �%3æ©úøâ ý©á*#

��C,Nÿê�P»ë�ê+Nnì�� � � P���í©î ô¥ñFó M�ìfíCë6ô©î ñFó¥ô Nnì �2í©î <UM�ìfí �%3 P Nnì¡õ ñFïQPüøè�äHû�é�è�#©æ©ú¥÷ ÷*) èfå�å�è�é�è

. ë ��C,N ��C,Nÿê�P"M ��C,Nÿê�P,P�ë�í©î ô¥ñFó M�ìfí �2í©î ô¥ñFó M�ìfíCë6í©î ì¡ìfð M�ìfí � Nnì¡õ ñHô�P÷*) è�é�é�ã¡é�è#å�ý©÷ ÷çú£ã¡ùqè�é ú&%�á ã¡ä©èKé�áçå�ý©÷ û ú�:

� � ë ����. ! . .

����ë

ë í©î ì¡ìfðHï¥ôHï¥í¡ðDM�ìfí � ! í©î ì¡ìfð M�ìfí �í©î ì¡ìfðHï¥ôHï¥í¡ðDM�ìfí � ë�í©î ïUM�ìfí ��æ©úøâ ý©á :

. ë .�Nnì � � � P���í©î ì¡ìfðUM�ìfí � ë N�í©î ì¡ìfðHï¥ôHï¥í¡ð4M�ìfí � P Nnì��2í©î ï M�ìfí �� P Nnì¡õ ñ¡ñ PKú¥÷ ÷çú Nnì¡õ ñHô�P�#Fû�è�ä©è�äxæ3ã�ù©é�èfå�è�äHû�è#÷ è Nnì¡õ ñFïQP�è Nnì¡õ ñ¡ñ P�#3èfå�è �¡ý©è�äxæ3ã£á�â�ú¥÷çâ ã¡÷ áhá ä�ù©é�èfâ áçå�á ã¡ä©è#á ä��xä©á û ú�#xå�ásã¥û�û�á è�ä©èQ:

ë ô©î ñFó¥ô Nnì!�2í©î <UM�ìfí �%3 P �2ô©î ñFó¥ô Nnì!�2í©î <UM�ìfí �%3 P�ëë 2 ô©î ñFó¥ô Nnì!�2í©î <UM�ìfí �%3 P"M�ô©î ñFó¥ô Nnì!�2í©î <UM�ìfí �%3 P 4 M Nnì��2í©î ï M�ìfí �� P�ëë ì¡ì¡î ðHï¥ôHï¥í¡ð Nnì �2í©î <UM�ìfí �%3 P �

� ��� �! � � ;�� � �Nnì �2í©î ïUM�ìfí �� P� ��� � � ;����

àFãHånû ú¥ä�%�áçú¥÷ üøè�äHû�è#é�áçå�ý©÷ û ú�:.� dë:ê� ��ê� �ë�ê � Nnì�� � � P � Nnì�� � � P â ã¡ä � � � � E � < � � � � E �

� áP'ýNá äBæ3è û�û ú&�¡÷ á ã :. ë ì¡ì¡î ðHï¥ôHï¥í¡ð Nnì��2í©î < M�ìfí �%3 P � Nnì��2í©î ï M�ìfí �� P�ë

ë Nnì¡ì¡î ðHï¥ôHï¥í¡ð�P �� Nnì �2í©î < M�ìfí �%3 P Nnì �2í©î ïUM�ìfí �� P �� MM Nnì¡ì¡î ðHï¥ôHï¥í¡ð�P �� Nnì �2í©î < M�ìfí �%3 P Nnì �2í©î ïUM�ìfí �� P ��

èD)Hý©èfånû ) ý©÷ û�á üøã 'è#á ÷�é�áçå�ý©÷ û útû�ãøæ3è�÷ ÷çúøüøã¡÷ û�á ù©÷ áçâ�ú&%�á ã¡ä©è#èfå�è �¡ý©á û ú á äBú¥é�á û�üøè û�áçâ�úøú£ù©é�èfâ áçå�á ã¡ä©è#á ä��xä©á û ú�:. ë 2 ê+Nnì�� � � P : ì � � � 4� ��� �! M 2 ê+Nnì � � � P : ì � � � 4� ��� �!

Page 62: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

� �

C�ã¡ä£á ÷3ù©é�ã3â èfæ3á üøè�äHû�ã<æ3èfå�â é�á û�û�ã #&�¡÷ á3è�é�é�ã¡é�áFá äHû�é�ã3æ3ã¥û�û�á©æ©ú¥÷ ÷çúKüøã¡÷ û�á ù©÷ áçâ�ú&%�á ã¡ä©èmèfå�è �¡ý©á û ú#úKù©é�èfâ áçå�á ã¡ä©è �xä©á û ú<å�ã¡ä©ãé�áçâ ã¡äxæ3ã¥û�û�á^ú¡æBè�é�é�ã¡é�á�å�ýYê�õ � è�éKèfå�è �¡ý©á é�èCý©äxúNånû�á ü�úNæ3è�÷ ÷çúXùqè�é�û�ý©é �xú&%�á ã¡ä©è é�áçâ ã¡äxæ3ã¥û�û ú\å�ý©÷pæ©útû�ã\á ä©á %�áçú¥÷ è£êVå�áù©ýQ'ã�â�ú¥÷çâ ã¡÷çú¥é�è�÷çú�æ3áçånû ú¥ä�%fú£é�è�÷çútû�á -túCû�é ú£ê�è#ê #3ãE-&-¡è�é�ã ÷*) è�é�é�ã¡é�èKé�è�÷çútû�á -¡ãøå�ý�ê�:� I! ë � ê ! ê �� ê � ë � ê+Nnì�� � � P @ ì � � � ! ê �� ê �

E � : ì � � � ! ì&�)� � � � : ì � � � � EE ï � � � � ë�ï � ����N � � P J ï �æ©ú\â ý©á^å�ápæ3èfæ3ýxâ è£â('©è ÷*) è�é�é�ã¡é�èCé�è�÷çútû�á -¡ãXá äYê 'è ü�ú&���¡á ã¡é útû�ã #xúXüøè�ä©ã�æ3á ��N � �RP�#�æ©úNï � Vså�ãHånû ú¥ä�%�áçú¥÷ üøè�äHû�èCê 'èý©ä+) ú¥ù©ù©é�ãHå�å�á ü�ú&%�á ã¡ä©èKæ3ásê�â('©è#æ3áçånû úøæ©ú£ê�æ3ásý©äxú )Hýxú¥äHû�á ûE'úøæ3è�÷ ÷ ã�ånû�èfå�å�ã£ã¡é æ3á ä©è#æ3á+�¡é ú¥äxæ3è % %fú æ3è�÷ ÷çú£ü�ú¡å�å�á ü�úú¡â�â ý©é útû�è % %fúCé�è�÷çútû�á -tú�#©â á ã*'èQ:� I! J ( I � â ã¡ä ( I ë6ï

ãE-&-¡è�é�ã�ê G",; I T �.P�P�F.>RG�G�� < �I��*>�I" � ��ê �QC�C(" K K �����! "U �Èä )Hý©èfånû�ãNâ�ú¡å�ã\÷çú\ùqè�é�û�ý©é �xú&%�á ã¡ä©èCá äHû�é�ã3æ3ã¥û�û úNæ©ú¥÷ ÷ è ã¡ùqèYé ú&%�á ã¡ä©ámèfå�è �¡ý©á û�è�æ©ú¥÷ ÷*) ú¥÷ �¡ã¡é�á û�üøã #pèNùqã¡á�é�áçâ ã¡äxæ3ã¥û�û úYå�ý©÷dæ©útû�ã�á ä©á %�áçú¥÷ è�# 'è�æ3è�÷ ÷ ã�ånû�èfå�å�ãYã¡é æ3á ä©è�æ3á �¡é ú¥äxæ3è % %fúæ3è�÷ ÷*) è�é�é�ã¡é�è<æ3á�é�ã¡ý©äxæ3ã 1Vå�ý©÷�æ©útû�ãXá ä©á %�áçú¥÷ è�#qùqè�éKâ ý©á ��� 6��*3qÝ5/tÜ�fÚ2Ý�Û�Ù'/NÜ�����6����¡Ý ��Ý 0�Ù�����8670�/16��fÝ �Ù�Ø �6��^Ü���Ù(�Ù�mØfÙ�(�ØfÝ 0�Ù���*6��mß Ùqß 6sß�õ Ö

ÖP×KØfÙFÚ�Û^Ü�Ý2ÞqßBA ��ß#àFáçú¥ä©ãøú¡å�å�è �¡äxútû�áhê #�# � � � #xæ3ãE-¡è R'è#ý©äNã¡ù©ùqã¡é�û�ý©ä©ã�å�áçånû�è�ü�úøú¥é�á û�üøè û�áçâ ã þnõ ù�õ #©ãE-&-¡è�é�ãå�áçú¥ä©ãêIë ��C,Nÿê�P»ë:ê+Nnì�� � � P�-ë ��C,N ��Ppë � Nnì � � � P��ë ��C,N� Ppë �%Nnì � � 2 Pâ ã¡ä � � ë6í < �pë?ì < ï < ô©õ�Cmú¥÷çâ ã¡÷çú¥é�è�#Fá ä � # ò3 ë�ê � ��� �,dè�÷�å�áçånû�è�ü�ú�ú¥é�á û�üøè û�áçâ ã � #»å�á�æ3è/-¡ã¡ä©ãBèfå�è �¡ý©á é�è�#pæ3ý©ä*)Hý©è�#pý©äxú�< > K*� P �5C(�I��*>�I" EJU P UVèfæ�ý©ä+)!����� � I��*>�I" EJU P U õ� ãHånû�ã

ò� ë � �%Nnì � � � P <á ÷�â�ú¥÷çâ ã¡÷ ãøæ3á�ò3 å�ásù©ýQ'ã�é�áçâ ã¡äxæ3ý©é�é�è#ú¥÷ ÷*) ã¡ùqè�é ú&%�á ã¡ä©èdþnõ ù�õ

ò3 ë�ê � ò� ë Nÿê �Õò� P Nnì�� � � Pè�# )Hý©á äxæ3á*#ò3 ë Nÿê � N � �%Nnì � � � P,P,P Nnì�� � � P

. ù©ù©÷ áçâ�ú¥äxæ3ãø÷çú ��õ è¡õ ú©õ»å�á 'xúò3 ë Nÿê � N � �%Nnì � � � P,P,P Nnì�� � � P�ë Nÿê � � � � � � � � P Nnì�� � � P»ëë ê � � � � � � � � ��ê � � � � � � � � � � � � � � ëë ê ��ê � � � � � � � � � � � � � � � � � � � � � � ëë ê+Nnì�� � � P � � Nnì � � � P �%Nnì � � � P Nnì¡õ ñFóQP

. ÷ ÷ ã¡é úNò3 å�ásù©ýQ'ãXâ ã¡äxå�áçæ3è�é ú¥é�èKâ ã¡üøè#á ÷�é�áçå�ý©÷ û útû�ã�"G ��K KM>#æ3è�÷ ÷ è#ã¡ùqè�é ú&%�á ã¡ä©ásèfå�è �¡ý©á û�èKû�é úê�ë:ê+Nnì�� � � P < ��ë � Nnì � � � P è �£ë �%Nnì � � � P®î

Ö

Page 63: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# � 1

021�� � JK@BT Q?;H5 =?;/J�9<L?=?;H5��OAC;FJK58EFLDMN;H587D@ =?; L?7 �@���LDG�MN;H587D@=?; JK@BWX587?=D5 �8A<GV=D5

Ô b�zH]h[^�fbj_^aHefb�iuo �haH[^aHefbjzFo�aH~�|poq�Hbj]h[^aY_^b��faHzH]h[^_^] �heyoq_^] �8h� � 5 �y� 5 � Ï�� ? ; �h� " �@?

_^]3�haq�<c�aHeY�faHk8c^ijbjzHb�rx}o��d�fb �f|^c^c�]h[ �h]h[^] 8p� ��a � efaFoqijb�������[^]qry]wzt�^aq�#�fa 8 <Ï ���Kija8eyoq_^bjzHbC_^b; �h� " �@?£�poq[^[^]2iuo��fa �h|^aH[sryaYaH�fc^efaH�f�fbj]h[^a��

� � Ï Ò � 5�� � � Ò " 8 �= 8 ? ; �h� "+, ?

� � Ï Ò �£Ò � � � Ò " 8 �= 8 Ó ; �h� " � ?

���p[poxry|^eyoqijaB�fzHefb��haHefa�|^[woqi*�h]hefb�ryk8]�efbj�f]hij|�ryb��h]�zt�^a�oqc^c^ijbjzt�^b�_^bjefa¡rfrtoqk8aH[srya ; �h� "+, ?£a ; �h� " � ?&�

���� �çÏ�Ò ����çÏ�� �� �çÏ " �-8 �çÏ � � �� �çÏ �VÒ � �çÏ � �� � �çÏ � 5 �� � �çÏ ��Ò �� �çÏ = � 8� � �çÏ � � � �� � �çÏ � � � ����

' (*),+.-./01(325416 8:9<; =>)1(@? A#B�)DC>-E(�; 2&(@? K )H; 0�� ? )HML- /H? 01M J -� 0E2�� ? )HML- /H?,K -.+.)HML/1)N=H(32L/1)

� bj�f�yoxry]bjiø�fbj��ryaHk`ooqefb�ryk8a¡rybjzH]wl � c�� zH]h[ � ; ���+? � ? � �+?FÒ � �@?¥�Cb�~�|poxrfryef]�aH�faHk8c^bX�fa �h|^aH[srybk8]h��ryeyoq[^]8i�� oqc^c^ijbjzFoq�Hbj]h[^aB_^aHiji�� oqi*�h]hefb�ryk8] �8oqijiuo�efbj�f]hij|^�Hbj]h[^aB_^b�oqijzH|^[^aYaH~�|poq�Hbj]h[^b�_^bd�faHzH]h[^_^]�heyoq_^]^��h�g�x� � 5 �©�`Ò " Ï������a�eyoq_^bjzHb��f]h[^] �

� � Ï>��Ó ��? � � Ï/ÒG�hÓ ��������������ÓFÓFÓ�?

Page 64: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

, �

~�|^aHijijaB]qrfryaH[�|�ryaYzH]hk8aBefbj�f|^i�rtoxry]�_^aHiji�� oqi*�h]hefb�ryk8] �V�f]h[^] �� � Ï>��Ó � �O�O�O�O�O�O� ? � � Ï/ÒG�hÓ ������������� B

bj[6~�|^aH��ry]`zFoq�f]8i�� oqi*�h]hefb�ryk8] ��ln]hef[^bj�fzHa�iuoVk`oq�f�fbjk`o�oqzHzH|^eyoxryaH�H�Fo8_^aHi�efbj�f|^i�rtoxry]^�=��ø��� Ò ��� 3 � 5 �NÏ�����<� aH~�|poq�Hbj]h[^aY�po�ijaB�fa �h|^aH[srybdeyoq_^bjzHb���eyoqc^c^efaH�faH[srtoxryaYzH]h[�|^[^_^bjzHb�zHb�lnefaB�fb*�h[^b��pzFoxryb��ha��

� � Ï 1�1�1�1�1 Ó 1�1�1�1�1 � ? � � Ï>��Ó �O�O�O�Q���O�O�O�O�O�O�O�O�Q�CÓ�<� aH�faHzH|^�Hbj]h[^aY_^aHiji�� oqi*�h]hefb�ryk8] �Vln]hef[^bj�fzHa��

�YÏ/Ò ��Ï���Ó����O�O�O�O�O�O� � ��� ��2Ï��E�øÏ���Ó����O�O�O�O�O�O� � ��� �®�� Ï " � 8�Ï>��Ó " �O�O�O�O�O�O� � ����� Ï � � � Ï���Ó " �O�O�O�O�O�O� � ������:Ï ��Ò Ï���Ó����O�O�O�O�O�O� � �����®�#Ò=��Ó " �O�O�O�O�O�O� � �����CÏ

Ï���Ó����O�O�O�O�O�O� � �����®���Ï � ��Ï���Ó����O�O�O�O�O�O� � ��� �� � Ï � 5 �VÏ���Ó>= �O�O�O�O�O�O� � ��� � B � � Ï ��Ò �VÏ���YÏ = � 8�Ï>��Ó>= �O�O�O�O�O�O� � ������ � Ï � � � ��Ï>��Ó����O�O�O�O�O�O� � ��� � B � � Ï � � � ��Ï��

� ijb�aHefef]hefb�efaHiuoxryb���bm_^aHijijaB�f]hij|^�Hbj]h[^b�zFoqijzH]hiuoxrya�_poqiji�� oqi*�h]hefb�ryk8]��f]h[^] �Î0:� Ï

Ð � � Ò � � ÐÐ � � ÐQ ����� � ? Î0:� Ï

Ð � � Ò � � ÐÐ � � ÐÏ �CÓ

��|^[^~�|^aNi�� oqi*�h]hefb�ryk8]\ln]hef[^bj�fzHaX|^[�� ]qrfrybjk`o�oqc^c^ef]h�f�fbjk`oq�Hbj]h[^a�_^bp� � ; k`oq�f�fbjk`oBoqzHzH|^eyoxryaH�¥��Fo@?¥�mk`o�ln]hef[^bj�fzHa�|^[�� oqc^c^ef]h�f�fbjk`oq�Hbj]h[^aY_^bd� � o Ama¡rfrto`_po8|^[DaHefef]hefa�efaHiuoxryb��h]`_^aHi ���O���:��£b }]`}a\_^]3��|�ry]�oqipl�oxrfry]Yzt�^aN[^aHi�zFoqijzH]hij]�_^b ����bj[`oqefb�ryk8a¡rybjzFo�oYc^efaHzHbj�fbj]h[^a(�p[^b�rto����fb�c�aHef_^azH]hk8c^ija¡rtoqk8aH[srya\bji»zH]h[sryefbjg^|�ry]V_^b aq��~�|^bj[^_^b���bji»zFoqijzH]hij]Y_^b � � _^b��haH[srto�|^[po��f]qrfryeyoq�Hbj]h[^aryeyo2_^|^a�[�|^k8aHefb�| �h|poqijb ; lnaH[^]hk8aH[^]8_^b�zFoq[^zHaHijiuoq�Hbj]h[^a�?¥�

���ø��� Ò " � 5 ��Ó 1�1�1�1�1�1�1 Ï������a�eyoq_^bjzHb���eyoqc^c^efaH�faH[srtoxrya�zH]h[6]qrfry]2zHb�lnefaB�fb*�h[^b��pzFoxryb��haq�^�f]h[^] �

� � Ï =�Ó �O�O���Q�=� =�? � � Ï�ÒG�hÓ 1�1�1 � � � � Ó3Nc^c^ijbjzFoq[^_^]2i�� oqi*�h]hefb�ryk8] �V�fb��po$�

�YÏ/Ò ��Ï���Ó " �O�O�O�O�O�O� � ������2Ï��E�øÏ���Ó��=�O�O�O�O�O�O� � ��� �� Ï " � 8�Ï>��Ó " �O�O�O�O�O�O� � �����

Page 65: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# , �

Ï � � � Ï ; ��Ó " �O�O�O�O�O�O� � ������?�� ; ��Ó � 1�1�1�1�1�1�1 � ������?<ÏÏ���Ó��=�O�O�O�O�O�O� � ��� �

��Ï ��Ò Ï���VÏ � �:Ï��� � Ï � 5 ��Ï � � Ï ��Ò �VÏ>��Ó " �O�O�O�O�O�O� � �������Ï = �-8�Ï���Ó>= �O�O�O�O�O�O� � ��� �� � Ï � � � ��Ï � � Ï � � � ��Ï���Ó>= �O�O�O�O�O�O� � �����

�<� oqi*�h]hefb�ryk8]VzFoqijzH]hiuo�|^[po8eyoq_^bjzHa�_^]hc^c^buo��»bj[s�haHzHa�_^bd_^|^aYeyoq_^bjzHb��faHk8c^ijbjzHb��pa�rtoqija�eyoq_^bjzHa�po " zHb�lnefa6_^aHzHbjk`oqijb£bj[>zH]hkV|^[^a�zH]h[>� � �£zHbj]^}a��f]hij]�k8a¡rx}o�_^aHijiuoDc^efaHzHbj�fbj]h[^a6_^b �2� ®[~�|^aH��ry]:zFoq�f]8iuo2_^b��8zH]hi�rx}o�efbj�fbjaH_^a�[^aHi�l�oxrfry]8zt�^aYc�aHe\eyoqc^c^efaH�faH[srtoqefa�iuo2k`oq[srybj�f�yo2_^b �fb�poq[^[^]8o2_^bj�fc�]h�fbj�Hbj]h[^a��f]hij] � zHb�lnefaq�

" �0��� � � � Ò � � ��� � � 5 � � ��� � Ï>�����a�eyoq_^bjzHb�aH�yoxrfrya�_^aHiji�� aH~�|poq�Hbj]h[^aB�f]h[^] �

� � Ï � ? � � Ï =�B

oqc^c^ijbjzFoq[^_^]�i�� oqi*�h]hefb�ryk8] �V�fbd�po$���Ï/Ò ��Ï���Ó � �O�O�O�O�O�O� � ��� � ��2Ï�� � Ï���Ó>= � �O�O�O�O�O� � ��� � � . ( �P��� � Ï ��� 3 ���

zHbj]^}aV�fb#�haHefb��pzFo�|^[]3�haHe)�p]�����3N[^zt�^a8�fa�b#zH]q�8zHbjaH[sryb<a�ija�eyoq_^bjzHbK�f]h[^]6eyoqc^c^efaH�faH[srtoqg^bjijbbj[ �2��i�� oqi*�h]hefb�ryk8]`c^ef]�_^|^zHa8iuo��fb�ry|poq�Hbj]h[^a2aHzHzHaH�Hbj]h[poqija2_^bC]3�haHe)�p]����#[^]h[�ln]hef[^aH[^_^]Doqiÿ�zH|^[po�bj[�ln]hefk`oq�Hbj]h[^a\�f|^ijiuo��f]hij|^�Hbj]h[^aq� ®[ �haH[^aHeyoqijaq�^_poxry]�zt�^a�ijaNln]hefkV|^ija ; �h� "+, ? a ; �h� " � ?efbjzt�^bjaH_^]h[^]�bjipzFoqijzH]hij]Y_^b��E�øa " 8 � ���fbmc^|�}]Y�haHefb��pzFoqefa�]3�haHe)�p]��Õ]�|^[^_^aHe)�p]���oq[^zt�^a�~�|poq[��_^]bXzH]�a¡�8zHbjaH[sryb\a�ija6eyoq_^bjzHb��f]h[^]^�øbj[Pk8]�_^|^ij]^����k8]hi�ry]�_^bj��rtoq[srybj��_po)( �P��� � a ( �P�� �efbj�fc�a¡rfryb��qoqk8aH[sryaq�

��eyoq[^[^aXzt�^a�[^aHipzFoq�f]��h�si�� oqc^c^ijbjzFoq�Hbj]h[^a _^bjefa¡rfrto�_^aHijija£ln]hefkV|^ija ; �h� "+, ?�a ; �h� " � ?d[^]h[�ln]hef[^bj�fzHaefbj�f|^i�rtoxrybm�f]�_^_^bj��l�oqzHaH[sryb���c^efbj[^zHbjcpoqijbqefaH~�|^bj�fb�rybs_^b�|^[D��g^|^]h[^��oqi*�h]hefb�ryk8]^�Hc�aHe�bjihzFoqijzH]hij]X_^aHijija<eyoq_^bjzHbs_^b�|^[Yc�]hijbj[^]hk8bj]

_^b��faHzH]h[^_^] �heyoq_^]�8h� � 5 �y� 5 � �^�f]h[^]8b��fa �h|^aH[sryb �t�X�fa 8�Ï���Ï � Ï>���i�� oqi*�h]hefb�ryk8]N_^a¡�haN�fa �h[poqiuoqefaXzt�^aNi�� aH~�|poq�Hbj]h[^aY}aX�f]�_^_^bj��l�oxrfrto�_po�]�h[^b»[�|^k8aHef]YzH]hk8c^ijaH�f�f]�DB

t�X�fa 8�Ï���Ï>�2a � <Ï����_^a¡�haY�fa �h[poqiuoqefaBzt�^aYi�� aH~�|poq�Hbj]h[^aB[^]h[woqk8k8a¡rfrya��f]hij|^�Hbj]h[^b B

Page 66: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

, =

t���fa 8�Ï>��a �$<Ï����_^a¡�ha��fa �h[poqiuoqefa8zt�^a:i�� aH~�|poq�Hbj]h[^aw}a`_^b c^efbjk8]��heyoq_^]wa:zFoqijzH]hiuoqef[^a2iuo��f]hij|^�Hbj]h[^a8zH]h[�iuoln]hefkV|^iuo��6Ï/Ò � � � B

�V�X�fa 8 <Ï>���_^a¡�ha`zFoqijzH]hiuoqefa�ija�eyoq_^bjzHb<zH]h[|^[�� oqzHzH|^eyoxryaH�H�FoDoq_^a �h|poxrto ; a¡��b�rtoq[^_^]^��~�|^bj[^_^b���zFoq[^zHaHijiuo©��Hbj]h[^b<a8�fb�ry|poq�Hbj]h[^b<_^bC]3�haHe)�p]��ÑaH_�|^[^_^aHe)�p]���[^aHijija�]hc�aHeyoq�Hbj]h[^b6?¥�d�fa8~�|^aH��rya:�f]h[^]�eyoqc��c^efaH�faH[srtoqg^bjijb�[^aHiK�fbj��ryaHk`o:oqefb�ryk8a¡rybjzH]�l � c���|�rybjijbj�H�Foxry]2aV_^a¡�haV�fa �h[poqiuoqefa�bji��haHefb��pzFoqef�fb#_^b�fb�ry|poq�Hbj]h[^b�aHzHzHaH�Hbj]h[poqijb�bj[6zFoq�f]8zH]h[sryeyoqefbj] 3 � �

��]6����bjij|^c^c�]�_^b<|^[oqi*�h]hefb�ryk8]`zt�^a8oqg^g^buo:ry|�rfryb#b#efaH~�|^bj�fb�ryb ��� �Ñ[^]h[P}a2o A�oxrfry]��faHk8c^ijbjzHaq���eyoqiuoq�fzHbuoq[^_^]�bjipefaH~�|^bj�fb�ry]�_^aHiji�� oqzHzH|^eyoxryaH�H�FoYbj[ �V�»}a\c�]h�f�fbjg^bjijaøry|�rfrtoF��buo�|�rybjijbj�H�Foqefa2���faHk8c^ijbjzHbj�oqzHzH]he �hbjk8aH[srybpc�aHe �f|^c�aHeyoqefa�oqijzH|^[^a\_^b��8zH]hi�rx}oBzH]hk8a\~�|^aHijijaXbj[^zH]h[sryeyoxrya\[^aHb�c^efaHzHaH_^aH[srybmaH�faHk8c^b=3� " ���aBln]hefkV|^ija ; �h� "+, ?£a ; �h� " � ?£c�]h�f�f]h[^]8aH�f�faHefaYefbj�fzHefb�rfryaq�^eyoq�Hbj]h[poqijbj�H�Foq[^_^]Vb�[�|^k8aHeyoxry]hefb���[^aHik8]�_^]��fa �h|^aH[srya��

� � Ï = �Ò �£Ò � � � Ò " 8 �

? ; �h� " 1 ?� � Ï = �

Ò � 5�� � � Ò " 8 �Ó ; �h� � �@?

Ô a�[^aHiji�� aH�faHk8c^bj]�=��fb�|�rybjijbj�H�Fo ; �h� � �@?£c�aHeXzFoqijzH]hiuoqefaN� � �fb�]qrfrybjaH[^a��� �� Ï

��Ó>= �O�O�O�O�O�O� � �������Ó����O�O�O�O�O�O� � ��� � 5 ��Ó����O�O�O�O�O�O� � ��� � Ï>��Ó����O�O�O�O�O�O� � ��� � � ?

zHbj]^}aV�fb#zFoqijzH]hiuo`|^[?�qoqij]hefaV_^bK� � zH]h[?ry|�rfrya�ija � zHb�lnefaV_^aHijiuo`k`oq[srybj�f�yo:zH]hefefa¡rfryaq� ®[ �haH[^aHeyoqijaq��f|^c^c�]h��ry]��$<Ï��2a � � Ò " 8 � . ���p�fa � � �2ija�ln]hefkV|^ija ; �h� � �@?£a ; �h� "+, ?£a¡��b�rtoq[^]8iuoV�f]qrfryeyoq�Hbj]h[^aÒ �KÒ � � � Ò " 8 � ��zt�^aBc^|�}]VaH�f�faHefaBzFoq|^�yoV_^b�|^[:lnaH[^]hk8aH[^]V_^b�zFoq[^zHaHijiuoq�Hbj]h[^a%B�oq[poqij]�h]�_^bj�fzH]hef�f]�qoqija�c�aHeXija�ln]hefkV|^ija ; �h� " � ?£a ; �h� " 1 ?£[^aHi�zFoq�f]��1. ���Z\[8oqi�ryef]Bk8]�_^]Bc�aHeCa¡��b�rtoqefa�|^[8a¡�haH[sry|poqija�lnaH[^]hk8aH[^]�_^bpzFoq[^zHaHijiuoq�Hbj]h[^aøzH]h[^�fbj��ryaX[^aHipzFoqijzH]x�iuoqefa�|^[po2_^aHijijaBeyoq_^bjzHb�zH]h[ ; �h� "+, ? ] ; �h� " � ?¥�po��faHzH]h[^_po8zt�^a��fbuo � � �2] �1. ���^a�[^aHi�zFoqijzH]hiuoqefai�� oqi�ryeyo�k8aH_^buoq[sryaBiuoVefaHiuoq�Hbj]h[^a

� � � � Ï � � 88Ó ; �h� ��� ?ZXrybjijbj�H�Foq[^_^] ; �h� ��� ?Cc�aHeøzFoqijzH]hiuoqefa\� � ��[^aHiji�� aH�faHk8c^bj] =����fbm�poVij]V��ryaH�f�f]��qoqij]hefaN]qrfryaH[�|�ry]�zH]h[�iuoln]hefkV|^iuo ; �h� � �@?&�

���� ������O�O�O�O�O�O� � �����

; ��Ó����O�O�O�O�O�O� � ��� � ?�� ; ��Ó����O�O�O�O�O�O� � ��� � ? Ï���Ó����O�O�O�O�O�O� � ����� � ÓN �)�däxú ù©é�èfâ áçå�ú£å�ùqèfâ á �qâ�ú&%�á ã¡ä©èKæ3áhû ú¥÷ èKé�è�)Hý©áçå�á û�ãK'è#æ3áçå�â ýxå�å�ú á ä 2 ó 4 #©æ3ãE-¡èKå�áhé�áçâ('©á èfæ3è#ú¡æXèfå�è�üøù©á ãøâ('©è�#3ý3û�á ÷ á %Y

%fú¥äxæ3ã<ý©äXå�áçånû�è�ü�úCú¥é�á û�üøè û�áçâ ã#þnõ ù�õpâ ã¡ä � N�A < � < ��T�� ) < ��T 3 ê�P�#¡÷*) è�é�é�ã¡é�è�å�ý©÷ ÷ èdý©÷ û�á üøè#ï<â á þÿé�è�æ3áqâ áçú¡å�â ý©äxú#é ú¡æ3áçâ èâ�ú¥÷çâ ã¡÷çútû úCê ä©ã¡ä�å�áçú£ù©áP'ý �¡é ú¥äxæ3è#æ3á A ��ì#ý©ä©á ûE'ú�#xâ á ã*'è� ê ! ê �� ê � E ï � A ��ìA î

Page 67: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# , �

�do2_^b��8zH]hi�rx}o�zt�^aY�fb�c^efaH�faH[srto`[^aHiji�� aH�faHk8c^bj] �2c^|�}]`aH�f�faHefa��f|^c�aHeyoxrto2aH�fa �h|^aH[^_^]:bj[6_^]hc^c^buoc^efaHzHbj�fbj]h[^aBijaB]hc�aHeyoq�Hbj]h[^b�efaHiuoxryb��ha�oqi�zFoqijzH]hij]�_^b �

��Ï/Ò �XÏ>��Ó " �O�O�O�O�O�O� � ������2Ï � � Ï>��Ó��=�O�O�O�O�O�O�O�O�O�O�O�O�O�O� � ��� �� Ï " � 8�Ï���Ó " �O�O�O�O�O�O�O�O�O�O�O�O�O�O� � ����� Ï � � � Ï>��Ó���� 1�1�1�1�1�1 �O�O�O�O�O�O�O� � ��� ���Ï ��Ò Ï���Ó " �O�O�O�O�O�O�O�O�O�O�O�O�O�O� � ��� � ��VÏ � �:Ï���Ó ��� = � � ��� � �� � Ï � 5 ��Ï>��Ó " �O�O����� = � � ������B � � Ï ��Ò ��Ï>��Ó � 1�1�1 ��� , � � �������Ï = �-8�Ï���Ó>= �O�O�O�O�O�O� � ������ � Ï � � � ��Ï>��Ó>= �O�O���Q�=��� � ����� B � � Ï � � � ��Ï���Ó�� 1�1�1 � � � � � �����

�Koqija�ryaHzH[^bjzFo2c�aHefk8a¡rfryaB_^b�zFoqijzH]hiuoqefa�_^|^a��f]hij|^�Hbj]h[^b�_^bj��rybj[sryaq�^zH]hk8k8a¡rfryaH[^_^]8oqi�c^by}|�|^[�aHefef]hefa�f|^iji�� |^i�rybjk`oYzHb�lneyoV�fb*�h[^b��pzFoxryb��qo��

���©_poX]h�f�faHe��qoqefa zt�^a ija<_^b��8zH]hi�rx}o�efaHiuoxryb��haCoqiji�� aH�faHk8c^bj] �X�f]h[^]\ija �soxryaq�©c^by}|�zt�^a£oqiji�� oqi*�h]hefb�ryk8]���^oqiji�� aH~�|poq�Hbj]h[^aq��]3���haHef]8oqim�f|^]2zH]h[^_^bj�Hbj]h[poqk8aH[sry]^� Ô aHzH]h[^_^]2|^[�oqc^c^ef]�zHzHbj]V_^bmrybjc�]Vg�� aq� o����^ijaeyoq_^bjzHb�zH]hbj[^zHbj_^aH[sryb � � Ï � � Ï =V�fb�c�]h�f�f]h[^]2�haH_^aHefaYzH]hk8a��f]hij|^�Hbj]h[^bmaH�yoxrfryaY_^aHiji�� aH~�|poq�Hbj]h[^a

��Ó 1�1�1�1�1�1�1�1 =©� � Ò ��Ó 1�1�1�1�1�1�1 � � � 5 ��Ó 1�1�1�1�1�1�1 � � Ï>��?

b<zH|^b zH]�a¡�8zHbjaH[srybC_^b5AmaHefbj�fzH]h[^]6_poqbCzH]�a¡�8zHbjaH[srybC]hk8]hij]�h�^bK_^aHiji�� oqi�ryeyo:aH~�|poq�Hbj]h[^a8c�aHeYk8aH[^]�_^b��|^[^b�rx}oV�f|^iji�� ]qrfrtoF�qo�zHb�lneyoV�fb*�h[^b��pzFoxryb��qo+B�~�|^bj[^_^b��^o�c^bjzHzH]hijaN�qoqefbuoq�Hbj]h[^b�[^aHb�_poxryb�zH]hefefbj�fc�]h[^_^]h[^]�faH[^�fbjg^bjijbm�qoqefbuoq�Hbj]h[^b�[^aHb�efbj�f|^i�rtoxryb��

8�aHe4�p[^bjefaq�s�fbpzH]h[^�fbj_^aHefbp[�|^]3�qoqk8aH[srya\i�� aH�faHk8c^bj] " �Cv�]hi�rybjc^ijbjzFoq[^_^]NbpzH]�a¡�8zHbjaH[srybp_^aHiji�� aH~�|po©��Hbj]h[^aBc�aHe0��� � � �fb��po$�

� � Ò �©� 5 ��Ï>� ?ija:zH|^b��f]hij|^�Hbj]h[^b£�fbøc�]h�f�f]h[^]�zFoqijzH]hiuoqefa:aH�yoxrfrtoqk8aH[srya�zH]h[>i�� oqi*�h]hefb�ryk8] ��� ®[>~�|^aH��ry]zFoq�f]�fb_^bjzHaYzt�^a`}a���rtoxry]`aQAma¡rfry|poxry]8|^[^]6¨3Ëy§qª¬ÃÈÄ�4�Æ�¢¡Ã£Ët¦3¢�� Ë¡Ã�¢¡Äp­�ÃÈ��8�aHeNa¡��b�rtoqefa��hijb�aHefef]hefb�_^bdef]h|^[^_^]%AzH]h[^[^aH�f�fbdzH]h[�iuo�k8]hi�rybjc^ijbjzFoq�Hbj]h[^a�l � c����^�fbd�fzHaHi*�h]h[^]�l�oxrfry]hefb�_^b��fzFoqijbj[ ��zt�^aY�f]h[^]2c�]qryaH[^�HaY_^aHijiuogpoq�fa:_^aHi£�fbj��ryaHk`owoqefb�ryk8a¡rybjzH]�|�rybjijbj�H�Foxry] ; ��� � � [^aHiji�� aH�faHk8c^bj]�?¥� Ô b [^]qryb���ry|�rfrtoF��buo��#zt�^a:[^]h[aH�fbj��rya�|^[po�ryaHzH[^bjzFo�_^b��fzFoqijbj[ �2oqc^c^ijbjzFoqg^bjijaNbj[ �haH[^aHeyoqijaq� 3 �N 2�ö�ãNå�â�ú¥÷ á ä��\æ3è�á^â ãFè�:Xâ á è�äHû�á�ä©ã¡ä�é�áçå�ã¡÷ -¡èCå�è�üøù©é�è á�ù©é�ã��©÷ è�üøá^æ3á�ãE-¡è�é��xã��wã3æBý©äxæ3è�é��xã��<õKàFápâ ã¡äxå�áçæ3è�é�á�ú¡æèfå�è�üøù©á ãø÷*) è�)Hýxú&%�á ã¡ä©è

ìfí ��2 d ê � ! ìfí 2 d ê ��ìfí 2 d ë6í <÷ èCâ ý©á�é ú¡æ3áçâ ásê � è<ê � å�ã¡ä©ã ��ù©é�ãHå�å�á üøè2�£úNìfí' d èCú¡æ�ì¡õ �dä �©ý©ã¡ä�ú¥÷ �¡ã¡é�á û�üøã£é�áçå�ã¡÷ ý3û�á -¡ã�æ3è/-¡è<èfå�å�è�é�è#á ä �¡é ú¡æ3ãæ3ádâ�ú¥÷çâ ã¡÷çú¥é�è\ý©ä+) ú¥ù©ù©é�ãHå�å�á ü�ú&%�á ã¡ä©èNæ3á�ê � â ã¡ä3þÿã¡é�üøè�üøè�äHû�è�ú¥÷ ÷ è�é�áçâ('©á èfånû�è�æ3è�÷�ù©ý©äHû�ã � � V»æ ) ú¥÷ û�é ú�ùxú¥é�û�èNã��¡ä©áû�è�äHû útû�á -¡ãBæ3á»å�â�ú¥÷çú¥é�è£á�â ãFè�:Xâ á è�äHû�á»æ3è�÷ ÷*) è�)Hýxú&%�á ã¡ä©èøüøã¡÷ û�á ù©÷ áçâ�ú¥äxæ3ã¡÷ ápùqè�é<ý©ä©ã�ånû�èfå�å�ã\þ�útû�û�ã¡é�èøä©ã¡äVüøá �¡÷ á ã¡é ú\÷çúå�á û�ýxú&%�á ã¡ä©è�#Fú¥ä�%�áhù©ýQ'ã ù©é�ãE-¡ã3â�ú¥é�è�ý©ä\ãE-¡è�é��xã��`ã3æ�ý©ä\ý©äxæ3è�é��xã��<õXW�ý©èfånû úCè�)Hýxú&%�á ã¡ä©è4'è�á ä3þ�útû�û�áhý©äNæ3á :Xâ á ÷ è�û�èfånûùqè�é�ý©äBú¥÷ �¡ã¡é�á û�üøã£é�áçå�ã¡÷ ý3û�á -¡ãxõ

Page 68: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

,5"

3Ni�ryefaXryaHzH[^bjzt�^a��f]h[^]V|�rybjijbj�H�Foqg^bjijb�c�aHe£a¡��b�rtoqefaNi�� ]3�haHe)�p]��ÕaNi�� |^[^_^aHe)�p]����D3N_:aH�faHk8c^bj]^���fa�bjizFoqijzH]hij]Y_^b��E��c^ef]3�h]�zFoV]3�haHe)�p]����pk8aH[sryefa 8p� �3� � a " 8 � �f]h[^]Veyoqc^c^efaH�faH[srtoqg^bjijb����fa �$<Ï>�����fbmc^|�}]zFoqijzH]hiuoqefa � � � Ò " 8 � �faHzH]h[^_^]8iuoVln]hefkV|^iuo$�

� � � Ò " 8 � ÏÑÐ �hÐ�� �XÒ 8 = �=� � B

�faq�mbj[s�haHzHaq��� <Ï ��� 8p���3� � �f]h[^]:eyoqc^c^efaH�faH[srtoqg^bjijb�a " 8 � c^ef]�_^|^zHaV|^[^_^aHe)�p]������fb�c^|�}]:zFoqijzH]hiuoqefa|^[po�_^aHijijaBeyoq_^bjzHb�zH]h[�iuo�ln]hefkV|^iuo��Ï/Ò �5� 8 ; �h� ��=O?

aBi�� oqi�ryeyoV��lnef|�rfrtoq[^_^] ; �h� ��� ?¥� 3 �3 zH]h[^zHij|^�fbj]h[^aø_^b�~�|^aH��rto�_^bj�fzH|^�f�fbj]h[^aq�s�fb�c^|�}]^�h~�|^bj[^_^b��qa¡��bj_^aH[^�HbuoqefaøzH]hk8aq�soq[^zt�^aøiuoNefbj�f]hij|���Hbj]h[^aN_^b�|^[�c^ef]hg^ijaHk`oYk`oxryaHk`oxrybjzH]��faHk8c^ijbjzHaq��~�|poqijaV}a�iuo�efbj�f]hij|^�Hbj]h[^a\_^bm|^[�� aH~�|poq�Hbj]h[^aB_^bm�fa¥�zH]h[^_^] �heyoq_^]^��c�]h�f�yoVefbj�f|^i�rtoqefa\aH��ryefaHk`oqk8aH[srya�zH]hk8c^ijaH�f�f]�|�rybjijbj�H�Foq[^_^]Y|^[��fbj��ryaHk`oVoqefb�ryk8a¡rybjzH]l � c��Co�c^efaHzHbj�fbj]h[^a �p[^b�rto��

�Èä �¡è�ä©è�é ú¥÷ è�#fá äCâ�ú¡å�áHâ ã¡üøèX)Hý©èfånû�ã #tå�á¡ù©ýQ'ã�â è�é â�ú¥é�è»æ3áHè1hè û�û�ýxú¥é�è»ý©ä©ã�G*C(� �MI H ��"��/CL> " � C�� " I K*� " ��"� ! �T �MI�CL> H�I �MK � #â á ã*'è#æ3áhû�é ú¡ånþÿã¡é�ü�ú¥é�è�Nnì¡õ ñ L�P�á äNý©ä+) è�)Hýxú&%�á ã¡ä©è3 I � � � I �$� e I ë�í <

â ã¡ä3 I ëDA � 3 < I ëWA � +< e I ëWA � e < êNëDA � � <

â('©è<å�áçúøæ3á�ù©áP'ýNþ�ú¡â á ÷ è#é�áçå�ã¡÷ ý�%�á ã¡ä©è 2 ìfô 4 õN,3)�Èä2û ú¥÷�üøã3æ3ã�å�ámû�é ú¡å�â ý©é úBñ 3 e�á ä@Nnì¡õ ñ�%�PCå�è �Pí�ã�á ä@Nnì¡õ ñ <�PCå�è �Pí©õ ) ú¥÷ û�é ú�ùxú¥é�û�è�#pä©è�÷�å�áçånû�è�ü�úú¥é�á û�üøè û�áçâ ãBæ3á»ý©äVè�÷çú&�qã¡é útû�ã¡é�è�#hý©ä�-tú¥÷ ã¡é�èøâ('©è�ù©é�ãE-¡ã3â�úXý©äxæ3è�é��xã�� -Fá è�ä©èøùqãHånû�ã # �¡è�ä©è�é ú¥÷ üøè�äHû�è�#�ý��¡ýxú¥÷ è�ú�íè�#�)Hý©á äxæ3á*#�÷çú Nnì¡õ ó¡ïQPKå�ápù©ýQ'ã -¡èfæ3è�é�èøâ ã¡üøè�â ã¡äxå�è �¡ý©è�ä�%fú\á üøüøèfæ3áçútû úBæ3è�÷ ÷*) ú¥ù©ù©÷ áçâ�ú&%�á ã¡ä©èøæ3è�÷ ÷çú�â ã¡é�é�áçå�ùqã¡äxæ3è�äHû�èþÿã¡é�ü ý©÷çú é�áçå�ã¡÷ ý3û�á -tú©õ

Page 69: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# , �

02130�� � JK@BACW\;HMN; J#L?EFE �GVAC;¡9<TU@N9<;FWXG 4 EH52GY9<;F7 � ����58;F7:9{\aHi�zH]hef�f] _^aHiji�� bj[sryaHef]*cpoqeyo��heyoxln]P_^aH_^bjzFoxry] o��hijb�aH�faHefzHbj�Hb����fb�bj[^_^bjzt�^aHe©}o>zH]h[��)|^[��fbj��ryaHk`ooqefb�ryk8a¡rybjzH] � ¦F§q­�ÃÈÄ�4$�jÇp¦xÃÈÄp­øo2c^efaHzHbj�fbj]h[^a �p[^b�rto

� Ï ; ( ?���?��& 6 .?��& 98h�"?zH]h[ (KÏ�gpoq�faq�C­fÏ�c^efaHzHbj�fbj]h[^aq�£¢¡Å�ÃÈÄ>ÏÑaH�fc�]h[^aH[srya`k8bj[^bjk8]^�<¢¡Å8§ � ÏÑaH�fc�]h[^aH[srya8k`oq�f�fbjk8]�B�b�qoqij]hefb�[�|^k8aHefbjzHb�_^aHb�cpoqeyoqk8a¡ryefbmzt�^a�ij]VzFoqeyoxrfryaHefbj�H�Foq[^]2�yoqeyoq[^[^]��fc�aHzHb��pzFoxryb��^iuoq_^_^]3�haB]�zHzH]hefefaq�

����� �� � ��" � & �/�'� �'� "'� �'� & � ���'����"'�¹B½H°�º��s³n´�³nµP¶ ZXrybjijbj�H�Foq[^_^]Vbjimryef]h[^zFoqk8aH[sry]2eyoqc^c^efaH�faH[srtoqefa��

� ®[� � (?Ï ������r¥Ï ��� �& 6 DÏ�Ò � ��� �& 98h��Ï 5 � �

o@? =�Ó , � � =�� � = � �g�?,ÒG��� , � ,5" � � = "z�? ��Ó � ,, =��=�_�?,ÒG��=$� � ��� � � 3

�³�½ �<µ�½¡À3¼

o@? ��Ó>= , � � =�� �����g�? Ò ��Ó���� , � , � ����� z�? ��Ó � ,, =�=�� ��� _�? Ò ��Ó���=$� � ��� � � �

� ®[� � (?Ï ������r¥Ï " � �& 6 DÏ�Ò 1 � �& 98h��Ï 5 1

o@? 1 � 1 = , �g�? ��Ó �O�O�O�O�O�O�O�O�O�O�O�Q�z�? Ò ��Ó � " � ��� 3_�?,Ò =$� � ��� � �

�³�½ �<µ�½¡À3¼

o@? ��Ó 1 � 1 � � ��� �g�?�6² �d°�º"�Kµ��z�? Ò ��Ó � " �O� � ��� 3_�? Ò ��Ó>=$�O�O� � ���

Page 70: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

, �

� ®[ � � (�Ï ������r¥Ï@��� �& 6 �Ï-Ò =�� �& 98h� Ï =��\�fbNeyoqc^c^efaH�faH[srybj[^]>bN�fa �h|^aH[sryb[�|^k8aHefb�a£�fb�zFoqijzH]hijbpÃȪ�Å�ÃÈÄpÃÈÅ`¦NÄp«�Å`¢¡¤y¦N¤f§¥ÇsÇ�¤y¢t¨3¢¡Äp­ § �HÃȪj¢ aNª�§BÅ8§©¨t¨¡ÃÈÅ8§B§�ËtË¡«�¤f§q­®¢yÉtÉ3§¤y¢¡ª�§q­�Ã65x§ �

� � Ï =�=�=�Ó��=�� � Ï���Ó>=� � Ï���Ó � �$�� � Ï ��Ó " � , � � "

�³�½ �<µ�½¡À3¼�� � � � 9�°�º"�Kµ�� {\]h[ \oqc^c^efaH�faH[srtoqg^bjija�bj[ ��� � Ï>��Ó>= �O� � ��� �<�yoxrfrtoqk8aH[srya�eyoqc^c^efaH�faH[srtoqg^bjijaBbj[ ��� � Ï>��Ó �$�O� � ��� � � �<�yoxrfrtoqk8aH[srya�eyoqc^c^efaH�faH[srtoqg^bjijaBbj[ ��� � Ï>��Ó � " � � ����� {\]h[6aH�yoxrfrtoqk8aH[srya�eyoqc^c^efaH�faH[srtoqg^bjija�bj[ �

®i�k8bj[^bjk8]�[�|^k8aHef]8eyoqc^c^efaH�faH[srtoqg^bjijaBbj[ � ; ryef]h[^zFoqk8aH[sry]�?N}a��(��P�� � � � Ï ����� � � � Ï ����� �

�do�k`oq�f�fbjk`o�oqzHzH|^eyoxryaH�H�Fo8efaHiuoxryb��qo`}a��( � � � Ï ��� � � � Ï ����� �

¹B½H°�º��s³n´�³nµ 7 Ô buo � �� � (?Ï ���+?��CÏ �+?��& 6 wÏ/Ò 1 ?��& 98h��Ï 1

�fb�zFoqijzH]hijbj[^] �; o@?øbjimk`oq�f�fbjk8]�[�|^k8aHef]8efaFoqijaBc�]h�fb�ryb��h]�eyoqc^c^efaH�faH[srtoqg^bjija%B; g�?øbji�k8bj[^bjk8]�[�|^k8aHef]8efaFoqijaBc�]h�fb�ryb��h]�eyoqc^c^efaH�faH[srtoqg^bjija%B; z�?øi�� ��� � bj[ � B; _�?��fbdaH�fa �h|po`i�� oq_^_^bj�Hbj]h[^aB�fa �h|^aH[sryaVaY�fbdzFoqijzH]hijb�i�� aHefef]hefa�efaHiuoxryb��h]8bj[sryef]�_^]qrfry]2[^aHidefbj�f|^iÿ�rtoxry] �

� , Ó " � 5 �hÓ �O�O�O��³�½ �<µ�½¡À3¼

; o@?øbjimk`oq�f�fbjk8]�[�|^k8aHef]8eyoqc^c^efaH�faH[srtoqg^bjija�}a���* 98h��Ï (��P��� � ; �XÒ7(�� � ?<Ï���Ó 1�1�1 � ��� �

Page 71: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# ,,

; g�?£bji�k8bj[^bjk8]�[�|^k8aHef]8eyoqc^c^efaH�faH[srtoqg^bjija2}a���* 6 wÏ (��P�� � � � Ï���Ó�� � ����� �

; z�?øi�� ��� � }a��( � � � Ï ��� � � � Ï ����� �

; _�?,�<�fa �h|^buoqk8]2i�� oq_^_^bj�Hbj]h[^a�|^[poV�h]hi�rto�eyoqc^c^efaH�faH[srtoxrybdbj[ ��b�[�|^k8aHefb �� � Ï � , Ó " � �� � Ï���Ó � ,5" � ��� �� � Ï �hÓ �O�O�O� �� � Ï���Ó�� �O� � ��� �

�� � 5 �� � Ï>��Ó � ,5" � ��� � 5 ��Ó �Q� � � ��� � Ï���Ó � 1 = � ��� ��<� aHefef]hefa�efaHiuoxryb��h]�}a��

Î � � ÏÐ ��Ó � 1 � " �O� � ��� � Ò ��Ó � 1 = � ��� � Ð

��Ó � 1 � " �O� � ��� � Q ��Ó �O� = � � , = Q ��Ó>= � � � ����� �

¹B½H°�º��s³n´�³nµ � Ô buo� � (?Ï ���+?��<Ï �+?��& 6 wÏ�Ò �+?��& 98h��Ï ��Ó

Ô b�eyoqc^c^efaH�faH[srybj[^]Bbj[ �*b�[�|^k8aHefb��fa �h|^aH[sryb^a£�fb�bj[^_^bjzt�^b�i�� aHefef]hefa£efaHiuoxryb��h]\zt�^aø�fb�zH]hk8k8a¡rfrya[^aHijiuoVij]hef]�eyoqc^c^efaH�faH[srtoq�Hbj]h[^aq�

o@? � � Ï ��=�Ó��=�g�? � � Ï �hÓ>=z�? � � Ï���Ó �O� �$� , �_�? � � Ï � " � , Ó � � "�³�½ �<µ�½¡À3¼

o@? �� � Ï>��Ó���=�� � ��� � {\]h[�eyoqc^c^efaH�faH[srtoqg^bjija�aH�yoxrfrtoqk8aH[srya�bj[ �g�? �� � Ï>��Ó���= � � ��� � \oqc^c^efaH�faH[srtoqg^bjija�aH�yoxrfrtoqk8aH[srya�bj[ �z�? �� � Ï>��Ó �$� , � ��� � � {\]h[�eyoqc^c^efaH�faH[srtoqg^bjija�aH�yoxrfrtoqk8aH[srya�bj[ �_�? � 5q¢¡¤ � ¦5#

Page 72: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

, �

8�aHe\~�|poq[sry]8efb*�h|poqef_po �hijb�aHefef]hefb�efaHiuoxryb���b��fb��po$�

Î � � � Ï ��� � � � � ��� � �� � � � � � � � �� � � � � ��� � � ���Ï>��Ó �O�O�O������� 1 Ï���Ó ����� � ��� � �

Î � � � Ï ��� � � ��� � �� � � � ��� � �� � � ��� � � ���Ï��

Î � � � Ï ��� � 3 � � � � � � � � � 3 � � � � � � � 3 � � � � � � � ���Ï>��Ó �O�O� " � " = " Ï���Ó " � " � ��� � �

¹B½H°�º��s³n´�³nµ � Ô buoq[^] �VaH_ � efbj�fc�a¡rfryb��qoqk8aH[srya�|^[`�fbj��ryaHk`o�oqefb�ryk8a¡rybjzH]Bbj[sryaHef]YaH_8|^[^]��p]soxrybj[ �x�c�]hbj[sr\o�c^efaHzHbj�fbj]h[^a �p[^b�rto��^_^b�cpoqeyoqk8a¡ryefb�efbj�fc�a¡rfryb��qoqk8aH[srya��

��� (?Ï ��� �CÏ � 6 6 DÏ�Ò 1�1�1 6 98h��Ï 1�1�1

� � (?Ï ��� �CÏ � �& 6 wÏ/Ò 1 �& 98h��Ï 1

Ô b�aH�fa �h|poq[^]`ijaB�fa �h|^aH[srybd]hc�aHeyoq�Hbj]h[^b��fa �h[poqiuoq[^_^]2a¡�haH[sry|poqijb��h�fb�ry|poq�Hbj]h[^b�aHzHzHaH�Hbj]h[poqijbj�^��h�B? �H²�� �

; o@? �� B; g�? � � �3 3 B; z�? � � � ; �J�O�O�@?¥�=��B? �H² � �; o@? �� B; g�?(��� �O�O�Q� � ��� �O�O�O�O�O� 1 � �+B; z�? =�� �$� � 5 � � ��� ���:��³�½ �<µ�½¡À3¼

�h�B? ®[ �:�fb��po$�

; o@? �� Ï �+B

; g�? � � �3 3 Ï\�E��B

; z�? � � � ; �J�O�O�@?fÏ��J�O�O�O�O� ; �N�haHe)�p]�� ?¥�=��B? ®[ ���fb��po$�

Page 73: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# , 1

; o@? �� Ï ��� ��� ��� B; g�? ; ��Ó�� � ��� � � ?�� ; ��Ó 1 � � � ��� � � ?CÏ ; Z\[^_^aHe)�p]�� ?�Ï �

; z�?£� � Ï =�Ó �$� � �� � Ï���Ó>= �$� � ��� �� � Ï � � ��Ó ��� �� � Ï���Ó � � � � ��� �

�� � 5 �� � Ï���Ó � � � � ��� � 5 ��Ó �O� = �$� � ��� � Ï>��Ó � � , � ��� �

¹B½H°�º��s³n´�³nµ � Ô buo$�� � (?Ï ���+?��CÏ �+? �& 6 DÏ/Ò 1 ? �& 98h��Ï 1 Ó

\oqc^c^efaH�faH[srtoqefa@�m��h�p��a ��_^]3�ha���6Ï �� ? �2Ï�Ò , � ����������� � ��Ó �O� , � 1 " 1 ? � Ï�Ò ��Ó �O�O�Q� � ��Ó �O�O�O�O�O� , � 1

� Ï " Ó �$� , 5 = � ��Ó � 1�³�½ �<µ�½¡À3¼

��6Ï���Ó � � ��� ��2Ï���Ó 1�1 , � ����� ; � � �� 0 � � ?� Ï�Ò ��Ó , � 1 � ��� � � ; � ��� �� 0 � � ?�� Ï���Ó " � , � ����� 5 ��Ó>= �$� � ��� � ÏÏ���Ó �O� " � , � ��� � 5 ��Ó>= �$� � ��� � ÏÏ���Ó>=$�O� � ��� �

¹B½H°�º��s³n´�³nµ � Ô buoq[^]� � a � � _^|^a`�fbj��ryaHk8b oqefb�ryk8a¡rybjzHb��p]soxrybj[ �x�®c�]hbj[srYo�c^efaHzHbj�fbj]h[^a �p[^b�rto�_^bcpoqeyoqk8a¡ryefb �� � � ( � Ï = � � Ï =$�� � � ( � Ï ��� � � Ï��

��a¡ryaHefk8bj[poqefa � � bj[�k`oq[^bjaHeyoNzt�^a�b�_^|^a��fbj��ryaHk8b^oqg^g^buoq[^]�iuoN��ryaH�f�yo�k`oq�f�fbjk`oNoqzHzH|^eyoxryaH�H�FoefaHiuoxryb��qo��

Page 74: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

� �

�³�½ �<µ�½¡À3¼

Ô buo � � iuo�k`oq�f�fbjk`oBoqzHzH|^eyoxryaH�H�Fo�efaHiuoxryb��qo�_^b � � aH_ � � iuoBk`oq�f�fbjk`o�oqzHzH|^eyoxryaH�H�FoYefaHiuoxryb��qo_^b�� � � Ô b��po$�

� � Ï�= �-( � � � �� T � � Ï

�= � = � � � � Ï =@� � �

8�]hbjzt���a��

� � Ï � � Ï�= � ��� � � � � T ��� � � � � Ï = � � Ï =@� � �

_po2zH|^b��� � Ï �XÒ�ij]� � = � � T � � Ï �XÒ*ij]� � =@� � � Ï � 5 =�=Vij]� � =

~�|^bj[^_^b� � Ï � 5 =�= � ij]� � ���ij]� � = Q � 5 �YÏ ,

¹B½H°�º��s³n´�³nµ � Ô buo$�� � (?Ï ���+?��<Ï =�? �& 98h��Ï �+? �& 6 DÏ�Ò ��Ó

~�|poq[srybd[�|^k8aHefb�k`oqzHzt�^bj[po�zH]h[srybjaH[^a � �

�³�½ �<µ�½¡À3¼

£[�|^k8aHefb�k`oqzHzt�^bj[po�c^efaH�faH[srybdbj[ �Õ�f]h[^] �= ; (wÒ � ?�( � � � ; �& 98h��Ò �& 6 5 � ? 5 �

a�_^|^[^~�|^a��= � 1 � ��� � , 5 �NÏ ��=$�Q�

Page 75: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# � �

¹B½H°�º��s³n´�³nµ Ô buo$�� � (�Ï ���+?��<Ï " ?��& 98h�6Ï 1 ?��& 6 wÏ/Ò 1 Ó

_^]qrtoxry]2_^aHijiuoVk`oq�f�fbjk`oVoqzHzH|^eyoxryaH�H�Fo8_^bj[poqk8bjzFo���\oqc^c^efaH�faH[srtoqefaYb�[�|^k8aHefb ��6Ï , � " Ó � � ��Ï���Ó �O�O�O�O�O�O�O� 1 � , �Q�

a�_^a¡ryaHefk8bj[poqefa �hijbdaHc^�fbjij]h[�k`oqzHzt�^bj[po � � � � a � � ��efaHiuoxryb���b�o8zHbuoq�fzH|^[^]:_^b�aH�f�fb ; �f|^c^c�]x�[^aH[^_^] �� � � . ��?¥�

�³�½ �<µ�½¡À3¼

��6Ï>��Ó , � " � � ��� � B ��2Ï���Ó 1 � , � � ����� �� ��� � Ï

�= ( � � � Ï>��Ó � � ��� � � � Ï>��Ó � � ����� �

� � � � Ï���Ó , � " � � ��� � � ��Ó � � ����� � Ï���Ó � � =$� � ��� � � ��Ï>��Ó 1 � , � ����� � � ��Ó � � ����� � Ï � ��� ; � ��� �� 0 � � ?

¹B½H°�º��s³n´�³nµ�� Ô buo$�� � (�Ï ���+?��<Ï =�?��& 98h�6Ï 1 ?��& 6 wÏ/Ò 1 Ó

_^a¡ryaHefk8bj[poqefa<~�|poq[srybs�f]h[^]\bh[�|^k8aHefbsk`oqzHzt�^bj[po�bj[ �aCeyoqc^c^efaH�faH[srtoqefa bjih[�|^k8aHef]��6Ï =�Ó���=bj[�rtoqija��fbj��ryaHk`o2oqefb�ryk8a¡rybjzH]^�

�³�½ �<µ�½¡À3¼

[�|^k8aHefb�k`oqzHzt�^bj[po�bj[ �Õ�f]h[^] �= ; (�Ò � ?�( � � � ; �& 98h�:Ò �& 6 5 � ? 5 �

]3���haHef]= � 1 � ��� � � � 5 �NÏ � = " �

bj[^]hi�ryefa�_poxry]��6Ï =�Ó���=��^iuoV�f|po2eyoqc^c^efaH�faH[srtoq�Hbj]h[^aYbj[ � }a����6Ï>��Ó>=�� � ��� �

_^|^[^~�|^a��D[^]h[}aBeyoqc^c^efaH�faH[srtoqg^bjija�aH�yoxrfrtoqk8aH[srya�bj[ ��a

Î � � ÏÐ ��Ó>=���= � �����KÒ ��Ó>=�� � �����3Ð

��Ó>=�� � ��� � Ï>��Ó �O� 1 " ��������Ó 1 " � � ����� �

Page 76: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

� =

¹B½H°�º��s³n´�³nµP¶ � Ô buo$�� � (�Ï ���+?��<Ï �+?��& 98h�6Ï = �+?��& 6 wÏ�Ò = ��Ó

aH�fa �h|^bjefa�bj[ ��ijaB�fa �h|^aH[srybd]hc�aHeyoq�Hbj]h[^b �

8�? � 5 ��� ��2? � 5 ��� �� ? � 5 ��� � �

�³�½ �<µ�½¡À3¼

�<�fa �h|^aH[^_^]:ijaB]hc�aHeyoq�Hbj]h[^b����fbd]qrfrybjaH[^a��

o@?/��Ó�� � ����� 5 ��Ó�� � ��� �øÏ>��Ó �O�O�O�O�O�O�Q� � ��� � 5 ��Ó�� � ��� �øÏ���Ó�� � ��� �g�?/��Ó�� � ��� � 5 ��Ó�� � �����CÏ>��Ó����O�O� � ��� � 5 ��Ó �O�O�Q� � ��� � Ï>��Ó����O�Q� � ��� �z�?/��Ó�� � ����� 5 ��Ó�� � ��� � �øÏ���Ó�� � ����� 5 ��Ó �O�O�O�O�O�O�Q� � �����CÏ>��Ó�� � �����

¹B½H°�º��s³n´�³nµP¶»¶ Ô buo$�� � (?Ï ���+?��<Ï " ? �& 98h��Ï 1 ? �& 6 DÏ�Ò 1 Ó

�fb��f|^c^c�]h[ �so�zt�^a�bj[�rtoqija��fbj��ryaHk`o�oqefb�ryk8a¡rybjzH]w�fbuo?c^efaH�faH[sryaw|^[�¤y¢�4qÃj¨¡­�¤y¦�c�aHe�aH�fa �h|^bjefab\zFoqijzH]hijb�bj[ _^]hc^c^buoc^efaHzHbj�fbj]h[^a�zH]h[ c^efaHzHbj�fbj]h[^a.�� � � Ï � � �CaHefb��pzFoqefawzt�^aDbji��fbj��ryaHk`ooqefb�ryk8a¡rybjzH]`zH]h[^�fbj_^aHeyoxry]6[^]h[ �h]�_^aV_^aHijiuo`c^ef]hc^efbja¡rx}o�oq�f�f]�zHbuoxryb��qo zFoqijzH]hiuoq[^_^] 8 5 � 5 � �zH]h[wo��pg��^a�z��

8VÏ���Ó � "+, = � ��� ��Ï���Ó " � � � � ��� � �� Ï���Ó � = 1 , � ��� � �

�³�½ �<µ�½¡À3¼Ô ]hk8k8buoqk8]�[^aHiji�� ]hef_^bj[^a ; o 5 g�? 5 z��

� 8���6 � �& � � 6�8 ��� 6 � � ��8 ��Ó � "+, = � ��� � ��Ó " � � � � ��� � � ��Ó �O�O�O� " � � � � ���

8 5 � ��Ó � "+, = " � � � � ��� 0 ; 8 5 �2? ��Ó � "+, = � ���

� ��Ó � = 1 , � ��� � � ��Ó �O�O�O��� = 1 , � ��� 0 ; 8 5 �2? 5 � ��Ó � "+, =$� = 1 , � ���

0 ; 0 ; 8 5 �2? 5 � ? ��Ó � "+, = � ���

Page 77: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# � �

®i�efbj�f|^i�rtoxry]�}aB| �h|poqija�oq_288c�aHefzt���a �Na � �f]h[^]2aH[sryeyoqkVg^b�k8]hi�ry]�c^by}|�c^bjzHzH]hijbm_^aHi�c^efbjk8]^�Ô ]hk8k8buoqk8]Voq_^aH�f�f]8[^aHiji�� ]hef_^bj[^a ; g 5 z�? 5 o$�

� 8���6 � �& � � 6�8 ��� 6 � � ��� ��Ó � = 1 , � ��� � � ��Ó �O�O�O��� = 1 , � ���

� 5 � ��Ó �O�O�O� , 1 � = � ��� 0 ; � 5 � ? ��Ó , 1 � = � ��� � � ��Ó �O�O�O� , 1 � = � ���

8 5 0 ; � 5 � ? ��Ó � "+, = , 1 � = � ��� 0 ; 8 5 0 ; � 5 � ?�? ��Ó � "+, � � ���

®i efbj�f|^i�rtoxry]6_^b ; � 5 � ? 5 8D_^b5AmaHefbj�fzHa`_po ; 8 5 �2? 5 � � XaH��rtow_^|^[^~�|^a:c^ef]3�qoxry]wzt�^a:bji�fbj��ryaHk`o2oqefb�ryk8a¡rybjzH]VzH]h[^�fbj_^aHeyoxry]6Äm¦xÄ/Í¢�������������� ����#�

¹B½H°�º��s³n´�³nµP¶�7 Ô buo$�� � (?Ï =�?��<Ï , ?��& 98h�6Ï , ?��& 6 DÏ�Ò , Ó

_^]qrtoxry]`_^bKk`oq�f�fbjk`o8oqzHzH|^eyoxryaH�H�Fo�_^bj[poqk8bjzFo�� \oqc^c^efaH�faH[srtoqefaV��Ï 1 Ó �`a � Ï " Ó>=8bj[ �B���ef]3�qoqefa�_^|^a�[�|^k8aHefb � * �Õa � * �Õrtoqijbmzt�^a�� 5 �2Ï��Da � 5 �CÏ � �

�³�½ �<µ�½¡À3¼

Ô b��po$�1 Ï ���O�Q� � ��Ó �YÏ���Ó ���O�Q� �

" Ï ���O� � ��Ó>=�Ï���Ó �O�Q�O� �c�aHeXzH|^b �

�6Ï>��Ó����O�Q�O���O� � = � �2Ï���Ó���= ��Ï>��Ó�� � =@� �®� � 5 �2Ï>��Ó����O�Q�O���O� � = � 5 ��Ó �O�O�O�O�O�O�Q� � = � Ï �

�6Ï>��Ó����O�Q�O���O� � = � � Ï���Ó ��� = ��Ï>��Ó�� � =@� � �� � 5 �CÏ���Ó����O�Q�O���O� � = �®� 5 ��Ó �O�O�O�O�O�O�O�Q� � = �®� Ï �

Page 78: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

� "

¹B½H°�º��s³n´�³nµP¶ ���Boxrybp�6Ï ��=$��Ó " � , � a �6Ï ��=$��Ó " � 1�1 zFoqijzH]hiuoqefaXi�� aHefef]hefa\oq�f�f]hij|�ry]VÎ�a\i�� aHefef]hefa

efaHiuoxryb��h]8Î F� �£]h[srtoqefaYbjim[�|^k8aHef]8_^b�zHb�lnefaB�fb*�h[^b��pzFoxryb��ha�a�_^aHzHbjk`oqijb��

�³�½ �<µ�½¡À3¼

Ô b<�po6Î�Ï)ÐB��=$��Ó " � 1�1 Ò���=$��Ó " � , � Ð^Ï ��Ó �O� =���Ï ��Ó>=�� � ��� � � _po�zH|^b<�fb#�po�zH]h[�lnaHefk`o

zt�^aBi�� oqc^c^ef]h�f�fbjk`oq�Hbj]h[^a�}aNzH]hefefa¡rfrtoVo�7 �s³� º©°R�d°��s³���¼p¾�³ �sk8aH[sryefaBÎ øÏÕÎ�����=$��Ó " � , � Ï

��Ó �O�O�O�Q� , Ó�Ó�Ï ��Ó�� , Ó�Ó�Ó � ��� � � ��_po�zH|^bC�fb<efbjzFoF�qo�zt�^a8i�� oqc^c^ef]h�f�fbjk`oq�Hbj]h[^a:}a2zH]hefefa¡rfrto6o �

�s³� º©°D½F³��²�³È± �h¼�À©³69�°»�

¹B½H°�º��s³n´�³nµP¶ � �Boxrybp�6Ï>��Ó �O�O� " � � a �6Ï>��Ó �O�O� " ��=BzFoqijzH]hiuoqefaXi�� aHefef]hefa\oq�f�f]hij|�ry]VÎ�a\i�� aHefef]hefa

efaHiuoxryb��h]8Î F� �£]h[srtoqefaYbjim[�|^k8aHef]8_^b�zHb�lnefaB�fb*�h[^b��pzFoxryb��ha�a�_^aHzHbjk`oqijb��

�³�½ �<µ�½¡À3¼

Ô b��po:Î�Ï Ð ��Ó �O�O� " � � Ò ��Ó �O�O� " ��=»Ð�Ï ��Ó �O�O�O�O����Ï ��Ó � �=��� � 3�_po8zH|^b��fbK�po8zH]h[�lnaHefk`o

zt�^aBi�� oqc^c^ef]h�f�fbjk`oq�Hbj]h[^a�}aNzH]hefefa¡rfrtoVo � �s³� º©°R�d°��s³���¼p¾�³ �sk8aH[sryefaBÎ øÏÕÎ�� ��Ó �O�O� " � � Ï

��Ó �Q� " � ��Ï ��Ó�� " � ��Ó�Ó�Ó � ��� � �¥�d_po6zH|^bC�fbCefbjzFoF�qo�zt�^a8i�� oqc^c^ef]h�f�fbjk`oq�Hbj]h[^a�}a2zH]hefefa¡rfrto�o 7�s³� º©°D½F³��²�³È± �h¼�À©³69�°»�

¹B½H°�º��s³n´�³nµP¶ � Ô buo$�� � (�Ï ���+?��CÏ ��? � �� � Ï�Ò 1 � ��� � Ï 1 Ó

Ô b�_^a¡ryaHefk8bj[^b£iuoDk`oq�f�fbjk`oDoqzHzH|^eyoxryaH�H�FoefaHiuoxryb��qo �#� \oqc^c^efaH�faH[srtoqefaq�£k8aH_^buoq[srya6oqefef]x�ry]h[^_poqk8aH[sry]^� bø[�|^k8aHefbø� � Ï ��=$��Ó " � , � �<� � Ï ��Ó � =���� 1 � " �<zFoqijzH]hiuoqefa:bj[^]hi�ryefa`i�� aHefef]hefaefaHiuoxryb��h]�_^b�ef]h|^[^_��®]%Awa\�haHefb��pzFoqefa�zt�^a�aH�f�f]�}aNk8bj[^]hefa\_^aHijiuo�k`oq�f�fbjk`oYoqzHzH|^eyoxryaH�H�FoVefaHiuo©�ryb��qo��

�³�½ �<µ�½¡À3¼

Ô b��po ��Ï���Ó � � ��� � � �®[^]hi�ryefa2}a�� � Ï>��Ó���=$� " � , � � ��� �\a � � Ï���Ó���=$� " � � ��� �F��<� aHefef]hefa�efaHiuoxryb��h]8oqijij]heyo`}a

Ð ��Ó���=$� " � , � � ��� � Ò ��Ó���=$� " � � ��� � Ð��Ó���=$� " � , � � ��� � Ï>��Ó �O�O�O�Q� , Ó�Ó�Ó�Ï>��Ó�� , Ó�Ó�Ó � ����� � � �

Page 79: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# � �

Ô b<�po�c�]hb#� � Ï ��Ó>=���� 1 � " � ��� � �Ba2~�|^bj[^_^b � � Ï ��Ó>=���� 1 � � ��� � �¥�:�<� aHefef]hefa2efaHiuoxryb��h]oqijij]heyo`}a

Ð ��Ó>=���� 1 � " � ��� � �KÒ ��Ó>=���� 1 � � ��� � �FÐ��Ó>=���� 1 � " � ��� � � Ï>��Ó �O�O�O�Q� � Ó�Ó�Ï>��Ó�� � Ó�Ó � ����� � � �

¹B½H°�º��s³n´�³nµP¶ � Ô ]hk8k`oqefaCija �fa �h|^aH[sryb�zH]hc^c^bja _^b�[�|^k8aHefb�bj[�|^[V�fbj��ryaHk`oNoqefb�ryk8a¡rybjzH]\o\k`oq�f�fbjk`ooqzHzH|^eyoxryaH�H�FoV_^bj[poqk8bjzFo$� ��Ó � " � =�� � ��� � � 5 ��Ó , � � " � � ��� �øa ��Ó " 1 � � " � ��� � 5 ��Ó>=�� " � � ���� � � ��³�½ �<µ�½¡À3¼

�do�c^efbjk`oV�f]hk8k`o:}aG��Ó , � � 1 , � ��� �F�pk8aH[sryefa�iuoV�faHzH]h[^_po�}aG��Ó " 1 � � " � ��� �F�¹B½H°�º��s³n´�³nµP¶ �>3N�f�f|^k8buoqk8]�|^[�fbj��ryaHk`o�l � c��2zH]h[gpoq�fa�( Ï ����� �BÏ " �dc^efaHzHbj�fbj]h[^a2efaHiuoxryb��qo

_^aHijiuo�k`oqzHzt�^bj[po ��� � Ï ��� � 3:aH_ aH�fc�]h[^aH[srya?k`oq�f�fbjk8]awk8bj[^bjk8] �& 98h� Ï 5 = ��aH_�& 6 ÏIÒ = ����efbj�fc�a¡rfryb��qoqk8aH[sryaq� \oqc^c^efaH�faH[srtoqefabji�efbj�f|^i�rtoxry]*_^aHijija?]hc�aHeyoq�Hbj]h[^bNl � c���fa �h|^aH[sryb

; o@? � 5 ��� � � ; g�? � 5 ��� �; z�? � 5 ��� � ; _�? ����� 5 ��� �; a�? ����� �������43 ; lE? ��� � � � ��� � �43

�³�½ �<µ�½¡À3¼

Ô b�]qrfrybjaH[^a; o@? ��Ó����O�O� � ����� ; g�?/��Ó����O�Q� � ��� �; z�?(��Ó����O�O� � ��� � ; _�?(��Ó����O�O� � �����®�; a�?(��Ó�� � ��� � � Ï ��� � ; lE? ��Ó�� � ��� � � � Ï��

¹B½H°�º��s³n´�³nµP¶ ®[D|^[?�fbj��ryaHk`o`oqefb�ryk8a¡rybjzH]`oF�haH[srya�|^[po:�f]�hijbuo2_^b � ��� Ï ��� � � �B�fb�aH�fa �h|poq[^]ijaB]hc�aHeyoq�Hbj]h[^b �

; /? 8�Ï�� � � 5 � � ; /? 8�Ï � � � 5 � � ; /? � Ï ; � � ?�� ; � � � ?

Page 80: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

� �

Ët¦xÄ �2Ï �xÊ Ï ��� � �43¡Ê � Ï ��� � � ¢ � Ï ��� � � 3

�³�½ �<µ�½¡À3¼

Ô b�]qrfrybjaH[^a��

; /? 8�Ï � � � 5 � � Ï�� � 5 ; ��� � � 3 ? � Ï � � 5 ��� � 3 Ï � � 5 �YÏ �; /? 8�Ï � � � 5 � � Ï � ��� � � 5 ��� � 3 Ï � � 5 ��Ï>�; /? � Ï � ����� Ï�� � �

Page 81: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# � ,

����� �� � ��" � & ��� � �'����� ��h���ø«^§qÄ�Æ�¦�ª�§�¨3¦xª¬«hÉHÃ�¦xÄm¢�Ëy§qªjËt¦xª�§q­ §VÆqÃ�«�Ä�Æs§q­®¦�Ç�¤y¦��Hªj¢¡Å8§�Ç�«\ͦ�¢t¨t¨3¢¡¤y¢�Ët¦xÄ�¨¡Ã�Æ�¢¡¤f§q­ §V§�ËtË¡«�¤f§q­ §Äm¢¡ª�¨3¢¡Ä�¨3¦`Æ�¢¡ªÈª�§Y�øoqz1���øoqef_ �<efef]he(3N[poqi*��fbj� ���������� �����

{\aHijiuo �øoqz1���øoqef_ �<efef]he 3N[poqi*��fbj�8�fb�oq�f�f|^k8awzt�^aDiuo�f]hij|^�Hbj]h[^a�zFoqijzH]hiuoxrto�_^b\|^[_poxry]>c^ef]hg^ijaHk`o��fbuo*oqzHzH|^eyoxrto��\�fa�aH�f�yo�}a?iuo��f]hij|^�Hbj]h[^aDaH�yoxrfrto ; ]qrfryaH[�|�rto*bj[ |^[�fbj��ryaHk`o`oqefb�ryk8a¡rybjzH]2o8c^efaHzHbj�fbj]h[^aYbj[Q�p[^b�rto@?£_^aHi�k8aH_^aH�fbjk8]8c^ef]hg^ijaHk`o�c�aHe�ry|^efgpoxry]^�]3���haHef]�_^aHijij]D��ryaH�f�f]c^ef]hg^ijaHk`oDbj[*zH|^b��fbuoD��rtoxrto?bj[sryef]�_^]qrfrtow|^[poD�qoqefbuoq�Hbj]h[^a:[^aHb_poxryb�zH]hk8cpoqeyoqg^bjijaNzH]h[�i�� aHefef]hefaB_^b<¤y¦x«�Ä�Æ$�f¦���_^b�eyoqc^c^efaH�faH[srtoq�Hbj]h[^a�_^aHb�_poxryb��®[^_^bjzFoxrtoNzH]h[ �8iuoXln|^[^�Hbj]h[^a£zt�^a�_^aH�fzHefb��ha�bji�c^ef]hg^ijaHk`o\_poNefbj�f]hi��haHefaq�q�8b�_poxryb�aH�yoxrfryb����*b<_poxrybCo Ama¡rfrybC_po6aHefef]hefaq� �*iuo:ln|^[^�Hbj]h[^a2zt�^a8_^aH�fzHefb��ha`bji#c^ef]hg^ijaHk`o�c�aHe�ry|^efgpoxry]efbj�f|^i�rto$� � ; �"?CÏ� ; ��"?CÏ ; � 5 � ?¥�©zH]h[�Ð �pÐ�� � �©_^]3�ha � }aCiuo�c^efaHzHbj�fbj]h[^aC_^aHi��fbj��ryaHk`o��®[ � �h|^eyow}aY_^aH�fzHefb�rfry]`bjid_^buo��heyoqk8k`oV]hc�aHeyoxryb��h]8_^aHijiuo �øoqz1���øoqef_ �<efef]heK3N[poqi*��fbj�H�zH]h[�lnef]h[srtoxry]6zH]h[i�� oqi�ryeyo8ryaHzH[^bjzFo�_^b<oq[poqijbj�fbK_^aHijiuo:c^ef]hcpo��soq�Hbj]h[^aV_^aHiji�� aHefef]hefaq��zHbj] �aiuo � ]he �øoqef_ 3N[poqi*��fbj�H�

� � �� ; ��?c^efaHzHbj�fbj]h[^aBbj[Q�p[^b�rto

� � � � � � � � � � � � � ���

c e a z b � b ] [ a l b [ b r o �� ; ��?

��°�º©º©µ»º©°D½¡À©³���¼�À3µ�d¼p¾�¾n¼ �y·ÿ°»·ÿ¼�·

�� �c^efaHzHbj�fbj]h[^aBbj[Q�p[^b�rto�

°�º©º©µ»º©°D½¡À©³���¼�À3µ�d¼p¾�¾n¼ � ·ÿ°»·ÿ¼�·

¹�º©º©µ»ºx³ ½¡À©³���¼�À©³ �hµ�² ¾n¼ � µ»º �V¼^º*� °�º©º©µ»º`¼p²d¼p¾ �d½F³�½2° �hµ�² ¾n¼ ��¼ ��� �V¼^º*�°�º©º©µ»º`¼p²d¼p¾ �d½F³�½q·

Page 82: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

�O�

=�� �¥Ä�«�Ĩ¡Ãj¨¡­®¢¡Å8§w§q¤¥ÃÈ­�Å`¢¡­�Ã�Ët¦w§VÇ�¤y¢tË¡Ãj¨¡Ã�¦xÄm¢�6CÄpÃÈ­ §?Ët¦3¨F§�Í¢�Ç�Ã�Í«:Çp¢¡¤¥Ã�Ët¦xªj¦3¨3¦6Æq«�¤f§qÄp­®¢2ª�� ¢t¨3Ë¡«��ÉHÃ�¦xÄm¢�ÆqÃ<«�Ä�� ¦tÇp¢¡¤f§3ÉHÃ�¦xÄm¢ � «�Ä?|^[^_^aHe)�p]�� ¦2«�Ä?]3�haHe)�p]�� � � Çp¢¡¤yËE���¢ �� ������� �����

Z\[�]3�haHe)�p]��"}a`|^[�a¡�haH[sry]?c^by}|�_^b��8zHbjija2_po��haH��rybjefa8_^|^eyoq[srya`bji<zFoqijzH]hij]^�W�<�f�f]Dbj[��l�oxrfryb��sc^|�}]VzH]hk8c�]he�rtoqefaXi�� bj[sryaHefef|^�Hbj]h[^aX_^aHiji�� aH�faHzH|^�Hbj]h[^a\_^aHi�zFoqijzH]hij]�B�oqijzH|^[^b»�fbj��ryaHk8b|�rybjijbj�H�Foq[^]?_^a �hijbø]hc^c�]he�ry|^[^b�bj[^_^bjzFoxry]hefb£_^børtoqijb£�fb�ry|poq�Hbj]h[^bøaHzHzHaH�Hbj]h[poqijb��CzH]hk8a�oq_aH�faHk8c^bj]Y[^aHiji�� J�����*bjip[�|^k8aHef]�k`oqzHzt�^bj[po� { � � � |poq[^_^]��fb»efaFoqijbj�H�Fo�|^[`|^[^_^aHe)�p]��bj[s�haHzHaq�^iuoYk`o���hbj]he<cpoqe�rya�_^aHb��fbj��ryaHk8bmoqefb�ryk8a¡rybjzHb»oq�f�fa �h[po�oqi�efbj�f|^i�rtoxry]Y_^aHimzFoqijzH]hij]bji 9^¼p¾nµ»º©° ��� � |^aH��ry]:�qoqij]hefa�c^|�}]�aH�f�faHefa2c^iuoq|^�fbjg^bjija�aV~�|^bj[^_^bK[^]h[?bj[Q�pzHbuoqefa�i�� aH�fa¥�zH|^�Hbj]h[^a2_^aHijijaV�f|^zHzHaH�f�fb��ha`]hc�aHeyoq�Hbj]h[^b�� ��|�rfrtoF��buo�oq[^zt�^a2bj[?rtoqi#zFoq�f]��qo��haHefb��pzFoxry]�faNbji^[�|^k8aHef]�zt�^a�}aNoq[^_poxry]�bj[8|^[^_^aHe)�p]���[^]h[`�fbuo�c�]hb»|�rybjijbj�H�Foxry]BzH]hk8aX_^b���bj�f]hefa\_^b|^[po��f|^zHzHaH�f�fb��qo�]hc�aHeyoq�Hbj]h[^a�aBzHefaFoqefaq�^_^|^[^~�|^aq�»|^[��^|^i�ryaHefbj]hefaN�fb�ry|poq�Hbj]h[^aN_�� aHefef]hefaq�

��� �¥Ä�«�Ĩ¡Ãj¨¡­®¢¡Å8§w§q¤¥ÃÈ­�Å`¢¡­�Ã�Ët¦ ��� Ç��6§VÇ�¤y¢tË¡Ãj¨¡Ã�¦xÄm¢�6CÄpÃÈ­ §©ÊXÃȪ»Ç�¤y¦FÆ�¦x­�­®¦wÆqÃXÆq«p¢�Äp«�Å`¢¡¤¥ÃøÅ8§�Ë��ËE��ÃÈÄ�§8Äm¦xÄ/Í¢tÊ£ÆqÃK¨3¦xª¬ÃÈ­®¦3ÊC¤f§¥ÇsÇ�¤y¢t¨3¢¡Äp­ § �HÃȪj¢V¢t¨F§q­�­ §qÅ`¢¡Äp­®¢ � � ¢¡¤yËE���¢��� ������� ������£]h[^�fbj_^aHefbuoqk8]?bjiøzFoq�f]bj[>zH|^b�[^]h[P�fb��haHefb��pzt�^bj[^]��fb�ry|poq�Hbj]h[^bøaHzHzHaH�Hbj]h[poqijb��CzHbj] �a6bjiefbj�f|^i�rtoxry]^�����X_^aHiji�� ]hc�aHeyoq�Hbj]h[^a��fbuo�oq[^zH]heyo�eyoqc^c^efaH�faH[srtoqg^bjijaDbj[ �B� Ô |^c^c�]h[^buoqk8]bj[^]hi�ryefa�zt�^a:bji<�fbj��ryaHk`o�oqefb�ryk8a¡rybjzH]�o6c^efaHzHbj�f]h[^a �p[^b�rto � �soqeyoq[srybj�fzFo�iuo6k`oq�f�fbjk`ooqzHzH|^eyoxryaH�H�Fo�_^bj[poqk8bjzFo��»]3���haHef]��fbuo8rtoqija�zt�^aVbji�efbj�f|^i�rtoxry]���`_^bK|^[�� ]hc�aHeyoq�Hbj]h[^a�l � c��_^b5AmaHefbj�fzFo�_poqiNefbj�f|^i�rtoxry]�zFoqijzH]hiuoxry]�bj[ oqefb�ryk8a¡rybjzFoo�c^efaHzHbj�fbj]h[^aDbj[Q�p[^b�rto���_^bN|^[po~�|poq[sryb�rx}o�zt�^a8}a�bjim�f]hij]2aHefef]hefa�_^b�eyoqc^c^efaH�faH[srtoq�Hbj]h[^a�_^b��Vbj[ �B�»oF����bjaH[^a�zHbj] �a��

��YÏ � � �2Ï� 0 ; �O?<Ï 0 ; ��� � ?zH]h[

�,Ï 0 ; �"?� Ï 0 ; � ?

®i efbj�f|^i�rtoxry]w_^b |^[�c^ef]�_^]qrfry]wryeyo�_^|^a:[�|^k8aHefb£aH�yoxrfrtoqk8aH[srya`eyoqc^c^efaH�faH[srtoqg^bjijb�� �ha¥�[^aHeyoqijk8aH[srya:[^]h[)Í¢�¢t¨F§q­�­ §qÅ`¢¡Äp­®¢�¤f§¥ÇsÇ�¤y¢t¨3¢¡Äp­ § �HÃȪj¢:bj[*~�|poq[sry]��faq� oq_*aH�faHk8c^bj]^�<�fbk8]hi�rybjc^ijbjzFoq[^]N_^|^a\[�|^k8aHefb»_^bpr zHb�lnefaXbjipefbj�f|^i�rtoxry]Bc^|�}]�aH�f�faHefa ; oqipc^by}|�?KzH]hk8c�]h��ry]Y_po=©røzHb�lnefa\aH_?}a�_^|^[^~�|^a�[^aHzHaH�f�yoqefbj]�|^[?­�¤y¦xÄmËy§qÅ`¢¡Äp­®¦ ; ]V|^[�§q¤¥¤y¦x­®¦xÄ�Æs§qÅ`¢¡Äp­®¦@?C�fa\rB}aiuoVc^efaHzHbj�fbj]h[^a�_^aHi��fbj��ryaHk`o2oqefb�ryk8a¡rybjzH]^�� bj�f�yoxry]6|^[��fbj��ryaHk`o6oqefb�ryk8a¡rybjzH]6o6c^efaHzHbj�fbj]h[^a��p[^b�rto�_^bCgpoq�fa�( Ï �����dc^efaHzHbj�fbj]h[^a�CÏ =�aH_��& 6 DÏ�Ò =�? �& 98h�6Ï =��^zt�^aY|�rybjijbj�H�Hbmbjimryef]h[^zFoqk8aH[sry]^�^�fbuo$�

�RÏ ��Ó>= � � ��� �� Ï ��Ó>= � � ��� �

Page 83: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# � 1

oqijij]heyo���Ï� 0 ;�; ��Ó>= � � ��� � ?�� ; ��Ó>= � � ��� � ?�?<Ï���Ó � = � ��� � <Ï ��Ï���Ó � = � � ��� �

" � ÃCÆqÃ�§`«�Ä�¢t¨3¢¡ÅNÇ�Ã�¦8Æqã¦tÇp¢¡¤f§3ÉHÃ�¦xÄm¢���� Ç���ËE�p¢øÇp¦3¨t¨F§BÇ�¤y¦FÆq«�¤¥¤y¢YÃD5x§qªj¦x¤¥Ã ��!� �¢ !�§L! ���������� �����

6�qoqij]hefbYbj[^_^bjzFoxrybY�f]h[^] bj[sryef]�_^]qrfrybYbj[�|^[��fbj��ryaHk`o oqefb�ryk8a¡rybjzH]P��rtoq[^_poqef_ J�����~�|poq[^_^]`�fb�aH�fa �h|^]h[^]8_^aHijijaB]hc�aHeyoq�Hbj]h[^bm_^aHi�rybjc�] �

�28 �)�� � ����������

� 6Ï �� ? � * �PÒ �

Ô a8�fb<zH]h[^�fbj_^aHeyo6|^[�fbj��ryaHk`o6oqefb�ryk8a¡rybjzH]:l � c��2zH]h[�f]�hijbuo:_^bCZ � ��Ï$��� � � � aH_|^[pozH]h��rtoq[sryaYefaFoqija ��Ï���Ó>= � ��� � � � �^i�� ]hc�aHeyoq�Hbj]h[^a��fa �h|^aH[srya�c^ef]�_^|^zHaY{Noh{ ���Ï �� Ï �28 �

v�aH[sryefa�bji��qoqij]hefa��!� /�fbd]qrfrybjaH[^a�~�|poq[^_^]`�fb�aQAma¡rfry|po8|^[po2_^b���bj�fbj]h[^aBc�aHe\�HaHef]`[^aHi�fbj��ryaHk`o2oqefb�ryk8a¡rybjzH]�l � c��C|�rybjijbj�H�Foxry] �

� � ��Ï �� ? � * �PÒ � �Ó

��� Ã<¨®Ç�Ã�¢�4 ��à ª�§�Æqà ��¢¡¤y¢¡Ä�É3§:­�¤f§6ª�� ¢®Çp¨¡ÃȪj¦xÄ�Å8§�ËtËE��ÃÈÄ�§ ��� � ¢yÆ�ÃȪKª¬Ã65q¢¡ªÈªj¦�Æqã«�Ä�Æ�¢¡¤ � ¦5# % :�ÃÈÄD«�Äw¨¡Ãj¨¡­®¢¡Å8§:§q¤¥ÃÈ­�Å`¢¡­�Ã�Ët¦ ��� Ç���Â�à ÌH«p¢t¨¡­®¢VÆq«p¢�ÌH«^§qÄp­�ÃÈ­�ͧ �� §! �ø«^§qªj¢VÆ�¢¡ªÈªj¢�Æq«p¢VÆqà Çp¢¡Ä�Æ�¢N¨3¦xªj¦:Æs§qª�Äp«�Å`¢¡¤y¦`Æqãˡà �¥¤y¢�Æ�¢¡ªÈª�§8Å8§qÄp­�Ãj¨t¨F§ ��J�J �ø«^§qªj¢VÆ�¢¡ªÈªj¢�Æq«p¢VÆqà Çp¢¡Ä�Æ�¢N¨3¦xªj¦:Æs§qªÈª�� ¢t¨®Çp¦xÄm¢¡Äp­®¢����ËS �ø«^§qªj¢VÆ�¢¡ªÈªj¢�Æq«p¢YÄm¦xÄ�Æqà Çp¢¡Ä�Æ�¢�Æs§qª�­�à Çp¦:ÆqÃC§q¤¥¤y¦x­®¦xÄ�Æs§qÅ`¢¡Äp­®¦2«�­�ÃȪ¬Ã�ÉtÉ3§q­®¦ �� Æ! �ø«^§qªj¢YÆ�¢¡ªÈªj¢YÆq«p¢�Äm¦xÄ�Ëy§qÅ �HÃ�§`ÌH«^§qÄ�Æ�¦�¨¡ÃK«�­�ÃȪ¬Ã�ÉtÉ3§qÄm¦`Æ�¢¡ÃKÄp«�Å`¢¡¤¥Ã<Æ�¢¡Äm¦x¤¥Å8§qª¬Ã�ÉtÉ3§q­�à ���������� �����

®isijb��haHijij]£_^bh|^[^_^aHe)�p]�� ; Z � � ?�}a<bjihc^by}|�c^bjzHzH]hij]�[�|^k8aHef]�k`oqzHzt�^bj[po�c�]h�fb�ryb��h]^�3]3���haHef] �

� ���>Ï>��Ó�� �-(��P�� � Ï (��P�� � � ��<� aHc^�fbjij]h[�k`oqzHzt�^bj[po ; ��� � ?�}a�bjiKc^by}|?c^bjzHzH]hij]:[�|^k8aHef]6k`oqzHzt�^bj[po:c�]h�fb�ryb��h]�zt�^a2_�}ozH]h[sryefbjg^|�ry]`oqijiuo��f]hk8k`o�l � c��CzH]h[ ��Bp]3���haHef] �

Page 84: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

1 �

���� ��� � Ï 0 ; � 5 ��� � ? .��Xbj�f|^i�rto$�

��� � Ï ( � � � � �� �� ��� � 8@ � ��; o@?ø�<� ��� � �; g�?��<� Z � �<�; z�?��<� Z � �<�; _�?#�doBc^efaHzHbj�fbj]h[^aXefaHiuoxryb��qoN_^aHijiuo�k`oqzHzt�^bj[po�[^]h[2�qoqefbuo��faX�fbp|�rybjijbj�H�Foq[^]�[�|^k8aHefbp_^a¥�[^]hefk`oqijbj�H�Foxryb�� 3NipzH]h[sryeyoqefbj]�i�� bj[sryef]�_^|^�Hbj]h[^a£_^aHbp[�|^k8aHefbp_^aH[^]hefk`oqijbj�H�Foxryb�c�aHefk8a¡rfryaoqi��fbj��ryaHk`o�_^bXoF�haHefaw|^[ % Ä�Æ�¢¡¤ � ¦5# 4q¤f§sÆq«^§qªj¢¡� zt�^a�}a6bjk8c^ijaHk8aH[srtoxry]?efbj�faHe��qoq[^_^]�qoqij]hefbm�fc�aHzHbuoqijb�oqi�zFoqk8c�]2_^aHiji�� aH�fc�]h[^aH[sryaq� ��|^[^~�|^aYiuoV�f]�hijbuo�_^bdZ � ���qoqefbuo��Ô a��fb�zH]h[^�fbj_^aHeyo2|^[��fbj��ryaHk`o2oqefb�ryk8a¡rybjzH]�o�c^efaHzHbj�fbj]h[^a �p[^b�rto

� Ï ; ( ?���?��& 6 .?��& 98h�"?zt�^a�oq_^]hc�aHefb�[�|^k8aHefb�_^aH[^]hefk`oqijbj�H�Foxryb�oqijij]heyo`}a�c�]h�f�fbjg^bjija�eyoqc^c^efaH�faH[srtoqefa��

��ÓE�Q�hÓ�Ó�Ó�Ó�Ó�Ó �� ��� � � ( �P�� � Ï ��Ó�� �-( �P�� � � �r

��ÓE�O�Q�hÓ�Ó�Ó �� ��� � � ( �P�� � Ï ��Ó�� �-( �P�� � � �rÓ�Ó�Ó Ó�Ó�Ó

��ÓE�O��Ó�Ó�Ó�Ó�Ó�Ó��� ��� � � ( �P�� � Ï ��Ó�� �2( �P�� � � �r

_^|^[^~�|^aYiuo��f]�hijbuo�_^bdZ � �_^b��haH[srto ��Ó�� �-( �P�� � � �®���� ÃK¨®Ç�Ã�¢�4 ��Ã#ÃȪ �F¢¡Äm¦xÅ`¢¡Äm¦`Æ�¢¡ªÈª�§�Ëy§qÄmËt¢¡ªÈª�§3ÉHÃ�¦xÄm¢ �

� ������� �����

�dowzFoq[^zHaHijiuoq�Hbj]h[^a�}a`|^[�lnaH[^]hk8aH[^]wrybjc^bjzH]w_^aHb �fbj��ryaHk8b£o6Ç�¤y¢tË¡Ãj¨¡Ã�¦xÄm¢ 6CÄpÃÈ­ §q� �<�f�f]�fb��haHefb��pzFo2o�zFoq|^�yo2_^aHi�[�|^k8aHef]8_^b�zHb�lnefa �p[^b�ry]�_^aHijiuo�k`oq[srybj�f�yo�c�aHe�eyoqc^c^efaH�faH[srtoqefa|^[�[�|^k8aHef]:l � c���a�c^|�}]6_^a¡ryaHefk8bj[poqefa�iuo:c�aHef_^b�rto:_^bKzHb�lnefaV�fb*�h[^b��pzFoxryb��haV~�|poq[^_^]6�fbaH�fa �h|^aY|^[po2�f]qrfryeyoq�Hbj]h[^a�ryeyo�[�|^k8aHefb\� ��bjzHbj[^bj�^�3N_2aH�faHk8c^bj]^�szH]h[^�fbj_^aHeyoxry]B|^[8�fbj��ryaHk`oBoqefb�ryk8a¡rybjzH]�oBc^efaHzHbj�fbj]h[^aU�p[^b�rto�zH]h[ (�Ï ������CÏ " � �& 6 DÏ/ÒG��=�� �& 98h��Ï ��=���aH�fa �h|^aH[^_^]�iuoY�f]qrfryeyoq�Hbj]h[^aXl � c��#_^aHb�_^|^aN[�|^k8aHefb�fa �h|^aH[sryb �

Page 85: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# 1 �

��Ó���=$� " � ��� � Ò=��Ó���=$��� � ��� � Ï>��Ó �O�O�Q� � ��� � Ï>��Ó����O�O� � ��� �fb�]qrfrybjaH[^aC|^[�[�|^k8aHef]NzH]h[�|^[po\�f]hiuoXzHb�lneyoX�fb*�h[^b��pzFoxryb��qo��3bj[�oqi�ryefaCcpoqef]hijaq�x�fb�c�aHef_^]h[^]�2zHb�lnefaB�fb*�h[^b��pzFoxryb��ha�[^aHijiuoVeyoqc^c^efaH�faH[srtoq�Hbj]h[^aY_^aHi�efbj�f|^i�rtoxry]^�®[��haH[^aHeyoqija��fa

� � � � � �

}aV|^[po:ln|^[^�Hbj]h[^aVzt�^a8oqijiuo8zH]hc^c^buo:_^b#[�|^k8aHefb#efaFoqijb ; �D?�� ?\oq�f�f]�zHbuo:bjiK[�|^k8aHef]6efaFoqija� Ï��:Ò ���d}aY[^]qry] ; z¡lneH��zFoqc���� ?Xzt�^a�i�� bj[^_^bjzHa�_^bdzH]h[^_^bj�Hbj]h[poqk8aH[sry]8_^aHi�c^ef]hg^ijaHk`o��}aB_poxry]2_po

� ; "?f�D?�� ?<Ï ;Ð �<Ð 5 Ð �dÐ ?Ð �`Ò �dÐ

�Koqija�~�|poq[sryb�rx}o�zHefaH�fzHa`oqi<_^bjk8bj[�|^bjefa�_^aHijiuo:_^bj��rtoq[^�Fo:ryeyoY��a h� �£bj] �a�iuo�zFoq[^zHaHijiuo©��Hbj]h[^aV}aNi�� aQAma¡rfry]�_^aHi ��¼p¾ �hµ�² ��³n´�³nµ�²d¼ ��°�²�À3µ:_^aHijiuoY�f]qrfryeyoq�Hbj]h[^aXryeyo�_^|^a�[�|^k8aHefbl � c���� ��bjzHbj[^bj�^�

, � � ¢¡¤Ëy§qªjËt¦xª�§q¤y¢DÃȪCÇ�«�Äp­®¦*Å`¢yÆqÃ�¦ ,ÆqÃ�«�Ä�ÃÈÄp­®¢¡¤ 5x§qªÈªj¦�� � Ê ��� Ê�ÌH«^§qªj¢Æ�¢¡ªÈªj¢�¨3¢�4q«p¢¡Äp­�Ã�Æq«p¢�F¦x¤¥Å�«�ªj¢ Í¢�Ç�¤y¢ �F¢¡¤¥Ã��HÃȪj¢`«�­�ÃȪ¬Ã�ÉtÉ3§q¤y¢`ÃÈÄ>«�Ä*¨¡Ãj¨¡­®¢¡Å8§�§q¤¥ÃÈ­�Å`¢¡­�Ã�Ët¦G��� Ç���§`Ç�¤y¢tË¡Ãj¨¡Ã�¦xÄm¢ 6CÄpÃÈ­ § �� ¢¡¤yËE���¢��

; 8�?� �Ï� � ; �D?�� ?<Ï ; � 5 � ?���=�? ; �2?� �Ï� � ; �D?�� ?<Ï�� 5 ; ��Ò��"?���=��������� �����

�øoqijzH]hijbuoqk8]�i�� bj[^_^bjzHaB_^bdzH]h[^_^bj�Hbj]h[poqk8aH[sry]2efaHiuoxryb��h] � ; "?f�D?�� ?ø_^aHijija�_^|^a�ln]hefkV|^ijaq�3 3 8�aHe\iuo�ln]hefkV|^iuo ; 8�?ø�fb��po$�� ; � ?f�D?�� ?#Ï

����

�� 7 �� ��=����5����

�� 7 �� ��=����Ï Ð �<Ð 5 Ð �dÐÐ � 5 �dÐ ; �h� �$�@?

3Nc^c^ijbjzFoq[^_^]*iuo�c^ef]hc^efbja¡rx}o�ryefbuoq[ �h]hiuoqefaw_^aHbN�qoqij]hefb�oq�f�f]hij|�ryb�oqi�_^aH[^]hk8bj[poxry]hefaD_^b� ; � ?f�D?�� ?�oqg^g^buoqk8]zt�^awiuo ; �h� �$�@?�efbj�f|^i�rto�ijbjk8b�rtoxrtowbj[�lnaHefbj]hefk8aH[srya6aw�f|^c�aHefbj]he �k8aH[srya�_poqijija�~�|poq[sryb�rx}o2�fa �h|^aH[sryb �

N N ��è�÷çútû�á -tú¥üøè�äHû�èXú¥÷»ù©é�ã��©÷ è�ü�ú�æ3á�-tú¥÷ ý3û ú¥é�èøý©äxúNþÿý©ä�%�á ã¡ä©è �$#�é�èfú¥÷ è�æ3á�-tú¥é�áçú&�©á ÷ èøé�èfú¥÷ è�#�á äVý©ä�ù©ý©äHû�ãBê #�å�áé�áçâ�ú -tú�#3â ã¡üøè��MI� �5C("�� ��CL>�I� � I��*>�I�1< " I KM> F "� ��K*�ML1> #x÷çú )Hýxú¥äHû�á ûE'ú�:� N � < ê�P�ë � � I Nÿê�PÈê � % � ��Nÿê�P

Page 86: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

1 =

Ð �<Ð 5 Ð �dÐÐ �<Ð 5 Ð �dÐ� ��� ��

� � � � 7 � � �� � 7 � � � � � � 7 � � �� � � � � � � � �Ô aø�`aH_ �Y�f]h[^]w� ��bjzHbj[^bj�\ryeyo�ij]hef]�bj[Vk8]�_^|^ij]^��Ð �<Ð ��Ð �dÐç� � ; � ?f�D?�� ?�[^]h[�}a�ijbjk8b�rtoxry]�f|^c�aHefbj]hefk8aH[srya��

ijbjk� � � � � � � Ð �<Ð 5 Ð �dÐÐjÐ �<ЩÒ�Ð �dÐjÐ Ï=»Ð �<Ð� Ï 5 �

~�|^bj[^_^bpbj[8~�|^aH��ry]�zFoq�f]�iuo�ln]hefkV|^iuo ; 8�?#[^]h[w}aXg�aH[`zH]h[^_^bj�Hbj]h[poxrto�aH_`bjipzFoqijzH]hij]B_^bZ zH]h[2rtoqija�ln]hefkV|^iuo��hbj[2|^[`�fbj��ryaHk`o�oqefb�ryk8a¡rybjzH] � o�c^efaHzHbj�fbj]h[^a �p[^b�rto���c^| �]YzH]h[^_^|^efefaoq_6|^[�efbj�f|^i�rtoxry]�[^]h[woqzHzHa¡rfrtoqg^bjijaq��doVln]hefkV|^iuo ; �2?ø�po�bj[s�haHzHaYbj[^_^bjzHaB_^b�zH]h[^_^bj�Hbj]h[poqk8aH[sry]�| �h|poqija�o$�

� ; � ?f�D?�� ?�Ï ����

�� 5 � � �� ��� �XÒ �=��������� ����

���� �������� ������� �� � ����

���� ���

������ �� � ���� ���� ���

������� �� ��� � � � ��� ���� ��� �� � �����! �"#%$&$&')()*,+.-&/&01'2+3$&450�$&45()687:9+;7<45(2+.-&=�0�'2+.456>/&6?(A@.+.')0�45(%+.B5B50�')CD7<(E+.'A/&6?-&0�FG()-H+!7<0�456I/&(J

�LK�,M M

"�N'2+� �����! �"

45()B5C&'O7P+Q')()FG(O7P+!7P+R()-DST6?45()0�45FG6?-U7<6V6WB5C&$�6?45()0�45FG6?-U7<6W/H+.')')6YX�CH+.-�Z7<(O7![+\B56?=�C&6?-U7<(L]^�^ _`^�^� ^

�^ _`^�� ��

^ba c� � ^ �^ _`^�^

� �_Vdfehgi � a ^

�^ _`^�^^ � ^

�^ ^�� ��

^ ^j ->X�C&6?Bf7<0k*,+.B50 N B56 6?/ B50�-&0;lm@�()*?()-&()n\7<4<+o')0�450 N qp N

+.')')0�4<+D]r ')()F�<sQ�

^�^ _`^�^� ^

�^ _`^�� ��

^ �� ^�^� ^�^ � �r ')()F

�<sQ�^�^ _`^�^^ � ^

�^ ^�� ��

^ ^ �� ^�^� ^�^ � �6t$�6?4u()'�7<6?0�456?Fk+o/&6?'�*?0�-DST450�-U7<0GB5(�vH+D]')()F�<sQ�

�� ��� � � � ��� ���� ��� �� � �xwy 6?4f7P+.-U7<0 N -&6?'�*,+.B50G()-q*?C&( 6?/ B5(2+.-&0;lm@�()*?()-&()n\7<4<+o')0�450 N '2+zST0�45F\C&'2+ �|{ "

45()B5C&'O7P+} 6?-~*?0�-&/&()�?()0�-H+!7P+>6�/&C&-&X�C&6q()'R*,+.')*?0�')0�/&(Q��*?0�-17P+.')6GST0�45F\C&'2+ N ()-1C&-1B5()Bf7<6?Fk++.45(O7<FG687<()*?0���+o$&456?*?()B5()0�-&6A�H-&(O7P+ N *?0�-&/&C&*?6z+./qC&->45()B5C&'O7P+!7<0�+.*?*?68757P+ } ()')6 �

Page 87: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# 1�2

3 0�-&B5()/&6?45(2+.FG0�C&-�B5()Bf7<6?Fk+q+.45(O7<FG687<()*?0q� +�$&456?*?()B5()0�-&6o�H-&(O7P+�*,+.4<+!757<6?45()�?�,+!7<0q/H+.(B56?=�C&6?-U7<( $H+.4<+.FG687<45(L]� ]54

� � 6 M�7 � M98 �;:=<

� > 1 M98 �@? ���

1 M6 B5C&$&$�0�-&(2+.FG0o*Pv&6z()-�� @�6?-&=U+GCD7<()')()�?�,+!7P+ N $�6?4u'2+o4<+.$&$&456?B56?-U7P+.�?()0�-&6 /&6?( -�C&FG6?45( N'2+ASTC&-&�?()0�-&6 /&(BA,CED F�GEH%I�JKFLAD � 3 +.')*?0�')(2+.FG0%()-k� ()'&$&C&-U7<0 FG6?/&()0E� $�6?4W')6 B56?=�C&6?-U7<(*?0�$&$&()6�/&( -�C&FG6?45(�()- $H+.456?-U7<6?B5(\[6�45()$�0�4f7P+!7<0 ()' @.+.')0�456>/&( � *Pv&6IB5(%0.757<()6?-&6>()-+.45(O7<FG687<()*,+\+�$&456?*?()B5()0�-&6E()-D�H-&(O7P+ "�N /&()Bf7<()-&=�C&(2+.FG0\/&C&6t*,+.B5(L]����" 6?/

lm@�()*?()-&()n\7<4<+o')0�450o()-qFG0�/&C&')0&]��� + " �

2 w M%NOM%P M �2 w M%N�N�P M �

��

6 w 2 M%NOM 2RQ � 6 c " S��� } " �2 w M%NOM%P M �

> 2 w M%N�N�P M ���

6 w�� Q � 6 T " S� ��" 6?/

n�/&()Bf7P+.-U7<()n&]� � + " �2 w M%NOM%P M �

� w N � ��� M ���

6 w 2 � 1 � �� Q � 6 c " S� � } " �� w 1 M%N�P M �

> N w M%NOM%P M ��� > 6 wO� P 1 ��� Q � 6 c " S� � * " �

2 w M%NOM%P M �6 w � N ��� Q � 6

� M���

6 wO� P�P � �%6 ��� Q � 6 c " S� � / " �� w 1 M�M � M �

> N ��w 2 � NOM M ��� > 6 w � 1 � M N Q � 6 � " S� � 6 " � > � w M � 6U� M �

> N w 2 N�P � � M ��� > 6 w ���� 1�2 N Q � 6 c "��j -&/&()*Pv&(2+.FG0I*?0�-VU 6WU

N'2+�4<+.$&$&456?B56?-U7P+.�?()0�-&6�/&( 6

()- �b6?/~+.-H+.')()�?�?(2+.FG0>')6B50�')C&�?()0�-&(XU�;Y M U�;ZQ0.757<6?-�CD7<6\()-q7P+.')6 B5()Bf7<6?Fk+G45()B5$�68757<(O@.+.FG6?-U7<6t*?0�-I')6tST0�45F\C&')6

�? "6

�|{ "]

U U

U�;Y U�;Z

��� + " 6 w 2 M%NOM Q � 6 c 6 w 2 M%N�N Q � 6 c 6 w 2 M%N�N Q � 6 c 6 w 2 M%NOM Q � 6 c��� } " 6 w 2 M%NOM Q � 6 c > 6 w 2 M%N�N Q � 6 c 6 w 6 Q � 6%[ 6 wO� Q � 6 T� � + " 6 w 2 M%NOM Q � 6 c 6 w � N � � Q � 6 c 6 w 2 � 1 Q � 6 c 6 w 2 � 1 Q � 6 c� � } " 6 w � 1 M%N Q � 6 c > 6 w NOM%NOM Q � 6 c > 6 wO� P 1 � Q � 6 c > 6 wO� P 1 � Q � 6 c� � * " 6 w 2 M%NOM Q � 6 c 6 w � N ��� Q � 6

� 6 wO� P�P Q � 6 c 6 wO� P�P � Q � 6 c� � / " 6 w � 1 M�M Q � 6 c > 6 w N � 2 � Q � 6 � > 6 w � 1 ��� Q � 6 � > 6 w � 1 � N Q � 6 �� � 6 " > 6 w � M � 6 Q � 6 c > 6 w N 2 N�P Q � 6 c > 6 w ���� 1�Q � 6 c > 6 w ���� 1�Q � 6 c\ (tB56?=�C&(O7<0 B50�-&0 45()$�0�4f7P+!7<(A=�')(E6?45450�45(t456?'2+!7<(O@�(^]_Y 6?/`]_Z N *Pv&6;B5( *?0�FGFG68757<0�-&0-&6?')'=a +.$&$&450�B5B5()Fk+.456A� N 45()B5$�68757<(O@.+.FG6?-U7<6E*?0�-bU�;YE6?/bU�;ZP]]_Y ]_Z

��� + " 6 w 2 �� Q � 6 T 6 w M%N 1RQ � 6

c

��� } " 6 wO� Q � 6 c 6 w P Q � 6%[� � + " 6 wO�, 6 P Q � 6 T 6 wO�, 6 P Q � 6

T� � } " 6 w � N 2 M Q � 6 T 6 w � N 2 M Q � 6

T� � * " 6 w N 2�1 Q � 6 T 6 wO� 6 1RQ � 6

T� � / " 6 wO��� M%N Q � 6 T 6 wO�, N Q � 6

T� � 6 " 6 w M%P 1�Q � 6 c 6 w M%P 1RQ � 6

c

Page 88: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

1 � B5B56?4f@.+.-&/&0z'2+A7P+ } 6?')'2+ N 6?-U7<4<+.F } 6u')6QST0�45F\C&')6 $&450�/&C&*?0�-&0 45()B5C&'O7P+!7<(H+.*?*?68757P+ } ()')( N ()-$H+.4f7<()*?0�'2+.456A$�6?4 (�*,+.B5(�*Pv&6z+ }&} (2+.FG0o/&()Bf7<()-U7<0&]����" 6?/

lm@�()*?()-&()nz7<4<+�')0�450o()-qFG0�/&C&')0� � � p � � "

]'2+ ST0�45F\C&'2+�|{ "

$&450�/&C&*?6Q45()B5C&'O7P+!7<(�$&(.9C\+.*?*?C&4<+!7<(� ���

+ "�N&��� } "V"�N ()-DS|+!757<(�=�')(�6?45450�45(0.757<6?-�CD7<(`B50�-&0o/&6?')'=a 0�45/&()-&6 /&6?')'2+ $&456?*?()B5()0�-&6A/&6?' B5()Bf7<6?Fk+ N 0�+.' $&(.9Ck()'�45()B5C&'O7P+!7<0/&(�� 6?45()B5*?6t/H+.')'2+zB50�')C&�?()0�-&6E6?B<+!757P+ N B5C&')'=a C&'O7<()Fk+ *?(OST4<+\B5()=�-&(O�H*,+!7<(O@.+ S� ��" B56 6?/ B50�-&0G/&()Bf7P+.-U7<(`7<4<+o')0�450

�*?0�-&*?0�45/&(�0�/&()B5*?0�45/&( " ]6?-U7<4<+.F } 6q')6�ST0�45F\C&')6�$&450�/&C&*?0�-&0 ')0 Bf7<6?B5B50;45()B5C&'O7P+!7<0 -&6?( *,+.B5( � � + "�N � � } " 6� � 6 "�N FG6?-U7<456t-&6?'�*,+.B50 � � * " 6 � � / " [6t$&(.9Cq+.*?*?C&4<+!7P+�'2+zST0�45F\C&'2+ �|{ "

S

P � ��� J�� JKF�G� � F�D� �LJ�� � A���H�� � D F ��� F G�� ��� �.GEH � G�D�� D�/J���� H�� D C�I���� H��

>

{���

{� > ?! � ?

$&C [0G/H+.456t')C&0�=�0�+./q6?45450�45( N ()->+.45(O7<FG687<()*,+#"H0U+!7<()-&=!Z $�0�()-U7 �$&%�'�(*)�+,(�-�.

�j " / 6 } [6kB5C10G*?()6?-U7<6?FG6?-U7<6�=�4<+.-&/&6G()'W*,+.')*?0�')0q/&( {

� $�0.7<456 }&} 6k*?0�FG$�0�4f7P+.456�C&-0h@�6?42"H0�3 ��j5j " 3 (H$�0.7<456 }&} 6?450 6?B5B56?456 6?45450�45(H/&(&*,+.-&*?6?')'2+.�?()0�-&6Q/&()Bf7<45CD757<(O@.+AX�CH+.-&/&0 B5(H6�� 68757<CH+()'`*,+.')*?0�')0o/&6?'�/&()B5*?45()FG()-H+.-U7<6 �3 0�-&B5()/&6?45(2+.FG0 N +./q6?B56?FG$&()0 N C&-qB5()Bf7<6?Fk+�+.45(O7<FG687<()*?0\+o$&456?*?()B5()0�-&6A�H-&(O7P+\*?0�- 4

�� 6 N 7 � &N 8 �;:=<

� > � 6 N 8 �@? � � 6 S +.')')0�4<+zB56.]� ��"

?

�6 wO� Q � 6 c { �

6 w � Q � 6 4

�6 w 2RQ � 6 c�

� " ? �6 wO� Q � 6 c { �

6 wO� Q � 6 c

�6 w 2RQ � 6 cB5(�vH+D]

� ��" {��

6 w Q � 6 c c �� @�6?42"H0�3 "�

� " 5 �6 wO� Q � 6 � > 6 wO� � Q � 6 � �

6 wO� Q � 6 � �3 +.-&*?6?')'2+.�?()0�-&6 "1 � ��� ����6�6LD F�7OH8 � C � � D��:9%JKCEJ;��F=< J?>@��H�� � D F�JA � � J�G�D FB/DA7%C0H�/DXG�D IC6B� JKA�H8/J��BA � 6LDD�

? �� {

�6

��A � � � �� H%FB/D#��F8� � �KAJKI HRH%C � A,I�JKA � G�DE�@F 6GF#H�F�D C�I H � � �� H%ADI G�D F�4

� � 6 7 �2 J ?

� ��w ����J{ �2 w 2 J �

� w � P FH HKIMLN��H �#OJ � �P9 H � D CEJ GEH � G�D�� H%ADQ/J��R � � GKC � I � F&H%FLAJ

{� > ?! BSHUTUI8LN��H ��OJ � ��9 H � D CEJ G�D C�CEJKA,ADV/J��= � � GKC � I � F&H%FLAJ � F H%C � A,I�JKA � GEHRH�6&CEJ�G � � � D F�J � FXW5F � A�HGS

H*GYIMLN��H �#OJ;�Z< JKC�CED CEJ CEJ�� H%A � 9%D�F�J��G9 H � D CEJRGEH � G�D�� H%ADQ/J��R � � GKC � I � F&H%FLAJ1S

Page 89: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# 1 �$ %�'�(*)�+,(�-�.

j -&/&()*,+.-&/&0�*?0�- U5 ()' /&()B5*?45()FG()-H+.-U7<6�*,+.')*?0�'2+!7<0 6q*?0�-&5 X�C&6?')')0 ()- +.45(O7<FG687<()*,+I+$&456?*?()B5()0�-&6t()-D�H-&(O7P+\B5(�0.757<()6?-&6.]�? "

U5

�6 wO��������� N Q � 6 � > 6 wO������� N Q � 6 � �

6

�|{ "5

�6 w � 1 � Q � 6 c�

" ] ����� �^ � ���^^ �V^ � � � � 6�6�

� 6 � ��� G�D F1� � /JKC � �Z< J�Y6&CEJ�� � D F�JX� �� > > �� �

MH HKI�_JKCV>@��H � J � FLAJKCK9 H ��� D8 � 9 H � D C � � � OJ� � � G � � JA��F GEH � G�D�� D;H/G�G���C0H%AD � F H%C � A,I�JKA � GEH�@F 6GF?SHUTUI ��� � H ��F=< J�Y6&CEJ�� � D F�J J?>@� � 9 H � JKFLAJ /J���� H � D C�I���� H � F I�DX/D G�LJ J 6LJKC � 9 H � D C � � �/J���6B��FLADMH HKI J � �.GEH � G�D�� DV� � H�6 � O� H/G�G���C0H%AD � F H%C � A,I�JKA � GEH �@F 6GF

$ %�'�(*)�+,(�-�.

�+ " / (V$�0�B5B50�-&0�+,@�6?456�$&450 } ')6?FG(�-&6?'Y*,+.')*?0�')0�/&6?')'=a 6?B5$&456?B5B5()0�-&6oX�CH+.-&/&0 K

�� " -&0�-/ [+�*?0�-U7<45( } CD7<0�-&6?')'2+\B50�FGFk+��!/&(�� 6?456?-&�,+\*?0�- ��� j -DS|+!757<(�B56 � K

�� "h� ����� Y����P]

�� > K�� "

> �� � K�� "

� 6

#%/k6?B56?FG$&()0 N B56 *?0�-&B5()/&6?45(2+.FG0tC&-kB5()Bf7<6?Fk+ +.45(O7<FG687<()*?0t+t$&456?*?()B5()0�-&6 �H-&(O7P+t/&( } +.B564

� � 6 N 7 �P 8 �;:=<

� > �%6 N 8 �@? ��%6 N B56 � � 6

� +.')')0�4<+D]�� > � 6

� > �� � � 6 � � � > � � 6

�} " / 6 N ()-U@�6?*?6 N B5(�*?0�-&B5()/&6?4<+�'=a 6?B5$&456?B5B5()0�-&6t()/&6?-U7<()*,+D]� �� >

> �� � � �

� � > �()-oX�C&6?Bf7<0t*,+.B50 N -&6?')')0ABf7<6?B5B50tB5()Bf7<6?Fk+E+.45(O7<FG687<()*?0 /&6?'&$&C&-U7<0 �

+ " B5*?6?=�')()6?-&/&0E+.-&*?0�4<+� � 6

� ] � � � 6 �

� > � 6 cc �

� � � 6 �

� > 6 �� � � 6

����

6 w

Page 90: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

1 N

����� � J qp G � � � H*�Y6LJKA,A�HQ��F JKC�CED CEJV � GEH%F�G�J���� H�� � D F�J F�J��.GEH � G�D�� D� � �

�������

� " > �����

� "� H � JN6&CEDDT@� JKI H�� � 6B��OD C � � D��:9%JKCEJ D��� JKCK9 H%FB/DXG�LJX�

�������

� " > �����

� " ������ �

� w� D F >@�LJ�KAD H/G�G�D C 7 � I�JKFLAD�� � 6B��OD H�� JKC�I H%CEJ G�LJ � �"GEH � G�D�� D; � � D?7 H ��� I OJ,6 � O� H/G�G���C0H%AD=S

$&%�'�(*)�+,(�-�./ ( N $�6?45*Pv 96 B56

� ���� "+.')')0�4<+

������ " �

������ " �

��������� "5" M6�()'E*,+.')*?0�')0~/&( �

������

� " > �����

� "$&C [0 *?456,+.456;$&450 } ')6?FG(A/&(t*,+.-&*?6?')'2+.�?()0�-&6/&()Bf7<45CD757<(O@.+ � 3 (m[0�-&0�-�+.*?*,+./&6tB56t�E@�()6?-&6E@.+.')CD7P+!7<0o-&6?'�FG0�/&0�B56?=�C&6?-U7<6.]

�+ " 3 +.')*?0�')0o/&(

���} " 3 +.')*?0�')0o/&( �

�������

� ���

3 0�-&B5()/&6?45(2+.FG0k()' B56?=�C&6?-U7<6�B5()Bf7<6?Fk+�+.45(O7<FG687<()*?0k+k$&456?*?()B5()0�-&6 �H-&(O7P+ N � ]_4

� � 6 M7 �M M"8 �;:=<

� > � 6 M"8 �@? � � 6 N B5(2+.-&0&]

�6 w � � 6 1�1�1�1 Q � 6 c M �

6 w � ��� 6�6�6 � Q � 6 c3 +.')*?0�')(2+.FG0z(�')0�=U+.45(O7<FG(:/&( 6]

������ " �

6 w 2�1�1 NOM 2 � Q � 6 [������ " �

6 w 2�1�1 NOM 2�1RQ � 6 [� 6?'�*,+.')*?0�')0G/&6?')'2+G/&(�� 6?456?-&�,+kST4<+GX�C&6?Bf7<(�@.+.')0�45( N B5(�$�6?45/&0�-&0k7<CD757<6\')6z*?(OST456 B5()=�-&(O�DZ*,+!7<(O@�6 N +o*,+.C&B<+�/&6?')'2+\*,+.-&*?6?')'2+.�?()0�-&6E/&()Bf7<45CD757<(O@.+ S ()-DS|+!757<( B5(�vH+D]������ " > �����

� " �6 w 2�1�1 NOM 2 � Q � 6 [ > 6 w 2�1�1 NOM 2�1 Q � 6 [ �� > 6 w 6�6�6�6�6�6�6 Q � 6 [ � > 6 w Q � 6

4

Page 91: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# 1 M

3 +.')*?0�')(2+.FG0z0�4<+o()'`4<+.$&$�0�4f7<0o7<4<+ 6]

�6 w 1�1�1�1�1�1 � Q � 6 [/H+�*?C&(L]

����� � � � > 6 w 2 M 2 � M Q � 6

�()'�45()B5C&'O7P+!7<0�0.757<6?-�CD7<0kvH+ M *?(OST456tB5()=�-&(O�H*,+!7<(O@�6t6?/;[6 $&(<[C>+.*?*?C&4<+!7<0G/&6?' $&456?*?6?/&6?-U7<6 N()-qX�CH+.-U7<0G()'`45()B5C&'O7P+!7<0 N ()->+.45(O7<FG687<()*,+\+o$&456?*?()B5()0�-&6E()-D�H-&(O7P+ N [6.]����� �

� �

������ " > �����

� " �> 6 w 2 M 2 � M � �� 6 N 1 6�6 ��wOwOw Q � 6

� � � � J � OJ#��F 9%JKA,AD CEJV ����� J � H�F�D C�I H�� �LG�� � /JEH�OJA/J W5F � A�H8OH��

�� � �

� � ��c �� c�� �

� D I�J OJN6LD��� � T � � J J�9 � A�H%CEJ#��F D�9%JKC��_D�� F�J�� GEH � G�D�� DQ � A�H � J F�D C�I HGS

$ %�'�(*)�+,(�-�.

3 +.')*?0�'2+.-&/&0 '2+ -&0�45Fk+��YC&*?')()/&6,+Q*?0�-t'=a +.*?*?0�45=�()FG6?-U7<0QB56?=�C&6?-U7<6 N $�0�Bf7<0 U

�Fk+�� �� c�� �

]�� � � U

������� ��c�U

� �

\ (O@�()/&6?-&/&0k$�6?4%()' $&(<[C>=�4<+.-&/&6z/&6?=�')( 45()/&C&*?(2+.FG0G'=a 0�45/&()-&6t/&( =�4<+.-&/&6?�?�,+�/&6?( -�C�ZFG6?45(�6EX�C&()-&/&(�68@�(O7<(2+.FG0o'=a 0h@�6?42"H0�3 �3 0�-&B5()/&6?45(2+.FG0;()' B5()Bf7<6?Fk+3+.45(O7<FG687<()*?03+;$&456?*?()B5()0�-&6��H-&(O7P+3��]94

� � 6 M07 � ��N8 �;:=<

� > M"8 �@? ��� &N

6?/>()' @�68757<0�456t/&( � T ] � �

6 wO� Q � 6� M 6 w � Q � 6

T M 6 w N Q � 6 c "3 0�-&B5()/&6?45(2+.FG0o()'`X�CH+./&4<+!7<0G/&6?')')6AB5C&6 *?0�FG$�0�-&6?-U7<(L]

�c �

6 wO� Q � 6T M�� �

6 w Q � 6�� M �T �

6 w 2 N Q � 6�

Page 92: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

1 P

'2+\*?0�FG$�0�-&6?-U7<6� +.'`X�CH+./&4<+!7<0G=�6?-&6?4<+�C&->0h@�6?42"H0�3 �/ *,+.')(2+.FG0o+./&6?B5B50G()' @�68757<0�456 $�6?4 '2+o*?0�FG$�0�-&6?-U7<6t/&(`@.+.')0�4QFk+.B5B5()FG0&]

U

c �6 wO� Q � 6 c6 w � Q � 6 T �

6 w�� Q � 6 c MU

� �

6 wO� Q � 6 c M UT �

6 w N Q � 6 c6 w � Q � 6 T �6 w 2�Q � 6 c3 +.')*?0�')(2+.FG0z()'`X�CH+./&4<+!7<0�/&6?')')6E*?0�FG$�0�-&6?-U7<( N U

]U

�c �

6 w � � Q � 6 T M U�� �

6 wO� Q � 6 c M U�T �

6 w 1�Q � 6 T

()-�X�C&6?Bf7<0�FG0�/&0�@�()6?-&6z68@�(O7P+!7<0G'=a 0h@�6?42"H0�3 � 3 +.')*?0�')(2+.FG0o'2+�-&0�45Fk+ N ()-�+.45(O7<FG687<()*,++o$&456?*?()B5()0�-&6t()-D�H-&(O7P+D] �� � � 6 w �%6�6 2�2�1 M � Q � 6

T

FG6?-U7<456t*?0�->'=a +.*?*?0�45=�()FG6?-U7<0\$&450�$�0�Bf7<0 N B5(�0.757<()6?-&6.]� U� � � 6 w �%6�6 2 Q � 6

T

*?0�FG6%B5( $&C [0\-&0.7P+.456 N '=a 6?45450�456%*Pv&6AB5( *?0�FGFG68757<6z[6%B5C&')'2+ X�C&()-U7P+\*?(OST4<+ B5()=�-&(O�H*,+!7<(O@.+ N6?/�[6E7<4<+.B5*?C&4<+ } ()')6E$�6?45*Pv 96t*?0�FG$H+!7<( } ()')6%*?0�->'2+\$&456?*?()B5()0�-&6t/&6?'�B5()Bf7<6?Fk+ �� 2 ��� �����I � H%I�D���FM� � �KAJKI HXH%C � A,I�JKA � G�D �@F 6GF(F�D C�I H � � �� H%ADXG�D FX4

� � 6!J 7 �2 J J 8 �;:=<

�> 1 P FH HKIMLN��H �#OJ;� H � D?7 � � H8 ������� 6LJKCA>@�LJ�KAD�� � �KAJKI HGSHUTUI � J

�N w POM Q � 6

� � J

�N w P � Q � 6

� � J >@��H �AOJ � �BC � ����� A�H%AD � >

S

H*GYIMLN��H � J � H%CEJXT�T�J � � C � ����� A�H%AD8 � � � � J � �=� � �KAJKI HXH 9%J�� J;��F�C&-&/&6?42"H0�3�=�4<+./&CH+.')6��$&%�'�(*)�+,(�-�.

�+ " ���� �

6 wO� Q � 6 � �

�} " >

�6 w N Q � 6

���������

�6

�* " j -qC&->B5()Bf7<6?Fk+o*Pv&6 $�6?45FG68757<6t'=a C&-&/&6?42"H0�3�=�4<+./&CH+.')6E45()B5C&'O7P+D] >

�6 w 6�6 N Q � 6

� ��

6 w N Q � 6 ���

�, &� ��� H�� ���� ��F&H ���LG�G�J�� � D F�J;I�D F�D AD F&H /J�GKCEJ� G�JKFLAJ�W5F � A�H � F ��I�JKC � 6LD�� � A � 9 � H ������ _ c 6LJKC D?7%F ��� IDF! �F >@��H � J D CZ � F�J 7 � � J�� JKI�JKFLA � /J���� H ���LG�G�J�� � D F�J /J�9%D F�D J�� JKCEJ� D IRI H%A � 6LJKC^I � F � I � �� H%CEJ��Z< JKC�CED CEJV � H%C�CED AD FBOH%I�JKFLAD=S

$&%�'�(*)�+,(�-�./ 0�FGFk+.-&/&0t(H-�C&FG6?45(:()-G0�45/&()-&6u*?456?B5*?6?-U7<6 N 0h@�@�6?450\/H+.':$&(<[Co$&()*?*?0�')0 +.':$&(<[Co=�4<+.-&/&6 NB5(:B50�FGFk+.-&0t-�C&FG6?45(�+.$&$&450�B5B5()Fk+!7<(O@.+.FG6?-U7<6 /&6?')')0tBf7<6?B5B50\0�45/&()-&6u/&(:=�4<+.-&/&6?�?�,+ � / (45()/&C&*?0�-&0�X�C&()-&/&(�68@�6?-U7<CH+.')(�6?45450�45(�/&(�*,+.-&*?6?')'2+.�?()0�-&6A/&0h@�CD7<( +.')'=a +.45450.7<0�-&/H+.FG6?-U7<0 �j -DS|+!757<(YB5(2+>� C&-�B5()Bf7<6?Fk+q+.45(O7<FG687<()*?0�+q$&456?*?()B5()0�-&6\�H-&(O7P+�/&(Y$H+.4<+.FG687<45(�4

� � 6 N

Page 93: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# 1�1

7 �M N 8 �;:=<

� > � 6 N 8 �@? ��� �

6 � / 0�FGFG(2+.FG0Q(�$&45()FG( � 6�6�6 7<6?45FG()-&(U-&6?')'=a 0�45/&()-&6������������� ]� c [[[ �

c [[[ � � [ ���� � ��" T � � � �2 T � �h�h� � ��%6�6 � T

� c [[[ �6 w�� � � P 1 P�N Q � 6 c

/ 6t()-U@�6?*?6 B5(�B50�FGFk+.-&0�( 7<6?45FG()-&(`()-�Fk+.-&()6?4<+�� ��������������� B5(�vH+D]� c [[[ �

6 w�� � � P 1�1 Q � 6 c/ ( +.B5B5C&FG6t*Pv&6 ()' @.+.')0�456qlf6?B<+!757<0�nGB5(2+D]���c [[[ �

6 w�� � � P 1�1 w,w,w Q � 6 c/&0h@�6t7<CD757<6t')6E*?(OST456E()-&/&()*,+!7<6EB50�-&0�6?B<+!757<6 �� C&()-&/&(t'=a 6?45450�456�+.B5B50�')CD7<0 *Pv&6;B5(t*?0�FGFG68757<6 CD7<()')()�?�,+.-&/&017P+.')6 Bf7<4<+!7<6?=�(2+~vH+ C&-0�45/&()-&6E/&(�=�4<+.-&/&6?�?�,+G/&( � 6 � N 0h@�@�6?450&]]

� � � c [[[ > � �c [[[ � �6 wO� � Q � 6

�� 6?-U7<456q'=a 6?45450�456 N -&6?'QB5()Bf7<6?Fk+ +.45(O7<FG687<()*?0 � N CD7<()')()�?�,+.-&/&0�'2+IB56?*?0�-&/H+ Bf7<4<+!7<6?=�(2+ N45()B5C&'O7P+\-�C&')')0 �

����� ��� �Y6 � J 7*� � 6LJKCEG���J�� H � JKC � JA � 9%JKC 7/JKFLAJX� � � � c �<6LD��� H H 9%JKCEJ ��F&H � D IRI H W5F � A�H � F`H%C � A,I�JKA � GEH �@F 6GF�

6�H%C�A � CEJ OH&>@��H � J � FB � G�J 7 � �H��/JKFB � F�D F OH%FLF�D 6 � O� G�D FLA,C � T@��AD;H ��� H�� D IRI H%AD C � H;G�D F � AJKC�I � F � 6&CEJ�G�J?/JKFLA � S

$ %�'�(*)�+,(�-�.� 0�')6?-&/&0 6?B56?=�C&()456%'2+EB50�FGFk+t()-�C&-�B5()Bf7<6?Fk+ +.45(O7<FG687<()*?0t+t$&456?*?()B5()0�-&6u�H-&(O7P+ N 6?B5()Bf7<6C&-k()-&/&()*?6_< [ $�6?4R*?C&(:()'D7<6?45FG()-&6 � � < [ $&Cu90 -&0�-k/H+.456u*?0�-U7<45( } CD7<0z+.')'2+AB50�FGFk+E*?0�-`]�

c � � c � $�6?4u/&C&6tFG0.7<(O@�(L] $�6?45*Pv 96 c��! � C&-&/&6?42"H0�3 N 6tX�C&()-&/&( +./q6?B5B50G@�()6?-&6 +.B5B56?=�-H+!7<0G()' @.+.')0�456A�?6?450 S

Page 94: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

� 6�6 $�6?45*Pv 96.]

���� �< [ ��������� Y�� �����

� c � � c � �����6 /&C&-&X�C&6u()'�7<6?FG()-&6�< [ Z 6?B5()FG0A/&6?')'2+%B56?45()6Q-&0�-�/ [+A*?0�-U7<45( } CD7<0t+.')'2+ B50�FGFk+ *?0�-(`7<6?45FG()-&( $&456?*?6?/&6?-U7<( �

� N � � J KOJ;��F&H��K��F�� � D F�J^CEJEH � JA/JKC � 9 H�T � � J J � 9 H%C � H�T � � J^CEJEH � J J � � ��H��

K�

� " � KV�

�� " > KV�

"� �]

� " M ]

� " �

�� � "

/D�9%JN�Z< JKC�CED CEJ�]

� " AJKFB/J_H���JKCED;>@��H%FB/D ��� 6 F � F&HC����H HK6�6&CED��� � I H�� � D F�J8OJ J >@� � FB � J

KV��� " > KV�

"�

� D A,AD;>@��H � � G�D FB � � � D F � J F�J���� � �KAJKI H^H%C � A,I�JKA � G�D H 6&CEJ�G � � � D F�J�W5F � A�Hu� J A�H � J_J�Y6&CEJ�� � D F�J��H � � 7%F � W�GEH%AD=S

$&%�'�(*)�+,(�-�.

j -�C&-oB5()Bf7<6?Fk+E+.45(O7<FG687<()*?0E� +A$&456?*?()B5()0�-&6 �H-&(O7P+%(&/&C&6Q-�C&FG6?45(&Fk+.*?*Pv&()-H+ 6 ��B50�-&0G/&()Bf7<()-U7<(`B56

��� ��� Y�� � � �*?()0�96 N 45()B5C&'O7P+.-&0A/&()Bf7<()-U7<(DB56 � [6 C&-�-�C&FG6?450EFk+.*?*Pv&()-H+A*Pv&6u/ [+A*?0�-U7<45( } CD7<0 +.')'2+ B50�FoZFk+�*?0�- � 3 0�-&B5()/&6?45(2+.FG0 N +./ 6?B56?FG$&()0 N C&- B5()Bf7<6?Fk+�+.45(O7<FG687<()*?0�� +�$&456?*?()B5()0�-&6�H-&(O7P+o*,+.4<+!757<6?45()�?�,+!7<0�/H+.(`$H+.4<+.FG687<45(� ]54� � 6 M07 � M�8 �;:=<

� > � 6 M�8 �@? ��� �

6 M6tB5(2+KV� " �

� w3 +.')*?0�')(2+.FG0z()'`$&(<[C�$&()*?*?0�')0o-�C&FG6?450 � *Pv&6 / [+o*?0�-U7<45( } CD7<0k+.')'2+zB50�FGFk+\*?0�-�

6 w 2�Q � 6 c wj -I� 45()B5C&'O7P+��� Y��

�6 w�� Q � 6

T M

Page 95: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# � 6 �6?B56?=�C&6?-&/&0%')6Y0�$�6?4<+.�?()0�-&(�()-E/&0�$&$&(2+ $&456?*?()B5()0�-&6Y6W*?()0�96 7 ����� �P S B5(U0.757<()6?-&6 N /&C&-&X�C&6.]��� ��� Y�� � � � �

6 w�� Q � 6 c Q 6 w 2�Q � 6 c �

6 wO��� Q � 6 T

$�6?4u*?C&(W[6EB5C10G*?()6?-U7<6 *Pv&6 B5(2+�

�6 wO��� Q � 6

T

6 N /&(�*?0�-&B56?=�C&6?-&�,+ N ��

�6 w 2 6�6�6 ��� Q � 6 c

�� M$�6?45*Pv 96 + }&} (2+\B5()=�-&(O�H*,+!7<0\'2+ @.+.')CD7P+.�?()0�-&6A/&6?')'2+ STC&-&�?()0�-&6K()-�/&C&6E@.+.')0�45(�/&()Bf7<()-U7<( �\ a +.'O7<4<+\$H+.4f7<6 N 45()B5C&'O7P+

� KV��� "h� � ��� Y�� � � KV�

"h� � KV��� "

�� KV�

"$�6?4f7P+.-U7<0G/&68@�6 45()B5C&'O7P+.456E*Pv&6.]

KV��� " � ��� Y�� � KV�

"$�6?45*Pv 96z+ }&} (2+oB5()=�-&(O�H*,+!7<0o()'`-�C&FG6?4<+!7<0�456E/&6?'�4<+.$&$�0�4f7<0�()-&*?456?FG6?-U7P+.')6 �� M � K���:���KA,C0H%CEJ ��F I�JKADX/DM6LJKC J�9 � A�H%CEJ � � � JKF�D I�JKF�D /J���� H GEH%F�G�J���� H�� � D F�J � �KA,CK��A,A � 9 H F�J��

GEH � G�D�� DQ � ��F&H � D IRI H W5F � A�H8/J�� A � 6LDD�� � � [ � > ��"

��� � � � ��"

$ %�'�(*)�+,(�-�.

y 6?4 68@�(O7P+.456W()'�ST6?-&0�FG6?-&0u/&6?')'2+ ���� ���� ����������� � � � ��� � ������ � 0�*?*?0�45456W4<+.=�=�45C&$&$H+.456(`7<6?45FG()-&( /&6?')')0oBf7<6?B5B50kB56?=�-&0k6EB50�FGFk+.45')( ()-qFk+.-&()6?4<+z/&6?*?456?B5*?6?-U7<6 �3 (m[0�B5()=�-&(O�H*,+D]�+ " ���� � � ����� � �� ( 7<6?45FG()-&(`/&6?')')0oBf7<6?B5B50kB56?=�-&0 S�} " �� � ������� '2+oB50�FGFk+\$H+.45�?(2+.')6A/&6?(�-�C&FG6?45(�()-qFk+.-&()6?4<+\/&6?*?456?B5*?6?-U7<6 S�* " �� � ������� '2+o/&(�� 6?456?-&�,+\7<4<+o')6EB50�FGFG6E$H+.45�?(2+.')( �/ C&$&$�0�-&(2+.FG0o/&(`@�0�')6?4 @.+.')CD7P+.456A'2+oB50�FGFk+

� ��"$�6?4_<

��%6

� [ � � [ � > ��"���

� � � � > �2 � ��> �M � �h�h� > � &�

Page 96: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

� 6U�� 6?(�$H+.B5B5(

�+ " 6 �

} " B5(�*,+.')*?0�'2+.-&0o')6EB50�FGFG6E$H+.45�?(2+.')( B56?=�C&6?-U7<(L]� � � [�� � � [� � � Y � � > ��"���

� � � �2�1 � �2 � � �h�h� � �1 � �� � �>� � c � � � � c� � � ��� Y � � > ��"

��� � � � � &� � �2 M � �h�h� � ���� � �M � �2-&6?'�$H+.B5B50

�* " B5(�*,+.')*?0�'2+\'2+\/&(�� 6?456?-&�,+�/&6?')')6EB50�FGFG6A$H+.45�?(2+.')(L]� [ � � [ � > ��"

��� � � �

� � c � � � � [� � � Y � � > ��"���

� � �� � � [�� � � c� � � ��� Y � � > ��"

��� � �

� P � K� G�D FB � � � D F&H%I�JKFLAD OJM��F`J�� JKA,AD /J���� H 6&CEJ�G � � � D F�J H%C � A,I�JKA � GEH G�D F G�� � � � C � � D��:9%J � �6&CEDDT@� JKI H�F

$&%�'�(*)�+,(�-�.���=.�!(*+ � � j 'R*?0�-&/&()�?()0�-H+.FG6?-U7<0�[6GC&-H+�*,+.4<+!757<6?45()Bf7<()*,+q()-U7<45()-&B56?*,+>/&6?'$&450 } ')6?Fk+ 6E/&()$�6?-&/&6AB50�')0\/H+.')'2+t-H+!7<C&4<+\/&6?'`$&450 } ')6?Fk+ � � +\$&456?*?()B5()0�-&6A+.45(O7<FG687<()*,+S|+oB,9� *Pv&6 B5(�68@�()/&6?-&�?()-&0G=�')( 6�� 68757<(�/&6?' Fk+.' *?0�-&/&()�?()0�-H+.FG6?-U7<0 �� 1 � K� G�D FB � � � D F&H%I�JKFLADV � ��F�6&CEDDT@� JKI H � 6LJKFB/J�OH ���Z< H � 7/D C � A,I�DA��A � � � �� H%AD�6LJKC(C � � D��:9%JKCK� DDF

$&%�'�(*)�+,(�-�.���=.�!(*+ ��� j '�*?0�-&/&()�?()0�-H+.FG6?-U7<0o/&()$�6?-&/&6tB50�')0o/H+.'�$&450 } ')6?Fk+ ��%6 � � A � � � �� H%FB/D�� H�I H*�� � I H�6&CEJ�G � � � D F�J# � ��F � � �KAJKI HXH%C � A,I�JKA � G�D �_D H%A � F�7 6LD � FLA H;6&CEJ�G � � � D F�J W5F � A�H J � � 6B� �D_I � 7 � � D C0H%CEJ � �+G�D FB � � � D F&H%I�JKFLAD� � ��FC6&CEDDT@� JKI H������������ �"! #%$&#�� �"�(')� �$&%�'�(*)�+,(�-�.*��=.�!(*+ ��� j '�S|+!757<0�456 /&(�+.FG$&')(O�H*,+.�?()0�-&6E/&6?=�')( 6?45450�45( B5C&( /H+!7<( /&()$�6?-&/&6/H+.' $&450 } ')6?Fk+>6�-&0�-3/H+.' B5()Bf7<6?Fk+I+.45(O7<FG687<()*?0>CD7<()')()�?�,+!7<0 � y 6?4f7P+.-U7<0 N +.-&*Pv&6�CD7<( Z')()�?�,+.-&/&0 C&-H+ Fk+.=�=�()0�456 $&456?*?()B5()0�-&6%+.45(O7<FG687<()*,+ N '=a +.FG$&')(O�H*,+.�?()0�-&6Q/&6?')'=a 6?45450�456uB5C&')'2+B50�')C&�?()0�-&6E-&0�->$&Cu90�6?B5B56?456z68@�(O7P+!7P+ �� ��� � F,+ T@�LD F.- H � 7/D C � A,I�D�6&CEDX �LG�J ��F&H�� D��:�D� � D F�J H/G�G���C0H%A�H � FB � 6LJKFB/JKFLAJKI�JKFLAJ OH �@6&CEDDT@� J I H@G�LJ � � �KA�H�C � � D��:9%JKFB/DDF

$&%�'�(*)�+,(�-�./��=.�!(*+ ��� � a +.*?*?C&4<+!7<6?�?�,+E/&6?')'2+%B50�')C&�?()0�-&6 /&(DC&-�$&450 } ')6?Fk+ /&()$�6?-&/&6Q0�' Z7<456u*Pv&6%/H+.')'=a +.')=�0�45(O7<FG0 +.-&*Pv&6 /H+.':*?0�-&/&()�?()0�-H+.FG6?-U7<0E/&6?':$&450 } ')6?Fk+ � y 6?4f7P+.-U7<0 N C&-n } C&0�-&n\+.')=�0�45(O7<FG0�*?()0&[6 C&-�+.')=�0�45(O7<FG0EBf7P+ } ()')6 "�N B56 @�()6?-&6 CD7<()')()�?�,+!7<0t$�6?4R45()B50�'O@�6?456C&->$&450 } ')6?Fk+\Fk+.' Z *?0�-&/&()�?()0�-H+!7<0 N $&C [0�ST0�45-&()456AC&-H+�B50�')C&�?()0�-&6E()-H+.*?*?C&4<+!7P+ ���� � � H � G�J�� A�HM � ��FWH � 7/D C � A,I�D#6LJKC C � � D��:9%JKCEJ���F 6&CEDDT@� JKI HQ��H J�� JKA,ADQ������� H�6&CED6�H�7OH�� � D F�J

/J 7 � � JKC�CED C � F�J � OH%A � F

Page 97: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# � 6 2$ %�'�(*)�+,(�-�. � ����� + ��� j -�=�6?-&6?4<+.')6%/&C&6E+.')=�0�45(O7<FG(&$�6?4 '2+ 45()B50�')C&�?()0�-&6 /&(�C&-&0oBf7<6?B5B50$&450 } ')6?Fk+A$�0�B5B50�-&0 $&450�/&C&45456 45()B5C&'O7P+!7<(D/&(O@�6?45B5(H()-oX�CH+.-U7<0 =�')(D6?45450�45(&/&( ���� � ��� �� N/&0h@�CD7<( +.'YFG0�/&0q/&(�� 6?456?-U7<6G*?0�-�*?C&(Y@�6?-&=�0�-&0�6�� 68757<CH+!7<6G')6�0�$�6?4<+.�?()0�-&( N $�0�B5B50�-&0$&450�$H+.=U+.45B5(�()-qFk+.-&()6?4<+\/&(O@�6?45B<+ �� 2 � � J �LJ F ��I�JKC � CEJEH � � � D F�D J� H%A,A�H%I�JKFLAJ C0HK6�6&CEJ� JKFLA�H�T � � � G�D I�J F ��I�JKC � �@F 6GF � F ��F

� � �KAJKI H^H%C � A,I�JKA � G�D^H 6&CEJ�G � � � D F�JBW5F � A�H J H ��� D C0H � ��C � ����� A�H%AD; � ��F=< D6LJKC0H�� � D F�J H%C � A,I�JKA � GEH��� � � D CED�6B��OD F�D F J�� JKCEJ J� H%A,A�H%I�JKFLAJ C0HK6�6&CEJ� JKFLA�H�T � � JXF

$ %�'�(*)�+,(�-�. � ����� + ��� #%/&0�$�6?4<+.-&/&03C&- B5()Bf7<6?Fk+;+.45(O7<FG687<()*?0;+ 6&CEJ�G � � � D F�J#W5F � A�HX�CH+.-U7<0�+�� 6?45Fk+!7<0�[6tB5()*?C&4<+.FG6?-U7<6A@�6?450 �#%/k6?B56?FG$&()0 N �HB5B<+!7P+ '2+ } +.B56 4 � � 6 N '2+E$&456?*?()B5()0�-&6 7 �M 6u=�')(H6?B5$�0�-&6?-U7<(:FG()-&()FG06QFk+.B5B5()FG0%45()B5$�68757<(O@.+.FG6?-U7<6 8 �;:=<

� > � N 8 �@? ���

� N *?0�-&B5()/&6?4<+!7<(D(D/&C&6Q-�C&FG6?45(Fk+.*?*Pv&()-H+ c �

6 w 2�2 �! &��� 6 Q � 6 6 � �6 w M%N�N�N�N 6 M Q � 6()'`$&450�/&0.757<0�45()B5C&'O7P+

c Q � �� ��w M �, N 2 �%6 M�M 6�6�6 �

6 w � � M �, N 2 �%6 M�M Q � 6�6 N X�C&6?Bf7 a C&'O7<()FG0;[6 N 6�� 68757<(O@.+.FG6?-U7<6kC&-3-�C&FG6?450I4<+.$&$&456?B56?-U7P+ } ()')6 N 6?B5B56?-&/&0 ()'WB5C&06?B5$�0�-&6?-U7<6 8 �;:=< a � a 8 �@? N Fk+I-&0�-~6?B<+!757P+.FG6?-U7<6 N +,@�6?-&/&0;C&-~-�C&FG6?450 /&(*?(OST456 B5()=�-&(O�H*,+!7<(O@�6uFk+.=�=�()0�456 /&( 7 S +.$&$&')()*,+.-&/&0tC&-&0zB5*Pv&6?Fk+z/&( +.45450.7<0�-&/H+.FG6?-U7<0 B5(7<450h@.+�()' 45()B5C&'O7P+!7<0

K�� c Q � " �

6 w � � M �, N Q � 6�w� &� F ��I�JKC � �@F 6GF � D F�D ��F � � D C�I�JKI�JKFLAJA � �KA,C � T@� � A � ������� H�CEJKA,A�H�CEJEH � J � F

$ %�'�(*)�+,(�-�. ��=.�!(*+ ��� �YB5B5(%-&0�- B50�-&03C&-&(OST0�45FG6?FG6?-U7<6�/&()Bf7<45( } C&(O7<( N Fk+�')0;B50�-&0B50�')0o7<4<+�B5C&*?*?6?B5B5(O@�6\$�0.7<6?-&�?6 /&6?')'2+ } +.B56 /&(�-�C&FG6?4<+.�?()0�-&6 N 4 N /&6?'�B5()Bf7<6?Fk+ � / 6 N +./6?B56?FG$&()0 N �HB5B5(2+.FG0�C&-3B5()Bf7<6?Fk+�+.45(O7<FG687<()*?0>+�$&456?*?()B5()0�-&6��H-&(O7P+>/&( } +.B56X4

� � 6 N$&456?*?()B5()0�-&6 7 �� 6?/ 8 �;:=<

� > � M"8 �@? �� N +.')')0�4<+ 7<CD757<(A6IB50�')(%(A-�C&FG6?45(4<+.$&$&456?B56?-U7P+ } ()')(�()-qX�C&6?Bf7<0GB5()Bf7<6?Fk+oB50�-&0GX�C&6?')')(�+.$&$H+.4f7<6?-&6?-U7<( +.')'=a ()-&B5()6?FG6A�E]

� ]�

> 6 w 1�1 � � 6� M > 6 wO� � � 6

���� � 6 � 6 wO� � � 6 � M 6 w 1�1 � � 6

���� +.')(&-�C&FG6?45( N *?0�FG6 B5(H$&C [0 0�B5B56?4f@.+.456 N -&0�-GB50�-&0t6?X�C&()/&()Bf7P+.-&�?(2+!7<(&B5C&')'=a +.B5B56 456,+.')6 � �

102

-rmax=-0.99

102

100

10-1

101

102

10-2

10-1

100

101

102

10 -0.1 -2

102

0.1 10-2

+rmin=

OverflowUnderflowOverflow

. =-rmin =+rmax. . . 0.99

�����������������������! #"$���&%'����(*)!�,+���-#-/. #���� #��01�32

Page 98: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

� 6 ����������� � ���������������� �������� !�"� #$��&%������'�(���)����( * �(�� ,+-�����

. ��"/ ��10 �32 / (QB5*?45(O@.+IC&-1$&450�=�4<+.FGFk+>()-54H0�4f7<4<+.-3$�6?4o()'R*,+.')*?0�')0�/&(QC&-3@.+.')0�456�+.$&$&450�B5B5()Fk+!7<0I/&6?'-�C&FG6?450G/&( � 6?$�6?450 8 N CD7<()')()�?�,+.-&/&0\'2+o/&68�H-&()�?()0�-&6.]8 �

')()F�s�

�� � �< �

�w

�+ " 3 +.')*?0�'2+.456�'2+QX�CH+.-U7<(O7![+�6 � � c��7 � +.'.*?456?B5*?6?456R/&(�<o$�6?4 < � � 6

� *?0�- � � M � M w,w,w M �%6

�} " $�0�45456�$H+.4f7<()*?0�'2+.456\+!757<6?-&�?()0�-&6 �

6�� 68757<CH+.-&/&0>C&-�0�$&$�0�4f7<C&-&0q*?0�-U7<450�')')0 " +.'Y*?0�-U7<45( Z} CD7<0G/H+!7<0G+.')'2+zB50�FGFk+\/&( c� *?0�- � S�* " Bf7P+.FG$H+.456 N ()-q*?0�4545()B5$�0�-&/&6?-&�,+�/&(�0�=�-&(�+.$&$&450�B5B5()Fk+.�?()0�-&6 N$ '=a ()-&/&()*?6A/&6?' $H+.B5B50G*?0�45456?-U7<6 N$ ()' @.+.')0�456EBf7<()Fk+!7<0 N$ '=a 6?45450�456E456?'2+!7<(O@�0�$&450�/&0.757<0o-&6?' 45()B5C&'O7P+!7<0 �*?0�-DST450�-U7P+.-&/&0 N +./ 0�=�-&(V$H+.B5B50 N ()' @.+.')0�456\/&6?/&0.757<0>*?0�- ()'�@.+.')0�456 8 c 0.757<6?-�CD7<0qCD7<( Z')()�?�,+.-&/&0o'2+\STC&-&�?()0�-&6�T@� � � A Y� F /&6?'�4H0�4f7<4<+.-q6 �D$

� "S

�/ " \ 0�$�0oX�CH+.-U7<( $H+.B5B5(�*?0�-U@�()6?-&6t7<6?45FG()-H+.456E'=a 6?B56?*?C&�?()0�-&6t/&6?'�$&450�=�4<+.FGFk+ � y 6?45*Pv 96���6 " j ->*?0�4545()B5$�0�-&/&6?-&�,+�/&(�X�CH+.')6E$H+.B5B50GB5(�4<+.=�=�()C&-&=�6E()' FG()-&()FG0\6?45450�456E456?'2+!7<(O@�0 ��S " 3 0�B<+�+.*?*,+./&6q-&6?(Q$H+.B5B5(uB5C&*?*?6?B5B5(O@�(u6q*?0�B<+ B5( $&C [0 /&6?/&C&45456>B5C&')'=a +.*?*?C&4<+!7<6?�?�,+ /&6?'45()B5C&'O7P+!7<0 �

. ��"/ ��10 �98 / (�*?0�-&B5()/&6?45(�'2+\STC&-&�?()0�-&6.]

KV� " �

8

� > �: 7<()')()�?�,+.-&/&0o'2+\456?=�0�'2+o/&6 � a ; �� B5$&(O7P+.'`B5(�$&450h@.+o*Pv&6.]

')()F�8s

[ KV� " � ��w

�+ " / (E$&450h@�(tX�C&6?Bf7<0 45()B5C&'O7P+!7<01()FG$&')6?FG6?-U7P+.-&/&01C&- $&450�=�4<+.FGFk+3()-�4H0�4f7<4<+.- $�6?4q()'*,+.')*?0�')0o/&(`S

�� " $�6?4 � � 6

�N < � ��N�w,w,w�NO�

N �j 45()B5C&'O7P+!7<(�7<450h@.+!7<(`*?0�-&*?0�45/H+.-&0G*?0�->X�C&6?')')( 7<6?0�45()*?( � y 6?45*Pv 96���} " � +.')CD7P+.456AS �

� " CD7<()')()�?�,+.-&/&0o'2+zST0�45F\C&'2+D] KV� " �� g c� = �?> � g?@ �� C&6?Bf7P+oST0�45F\C&'2+\/ [+o45()B5C&'O7P+!7<( $&(.9Cq+.*?*?C&4<+!7<( � y 6?45*Pv 96��

. ��"/ ��10 �9A / (�B5C&$&$�0�-&=U+ /&(�=�6?-&6?4<+.456%-� �

$&C&-U7<(�6?X�C&()/&()Bf7P+.-&�?(2+!7<(H/&( ��

Z

Y� N -&6?')'=a ()-U7<6?4f@.+.')')0 + N }

Page 99: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# � 6 ��+ " j - C&- B5()Bf7<6?Fk+ +.45(O7<FG687<()*?0 S � $ � � X�CH+.')61/&6?(�/&C&6 +.')=�0�45(O7<FG( B56?=�C&6?-U7<(>[6~$&(.9C+.*?*?C&4<+!7<0 N [ �

? M ����� c �

� M

� � M w,w,w M <0�$&$&C&456���

?

�� M

�6 M w,w,w M <��

�} " / (VB5*?45(O@.+�C&- $&450�=�4<+.FGFk+G*Pv&6o()FG$&')6?FG6?-U7<( 6?-U7<4<+.F } (�=�')(�+.')=�0�45(O7<FG( 6\B5(�7<450h@�(VC&-6?B56?FG$&()0 N *?0�- ?

�6 6

{ � ��N *Pv&6 -&6t()')')C&Bf7<45(�')6E/&(�� 6?456?-&�?6 �. ��"/ ��10 ��� / (�B5*?45(O@.+uC&-z$&450�=�4<+.FGFk+Q$�6?4 ()'U*,+.')*?0�')0u/&6?')'2+ /&6?45(O@.+!7P+ /&(�C&-H+uSTC&-&�?()0�-&6YS�� " CD7<()')()�?�,+.-&/&0'=a +.$&$&450�B5B5()Fk+.�?()0�-&6%*Pv&6 B56?=�C&6.]

K�

� " p KV�

�� " > KV�

"� w/ (�$&450h@�(�()'`$&450�=�4<+.FGFk+zC&B<+.-&/&0G'2+zSTC&-&�?()0�-&6 � <� "�N $�6?4 �

� ��+ " / (�45()$�687P+o'=a 6?B56?45*?()�?()0oCD7<()')()�?�,+.-&/&0o'=a +.$&$&450�B5B5()Fk+.�?()0�-&6.]

K�

� " p KV�

�� " > KV�

> � "� �. ��"/ ��10 ��� / (�*?0�-&B5()/&6?45(�'2+\B56?45()6 +.45FG0�-&()*,+D]

� � � c �<�+ " y 450h@.+.456t*Pv&6 '2+oB56?45()6�[6E/&(O@�6?45=�6?-U7<6 ��} " / $&()6?=U+.456t$�6?45*Pv 96 '2+\B56?45()6tvH+oB50�FGFk+z�H-&(O7P+o()->+.45(O7<FG687<()*,+ S � $ ��* " / (%/&()*,+;X�CH+.-&/&01( 7<6?45FG()-&(u/&6?')'2+�B50�FGFk+!7<0�45(2+ -&0�- /H+.-&-&01$&(<[C *?0�-U7<45( } CD7<0 N -&6?'B5()Bf7<6?Fk+�+.45(O7<FG687<()*?0\Bf7P+.-&/H+.45/ j � � � N ()-qB5()-&=�0�'2+\6?/>()-q/&0�$&$&(2+\$&456?*?()B5()0�-&6 ��/ " / ( B5*?45(O@.+.-&0�/&C&6z$&450�=�4<+.FGFG(�$�6?4%()'�*,+.')*?0�')0G/&6?')'2+GB56?45()6z+.45FG0�-&()*,+�C&-&0�()-�B5()-&=�0�'2+6�'=a +.'O7<450>()-1/&0�$&$&(2+�$&456?*?()B5()0�-&6 � / (Q*?0�-U7<450�')')(W()'R$&450�*?6?B5B50 /&(Q*,+.')*?0�')0�45()$�0�4f7P+.-&/&0'=a ()-&/&()*?6%6E'2+\B50�FGFk+z$H+.45�?(2+.')6A+./�0�=�-&( (O7<6?4<+.�?()0�-&6 � � CH+.')6A*?45(O7<6?45()0\/&(�+.45456?Bf7<0�$&CY[C6?B5B56?456zCD7<()')()�?�,+!7<0 � / (�*?0�-DST450�-U7<()-&0�(`45()B5C&'O7P+!7<( *?0�->(`@.+.')0�45(�7<450h@.+!7<( �. ��"/ ��10 ��� / ( B5*?45(O@.+ C&-1$&450�=�4<+.FGFk+�$�6?4\()' *,+.')*?0�')0�/&6?')'2+�FG6?/&(2+�� 6�/&6?')'2+>@.+.45(2+.-&�,+�� � /&(QC&-H+B5C&*?*?6?B5B5()0�-&6A�H-&(O7P+ � � [6%C&-�@�68757<0�456E/&(�/&()FG6?B5()0�-&6(< N $�6?4Q()':*,+.')*?0�')0z/&6?')'2+t@.+.45(2+.-&�,+zB5(CD7<()')()�?�?()-&0\6?-U7<4<+.F } 6E')6AST0�45F\C&')6.]

� �� �< > �

� ��c � > � " �

Page 100: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

� 6 N

� �� �< > �

�� � ��c � > �<

� � ��c

����/ (�*?0�-DST450�-U7<()-&0�(�45()B5C&'O7P+!7<( $�6?4 ')6E/&C&6 B5*?6?'O7<6 ������������ � � #-� ��# +-�

\ (HB56?=�C&(O7<0 @�6?454<+.-&-&0o$&450�$�0�Bf7<(:(:$&450�=�4<+.FGFG(H()- 4H0�4f7<4<+.-k6%')6u45()B5$�0�Bf7<6A+.(:456?'2+!7<(O@�(H$&450 } ')6?FG(H()' Z')C&Bf7<4<+!7<(D-&6?'&$H+.4<+.=�4<+!ST0E$&456?*?6?/&6?-U7<6 � j 45()B5C&'O7P+!7<(�-�C&FG6?45()*?(&B50�-&0tBf7P+!7<(&0.757<6?-�CD7<(H()FG$&')6?FG6?-U7P+.-&/&0=�')(`+.')=�0�45(O7<FG(:()- 4H0�4f7<4<+.-qB5C>C&->*,+.')*?0�'2+!7<0�456 j� ��uj /�3 / � Bf7<6?F � N 6�6�6 �

Page 101: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# � 6 M. ��"/ ��10 � 2�� ���� � ���� � �� � ��0 �� � ��� �� ��$&450�=�4<+.F ���� � ���� � ��0 �� � ��� �� ��

3 4:+.B56E/&(�/&()*Pv&(2+.4<+.�?()0�-&6A/&6?')')6A@.+.45(2+ } ()')(()-U7<6?=�6?4 < N -456,+.' � ��N -&6?$ N 6456,+.'�6?454 ��N 6?$&B5()')0�-3u45(O7<6�r N r "3u45(O7<6�r N r "3u45(O7<6�r N r "3u45(O7<6�r N r "3u45(O7<6�r N r "

aaaaa

3 +.')*?0�')0o/&6?'�-�C&FG6?450 /&( � 6?$�6?450GFG6?/&(2+.-U7<6E()'`')()FG(O7<6� ��"

< � :=<

K/&(

� � � � � < " �aaaaa

3 3 +.')*?0�')0o/&6?'�-�C&FG6?450�/&( � 6?$�6?450�7<4<+.FG(O7<6%STC&-&�?()0�-&6E()-U7<6?45-H+-� �6�6 �D$

� ���6 "3 3 +.')*?0�')0o/&6?')'=a 6?$&B5()')0�-�Fk+.*?*Pv&()-H+6?$&B5()')0�-�6?$&B5()')0�-

� ���6 "$&45()-U7 r N a �Y$&B5()')0�-qFk+.*?*Pv&()-H+ � a N 6?$&B5()')0�-<

�6$&45()-U7 r N a � C&FG6?450�/&( � 6?$�6?450GFG6?/&(2+.-U7<6ASTC&-&�?()0�-&6E()-U7<6?45-H++a N 63 4:+.B56E/&(�*,+.')*?0�')0

� *?0�-U7<()-�C&6<�<� �-

� �6 r�r <� � � ���

6 �!-(OS�� �G� =�6 � 6?$&B5()')0�- " 7<v&6?-� � � ���

6

�� �-&6?$

�� � r�r -3 3 +.')*?0�')0o/&6?')'=a 6?45450�456A456?'2+!7<(O@�06?454 � �+ } B

�6�Z -&6?$ " �.+ } B �

6 "3 / 7P+.FG$H+\/&6?'�@.+.')0�456A*,+.')*?0�'2+!7<0o/&6?'�-�C&FG6?450�/&( � 6?$�6?450$&45()-U7 r N a��������������������������������������������Xa$&45()-U7 r N a (O7<6?4<+.�?()0�-&6A-�C&FG6?450�a N <$&45()-U7 r N a /&68S � ��" �a N -&6?$$&45()-U7 r N a �Y45450�456E456?'2+!7<(O@�0@a N 6?454 �$&45()-U7 r N a��������������������������������������������Xa$&45()-U7 r6?')B56 $&45()-U7 r N a � �!-q6?nGFG()-&0�456A/&6?')')n�6?$&B5()')0�-�Fk+.*?*Pv&()-H++a$&45()-U7 r N a /&68S � ��" �a N -&6?$6?-&/&(OS

3 � 6?45FG()-H+.�?()0�-&6A/&6?'�*?()*?')0o/&(�*,+.')*?0�')0(OS�< � 'O7 � � �z" =�0.7<0 �Bf7<0�$6?-&/

Page 102: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

� 6 P. �� � � � � �� �� ���������

3 +.')*?0�')0\/&6?' -�C&FG6?450 /&( � 6?$�6?450GFG6?/&(2+.-U7<6E()' ')()FG(O7<6� ��"

< � :=<

K/&(

� � � � � < " ��Y$&B5()')0�-qFk+.*?*Pv&()-H+

� ���O� 1 �%6 1 � 1 ��Z 6 M� C&FG6?450�/&( � 6?$�6?450GFG6?/&(2+.-U7<6ASTC&-&�?()0�-&6E()-U7<6?45-H+ � � M � P � P � M �

Page 103: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# � 6 1�������������������������������������������(O7<6?4<+.�?()0�-&6A-�C&FG6?450 �/&68S

� ��" �� ��� 1�2 M 2 6 1�Y45450�456t456?'2+!7<(O@�0 6 � 6 �� P ��� � �, ���������������������������������������������

�������������������������������������������(O7<6?4<+.�?()0�-&6A-�C&FG6?450 �/&68S� ��" �

� � M 6 P � 2�Y45450�456t456?'2+!7<(O@�0 6 � 6�6 1 �! POM 2 � P�������������������������������������������

�������������������������������������������(O7<6?4<+.�?()0�-&6A-�C&FG6?450 2/&68S� ��" �

� � M � M 6 &� 1 M�Y45450�456t456?'2+!7<(O@�0 6 � 6�6�6 �� N 6 P�P 2�2�1�������������������������������������������

�������������������������������������������(O7<6?4<+.�?()0�-&6A-�C&FG6?450 /&68S� ��" �

� � M � P 6U� 2�2�Y45450�456t456?'2+!7<(O@�0 1 ��� 6 M%N�P�P 6 P ��Z 6 ��������������������������������������������

�������������������������������������������(O7<6?4<+.�?()0�-&6A-�C&FG6?450 �/&68S� ��" �

� � M � ��� � 2 � ��Y45450�456t456?'2+!7<(O@�0 6 � 6�6 ��� 1 � � P 2�2 ��������������������������������������������

�������������������������������������������(O7<6?4<+.�?()0�-&6A-�C&FG6?450 N/&68S� ��" �

� ��� 1 N 6 1 M�Y45450�456t456?'2+!7<(O@�0 6 � 6 ��! 1 P �, 2�1�������������������������������������������

� �!-�[6EFG()-&0�456E/&6?')'=a 6?$&B5()')0�-�Fk+.*?*Pv&()-H+/&68S� ��" �

� ��� 1 N 6 1 M� B5B56?4f@.+.�?()0�-&6 �

Page 104: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

��� 6#%$&$&450�B5B5()Fk+.-&/&0;()' -�C&FG6?4503/&( � 6?$�6?4503*?0�- '2+ ST0�45F\C&'2+� ��"�N

'=a +.')=�0�45(O7<FG0I$&450�/&C&*?6�'=a 6?45450�456FG()-&()FG0\+.')'=a (O7<6?4<+.�?()0�-&6 -�C&FG6?450 /&0h@�6.]'=a +.$&$&450�B5B5()Fk+.�?()0�-&6 @.+.')6 � � M � P 6U� 2�2'=a 6?45450�456E456?'2+!7<(O@�0 1 ��� 6 M%N�P�P 6 P ��Z 6 �# $H+.4f7<()456W/H+.')'2+ X�C&()-U7P+ (O7<6?4<+.�?()0�-&6W'=a 6?45450�456 +.C&FG6?-U7P+ 6RX�C&()-&/&(�'=a�+.$&$&450�B5B5()Fk+.�?()0�-&6W/&6?'�-�C&FG6?450/&( � 6?$�6?450�/&(O@�6?-U7P+�B56?FG$&456tFG6?-&0�+.*?*?C&4<+!7P+ � 3 (m[0�[6t/&0h@�CD7<0G+.')'2+z$&450�$H+.=U+.�?()0�-&6A/&6?')'=a 6?45450�456E/&(450�C&-&/�Z 0 � /&C&4<+.-U7<6t'=a 6?B56?*?C&�?()0�-&6t/&6?')'=a +.')=�0�45(O7<FG0 � j -DS|+!757<( N $�0�Bf7<0 > � @� � � �< M*?0�-&B5()/&6?45(2+.FG0\'=a 6?45450�456E456?'2+!7<(O@�0�/&(`4<+.$&$&456?B56?-U7P+.�?()0�-&6 N -&6?'�/H+!7<0 N +.')'2+ < Z 6?B5()Fk+\(O7<6?4<+.�?()0�-&6.]� > � @

� > � @

� � �� � " *?0�- � � � � a�� w[�1-&0.7<0

�*8ST4 � *,+.$ �O��" *Pv&6A'=a ()-&/&()*?6u/&(�*?0�-&/&()�?()0�-H+.FG6?-U7<0 456?'2+!7<(O@�0z/&6?')'2+ESTC&-&�?()0�-&6 �

> � @ " �G[6%/H+!7<0/H+D]����� � < � � c� � ������

<

$�0�-&6?-&/&0 �� > � @ N $�6?4z*?C&( ()'RFk+.'W*?0�-&/&()�?()0�-H+.FG6?-U7<0�/&6?'R$&450 } ')6?Fk+>*?456?B5*?6�*?0�- < � y 0�Bf7<0 N/&C&-&X�C&6 N < � � 6� *,+.')*?0�'2+.-&/&0 '2+I$�0.7<6?-&�,+ -�Z 6?B5()Fk+I'=a 6?45450�456�456?'2+!7<(O@�0 /&( 4<+.$&$&456?B56?-U7P+.�?()0�-&6 N-&6?')'2+\B50�')C&�?()0�-&6 N B<+.4�[+D]

� ] � Y � � � �< � � � � � � � 6

� � � � � � *?0�- � � � � a��y 6?4f7P+.-U7<0 N +.')'2+zX�C&()-U7P+�(O7<6?4<+.�?()0�-&6A*?()0&[6E$�6?4 � ��N B5(�0.757<()6?-&6.]� ] � Y � � � � � 6�� � � � � � a � 6�� � � � � 6�� � � 6 �� � 6

�FG()-&0�456t/&6?')'2+GFk+.B5B5()Fk+k+.*?*?C&4<+!7<6?�?�,+�456?'2+!7<(O@.+ N ()-IC&-IB5()Bf7<6?Fk+�+.45(O7<FG687<()*?0k+k$&456?*?()B5()0�-&6 �H-&(O7P+*?0�- } +.B56�4

� � 6 6Q$&456?*?()B5()0�-&6 7 �P N $�6?4�*?C&( � �

4 c �

� � 6 � � � CH+.-U7<0A/&68757<0 N [6Q*?0�-DST6?45Fk+!7<0/H+.'`45()B5C&'O7P+!7<0o/&6?')'2+\B5$�6?45()FG6?-U7P+.�?()0�-&6E-�C&FG6?45()*,+ N *Pv&6 $�6?4 � � $&450�/&C&*?6.]@.+.')0�456A*,+.')*?0�'2+!7<0 � w M � ��� � 2 � � �

6 w � M � ��� � 2 � � � 66?45450�456E456?'2+!7<(O@�0 6 w 6�6 ��� 1 � � P 2�2 � �6 wO��� 1 � � P 2 � � 6

�0h@�@�6?450GX�C&6?Bf7 a C&'O7<()FG0�[6E/&6?')')0\Bf7<6?B5B50G0�45/&()-&6E/&(`=�4<+.-&/&6?�?�,+o/&6?'�Fk+.B5B5()FG0\6?45450�456E/&(`4<+.$&$&456?B56?-�Z7P+.�?()0�-&6 �� a +.')=�0�45(O7<FG0z/&6?B5*?45(O757<0q[6t$�6?4f7P+.-U7<0 � � � ���-/ � ����� B5B56?4f@.+.�?()0�-&6 �

Page 105: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# �����\ +.')'=a (O7<6?4<+.�?()0�-&6 -�C&FG6?450 M ()-q$�0�( N B5(�0.757<6?-&=�0�-&0�(�45()B5C&'O7P+!7<( B56?=�C&6?-U7<(L]@.+.')0�456E*,+.')*?0�'2+!7<0 � w�� 1 N 6 1 M6?45450�456t456?'2+!7<(O@�0 6 w 6 ��! 1 P �, 2�1*?()0&[6z()'�@.+.')0�456 /&6?')'=a +.$&$&450�B5B5()Fk+.�?()0�-&6E-&0�-I@.+.45(2+�()-IX�CH+.-U7<0 � � < N -&0�-I/ [+k$&(<[C�*?0�-U7<45( } CD7<0�+.')'2+B50�FGFk+\*?0�- ��N $�6?45*Pv 96 � � < 45()B5C&'O7P+oFG()-&0�456A/&6?')'=a 6?$&B5()')0�-�Fk+.*?*Pv&()-H+ �

Page 106: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

��� �. ��"/ ��10 � 8 � ����� ������ � 0 ��������� � � � � ��� 0 � ��$&450�=�4<+.F ����� ������ � 0 ��������� � � � � ��� 0 � ��

3 4:+.B56 /&(�/&()*Pv&(2+.4<+.�?()0�-&6A/&6?')')6A@.+.45(2+ } ()')(()-U7<6?=�6?4 < N -456,+.' � ��N -&6?$ N -&6?$ ��N +.C&B N 6 N 6?454 ��N 6?454 �3 / 7P+.FG$H+�+\@�()/&6?0G/&6?'�$&450 } ')6?Fk+\/H+o45()B50�'O@�6?4563u45(O7<6

�r N r "3u45(O7<6�r N r "3u45(O7<6�r N r "3u45(O7<6�r N r "3u45(O7<6�r N r "3u45(O7<6�r N r "

aaaaaa

3 +.')*?0�')0o/&6?'�')()FG(O7<6� ��"

� 6 /&(���� y

�� " Z ��" � ��

� " � 6 /&(���� y

�� " Z ��" �!')0�= �

��� y�� "5"

aaaaaa

3 4:+.B56t/&(�*,+.')*?0�')0<�

6

� *?0�-U7<()-�C&6<�<� �-

� �6 r�r

�< "� � � ���

6 �!-+.C&B�6 �D$

�� ��"-&6?$ � �

+.C&BmZ �(OS�-&6?$ ��� -&6 � 6 � 6 " 7<v&6?-6�-&6?$ � � �

')0�=�+.C&B "5"+.C&B

�-&6?$ � � � �6?454 � �+ } B

� �Zm+.C&B "6?454 �

�+ } B

� �Z 6 "3 / 7P+.FG$H+�+\@�()/&6?0G/&6?')'2+\B50�')C&�?()0�-&6E*,+.')*?0�'2+!7P+$&45()-U7 r N a < �

a N <$&45()-U7 r N a ')()FG(O7<6 � ��" �a N +.C&B N a>6?45450�456 +.B5B50�')CD7<0 �

a N 6?454 �$&45()-U7 r N a ')()FG(O7<6 �� " �

a N 6 N a>6?45450�456 +.B5B50�')CD7<0 �a N 6?454 �$&45()-U7 r N a��������������������������������������������Xa6?')B56

3 ()B5C&'O7P+!7<0o/&0h@�CD7<0k+.')'2+z$&456?*?()B5()0�-&6E�H-&(O7P+$&45()-U7 r N a < �a N <$&45()-U7 r N a ')()FG(O7<6 � ��" �

')()FG(O7<6�� " �

6 a$&45()-U7 r N a��������������������������������������������Xa6?-&/&(OS3 � 6?45FG()-H+.�?()0�-&6E/&6?')'2+zS|+.B56t/&(`*,+.')*?0�')0(OS

�< � 'O7 � � N " =�0.7<0 �Bf7<0�$6?-&/

Page 107: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# ��� 2. �� � � � � �� �� ���������

3 +.')*?0�')0o/&6?'�')()FG(O7<6� ��"

� 6 /&(���� y

�� " Z ��" � ��

� " � 6 /&(���� y

�� " Z ��" �!')0�= �

��� y�� "5"

<� �')()FG(O7<6

� ��" � ��� 6 ��� M 6 1 N � 6?45450�456 +.B5B50�')CD7<0�

6 � 6 ��� M 6 1 N ��� 1')()FG(O7<6�� " � ��� 6 ��� M 6 1 � P 6?45450�456 +.B5B50�')CD7<0

�6 � 6 ��� M 6 1 � M ����������������������������������������������<

��')()FG(O7<6� ��" � ��� 6�6 � 6 � M�M%N 6?45450�456 +.B5B50�')CD7<0

�6 � 6�6 � 6 � M�M � M �')()FG(O7<6

�� " � ��� 6�6 � 6 � N�P 6?45450�456t+.B5B50�')CD7<0

�6 � 6�6 � 6 � N�P 6 2 M �������������������������������������������<

�2')()FG(O7<6� ��" � ��� 6�6�6 � � 2 � M 6?45450�456 +.B5B50�')CD7<0

�6 � 6�6�6 � � 2 � NOM �')()FG(O7<6

�� " � ��� 6�6�6 � 6�6U� 6?45450�456t+.B5B50�')CD7<0

�6 � 6�6�6 � 6�6U�%6U� � M 1�������������������������������������������<

� ')()FG(O7<6� ��" � ��� 6�6�6 � N � 1 6?45450�456 +.B5B50�')CD7<0

�6 � 6�6�6 � N � 1�2�1�2�2 �')()FG(O7<6

�� " � ��� 6�6�6�6 1�1 � 6?45450�456 +.B5B50�')CD7<0

� &� 1�1 P�N 1 � 2 ��Z 6 ��������������������������������������������<� �')()FG(O7<6

� ��" � ��� 6�6 � 2 � P 6 2 6?45450�456 +.B5B50�')CD7<0�

6 � 6�6 � 2 � P 6 2 ��� 2')()FG(O7<6�� " � ��� 6�6�6�6�6 � 6 � 6?45450�456 +.B5B50�')CD7<0

� ��� 6�6 NOM 1 6 � N ��Z 6 N�������������������������������������������<�

N')()FG(O7<6� ��" �

6 � 1 � 2 NOM 2 � N 6?45450�456 +.B5B50�')CD7<0�

6 � 6 N 2 � � N�P 2 N')()FG(O7<6�� " � ��� 6�6�6�6�6�6 P 6?45450�456 +.B5B50�')CD7<0

� &� M%N�P 2 M ��� P ��Z 6 M�������������������������������������������<�

M')()FG(O7<6� ��" � ���O� 1 �%6 1 � 1 6?45450�456t+.B5B50�')CD7<0

�6 �O� 1 �%6 1 � P 1 N')()FG(O7<6

�� " � ��� 6�6�6�6�6�6 � � 6?45450�456 +.B5B50�')CD7<0

�6 � 6�������������������������������������������<

�P')()FG(O7<6� ��" �

')()FG(O7<6�� " �

6�������������������������������������������

� B5B56?4f@.+.�?()0�-&6 �

Page 108: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

���, � a 6?45450�456 FG()-&()FG0A*Pv&6uB5(:*?0�FGFG68757<6 +.$&$&450�B5B5()Fk+.-&/&0t()'&')()FG(O7<6 +.B5B56?=�-H+!7<0 FG6?/&(2+.-U7<6 '2+AST0�45F\C&'2+� ��"�N

B5(�0.757<()6?-&6t+.')'=a (O7<6?4<+.�?()0�-&6 -�C&FG6?450 ]@.+.')0�456A*,+.')*?0�'2+!7<0o/&6?'�')()FG(O7<6 ��� 6�6�6 � N � 1 6?45450�456t+.B5B50�')CD7<0 6 � 6�6�6 � N � 1�2�1�2�2 �0.757<6?-&6?-&/&0GC&-Ba +.*?*?C&4<+!7<6?�?�,+��H-&0k+.')'2+zX�CH+.4f7P+o*?(OST4<+o/&6?*?()Fk+.')6 �#%'�*?456?B5*?6?456\/&6?' -�C&FG6?450k/&( (O7<6?4<+.�?()0�-&(`()'�45()B5C&'O7P+!7<0�/&(O@�6?-U7P+GB56?FG$&456 FG6?-&0�+.*?*?C&4<+!7<0 N +�*,+.C&B<+/&6?')'2+\$&450�$H+.=U+.�?()0�-&6A/&6?')'=a 6?45450�456E/&(�450�C&-&/�Z 0 � � j -&0�'O7<456 N /H+.')'=a 0.757P+,@.+z(O7<6?4<+.�?()0�-&6E()-�$�0�(�()' 45()B5C&' Z7P+!7<0q[6E/&6?'`7<CD757<0 ������� � N ()-DS|+!757<(`'=a +.')=�0�45(O7<FG0 $&450�/&C&*?6.]8

� > � �8s[� 6

#%-H+.')()�?�?(2+.FG0z-&6?'�/&68757P+.=�')()0o*?0�B<+�+,@�@�()6?-&6 �j -qC&->B5()Bf7<6?Fk+�+.45(O7<FG687<()*?0o+�$&456?*?()B5()0�-&6A�H-&(O7P+ N /&C&6t-�C&FG6?45( 6 7 *?0�()-&*?()/&0�-&0�X�CH+.-&/&0&]

� 7�� �

�7

� � �� " M � � � � �

�/&0h@�6 �� �

� � � ��N6 � [6t'=a 6?$&B5()')0�-�Fk+.*?*Pv&()-H+ �� 6?'VB5()Bf7<6?Fk+q+.45(O7<FG687<()*?0 j � � � N CD7<()')()�?�,+.-&/&0�'2+�B5()-&=�0�'2+�$&456?*?()B5()0�-&6 N ()'�@.+.')0�456\/&( 8 � *?0�- �� 6

� N *,+.')*?0�'2+!7<0\FG6?/&(2+.-U7<6E'2+\450�CD7<()-&6�T@� � � A Y� F /&6?'�4 � � # � 456?Bf7<(O7<C&()B5*?6t()'�@.+.')0�456 � ]8 c [ e�� � �X�C&()-&/&(L]8 c [ e�� > � � � > � �

6

$�6?4f7P+.-U7<0 N +�$H+.4f7<()456A/H+.')'=a (O7<6?4<+.�?()0�-&6 �P N ()'�@.+.')0�456E*,+.')*?0�'2+!7<0\/&6?'�-�C&FG6?4<+!7<0�456�[6 6 �� B5B56?4f@.+.�?()0�-&6 �

� +IST0�45F\C&'2+�� " ST0�45-&()B5*?6qC&- 45()B5C&'O7P+!7<0 $&(<[C~+.*?*?C&4<+!7<0�/&6?')'2+IST0�45F\C&'2+ � ��"

�H-&0;+.')'=a (O7<6?4<+.�?()0�-&6-�C&FG6?450 M � j -&0�'O7<456 N *?0�FG6\B50.757<0�')()-&6,+!7<0�-&6?')'=a � B5B56?4f@.+.�?()0�-&6 ��N +.')'=a +.C&FG6?-U7P+.456z/&6?'Y-�C&FG6?450q/&((O7<6?4<+.�?()0�-&( N ()' -�C&FG6?4<+!7<0�456�45()B5C&'O7P+ -�C&')')0 N $�6?4k*?C&( N $�6?4 � P N '2+ ST0�45F\C&'2+ � ��"456?Bf7<(O7<C&()B5*?6@.+.')0�456��?6?450 N 6?B5B56?-&/&01-&0�- -�C&')')0 ()' /&6?-&0�FG()-H+!7<0�456 N N FG6?-U7<456>'2+ �

� " 456?Bf7<(O7<C&()B5*?6>C&- � + � N*Pv&6z/&6?-&0.7P+k'2+�$&456?B56?-&�,+�/&( C&-H+kB5(O7<CH+.�?()0�-&6t6?*?*?6?�?()0�-H+.')6 N /&0h@�CD7P+�+.'�7<6?-U7P+!7<(O@�0�/&( 6?B56?=�C&()456z'2+/&(O@�()B5()0�-&6 6 � 6 � j -�6�� 68757<( N +q$H+.4f7<()456\/H+.')'=a (O7<6?4<+.�?()0�-&6 P N +.-&*Pv&6�()'V/&6?-&0�FG()-H+!7<0�456o')0�= �6 �D$

� "5"45()B5C&'O7P+I-�C&')')0 � 4�()-&0�+.')'=a (O7<6?4<+.�?()0�-&6 M N ()-&0�'O7<456 N $&C&4G6?B5B56?-&/&03C&=�CH+.')6 N $�6?4�6?-U7<4<+.F } 6>')6�ST0�4mZF\C&')6 N ()' @.+.')0�456\+.B5B5C&-U7<0>/H+.'�-�C&FG6?4<+!7<0�456\6 N +�FG6?-&0�/&6?')'=a 6?45450�456\/&(�450�C&-&/ 0 � N +.-&*Pv&6�X�C&6?')')0+.B5B5C&-U7<0 /H+.' /&6?-&0�FG()-H+!7<0�456 N ')6k/&C&6kST0�45F\C&')6GST0�45-&()B5*?0�-&0 N +.' *?456?B5*?6?456�/&( �

0 N 6?X�C&(O@.+.')6?-U7<6�ZFG6?-U7<6G+.'V7<6?-&/&6?456G/&( +q�?6?450 "�N ()'�@.+.')0�456�/&6?'Y')()FG(O7<6z*?0�-;C&-Ba +.*?*?C&4<+!7<6?�?�,+>/&(O@�6?45B<+>6GX�C&6?Bf7<0

Page 109: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# �����[6 /&0h@�CD7<0E6?B5*?')C&B5(O@.+.FG6?-U7<6Q+.')'=a 0�$�6?4<+.�?()0�-&6Y/&(D/&(O@�()B5()0�-&6Y7<4<+%/&C&6 -�C&FG6?45(D456,+.')(�"H0U+!7<()-&= $�0�()-U7 � � 4/ C&$&$�0�-&(2+.FG0 *Pv&6 8�6 ����� @�6?-&=U+.-&0 *,+.')*?0�'2+!7<(.*?0�- C&- 6?45450�456W456?'2+!7<(O@�0 FG()-&0�456V/&6?')'2+Q$&456?*?()B5()0�-&6/&6?')'2+\Fk+.*?*Pv&()-H+

� � � 0�=�')(2+.FG0z*,+.')*?0�'2+.456A'2+\B56?=�C&6?-U7<6zX�CH+.-U7<(O7![+D]KV� " �

8

� > ������ 8�()-&/&()*,+.-&/&0o*?0�- U

�8

� � � �� " *?0�- � � � � � N ()' @.+.')0�456A*,+.')*?0�'2+!7<0z/&( 8 � N

6?/q6?B5B56?-&/&0K������� U " �

� � �� " *?0�- � � � � � N B5(�vH+D]U

KV� " � K

� � K��U

> ��"K

������� U "� � K

� � �U

> ��" � � �

� c "� � �

� " � �� �

U

> ��" � � �

� c "� � �

� " � � �� � " �

� �8

� � � �� " > ��" � � �

� c " � � �� " c � � �

� � " pp 8

� > � � � �� c " � � �

� " c � � �� � " � KV�

" � � �� T "*?0�- � �

� � �

6 � � T � a T��c T�� �

� � "��/ 6?=�C&61*Pv&6~'=a 6?45450�456;456?'2+!7<(O@�0 *Pv&61B5(\*?0�FGFG68757<6-&6?')'=a +.$&$&450�B5B5()Fk+.456

KV� " *?0�-

U

KV� " [6.]� KV� " >

U

KV� "h�� KV�

"h�p �

� � "�w

������ �� ��������� ����� ��� �������� �"!#� þÿý ��$�� !%���'&(")+* ,�-�)+./�" ��1012431563'798:� ! �<;>= ? @ A"� ;>= ? ��� �CB�� � � !%�D����E� GF�DB�� !%�� %�H I�<!�B�B�� J�� � K!�B�B� �E�LM��E� IF�N� #�D� þÿý ��$�� !%�� K� ! �<; A"� ;PO�?�? �QF�N� #E�������E� GB� ������R� �6�" ���!%��� � �M��!%E� #�" �� �D� þ !%E�� ý �D� ;�S%? � T�D� O �U�" ���!%��� � �M��!%E� T�" �� �D� þ !%E�� ý �D� ;>VW? � þ !%E����DB�F!%��! ����E��M���DF����� ����� %��� !KB>�� GB�B�!K� ý �� �E�!#�X�%F�F�N�� � ���Y�I�� ���!�" �� ��Z �E�E�!%E� [�"�UE�! ý � �]\^!��1_

Page 110: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

��� N. ��"/ ��10 � A � ��� ����������� � � � � ���� �� ��� � � � �������������$&450�=�4<+.F � � ���� �� ��� � � � �������������

3 4:+.B56 /&(�/&()*Pv&(2+.4<+.�?()0�-&6A/&6?')')6A@.+.45(2+ } ()')(()-U7<6?=�6?4 < N - N ( N �456,+.'3+ N } N v N � �6 ] � 6�6�6�6�6 "�N � �

6 ] � 6�6�6�6�6 "3 / 7P+.FG$H+�+\@�()/&6?0G/&6?'�$&450 } ')6?Fk+\/H+o45()B50�'O@�6?4563u45(O7<6

�r N r "3u45(O7<6�r N r "3u45(O7<6�r N r "3u45(O7<6�r N r "3u45(O7<6�r N r "3u45(O7<6�r N r "

aaaaaa

4H0�45F\C&')6A*Pv&6 =�6?-&6?4<+.-&0�(�$&C&-U7<(� ��"

6�

? � " �

� > ��" �

� � M w,w,w M <�� "

� " �

?

�� �

�6 M w,w,w M <

aaaaaa

3 4:+.B56t/&(�*,+.')*?0�')0$&45()-U7 r N a j -&B56?45()456t=�')(`6?Bf7<456?FG( + N } a456,+./ r N + N }$&45()-U7 r N + N }$&45()-U7 r N a j -&B56?45()456t()'`-�C&FG6?450�/&6?( $&C&-U7<(�/&6?B5()/&6?4<+!7<(=a456,+./ r N -$&45()-U7 r N -��6 " �

+��6 " �

+v� �} Zm+ " �!-3�� 6?-&6?4<+.�?()0�-&6t/&6?(�-�C&FG6?45(�FG6?/&(2+.-U7<6E')6E/&C&6EST0�45F\C&')6

/&0 � 6 < � ��N-�

�< " �

��< Z ��" �

v��< " �

��6 " �

v r <� 6 *?0�-U7<()-�C&63 / 7P+.FG$H+�+\@�()/&6?0G/&6?')'2+\B50�')C&�?()0�-&6E*,+.')*?0�'2+!7P+

/&0 �%6 <�6 N -$&45()-U7 r N a y C&-U7<0 a N <$&45()-U7 r N a y C&-U7<(�*?0�->'2+zST0�45F\C&'2+ � ��" �

a3u45(O7<6

�N N r " � �

< "$&45()-U7 r N a y C&-U7<(�*?0�->'2+zST0�45F\C&'2+ �� " �

a3u45(O7<6

�N N r " � �

< "$&45()-U7 r N a������������������������������������������������ ���Xa�%6 *?0�-U7<()-�C&6Bf7<0�$6?-&/

Page 111: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# ��� M. �� � � � � �� �� ���������

4H0�45F\C&')6A*Pv&6 =�6?-&6?4<+.-&0G(`$&C&-U7<(� ��"

6�

? � " �

� > ��" �

� � M w,w,w M <�� "

� " �

?

�� �

�6 M w,w,w M <

j -&B56?45()456t=�')( 6?Bf7<456?FG( + N }6 � ���j -&B56?45()456t()'`-�C&FG6?450�/&6?(�$&C&-U7<( /&6?B5()/&6?4<+!7<(� 6�6 �y C&-U7<0 6y C&-U7<(�*?0�->'2+\ST0�45F\C&'2+

� ��" �6 �y C&-U7<(�*?0�->'2+\ST0�45F\C&'2+

�� " �

6 ������������������������������������������������ ���y C&-U7<0 �y C&-U7<(�*?0�->'2+\ST0�45F\C&'2+� ��" �

6 � 6�6�6 1�1�1 6�6�6 1 M �y C&-U7<(�*?0�->'2+\ST0�45F\C&'2+�� " �

6 � 6�6�6 1�1�1 6�6�6 1 M ������������������������������������������������ ���y C&-U7<0 �y C&-U7<(�*?0�->'2+\ST0�45F\C&'2+� ��" �

6 � 6�6 � 1�1 P 6�6 � 1 y C&-U7<(�*?0�->'2+\ST0�45F\C&'2+�� " �

6 � 6�6 � 1�1 P 6�6 � 1 ����������������������������������������������� ���y C&-U7<0 2y C&-U7<(�*?0�->'2+\ST0�45F\C&'2+� ��" �

6 � 6�6U� 1�1 M 6�6 2 6 2y C&-U7<(�*?0�->'2+\ST0�45F\C&'2+�� " �

6 � 6�6U� 1�1 M 6�6 2 6 2����������������������������������������������� ���y C&-U7<0 y C&-U7<(�*?0�->'2+\ST0�45F\C&'2+� ��" �

6 � 6�6 2�1�1 N 6�6 2 P�Py C&-U7<(�*?0�->'2+\ST0�45F\C&'2+�� " �

6 � 6�6 2�1�1 N 6�6 2 P�P����������������������������������������������� ���y C&-U7<0 �y C&-U7<(�*?0�->'2+\ST0�45F\C&'2+� ��" �

6 � 6�6 1�1 � 6�6 M y C&-U7<(�*?0�->'2+\ST0�45F\C&'2+�� " �

6 � 6�6 1�1 � 6�6 M ����������������������������������������������� ���

Page 112: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

��� P

��� ������������������������������������������� ����� �y C&-U7<0 1�1 Ny C&-U7<( *?0�->'2+zST0�45F\C&'2+� ��" �

6 � 1�1 1�1 � NOM 2y C&-U7<( *?0�->'2+zST0�45F\C&'2+�� " �

6 � 1�1 � 6�6 1 � �������������������������������������������� ����� �y C&-U7<0 1�1 My C&-U7<( *?0�->'2+zST0�45F\C&'2+� ��" �

6 � 1�1 � 1�1 � N My C&-U7<( *?0�->'2+zST0�45F\C&'2+�� " �

6 � 1�1 N 6�6 2�1 P �������������������������������������������� ����� �y C&-U7<0 1�1 Py C&-U7<( *?0�->'2+zST0�45F\C&'2+� ��" �

6 � 1�1 N 1�1 6 N � �y C&-U7<( *?0�->'2+zST0�45F\C&'2+�� " �

6 � 1�1 M 6�6U� 1 � 1������������������������������������������� ����� �y C&-U7<0 1�1�1y C&-U7<( *?0�->'2+zST0�45F\C&'2+� ��" �

6 � 1�1 M 1 P 1 � 1 �y C&-U7<( *?0�->'2+zST0�45F\C&'2+�� " �

6 � 1�1 P 6�6 � 1�1�2������������������������������������������� ����� �y C&-U7<0 � 6�6�6y C&-U7<( *?0�->'2+zST0�45F\C&'2+� ��" �

6 � 1�1 P 1 P�P � N 1y C&-U7<( *?0�->'2+zST0�45F\C&'2+�� " �

6 � 1�1�1 6�6�6 1 NOM������������������������������������������� ����� �y C&-U7<0 � 6�6 �y C&-U7<( *?0�->'2+zST0�45F\C&'2+� ��" �

6 � 1�1�1�1 POM �! 2y C&-U7<( *?0�->'2+zST0�45F\C&'2+�� " � ���������������������������������������������� ����� �

� B5B56?4f@.+.�?()0�-&6 �/ C&$&$�0�-&(2+.FG0;/&(%=�6?-&6?4<+.456 � 6�6 � $&C&-U7<(A-&6?')'=a ()-U7<6?4f@.+.')')0 6 NO��

� y 6?4�B56?FG$&')()*?(O7![+ 6 } 4568@�(O7![+3/&(B5*?45(O757<C&4<+o45()$�0�4f7<(2+.FG0zB50�')0o(�$&45()FG( *?()-&X�C&6t6t=�')(`C&'O7<()FG(�*?()-&X�C&6t6?')FG6?-U7<(`/H+�=�6?-&6?4<+.456.]�6 6 w 6�6 � 6 w 6�6U�R6 w 6�6 2 6 w 6�6 �wOwOw 6 w 1�1 N 6 w 1�1 M 6 w 1�1 P 6 w 1�1�1 ��"�w/ 6?FG$&456�$�6?4 } 4568@�(O7![+q/&(YB5*?45(O757<C&4<+ N + }&} (2+.FG0�45()$�0�4f7P+!7<0q(Y45()B5C&'O7P+!7<(�456?'2+!7<(O@�(Y+.')')6\/&C&6�ST0�45F\C&')6B50�')0o/&6?')')6E$&45()FG6A*?()-&X�C&6t(O7<6?4<+.�?()0�-&( /&6?')'=a +.')=�0�45(O7<FG0 �3 0�FG6%B5(�$&C [0o0�B5B56?4f@.+.456A/H+.')'2+t$&450h@.+\/&(�6?B56?*?C&�?()0�-&6 N '2+tST0�45F\C&'2+ �

� " [6%$&(<[C�+.*?*?C&4<+!7P+z/&6?')'2+ � ��"����� 68757<C&(2+.FG0 $H+.B5B5(�/&6?')'=a +.')=�0�45(O7<FG0tCD7<()')()�?�,+.-&/&0\'2+ ST0�45F\C&'2+� ��"

6AS|+.*?*?(2+.FG0\C&-Ba +.-H+.')()B5(�/&6?')'2+$&450�$H+.=U+.�?()0�-&6o/&6?')'=a 6?45450�456o/&(Y450�C&-&/�Z 0 � N ()-&/&()*Pv&(2+.FG0�*?0�- � Yo6 ��� N '=a 6?45450�456�*Pv&6GB5(W*?0�FGFG68757<6-&6?' 4<+.$&$&456?B56?-U7P+.456�45()B5$�68757<(O@.+.FG6?-U7<6X? 6?/ � ()-3C&-1B5()Bf7<6?Fk+I+.45(O7<FG687<()*?0�+I$&456?*?()B5()0�-&6k�H-&(O7P+ NFG6?-U7<456A*?0�- ����'=a 6?45450�456%*Pv&6tB5( *?0�FGFG68757<6%6?B56?=�C&6?-&/&0G'2+ B50�FGFk+\+.'�$H+.B5B50 > 8�� :L� � � / ( vH+D]

Page 113: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# ��� 1U [ � K

��? " �

?

� � �� Y "U c � K

��U [ �

U� " � �?

� � �� Y " �

� � ���� "5" � � �

���� "��� U�� � K

��U�� c �

U� " � �U�� c �

� � ���� "5" � � �

���� " ��

?

� � �� Y����

� � �h�h� ����

� " ��

� � ��������

� � �h�h� ����

� "*?0�- � � Y � a�� � ��� � a � � ��� � � a��

� � M wOw M </ ( -&0.7<( *Pv&6 +.' $H+.B5B50 > 8�� :L� � B5C@?G6AB5C � B5(�+.*?*?C&F\C&'2+.-&0z7<CD757<(`=�')(�6?45450�45( /&( 450�C&-&/�Z 0 � ����/&0h@�CD7<( +.')')6�B50�FGFG6�6�� 68757<CH+!7<6k-&6?( > � $H+.B5B5(R$&456?*?6?/&6?-U7<( � �YB56?=�C&(2+.FG0�=�')(YBf7<6?B5B5( $H+.B5B5( NCD7<()')()�?�,+.-&/&0z()-U@�6?*?6t'2+ ST0�45F\C&'2+

�� " 6A6�� 68757<C&(2+.FG0\'=a +.-H+.')()B5(:/&6?')'2+z$&450�$H+.=U+.�?()0�-&6%/&6?')'=a 6?45450�456A/&(450�C&-&/�Z 0 ��]

U [ � K��? " �

?

�� [U��

� K��U?

�U� � U " � �

?

� � �� [ " � �

� � ���� " � � � �

� � "5" � � ���� "5" � � �

���� "+ }&} (2+.FG0q()-&/&()*,+!7<0>*?0�- � Y N ��� N � � =�')(W6?45450�45(W*Pv&6�B5(R*?0�FGFG68757<0�-&0>-&6?'R4<+.$&$&456?B56?-U7P+.456�45()B5$�687fZ7<(O@.+.FG6?-U7<6R? N � 6 ()- C&-�B5()Bf7<6?Fk+q+.45(O7<FG687<()*?0�+�$&456?*?()B5()0�-&6\�H-&(O7P+ N FG6?-U7<456o*?0�- ���~'=a 6?45450�456*Pv&6tB5(�*?0�FGFG68757<6E6?B56?=�C&6?-&/&0k()'`$&450�/&0.757<0 � � 6t*?0�- ��� � '=a 6?45450�456E*Pv&6tB5(�*?0�FGFG68757<6E6?B56?=�C&6?-�Z/&0�'2+kB50�FGFk+�+.'�$H+.B5B50 > 8�� :L� � � / (�0�B5B56?4f@.+�S|+.*?()')FG6?-U7<6z*Pv&6�CD7<()')()�?�,+.-&/&0k'2+GST0�45F\C&'2+ �

� "�N+.'W=�6?-&6?45()*?0�$H+.B5B50 B50�-&0I$&456?B56?-U7<( B5(2+ ��� *Pv&6 ���� N Fk+q-&0�-3B5(RvH+>'=a +.*?*?C&F\C&')0q/&6?=�')(W6?45450�45(���

�:

� � M �h�h� M > � /&6?')')6�B50�FGFG6�6�� 68757<CH+!7<6q-&6?( > � $H+.B5B5(u$&456?*?6?/&6?-U7<( N *Pv&6qB5(Q@�6?45(O�H*,+()-U@�6?*?6 CD7<()')()�?�,+.-&/&0\'2+\ST0�45F\C&'2+� ��"��

Page 114: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

� �%6. ��"/ ��10 ��� � ���� � ���� � �� ��� � � ��� �����$&450�=�4<+.F ���� � ���� � �� ��� � � ��� �����

3 4:+.B56 /&(�/&()*Pv&(2+.4<+.�?()0�-&6A/&6?')')6A@.+.45(2+ } ()')(()-U7<6?=�6?4u- N ( N �456,+.' � N v N / ��N / � N <3 / 7P+.FG$H+�+\@�()/&6?0G/&6?'�$&450 } ')6?Fk+\/H+o45()B50�'O@�6?4563u45(O7<6

�r N r "3u45(O7<6�r N r "3u45(O7<6�r N r "3u45(O7<6�r N r "3u45(O7<6�r N r "3u45(O7<6�r N r "

aaaaaa

4H0�45F\C&')6 $�6?4u()'`*,+.')*?0�')0o/&6?')'2+\/&6?45(O@.+!7P+� ��" �LKV�

�� " > KV�

"5" � ��� " �LKV�

�� " > KV�

> � "5" � � �aaaaaa

3 4:+.B56t/&(�*,+.')*?0�')0$&45()-U7 r N a j -&B56?45()456t()'`-�C&FG6?450�/&(�$H+.B5B5( 6?/>()'`$&C&-U7<0G/&( @.+.')CD7P+.�?()0�-&6%a456,+./ r N - N �$&45()-U7 r N - N �<�� �� 6 *?0�-U7<()-�C&6v

� ���� <*,+.')' /&6?4�� N v N / ��N / � "$&45()-U7 r N a ()-&*?456?FG6?-U7<0�v �

a N v$&45()-U7 r N a \ 6?45(O@.+!7P+z*?0�->'2+zST0�45F\C&'2+ � ��" �a N / �$&45()-U7 r N a \ 6?45(O@.+!7P+z*?0�->'2+zST0�45F\C&'2+ �

� " �a N / �$&45()-U7 r N a�������������������������������������������������� �Xa<

�� r <(OS

�< � ')6 � - " =�0.7<0 � 6Bf7<0�$6?-&/

3 0�CD7<()-&6E$�6?4u()'`*,+.')*?0�')0\/&6?')'2+\/&6?45(O@.+!7P+o/&6?')'2+zSTC&-&�?()0�-&6t')0�=U+.45(O7<FG0B5C } 450�CD7<()-&6t/&6?4�� N v N / ��N / � "456,+.' � N v N / ��N / �/ � � �

')0�=���v " Z ')0�= �

� "5" �!v/ �� �')0�=

���v " Z ')0�= �

��Z v "5" � �� r v "45687<C&45-6?-&/

Page 115: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# � � �. �� � � � � �� �� ���������

4H0�45F\C&')6 $�6?4 ()'`*,+.')*?0�')0o/&6?')'2+\/&6?45(O@.+!7P+� ��" �LKV�

�� " > KV�

"5" � ��� " �LKV�

�� " > KV�

> � "5" � � �j -&B56?45()456t()'`-�C&FG6?450�/&(�$H+.B5B5(�6?/>()'`$&C&-U7<0G/&(`@.+.')CD7P+.�?()0�-&6� 6�6�6�6 � �()-&*?456?FG6?-U7<0�v

�6 ���\ 6?45(O@.+!7P+\*?0�->'2+zST0�45F\C&'2+

� ��" �6 � � N � POM �����\ 6?45(O@.+!7P+\*?0�->'2+zST0�45F\C&'2+

�� " �

6 ����� 6 P � � N 2 ����������������������������������������������� ����� ����� ������� ���()-&*?456?FG6?-U7<0�v�

6 � � �\ 6?45(O@.+!7P+\*?0�->'2+zST0�45F\C&'2+� ��" �

6 � M ��� 2 �%6 \ 6?45(O@.+!7P+\*?0�->'2+zST0�45F\C&'2+�� " �

6 ��� 6U� N � P�P 6 2����������������������������������������������� ����� ����� ������� ���()-&*?456?FG6?-U7<0�v�

6 �O� � �\ 6?45(O@.+!7P+\*?0�->'2+zST0�45F\C&'2+� ��" �

6 � P 1�1 NOM 1 N\ 6?45(O@.+!7P+\*?0�->'2+zST0�45F\C&'2+�� " �

6 ��� 6�6 N � � ��� ������������������������������������������������ ����� ����� ������� ���()-&*?456?FG6?-U7<0�v�

6 � 6 N � �\ 6?45(O@.+!7P+\*?0�->'2+zST0�45F\C&'2+� ��" �

6 � 1 � 2 NOM%N \ 6?45(O@.+!7P+\*?0�->'2+zST0�45F\C&'2+�� " �

6 ��� 6�6 � N 2 6 M%P����������������������������������������������� ����� ����� ������� ���()-&*?456?FG6?-U7<0�v�

6 � 6 2 � � �\ 6?45(O@.+!7P+\*?0�->'2+zST0�45F\C&'2+� ��" �

6 � 1 N � 2�2 P 6 \ 6?45(O@.+!7P+\*?0�->'2+zST0�45F\C&'2+�� " �

6 ��� 6�6�6 &� 6�6 P����������������������������������������������� ����� ����� ������� ���()-&*?456?FG6?-U7<0�v�

6 � 6 ��� N � �\ 6?45(O@.+!7P+\*?0�->'2+zST0�45F\C&'2+� ��" �

6 � 1 P 6 � P 2 � 1\ 6?45(O@.+!7P+\*?0�->'2+zST0�45F\C&'2+�� " �

6 ��� 6�6�6 ���, � � ����������������������������������������������� ����� ����� ������� ���()-&*?456?FG6?-U7<0�v�

6 � 6�6 M%P � � �\ 6?45(O@.+!7P+\*?0�->'2+zST0�45F\C&'2+� ��" �

6 � 1�1 ����� M � 1\ 6?45(O@.+!7P+\*?0�->'2+zST0�45F\C&'2+�� " �

6 �������������������������������������������������� ����� ����� ������� ���

Page 116: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

� ���������������������������������������������� ����� ����� ������� ����� �()-&*?456?FG6?-U7<0Gv

�6 � 6�6 2�1 6 N � �\ 6?45(O@.+!7P+\*?0�->'2+\ST0�45F\C&'2+

� ��" �6 � 1�1 6U� 2 2 P\ 6?45(O@.+!7P+\*?0�->'2+\ST0�45F\C&'2+

�� " �

6 ���������������������������������������������� ����� ����� ������� ����� �()-&*?456?FG6?-U7<0Gv�

6 � 6�6 � 1 � 2 � � �\ 6?45(O@.+!7P+\*?0�->'2+\ST0�45F\C&'2+� ��" �

6 � 1�1 M ��� P � 1\ 6?45(O@.+!7P+\*?0�->'2+\ST0�45F\C&'2+�� " �

6 ���������������������������������������������� ����� ����� ������� ����� �()-&*?456?FG6?-U7<0Gv�

6 � 6�6�6 1 M%N � N � �\ 6?45(O@.+!7P+\*?0�->'2+\ST0�45F\C&'2+� ��" �

6 � 1�1 POM�M 1�2\ 6?45(O@.+!7P+\*?0�->'2+\ST0�45F\C&'2+�� " �

6 ���������������������������������������������� ����� ����� ������� ����� �()-&*?456?FG6?-U7<0Gv�

6 � 6�6�6 P�P � P � � �\ 6?45(O@.+!7P+\*?0�->'2+\ST0�45F\C&'2+� ��" �

6 � 1�1 POM�M 1�2\ 6?45(O@.+!7P+\*?0�->'2+\ST0�45F\C&'2+�� " �

6 ���������������������������������������������� ����� ����� ������� ����� �()-&*?456?FG6?-U7<0Gv�

6 � 6�6�6U� � &�, 6 N � �\ 6?45(O@.+!7P+\*?0�->'2+\ST0�45F\C&'2+� ��" �

6 ���\ 6?45(O@.+!7P+\*?0�->'2+\ST0�45F\C&'2+�� " �

6 ���������������������������������������������� ����� ����� ������� ����� �()-&*?456?FG6?-U7<0Gv�

6 � 6�6�6 � ���%6 M 6 2 � �\ 6?45(O@.+!7P+\*?0�->'2+\ST0�45F\C&'2+� ��" �

6 ���\ 6?45(O@.+!7P+\*?0�->'2+\ST0�45F\C&'2+�� " �

6 ���������������������������������������������� ����� ����� ������� ����� �

� B5B56?4f@.+.�?()0�-&(/ (2+�'2+kST0�45F\C&'2+

� ��"*Pv&6G'2+kST0�45F\C&'2+

�� "�N *?0�Bf7<(O7<C&()B5*?0�-&0qC&-Ba +.$&$&450�B5B5()Fk+.�?()0�-&6o/&6?')'2+�/&6?45(O@.+!7P+ �y 6?4u6?-U7<4<+.F } 6�[6E$�0�B5B5( } ()')6ABf7<()Fk+.456A()'`Fk+.B5B5()FG0\6?45450�456t/&(�/&()B5*?45687<()�?�,+.�?()0�-&6E*Pv&6tB5(�*?0�FGFG68757<6 �� +\ST0�45F\C&'2+z/&( � + � ')0�4u+.'�B56?*?0�-&/&0G0�45/&()-&6 N /&(`$&C&-U7<0k()-&()�?(2+.')6���� �

M �� "�N [6.]

KV� " � KV�

� " � K�

�� " � > � " � K

� �

�� " �

> � " �� � ��

�5� > � "

T

"B50.757<4<+.6?-&/&0G+./�+.F } 0o(�FG6?F } 45(KV�� " 6 K

�� " � > � " 0.757<6?-&(2+.FG0&]

KV� " > KV�

� " > K�

�� " � > � " � K

� �

�� " �

> � " �� � ��

�5� > � "

T

"� 4<+.B5*?C&4<+.-&/&0�=�')( ()-D�H-&(O7<6?B5()FG(�/&(�0�45/&()-&6zB5C&$�6?45()0�456\+.'�B56?*?0�-&/&0�6\/&(O@�()/&6?-&/&0�+.F } 0k( FG6?F } 45($�6?4� > � " ]

KV� " > KV�

� "� > � "

>

K�

�� " � K

� �

�� " �

> � "�

Page 117: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# � � 2$H+.B5B<+.-&/&0k+.(`@.+.')0�45(`+.B5B50�')CD7<(L]���� KV� " > KV�

� "� > � "

>

K�

�� " ������ ���� K

� �

�� " �

> � "� ����� +.=�=�()0�45(2+.FG0 ()'�B56?*?0�-&/&0EFG6?F } 450 N B50�Bf7<(O7<C&6?-&/&0E+ � > � " '=a +.FG$&()6?�?�,+ � /&6?')'=a ()-U7<6?4f@.+.')')0 �

M �� "�N 0.757<6?-&(2+.FG0&] ���� KV� " > KV�

� "� > � "

>

K�

�� " ���� a ���� K

� �

�� "� � ����#%' $&45()FG0GFG6?F } 450k+ }&} (2+.FG0G'=a 6?45450�456z+.B5B50�')CD7<0k*Pv&6zB5( *?0�FGFG68757<6 -&6?')'=a +.$&$&450�B5B5()Fk+.456E'2+�/&6?45( Z@.+!7P+t$&45()Fk+

K�

� " *?0�-k'2+AST0�45F\C&'2+ � ��"�N

-&0.7P+t*?0�FG6 ST0�45F\C&'2+E/&6?')'2+� � � JKCEJKF�� H � F H 9 H%FLA � ��� �LJ6B��FLA � N $�6?4f7P+.-U7<0&] ] a ����

� � K� �

�� "� ���� M

� � � ��

� +�ST0�45F\C&'2+�� " @�()6?-&6�/H+.'Y*,+.')*?0�')0�/&6?')'2+kST0�45F\C&'2+k/&6?')'2+ � � JKCEJKF�� H G�JKFLA,C0H � J � � +�ST0�45F\C&'2+�/&(� + � ')0�4u+.'`7<6?45�?0�0�45/&()-&6 N /&(`$&C&-U7<0G()-&()�?(2+.')6 [6.]KV� � � " � KV�

" � K�

� " � � K

� �

� "� �� � � K

� � �

� "� T2 � �

�� c " � ��������"/H+o*?C&( B5(�vH+o'2+\ST0�45F\C&'2+z*?0�B5()/&/&68757P+�/&6?')'2+ � � JKCEJKF�� H � F H 9 H%FLA � ]

KV��� " > KV�

"� �

K�

� " � K

� �

� "�� � � K

� � �

� "� �2 � �

�� T "6tX�C&6?')'2+\/&6?')'2+\*?0�B5()/&/&68757P+�/&6?')'2+ � � JKCEJKF�� HXH ���Z< � FB � JKA,CED!]

KV� " > KV�

> � "� �

K�

� " > K

� �

� "�� � � K

� � �

� "� �2 � >

�� T "/ 0�FGFk+.-&/&0\FG6?F } 450�+\FG6?F } 450o6E/&(O@�()/&6?-&/&0o$�6?4 � ')6A6?B5$&456?B5B5()0�-&(�()-q+,@.+.-U7<(`6?/>+.')'=a ()-&/&()687<450 N7<4<+.B5*?C&4<+.-&/&0G=�')( ()-D�H-&(O7<6?B5()FG( /&(�0�45/&()-&6EB5C&$�6?45()0�456t+.'`7<6?45�?0 N B5(�0.757<()6?-&6.]

KV��� " > KV�

> � "� � � � K�

� " � K

� � �

� "� �N

/ 0.757<4<+.6?-&/&0KV� " +./�+.F } 0o(�FG6?F } 45(`6t$H+.B5B<+.-&/&0k+.( @.+.')0�45(�+.B5B50�')CD7<(`B5(�vH+D]

���� KV��� " > KV�

> � "� � >

K�

� " ���� � � ���� K

� � �

� "� �N ���� w#%'R$&45()FG0�FG6?F } 450I+ }&} (2+.FG0�'=a 6?45450�456k+.B5B50�')CD7<0I*Pv&6�B5( *?0�FGFG68757<6k-&6?')'=a +.$&$&450�B5B5()Fk+.456�'2+>/&6�Z45(O@.+!7P+q$&45()Fk+

K�

� " *?0�-3'2+qST0�45F\C&'2+ �

� "�N -&0.7P+>*?0�FG6#� D C�I���� H /J���� H � � JKCEJKF�� H G�JKFLA,C0H � J N +.'B56?*?0�-&/&0kC&-H+oB5CH+�Bf7<()Fk+ �

Page 118: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

� � j -I=�6?-&6?4<+.')6z')6 ST0�45F\C&')6 $�6?4A+.$&$&450�B5B5()Fk+.456 ')6z/&6?45(O@.+!7<6z*Pv&6\*?0�()-U@�0�')=�0�-&0k$&(<[C�$&C&-U7<(�B50�-&0�$&(<[C+.*?*?C&4<+!7<6 �� +�STC&-&�?()0�-&6t7<6?Bf7ACD7<()')()�?�,+!7P+o$�6?4A+.$&$&450�B5B5()Fk+.456 ()'�*,+.')*?0�')0G/&6?')'2+�/&6?45(O@.+!7P+>[6KV� " �

� <� "�N '2+*?C&( /&6?45(O@.+!7P+>[6

K�

� " � c

�S -&6?' $&C&-U7<0

�� 45()B5C&'O7P+

K�

�� " � c� �

6 w���� ()$�0�4f7<(2+.FG0o$�6?4 0�=�-&(()-&*?456?FG6?-U7<0 � '=a 6?45450�456E456?'2+!7<(O@�0�*Pv&6tB5( *?0�FGFG68757<6ACD7<()')()�?�,+.-&/&0o'2+zST0�45F\C&'2+� ��"

0�'2+�� " ]� � >

�_ � @

� >�@

� �Y454 � 456?'2+!7<(O@�0 � >�_ � @

� >�,

� @� � �Y454 � 456?'2+!7<(O@�06 ��� 6 � � N � POM ����� 6 �O� 6 M Q � 6%[ 6 ����� 6 P � � N 2 6 � � � M 6 Q � 6 c6 � � � 6 � M ��� 2 �%6 6 ��� M�M Q � 6 c6 ��� 6U� N � P�P 6 2 6 ��� � � M Q � 6

�6 �O� � � 6 � P 1�1 NOM 1 N 6 � 2 6�6 N Q � 6 c

6 ��� 6�6 N � � ��� � 6 �O� 2 6 � Q � 6 �

6 � 6 N � � 6 � 1 � 2 NOM%N 6 �O��� 2 � Q � 6 c6 ��� 6�6 � N 2 6 M%P 6 � 2 � N 6 Q � 6

T6 � 6 2 � � � 6 � 1 N � 2�2 P 6 6 � M�M 2 � Q � 6

� 6 ��� 6�6�6 &� 6�6 P 6 � P �%6�6 Q � 6 c

6 � 6 ��� N � � 6 � 1 P 6 � P 2 � 1 6 � 2 P�P 2BQ � 6 � 6 ��� 6�6�6 ���, � � 6 � ��� P Q � 6

c6 � 6�6 M%P � � � 6 � 1�1 6U� 2 2 P 6 �O� 1 � 2BQ � 6

� 6 ��� 6 �6 � 6�6 2�1 6 N � � 6 � 1�1 ����� M � 1 6 � 1 M%N�N Q � 6 T 6 ��� 6 �6 � 6�6 � 1 � 2 � � � 6 � 1�1 M ��� P � 1 6 � P�P Q � 6 T 6 ��� 6 �6 � 6�6�6 1 M%N � N � � 6 � 1�1 POM�M 1�2 6 � � � � Q � 6 T 6 ��� 6 �6 � 6�6�6 P�P � P � � � 6 � 1�1 POM�M 1�2 6 � � � � Q � 6 T 6 ��� 6 �6 � 6�6�6U� � &�, 6 N � � 6 ��� 6 � 6 ��� 6 �6 � 6�6�6 � ���%6 M 6 2 � � � 6 ��� 6 � 6 ��� 6 �3 0�FG6WB5(�$&C [0 -&0.7P+.456W'2+QST0�45F\C&'2+

�� " [6R$&(<[Ct+.*?*?C&4<+!7P+u/&6?')'2+ � ��"

6W*?0�-U@�6?45=�6 $&(<[CE@�6?')0�*?6?FG6?-U7<6 �

Page 119: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# � � �. ��"/ ��10 ��� � ���� � ���� � �� ����� � ���� � 0 ����� ���$&450�=�4<+.F ���� � ������ � ���� � 0 ����� ���

3 4:+.B56E/&(�/&()*Pv&(2+.4<+.�?()0�-&6A/&6?')')6A@.+.45(2+ } ()')(()-U7<6?=�6?4 ( N ":+.=456,+.'�6 N 6?$&B5()')0�- N � Fk+��456,+.' +./&/ N / C&F N / C&FG0�')/ N -3 / 7P+.FG$H+o+o@�()/&6?0�/&6?'�$&450 } ')6?Fk+3u45(O7<6

�r N r "3u45(O7<6�r N r "3u45(O7<6�r N r "3u45(O7<6�r N r "3u45(O7<6�r N r "

aaaaa

3 +.')*?0�')0o/&6?')'2+ / 6?45()6t#%45FG0�-&()*,+< � :=<

K/&( /

� ��!- "

aaaaa

3u45(O7<6�r N r " a j -&B56?45()456EFk+.B5B5()FG0o-�C&FG6?450G/&(`(O7<6?4<+.�?()0�-&(=a456,+./

�r N r " � Fk+��3u45(O7<6�r N r " � Fk+��3 j -&()�?(2+.')()�?�,+.�?()0�-&6%/&6?')'=a +./&/&6?-&/&0�-�Z 6?B5()FG0-

� ��� 6+./&/�-(

�6

":+.=�

63 j -&()�?(2+.')()�?�,+.�?()0�-&6%/&6?')'2+ / 0�FGFk+

/ C&F� ��� 63 3 +.')*?0�')0o/&6?')'=a 6?$&B5()')0�-�Fk+.*?*Pv&()-H+6?$&B5()')0�-�6?$&B

� "$&45()-U7 r N a �Y$&B5()')0�-qFk+.*?*Pv&()-H+ � a N 6?$&B5()')0�-3 4:+.B56E/&( 3 +.')*?0�')0� *?0�-U7<()-�C&6(

�(� �-

�-� ��� 6+./&/

� ���6 �!-(OS

�+./&/ � =�6 � / C&F r 6?$&B5()')0�- " 7<v&6?-/ C&FG0�')/

�/ C&F

/ C&F�/ C&F

�+./&/

Page 120: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

� � N3 / 7P+.FG$H+\/&6?'`@.+.')0�456E*,+.')*?0�'2+!7<0\/&6?')'2+ / 6?45()6$&45()-U7 r N a��������������������������������������������Xa$&45()-U7 r N a #%$&$&450�B5B5()Fk+.�?()0�-&6A/&6?')'2+\B56?45()6 �

a N / C&F$&45()-U7 r N a j 7<6?4<+.�?()0�-&(`6?B56?=�C&(O7<6 �a N ($&45()-U7 r N a��������������������������������������������Xa6?')B56 $&45()-U7 r N a � �!- [6tFG()-&0�456A/&( �

/ -�Z ��" r 6?$&B5()')0�-�Fk+.*?*Pv&()-H++a$&45()-U7 r N a #%$&$&450�B5B5()Fk+.�?()0�-&6A/&6?')'2+\B56?45()6 �a N / C&F":+.=

� �6?-&/&(OS3 � 6?45FG()-H+.�?()0�-&6A/&6?'�*?()*?')0o/&(�*,+.')*?0�')0(OS

�( � ')6 � � Fk+�� � +.-&/ � ":+.= � 6?X � 6 " =�0.7<0 �$&45()-U7 r N a j 7<6?4<+.�?()0�-&(`6?B56?=�C&(O7<6 �

a N (Bf7<0�$6?-&/y 45()Fk+�/&(�/&6?B5*?45(O@�6?456oC&-H+�$&450h@.+k/&(�6?B56?*?C&�?()0�-&6o/&6?')'=a +.')=�0�45(O7<FG0\/&6?B5*?45(O757<0 N B5(�45()$�0�4f7P+.-&0G')6z45( ZB5$�0�Bf7<6 +.(`X�C&6?B5(O7<( $&450�$�0�Bf7<(`-&6?'�$H+.4<+.=�4<+!ST0 ���O� 6 � 2 ]

�+ " � +oB56?45()6 +.45FG0�-&()*,+D] � � � c �<[6�C&-H+IB56?45()6q+�7<6?45FG()-&(R$�0�B5(O7<(O@�( N $�6?4f7P+.-U7<0 '2+�B5C&*?*?6?B5B5()0�-&6q/&6?')')6�B50�FGFG6k$H+.45�?(2+.')( � � � � N *?0�-� � � � � � � c c� [6EBf7<4568757P+.FG6?-U7<6 *?456?B5*?6?-U7<6 N 6tX�C&()-&/&(�'2+\B56?45()6t/&(O@�6?45=�6t$�0�B5(O7<(O@.+.FG6?-U7<6.]')()F�

s�

� � � ��

�} " j ->C&->B5()Bf7<6?Fk+�+.45(O7<FG687<()*?0o+�$&456?*?()B5()0�-&6A�H-&(O7P+o'2+\B56?45()6tvH+oB50�FGFk+z�H-&(O7P+o()-qX�CH+.-U7<0&]

� � _ c � � � � �

� ��� � � � � �

� �� � � � ��� Y��/&0h@�6 ��� Y��A[6t'=a 6?$&B5()')0�-�Fk+.*?*Pv&()-H+ �

�* " y 6?4�()'�$&C&-U7<0 �

} "�N ()- C&- B5()Bf7<6?Fk+ +.45(O7<FG687<()*?0u+u$&456?*?()B5()0�-&6W�H-&(O7P+ N '2+ B50�FGFk+ -&0�-z*,+.F } (2++.')'=a (O7<6?4<+.�?()0�-&6 > 8�� :L�@? N $�6?4u*?C&(�B5(`@�6?45(O�H*,+�'2+o/&()B5C&=�CH+.=�')(2+.-&�,+D]�

� �� � � � ��� Y��6E$�6?4u0�=�-&(B< � 45()B5C&'O7P+D]

� � � � �

Page 121: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# � � M

�/ " y 0�()*Pv 96o'2+kB56?45()6�[6\/&(O@�6?45=�6?-U7<6 N ()'�*,+.')*?0�')0k/&6?')'2+kB56?45()6o$&C [0�6?B5B56?456G+.45456?Bf7P+!7<0q0�X�CH+.-&/&0 4<+.=�=�()C&-&=�6EC&-�Fk+.B5B5()FG0�-�C&FG6?450k/&( (O7<6?4<+.�?()0�-&(�+.B5B56?=�-H+!7<0k0kX�CH+.-&/&0 4<+.=�=�()C&-&=�6E()'`@.+.')0�456*Pv&6t@�6?45(O�H*,+�()' $&C&-U7<0

�* "�� ()$�0�4f7<(2+.FG0 /&(%B56?=�C&(O7<0~+.')*?C&-&6I$&450h@�6 /&(%6?B56?*?C&�?()0�-&6 /&6?')'=a +.')=�0�45(O7<FG0 *Pv&6 *,+.')*?0�'2+�()'u@.+.')0�456/&6?')'2+kB56?45()6�+.45FG0�-&()*,+G()- $&456?*?()B5()0�-&6\�H-&(O7P+ � y 6?4tB56?FG$&')()*?(O7![+G/&(VB5*?45(O757<C&4<+ N /H+!7<0�'=a 6?')68@.+!7<0�-�C�ZFG6?450k/&(�(O7<6?4<+.�?()0�-&( 6?B56?=�C&(O7<6z/H+.')'=a +.')=�0�45(O7<FG0 N @�6?-&=�0�-&0�45()$�0�4f7P+!7<6 ')6 B50�FGFG6 $H+.45�?(2+.')( 456?'2+!7<(O@�6+.')')6AC&'O7<()FG6 (O7<6?4<+.�?()0�-&( 6?B56?=�C&(O7<6 �

Page 122: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

� � P. �� � � � � �� �� ���������

3 +.')*?0�')0\/&6?')'2+ / 6?45()6E#%45FG0�-&()*,+< � :=<

K/&( /

� ��!- "

j -&B56?45()456tFk+.B5B5()FG0o-�C&FG6?450G/&(`(O7<6?4<+.�?()0�-&(� 6�6�6�6�6�6�Y$&B5()')0�-qFk+.*?*Pv&()-H+

� ���O�1 �%6U� 1 ��Z 6 M

�������������������������������������������#%$&$&450�B5B5()Fk+.�?()0�-&6E/&6?')'2+\B56?45()6� � 2 � P�P 1 6�6�6 1j 7<6?4<+.�?()0�-&(�6?B56?=�C&(O7<6

�N 6 2�1 M 6

�������������������������������������������

�������������������������������������������#%$&$&450�B5B5()Fk+.�?()0�-&6E/&6?')'2+\B56?45()6� � 2 � P�P 1 6�6U� Pj 7<6?4<+.�?()0�-&(�6?B56?=�C&(O7<6

�N 6 2�1 M ��������������������������������������������

�������������������������������������������#%$&$&450�B5B5()Fk+.�?()0�-&6E/&6?')'2+\B56?45()6� � 2 � P�P 1 6�6 Mj 7<6?4<+.�?()0�-&(�6?B56?=�C&(O7<6

�N 6 2�1 M �

�������������������������������������������

�������������������������������������������#%$&$&450�B5B5()Fk+.�?()0�-&6E/&6?')'2+\B56?45()6� � 2 � P�P 1 6�6 N�Nj 7<6?4<+.�?()0�-&(�6?B56?=�C&(O7<6

�N 6 2�1 M 2

�������������������������������������������

� �!-q6%aHFG()-&0�456A/&( �/ -�Z ��" r 6?$&B5()')0�-�Fk+.*?*Pv&()-H+#%$&$&450�B5B5()Fk+.�?()0�-&6E/&6?')'2+\B56?45()6

� � 2 � P�P 1 6�6 N�Nj 7<6?4<+.�?()0�-&(�6?B56?=�C&(O7<6�

N 6 2�1 M � B5B56?4f@.+.�?()0�-&6� a +.')=�0�45(O7<FG0\B5(�+.45456?Bf7P+�+.')'=a (O7<6?4<+.�?()0�-&6A-�C&FG6?450

� � �N 6 2�1 M � � �

N 6 2�1 M ��N *?()0&[6 X�CH+.-&/&045()B5C&'O7P+D]�

� �� � � � ��� Y��

Page 123: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# � � 1� C&()-&/&(`()'`$&C&-U7<0

�* " B5(�456,+.')()�?�,+o+.')'=a (O7<6?4<+.�?()0�-&6 -�C&FG6?450 N 6 2�1 M � ]#%$&$&450�B5B5()Fk+.�?()0�-&6E/&6?')'2+\B56?45()6 � 2 � P�P 1 6�6 N�N�Y45450�456t456?'2+!7<(O@�0 ��� 1 6 M 2 P�N 2 ��Z 6 N ()$�0�4f7<(2+.FG0\/&(�B56?=�C&(O7<0G')0�Bf7<6?B5B50�+.')=�0�45(O7<FG0\$�6?4u()'`*,+.')*?0�')0�/&6?')'2+\B56?45()6z+.45FG0�-&()*,+ N Bf@�()')C&$�Z$H+!7<0�()-q/&0�$&$&(2+o$&456?*?()B5()0�-&6 �

Page 124: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

� 2 6 $&450�=�4<+.F ���� � ������ � ���� � 0 ����� ���

3 4:+.B56E/&(�/&()*Pv&(2+.4<+.�?()0�-&6A/&6?')')6A@.+.45(2+ } ()')(()-U7<6?=�6?4 ( N ":+.=/&0�C } ')6E$&456?*?()B5()0�-q6 N 6?$&B5()')0�- N � Fk+��/&0�C } ')6E$&456?*?()B5()0�->+./&/ N / C&F N / C&FG0�')/ N -3u45(O7<6

�r N r "3u45(O7<6�r N r "3u45(O7<6�r N r "3u45(O7<6�r N r "3u45(O7<6�r N r "

aaaaa

3 +.')*?0�')0o/&6?')'2+ / 6?45()6t#%45FG0�-&()*,+< � :=<

K/&( /

� ��!- "

aaaaa

3u45(O7<6�r N r " a j -&B56?45()456EFk+.B5B5()FG0o-�C&FG6?450�/&(�(O7<6?4<+.�?()0�-&(=a456,+./

�r N r " � Fk+��3u45(O7<6�r N r " � +.F��

3 j -&()�?(2+.')()�?�,+.�?()0�-&6%/&6?')'=a +./&/&6?-&/&0�-�Z 6?B5()FG0-� ��� 6+./&/�-(

�6

":+.=�

6

3 j -&()�?(2+.')()�?�,+.�?()0�-&6%/&6?')'2+ / 0�FGFk+/ C&F

� ��� 63 3 +.')*?0�')0\/&6?')'=a 6?$&B5()')0�-�Fk+.*?*Pv&()-H+o$&456?*?()B50�-&6t/&0�$&$&(2+6?$&B5()')0�-�6?$&B5/&0�C } ')6

� "$&45()-U7 r N a �Y$&B5()')0�-qFk+.*?*Pv&()-H+ � a N 6?$&B5()')0�-3 4:+.B56E/&(�*,+.')*?0�')0� *?0�-U7<()-�C&6(

�(� �-

�-� ��� 6+./&/

� ���6 �!-(OS

�+./&/ � =�6 � / C&F r 6?$&B5()')0�- " 7<v&6?-/ C&FG0�')/

�/ C&F

/ C&F�/ C&F

�+./&/

3 3 +.')*?0�')0\/&6?')'=a 6?45450�456t+.B5B50�')CD7<06?454 � �+ } B

�/ C&F Z / C&FG0�')/ "

Page 125: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# � 2 �3 / 7P+.FG$H+o/&6?'`@.+.')0�456A*,+.')*?0�'2+!7<0o/&6?')'2+ / 6?45()6$&45()-U7 r N a��������������������������������������������Xa$&45()-U7 r N a #%$&$&450�B5B5()Fk+.�?()0�-&6E/&6?')'2+\B56?45()6 �

a N / C&F$&45()-U7 r N a j 7<6?4<+.�?()0�-&(`6?B56?=�C&(O7<6�a N ($&45()-U7 r N a��������������������������������������������Xa$&45()-U7 r6?')B56 $&45()-U7 r N a � �!-q6?nGFG()-&0�456A/&( / �-�Z ��" r 6?$&B5()')0�-�Fk+.*?*Pv&()-H++a$&45()-U7 r N a #%$&$&450�B5B5()Fk+.�?()0�-&6E/&6?')'2+\B56?45()6 �

a N / C&F":+.=� �6?-&/&(OS

3 � 6?45FG()-H+.�?()0�-&6A/&6?'�*?()*?')0�/&(`*,+.')*?0�')0(OS�( � ')6 � � Fk+�� � +.-&/ � ":+.= � 6?X � 6 " =�0.7<0 �$&45()-U7 r N a j 7<6?4<+.�?()0�-&(`6?B56?=�C&(O7<6 �

a N (Bf7<0�$6?-&/

. �� � � � � �� �� ���������

/ (%/&6?B5*?45(O@�6 N /&(%B56?=�C&(O7<0 N C&- 6?B56?FG$&()01/&( 6?B56?*?C&�?()0�-&6 /&(%*?C&(%B5(%45()$�0�4f7P+.-&0 N $�6?4kB56?FG$&')()*?(O7![+ N/H+!7<0 '=a 6?')68@.+!7<0�-�C&FG6?450 /&(Q(O7<6?4<+.�?()0�-&(W6?B56?=�C&(O7<6 N B50�')0I(Q45()B5C&'O7P+!7<(W456?'2+!7<(O@�( +.')')6GC&'O7<()FG6k*?()-&X�C&6(O7<6?4<+.�?()0�-&( �3 +.')*?0�')0o/&6?')'2+ / 6?45()6E#%45FG0�-&()*,+< � :=<

K/&( /

� ��!- "j -&B56?45()456tFk+.B5B5()FG0o-�C&FG6?450�/&(�(O7<6?4<+.�?()0�-&(

� 6�6�6�6�6�6�6�Y$&B5()')0�-�Fk+.*?*Pv&()-H+�� � ���%6 � N 6 � ��Z � N�������������������������������������������#%$&$&450�B5B5()Fk+.�?()0�-&6A/&6?')'2+\B56?45()6

� � N � N 1 � 2 �����j 7<6?4<+.�?()0�-&(`6?B56?=�C&(O7<6 1�1�1�1�1�1 N�������������������������������������������

Page 126: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

� 2 ��������������������������������������������#%$&$&450�B5B5()Fk+.�?()0�-&6E/&6?')'2+\B56?45()6

� � N � N 1 � 2 ��� �j 7<6?4<+.�?()0�-&(�6?B56?=�C&(O7<6 1�1�1�1�1�1 N�������������������������������������������

�������������������������������������������#%$&$&450�B5B5()Fk+.�?()0�-&6E/&6?')'2+\B56?45()6� � N � N 1 � 2 ��� 2j 7<6?4<+.�?()0�-&(�6?B56?=�C&(O7<6 1�1�1�1�1�1 M

�������������������������������������������

�������������������������������������������#%$&$&450�B5B5()Fk+.�?()0�-&6E/&6?')'2+\B56?45()6� � N � N 1 � 2 ���, j 7<6?4<+.�?()0�-&(�6?B56?=�C&(O7<6 1�1�1�1�1�1 P

�������������������������������������������

�������������������������������������������#%$&$&450�B5B5()Fk+.�?()0�-&6E/&6?')'2+\B56?45()6� � N � N 1 � 2 �����j 7<6?4<+.�?()0�-&(�6?B56?=�C&(O7<6 1�1�1�1�1�1�1

�������������������������������������������

j 7<6?4<+.�?()0�-&(�6?B56?=�C&(O7<6� � 6�6�6�6�6�6�6� B5B56?4f@.+.�?()0�-&6�YB56?=�C&6?-&/&0 (z*,+.')*?0�')(E()-�/&0�$&$&(2+ $&456?*?()B5()0�-&6;'=a +.')=�0�45(O7<FG01B5(z+.45456?Bf7P+ +.')'=a (O7<6?4<+.�?()0�-&6 -�C&FG6?450

� 6�6�6�6�6�6�6 N ST0�45-&6?-&/&0;*?0�FG6>45()B5C&'O7P+!7<0&] � N � N 1 � 2 ������� #%-&*Pv&6I()- $&456?*?()B5()0�-&6q/&0�$&$&(2+ @.+.')=�0�-&0')6kBf7<6?B5B56�*?0�-&B5()/&6?4<+.�?()0�-&(RS|+!757<6k$�6?4\(R45()B5C&'O7P+!7<(R0.757<6?-�CD7<( /H+.')'=a +.')=�0�45(O7<FG0�()-3B5()-&=�0�'2+>$&456?*?()B5()0!Z-&6 � j -�X�C&6?Bf7<0k*,+.B50G'=a +.')=�0�45(O7<FG0 B5( +.45456?Bf7P+�$�6?45*Pv 96G[6tBf7P+!7<0G4<+.=�=�()C&-U7<0�()'`Fk+.B5B5()FG0o-�C&FG6?450G/&((O7<6?4<+.�?()0�-&( �

Page 127: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# � 2�2. ��"/ ��10 ��� � ���� � ���� � �� ����� � � ������ �$&450�=�4<+.F ���� � ���� � �� ����� � � ������ �

3 4:+.B56 /&(�/&()*Pv&(2+.4<+.�?()0�-&6A/&6?')')6%@.+.45(2+ } ()')(()-U7<6?=�6?4 - ��N <456,+.' �� �6�6 "�N B5()=�Fk+ ��N B5()=�Fk+ N FG6?/ N -3 / 7P+.FG$H+o+o@�()/&6?0�/&6?'�$&450 } ')6?Fk+\/H+�45()B50�'O@�6?456

3u45(O7<6�r N r "3u45(O7<6�r N r "3u45(O7<6�r N r "3u45(O7<6�r N r "3u45(O7<6�r N r "3u45(O7<6�r N r "

aaaaaa

3 +.')*?0�')0o/&6?')'2+ � +.45(2+.-&�,+� ��"%�

��-�Z ��" B5C&F ��(

� ��N- �

���( " Z FG6?/ " � �

�� "%� � �

-�Z ��" B5C&F ��(� ��N

- � ��( " � � �Z � �!- �

B5C&F ��(� ��N

- � ��( "5" � �

aaaaaa

3 4:+.B56E/&(;*,+.')*?0�')0$&45()-U7 r N a j -&B56?45()456t'2+\')C&-&=�v&6?�?�,+�/&6?'`@�68757<0�456%a456,+./ r N - �$&45()-U7 r N - �$&45()-U7 r N a j -&B56?45()456t')6E*?0�FG$�0�-&6?-U7<(`/&6?'`@�68757<0�456%a/&0 � 6 ( � ��N- �456,+./ r N � �( "$&45()-U7 r N � �( "� 6 *?0�-U7<()-�C&6FG6?/

�6/&0 �%6 <� ��N

- �FG6?/�FG6?/

���< "�%6 *?0�-U7<()-�C&6-

�- �FG6?/�FG6?/ �!-B5()=�Fk+�6B5()=�Fk+ � �6/&0 2 6 <� ��N

-B5()=�Fk+�B5()=�Fk+

� �5���< " Z FG6?/ " r�r � "B5()=�Fk+ � �

B5()=�Fk+ � � ���< " r�r � "2 6 *?0�-U7<()-�C&6B5()=�Fk+

� � �����-�Z ��"5" r B5()=�Fk+B5()=�Fk+ � � � ���

��-�Z ��"5" r �

B5()=�Fk+ � Z - r �F�6?/ r�r � "5"3 / 7P+.FG$H+o+o@�()/&6?0�/&6?(�45()B5C&'O7P+!7<($&45()-U7 r N a � 6?/&(2+ �

a N FG6?/$&45()-U7 r N a � +.45(2+.-&�,+\*?0�->'2+zST0�45F\C&'2+ � ��" �a N B5()=�Fk+$&45()-U7 r N a � +.45(2+.-&�,+\*?0�->'2+zST0�45F\C&'2+ �

� " �a N B5()=�Fk+ �$&45()-U7 r N a �t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�

aBf7<0�$6?-&/

Page 128: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

� 2 . �� � � � � �� �� ���������

3 +.')*?0�')0\/&6?')'2+ � +.45(2+.-&�,+� ��"%�

��-�Z ��" B5C&F ��(

� ��N- �

���( " Z FG6?/ " � �

�� "%� � �

-�Z ��" B5C&F ��(� ��N

- � ��( " � � �Z � �!- �

B5C&F ��(� ��N

- � ��( "5" � �

j -&B56?45()456t'2+o')C&-&=�v&6?�?�,+�/&6?'`@�68757<0�456�j -&B56?45()456t')6E*?0�FG$�0�-&6?-U7<(�/&6?'`@�68757<0�456� � &�M �P �� 6 �� 6?/&(2+

�N �O� 1�1�1�1�1 P �� +.45(2+.-&�,+o*?0�->'2+zST0�45F\C&'2+

� ��" � � 6 �O� 1�1�1�1�1 P� +.45(2+.-&�,+o*?0�->'2+zST0�45F\C&'2+�� " � � 6 � �%6�6�6�6�6 P�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�

3 +.')*?0�')0\/&6?')'2+ � +.45(2+.-&�,+� ��"%�

��-�Z ��" B5C&F ��(

� ��N- �

���( " Z FG6?/ " � �

�� "%� � �

-�Z ��" B5C&F ��(� ��N

- � ��( " � � �Z � �!- �

B5C&F ��(� ��N

- � ��( "5" � �

j -&B56?45()456t'2+o')C&-&=�v&6?�?�,+�/&6?'`@�68757<0�4562j -&B56?45()456t')6E*?0�FG$�0�-&6?-U7<(�/&6?'`@�68757<0�456� 6�6�6�6 �� 6�6�6 ���� 6�6�6U� �� 6?/&(2+

� � 6�6�6 ���� +.45(2+.-&�,+o*?0�->'2+zST0�45F\C&'2+� ��" � ���� +.45(2+.-&�,+o*?0�->'2+zST0�45F\C&'2+�� " �

6 ��t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�� B5B56?4f@.+.�?()0�-&(

Page 129: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# � 2 �� 6EST0�45F\C&')6ACD7<()')()�?�,+!7<6A/H+.')'=a +.')=�0�45(O7<FG0tB50�-&0&]

� ��"� �� �< > �

� ��c � > � " �

�� " �

�� �< > �

�� � ��c � > �<

� � ��c

� ��+.-H+.')()�?�?(2+.FG0z')6E/&C&6t$&450h@�6 /&(�6?B56?*?C&�?()0�-&6 B56?$H+.4<+!7P+.FG6?-U7<6 �� 6?')'2+�$&45()Fk+�$&450h@.+k()'�@�68757<0�456�[6

� �� M M M M P M � 6 "�� j - +.45(O7<FG687<()*,+�+k$&456?*?()B5()0�-&6 ()-D�H-&(O7P+ vH+�FG6?/&(2+k�

�N w � 6z@.+.45(2+.-&�,+ � � � � 6 w � � � a 6?45450�456\456?'2+!7<(O@�0�*Pv&6oB5(V*?0�FGFG68757<6zCD7<()')()�?�,+.-&/&0'2+zST0�45F\C&'2+

� ��"[6.]

� � � > U� � �� � � � � ��� 6 w � > � 6 wO� 1�1�1�1�1 P �� 6 w � �6 w � Q � 6

�CD7<()')()�?�,+.-&/&0o()-U@�6?*?6t'2+oB56?*?0�-&/H+oST0�45F\C&'2+

� � � > U� � �� � � � � ��� 6 w � > � 6 w �%6�6�6�6�6 P �� 6 w � �6 w M Q � 6

�()- 6?-U7<4<+.F } (V(V*,+.B5(Y()'�45()B5C&'O7P+!7<0I[6�+.*?*?68757P+ } ()')6 N B5(V0.757<()6?-&6oC&-Ba�+.*?*?C&4<+!7<6?�?�,+��H-&0>+.')'2+GB568757<()Fk+*?(OST4<+\B5()=�-&(O�H*,+!7<(O@.+ �#%-H+.')()�?�?(2+.FG0u0�4<+%'2+ B56?*?0�-&/H+A$&450h@.+ � j '�@�68757<0�456E[6 � � �

6�6�6�6 M � 6�6�6 � M � 6�6�6U� "�� j -�+.45(O7<FG687<()*,++k$&456?*?()B5()0�-&6\()-D�H-&(O7P+ vH+�FG6?/&(2+G�� � 6�6�6 � 6z@.+.45(2+.-&�,+ � � � ��� � a 6?45450�456\456?'2+!7<(O@�0�*Pv&6oB5(*?0�FGFG68757<6ECD7<()')()�?�,+.-&/&0\'2+\ST0�45F\C&'2+

� ��"[6.]

� � � > U� � �� � � � � ����w 6 > ��w 6 ���w 6 �6

CD7<()')()�?�,+.-&/&0o()-U@�6?*?6t'2+oB56?*?0�-&/H+oST0�45F\C&'2+� � � > U� � �� � � � � ����w 6 > 6 w 6 ���w 6 �

6 wO� Q � 6 c� C&6?Bf7<0IB56?*?0�-&/&0I6?B56?FG$&()0 N FG68757<6G()-�')C&*?6k()'VS|+!757<0>*Pv&6k()-3+.45(O7<FG687<()*,+�+>$&456?*?()B5()0�-&6��H-&(O7P+q'2+ST0�45F\C&'2+

�� " $&C [0q=�6?-&6?4<+.456\6�� 68757<(V/&(�*,+.-&*?6?')'2+.�?()0�-&6 /&()Bf7<45CD757<(O@.+k6\*?0�-&/&C&45456�+./ C&- 45()B5C&'O7P+!7<0/&6?'`7<CD757<0�B5*?0�454568757<0 �

Page 130: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

� 2 N������� �������� ��� ������������� ��������������������� ��!"�#�������$�&%����'�(�� )�+*�,

- �/. - � �)�0��'�1��������*2�+���)*2���

����������� 3 �� *���������547698:8:;=<�; ����������� ���?>� ��#$#$�����A@ ,+- * �\ (AB56?=�C&(O7<0~@�6?-&=�0�-&0 45()$�0�4f7P+!7<(%/&C&6�6?$&()B50�/&(E456?*?6?-U7<6?FG6?-U7<63+.*?*,+./&CD7<( � � 0 B5*?0�$�0 [6 X�C&6?')')0/&(R/&()FG0�Bf7<4<+.456�*Pv&6�'2+,@�0�4<+.-&/&0�()-;C&-1B5()Bf7<6?Fk+�+.45(O7<FG687<()*?0>+�$&456?*?()B5()0�-&6��H-&(O7P+ N '2+qS|+.*?()')(O7![+�/&(*?0�FGFG68757<6?456z6?45450�45( N B5(2+�$&C&4�6 � G�G�D�� � N $&C [0�*?0�FG$&450�FG68757<6?456\C&- 45()B5C&'O7P+!7<0 N ')6\*?C&(�*?0�-&B56?=�C&6?-&�?6-&0�-qB5(�')()FG(O7P+.-&0z$&C&4f7<450�$&$�0 N +.'`B50�')0�6?45450�456E-�C&FG6?45()*?0 ����������CB D�EGFH6+IJIJ6�<�KMLON=P�QR69;=PSFTNUE�8:NT6�<WV:KXQRIXN�YU<C69;j ' � � 4H6 }&} 4<+.()0 � 1�1 ��N /&C&4<+.-U7<6Q'2+u=�C&6?454<+%/&6?'D=�0�'OST0 N C&-H+ } +!757<6?45(2+%+.FG6?45()*,+.-H+u/&(DFG()B5B5()')( H%A,C � D A+ \ vH+.4<+.- N #%4<+ } (2+ / +.C&/&(O7P+ N -&0�-k45()C&B5*P[ � +./k()-U7<6?45*?68757P+.456uC&-kFG()B5B5()')6 / *?C&/�()4<+.*Pv&6?-&0 � � 0 / *?C&/*?0�')$�[ � C&-1+.*?X�CH+.4f7<()6?4<+.FG6?-U7<0�/&6?')'=a 6?B56?45*?(O7<0I+.FG6?45()*,+.-&0>C&*?*?()/&6?-&/&0 � P B50�')/H+!7<( � : -34<+.$&$�0�4f7<0/&6?' � # � Z j � � � 3 Z 1 � Z � N N '=a C10G*?()0 =�6?-&6?4<+.')6q/&( *?0�-U7P+ } ()')(O7![+ N /H+.'Q7<(O7<0�')0 n[Z � �� � � J]\ � � JKF1� � 9%D H%A,C � D A ����F 6&CEDDT@� JKI HA� DU��A �9H%CEJC��H�G�D FB/D A,AD H� ��F I H � �K��F�� � D F&H%I�JKFLAD /J�� � � �KAJKI H�H�\ ��H C0H%F J

�C0H�T � H � H � � A�HUn N B56?=�-H+.'m[0o'2+ *,+.C&B<+\/&6?' S|+.')')()FG6?-U7<0 � � +\*,+.C&B<+\6?4<+\/H+\+!757<45( } C&()45B5( +./�C&-*,+.')*?0�')0�()-&6?B<+!757<0>/&6?'W7<6?FG$�0q/&(CT�D D AE/&(W*,+.45()*,+.FG6?-U7<0>/&6?'WB5()Bf7<6?Fk+ N ()-&6?B<+!757<6?�?�,+>/&0h@�CD7P+�+.')'2+$&456?*?()B5()0�-&6z�H-&(O7P+�/&6?'VB5()Bf7<6?Fk+�+.45(O7<FG687<()*?0k/&6?'V*,+.')*?0�'2+!7<0�456 � � +.')6z7<6?FG$�0 N FG()B5C&4<+!7<0�/H+.')'=a 0�450!Z')0�=�()0�()-U7<6?45-&0q()- /&6?*?()FG(�/&(YB56?*?0�-&/&0 N 6?4<+qFG0�'O7<()$&')()*,+!7<0G$�6?4 � � � 6 $�6?4E$&450�/&C&45456�()'�7<6?FG$�0q()-B56?*?0�-&/&( � � +�4<+.$&$&456?B56?-U7P+.�?()0�-&6 } ()-H+.45(2+\/&( � � � 6 vH+�()-D�H-&(O7<6A*?(OST456 N ()-&/&()*Pv&(2+.FG0�'2+t*?0�- ]

� �� 6 � �6 wO��" c [ � �

6 w 6�6�6 ��� 6�6 ��� 6�6 ����wOwOw�" � � �6 w 6 6�6 ���h" �j 456?=�()Bf7<45(VCD7<()')()�?�,+!7<(V+,@�68@.+.-&0�C&-H+q')C&-&=�v&6?�?�,+>/&( � } (O7 N CD7<()')()�?�,+.-&/&0�C&- } (O7t$�6?4t()'VB56?=�-&0�'2+4<+.$&$&456?B56?-U7P+.�?()0�-&6 /&( � � � 6 STC\7<450�-&*,+!7P+A+.')'2+u@�6?-U7<(O7<456?6?B5()Fk+A*?(OST4<+%/&0�$�0A'2+ @�()45=�0�'2+ � j -&/&()*Pv&(2+.FG0'2+\4<+.$&$&456?B56?-U7P+.�?()0�-&6 /&( � � � 6 -&6?(�456?=�()Bf7<45(`*?0�- U ]

U�

6 w 6�6�6 ��� 6�6 ��� 6�6 ��� 6�6 ��� 6�6 ��� 6�6� a 6?45450�456E()-U7<450�/&0.757<0o/H+.'`7<450�-&*,+.FG6?-U7<0>[6.]]

� � > U � � �6 w 6�6�6�6�6�6�6�6�6�6�6�6�6�6�6�6�6�6�6�6�6�6�6 ��� 6�6 ����wOwOw�" �p

�6 w 6�6�6�6�6�6�6 1 ��" c [ � �

6 w 1 � Q � 6 �" c [� C&6?Bf7<0V6 � G�G�D�� Dz6?45450�456E/&(�7<450�-&*,+.FG6?-U7<0 N FG0�'O7<()$&')()*,+!7<0t$�6?4Q()'�7<6?FG$�0o()-�/&6?*?()FG( /&(`B56?*?0�-&/&0G()-*?C&(U6?4<+uBf7P+!7P+ +!757<(O@.+ '2+ } +!757<6?45(2+ /&6?( H%A,C � D A N 6R*?()0�96Y*?()45*,+ � 6�6 0�456RC&=�CH+.')(U+ � 6�6 Q N 6 Q N 6 Q � 6 �2 N 6�6�6�6 /&6?*?()FG( /&(�B56?*?0�-&/&0 N $&450�/&C&*?6tC&->6?45450�456EB5C&'`7<6?FG$�0�45()B5C&'O7P+.-U7<6E/&(�*?()45*,+ 6 � 2 B56?*?0�-&/&(L]6 w 6�6�6�6�6�6�6 1 � Q � 6�6 Q N 6 Q N 6 Q � 6 �

6 w 2 �� 0�'O7<()$&')()*,+.-&/&0zX�C&6?Bf7<0�7<6?FG$�0 N $�6?4 '2+z@�6?')0�*?(O7![+o*?0�-q*?C&(`@�(2+.=�=�(2+zC&-&0 � G�� N 6E*?()0�96E*?()45*,+ � NOM%NFG687<45(�+.'`B56?*?0�-&/&0 N 0.757<6?-&(2+.FG0&]/ $H+.�?()0�@�6?')0�*?(O7![+ Q 7<6?FG$�0

� � NOM%N � � � Q 6 w 2 � � � N 1 w P �

Page 131: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# � 2 M*?()0&[6tC&-&0 � G�� q*Pv&6 @�(2+.=�=�(2+o+ � NOM%N FG687<45( +.'�B56?*?0�-&/&0 N $�6?45*?0�45456 ()- 6 w 2 B56?*?0�-&/&( C&-&0kB5$H+.�?()0/&(�*?()45*,+ FG6?�?�?0z*Pv&()')0�FG687<450 ()-�+,@.+.-U7<(:-&6?')'2+tB5CH+z/&()456?�?()0�-&6 N 6%X�C&6?Bf7<0\B5$H+.�?()0zSTCkB5C10G*?()6?-U7<6A$�6?4C&B5*?()456ESTC&0�45(`/H+.')'2+\�?0�-H+o()-&/&(O@�()/&CH+!7P+\/H+.' H%A,C � D A ������������ � I�4�<�;UIJ69;UE�K���K�< <���� QR69NUE�K��j ' � ()C&=�-&0 � 1�1 N '=a �Y-U7<6 / $H+.�?(2+.')6 �YC&450�$�6?0o'2+.-&*?(m[0 -&6?')')0 B5$H+.�?()0\C&-�4<+.�?�?0\B56?-&�,+o6?X�C&()$H+.=�=�()0 N'=a�C � H%F�J �

4�()=�C&4<+ ��� P ] � a #%45(2+.-&6 �� D���9 � �:��6�6LD; �

�C � H%F�J��9%JKFLF�J � F � � � H%AD(F�J�� �������;G�D F �Z< � /JEH J F�D F CEJEH � � �� H%A�H J G�LJ /D�9%J�� J

� JKCK9 � CEJ H � IRI�JKA,AJKCEJ � F D C@T � A�HV� H�F&H 9%JKA,A�H J���CED6LJEH�� JKC�I�J�^J�6LJKC >@�LJ�KAD�I�D A � 9%D H 9%J�9 H � �LJT�D D��KAJKC � H%AJKC0H � � � D�9 C0H� � I�JKF1� � D F&H%A � C � �Y6LJKA,AD H � F�D C�I H � J GEH%C � G�D � � H%AJ���� � A � G�D IRI�JKCEG � H � � F K� ���LDM6&C � I�D � H%F�G � D&H

�C � H%F�J������ ����KI J � ���&7 � �D7%F�D /J���������� J AJKC�I � F�OD /D6LDWG � CEGEH����

� J�G�D FB � OH �E/J�G�D���� D J >@��H%FB/D � ��C0H����D C0H�7�7 � ��F1� J ��F=< H � AJ?�� H � � w�� I � 7 � � H! C&-�FG6?*?*,+.-&()B5FG0/&(%+.CD7<0�/&()Bf7<45C&�?()0�-&6 6LD�� J#W5F�J H ���Z<�#%45(2+.-&6 � J � F1� � JKI�J G�D F � H 6LJKCZ � A�H � G�D��KAD�� � J F�D FH*�� � G���C0H%A � � H%AJ���� � A � � G � JKFLA � W�G � � �:���KAJKC�/J����Z< � �

�G�LJ#� � A,CED�9 H 9 H%F�D;H T�D CZ/DDF� K� 6&CED?7/JKA,AD@JKC0H

��C0H%AD�G � CEGEH"��� H%FLF � J(G�D��KA�H%AD$# I � � � H%CZ � � /D���� H%C � F�

A�H � J � D IRI H�9 H%FLF�D H�7�7 � ��FLA �=� �"G�D��KAD � G�D��KA,CK�D� � D F�J /J��LC0H����DRJ�>@�LJ���� DV/J���GEH%C � G�D J G�LJR�K��CED F�D��KA � I H%A � G�D F ��F 9 H � D CEJ G�D IC6B� J�� � 9%D � �����I � � � D F � � /D���� H%C � F�\ D6LDQ��F G�D FLA,CED���� D@G�D IC6B� JKAD OJ�AD C�F&H%AD � F � J�9 � � � D � �%��� D A,ADDT�CEJ����� #RJ#OHQ>@�LJ���I�D I�JKFLADA��HXJ�� JKA,A���H%AD�� H%F�G � � JKF�� H � FLAD6�6 � F&� � �KAD � HV����HXJKF�D C�I�J 6LD AJKF�� H � � D�� � AD#9 � JKF�J���� H%AD 6LJKC I�JKA,AJKCEJ � F D C@T � A�H� �LJ � H%AJ���� � A � 6LJKC�9%D�� A�H�J�OH �(')���*� � H%F�G�JKCROHA9%JKC?� D� H � ��

�� 6 ��H � H;GEHK6 ����� H8 � C � W�D C � I�JKFLAD�H ��AD I H%A � GEH

�� � � � ��,+*- .0/

Page 132: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

� 2 P

4�()=�C&4<+ ��� 1 ] j T�D D��KAJKC?�: -H+ *?0�FGFG()B5B5()0�-&6�/Ba ()-&*Pv&()6?Bf7P+ Bf7<C&/&(m[0 ')6�*,+.C&B56q/&6?')'=a 6?B5$&')0�B5()0�-&6�6q/&0�$�0 /&C&6qB568757<()Fk+.-&6$&C }&} ')()*�[0qC&-�4<+.$&$�0�4f7<0 � � � � < JY6B� D�� � D F�J H 9�9%JKFLF�J�H GEH ��� H � ��F JKC�CED CEJ8/J�� � DU��A �9H%CEJ8/J�� � �

�KAJKI H � F�JKCZ� � H � J# � C � � JKC � I�JKFLAD � j '`*?0�FG$�0�4f7P+.FG6?-U7<0o/&6?' � � � 7*��A � D FLA,CED�� � � �KAJKI /&6?')'=a #%45(2+.-&6� 6?/~()' B5C&0 FG0.7<0I-&6?')')0IB5$H+.�?()0 N 6?4<+.-&0 FG()B5C&4<+!7<(R/H+IC&- �F�JKC�A � H ���_J � JKCEJKF�G�J � � �KAJKI

�/ uj "��� C&6?Bf7 a C&'O7<()FG0�/&()B5$�0�-&68@.+�/&(VC&-�*?0�FG$&CD7<6?4E()-U7<6?45-&0q()- *?C&(V@�6?-&(O@.+.-&0>FG()B5C&4<+!7<(�B5(2+�=�')(V+.-&=�0�')(/&(�()-&*?')()-H+.�?()0�-&6 +.B5B5C&-U7<(V/H+.'�4<+.�?�?0�/&C&4<+.-U7<6o()' @�0�')0�*Pv&6o')6z@�6?')0�*?(O7![+G4<+.=�=�()C&-U7<6 N B5C&')'2+ } +.B56o/&(()-DST0�45Fk+.�?()0�-&( $&450h@�6?-&()6?-U7<(Y/H+.')'2+k$&(2+!757P+!ST0�45Fk+k/&(V'2+.-&*?()0 � j /H+!7<(V/&6?' / uj @�6?-&(O@.+.-&0�7<4<+.B5FG6?BmZB5(A+.' *?0�FG$&CD7<6?4�/&( } 0�45/&0

�� � 3 N � F�� D H%CZ � D IC6B��AJKC "�N *Pv&6I*?0�-U7<450�')'2+,@.+�()' $&450�=�4<+.FGFk+/&(Y@�0�')0�6?/3(W/&()B5$�0�B5(O7<(O@�(W+./36?B5B50I*?0�45456?'2+!7<( � y 6?4zFG()=�')()0�4<+.456\'=a +�0G/H+ } ()')(O7![+�6?4<+>$&4568@�()Bf7P+�C&-H+X�CH+.-U7<(O7![+kB50h@�4<+ }&} 0�-&/H+.-U7<6�/&(Y+.$&$H+.456?*?*Pv&(2+!7<C&456 � 3 a 6?4<+.-&0 N ()-DS|+!757<( N /&C&6 / uj *Pv&6o'2+,@�0�4<+,@.+.-&0()-�$H+.4<+.')')6?')0 N } +.B<+!7<( B5C B50.S 7Y3Q+.456\6?/ vH+.45/13Q+.456o()/&6?-U7<()*?( � : -&0�/&(�6?B5B5(V6?4<+�+!757<(O@�0 N ()' B56?*?0�-&/&0()-�Bf7P+.-&/�Z } � N ()-IFG0�/&0G7P+.')6 *Pv&6 N X�CH+.')0�4<+G()' *?0�FG$&CD7<6?4 � � 3 ()-&/&(O@�()/&CH+.B5B56 C&- =�CH+.Bf7<0�0�/IC&-6?45450�456 -&6?' B5()Bf7<6?Fk+ / uj N ()FGFG6?/&(2+!7P+.FG6?-U7<6AB5( *?0�')')6?=U+.B5B56z+.'�B56?*?0�-&/&0 N +�*?0�-&/&()�?()0�-&6t*Pv&6zX�C&6�ZBf7 a C&'O7<()FG0>STC&-&�?()0�-H+.B5B56�*?0�454568757P+.FG6?-U7<6 � j -&0�'O7<456�*%a 6?4<+.-&0 /&C&6 � � 3 *?0�B [ � *?0�FG6�6?4<+.-&0 Bf7P+!7<6/&C&$&')()*,+!7<6 +.-&*Pv&6\C&-I*?6?4f7<0k-�C&FG6?450�/&( C&-&(O7![+�/&6?' � � � 7*��A � D FLA,CED�� � � �KAJKI � j ' 4<+.�?�?0�*?0�FG()-&*?(m[0+�/&()B5()-U7<6?=�4<+.45B5(u*?()45*,+ 2�1 B56?*?0�-&/&(%/&0�$�03()' '2+.-&*?()0 N +;*,+.C&B<+;/&( /H+!7<( 7<4<+.B5FG6?B5B5(%/H+.'uB5()Bf7<6?Fk++!757<(O@�0 / uj � N *Pv&6 N ()-IX�C&6?'VFG0�FG6?-U7<0 N -&0�- *?0�-U7<6?-&68@.+.-&0q()-DST0�45Fk+.�?()0�-&(�*?0�6?456?-U7<(V*?0�- ()'�@�0�')0Fk+>STC&450�-&0I()-U7<6?45$&45687P+!7<6�*?0�FG6k7P+.')( S ()' FG0.7<(O@�0�$�6?4\*?C&(Q()'RB5()Bf7<6?Fk+ / uj � +!757<(O@�0 -&0�-1()-U@�(m[0()-DST0�45Fk+.�?()0�-&(D*?0�454568757<6 *?0�-&B5()Bf7<68@.+ -&6?':S|+!757<0 *Pv&6%'=a C&-&(O7![+t+,@�68@.+z/&()*Pv&(2+.4<+!7<0t'2+t$&456?B56?-&�,+\/&(:C&-H+B5(O7<CH+.�?()0�-&6\6?*?*?6?�?()0�-H+.')6 N /&C&4<+.-U7<6�'=a 6?B56?*?C&�?()0�-&6�/&6?'YB50.S 7Y3Q+.456 N $�6?4E*?C&( / uj � B5(Y6?4<+�()-U7<6?45450.7fZ7<0 � ��� ���� 0 ������ ���� � �� �� � �� � ��� � ���� � ��� � � �� � ��� ���� � ��������� ��� � ��� �� � � ����� � �

� � � ��� � � ��� � �(' # ��������� # � ' ������� # ' � � � � � ��� �� # � '! #" � �%$&�'� # ')( � ��� ��0 ��

�+**- V /

Page 133: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# � 2�1� � ���� �� � � � � ����� ��� �� ���� � � � ���� ������ � � � � � ��� �� � � �� ���� � � � � ���� �� ���� � � �� / �/ � ���� � � ���� �� ������� ���� �������� � � � � � � ���� �� � 2 � / � ��� � ������ ����0 ������� � �� � ��� ����� � ���� �� � � � �� � � � �� � � a 6?45450�456EB5(`@�6?45(O�H*�[0G+\*,+.C&B<+o/&(`C&-q45()B5C&'O7P+!7<0\()-H+.B5$�68757P+!7<0o/&(`C&-H+STC&-&�?()0�-&6 ()-U7<6?45-H+G/&( +.')')()-&6,+.FG6?-U7<0 N *Pv&(2+.Fk+!7P+�; 0�45()�?0�-U7P+.' � (2+.B N � ; N *?0�45456?'2+!7P+k+.')'2+z@�6?')0�*?(O7![+0�45()�?�?0�-U7P+.')6 $�6?45*?6?$&(O7P+�/H+.')'2+G$&(2+!757P+!ST0�45Fk+o/&(�'2+.-&*?()0 S ()' @.+.')0�456 /&( � ; B5(�$&456?B56?-U7![0qFG0�'O7<0k$&(<[C+.'O7<0A/&(&X�CH+.-U7<0t*?(DB5(H+.B5$�68757P+.B5B56 � � C&6?Bf7<0t$&450h@�0�*�[0tC&-�6?45450�456 N D6LJKC0H%FB JKC�CED C N *Pv&6 ')6Q()Bf7<45C&�?()0�-&(/&(Y*?0�-U@�6?45B5()0�-&6G/&6?(W/H+!7<(Y-&0�-�6?4<+.-&0q$&456?/&()B5$�0�Bf7<6k+./�68@�(O7P+.456 � / 6 }&} 6?-&6G-&6GB5(2+�Bf7P+!7P+q()-&/&(O@�( Z/&CH+!7P+ '=a 0�45()=�()-&6 N ()-~456,+.'O7![+I-&0�-1STC~6?B<+!757P+.FG6?-U7<6qX�C&6?Bf7<03+ /&687<6?45FG()-H+.456�()' S|+.')')()FG6?-U7<0�/&6?')'2+FG()B5B5()0�-&6 � j ' FG6?*?*,+.-&()B5FG0 /&6?Bf7<()-H+!7<0;+.' 7<4<+!757P+.FG6?-U7<0 /&( B5(O7<CH+.�?()0�-&(Q6?*?*?6?�?()0�-H+.')(Q$&4568@�6?/&68@.+/&( ()-&/&(O@�()/&CH+.456t'=a 6?45450�456 -&6?' "HC&B5B50�/&( /H+!7<( N ()FGFk+.=U+.�?�?()-H+.456A()' *?0�FG$&')6?B5B50�/&6?( /H+!7<(�+�� 68757<( /H+6?45450�456z()-�C&-Ba +.456,+k/&( FG6?FG0�45(2+�6?Bf7<6?45-H+�+.' B5()Bf7<6?Fk+G6?/ N ()-D�H-&6 N /&687<6?45FG()-H+.456 ')0kB5$�6?=�-&()FG6?-U7<0/&6?'�B5()Bf7<6?Fk+ / uj � � STC>$&450�$&45()0o'2+\/&6?*?()B5()0�-&6t/&(`()-U7<6?45450�FG$�6?456E')6E0�$�6?4<+.�?()0�-&(`/&(`X�C&6?Bf7 a C&'O7<()FG0*Pv&6 N 6?B5B56?-&�?(2+.')FG6?-U7<6 N B5(H/&()FG0�Bf7<4�[0ES|+!7P+.')6 N /H+.'HFG0�FG6?-U7<0E*Pv&6%B<+.456 }&} 6 Bf7P+!7<0 ()FG$�0�B5B5( } ()')6R45()*,+.' Z*?0�'2+.456z'2+k$�0�B5()�?()0�-&6 N /&0�$�0�')0kB5$�6?=�-&()FG6?-U7<0 � \ +kX�C&6?'VFG0�FG6?-U7<0�()-I$�0�( N /&C&-&X�C&6 N ()' B5()Bf7<6?Fk+/ uj 45()B5C&'O7![0k()-�CD7<()')6 � j '�FG0.7<(O@�0k/&( X�C&6?')'2+�7<4<+.=�()*,+G/&()B5$�0�B5()�?()0�-&6t@.+k45()*?6?45*,+!7<0�-&6?' S|+!757<0�*Pv&6\()'$&450�=�4<+.FGFk+z/&6?')'2+\FG()B5B5()0�-&6E#%45(2+.-&6E$&4568@�6?/&68@.+G/&( +�� 450�-U7P+.456EB50�')0oS|+.')')()FG6?-U7<(:vH+.45/13Q+.456 �LN�LJ�KAD OJ8��F�D /J � I�D A � 9 � 6LJKC G�� � � � /J�9%D F�D � JKIC6&CEJ#6&CEJ�9%J?/JKCEJ�� � A���H�� � D F � J�G�G�J?� � D F&H � �F�J���� H CEJEH � � �� H�� � D F�J# � ��FM� DU��A �9H%CEJ �

Page 134: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

�, 6������� �X�O��������*2�+* - ��!"�'������*"����

()$�0�4f7<(2+.FG0k+.')*?C&-&(Y*?'2+.B5B5()*?(VFG687<0�/&(V$�6?4E()'V*,+.')*?0�')0�-�C&FG6?45()*?0q/&(�� N $�0�()*Pv 96o45(O7<6?-&(2+.FG0�CD7<()')6 N/H+.' $&C&-U7<0�/&( @�()Bf7P+k/&()/H+!757<()*?0 N 68@�()/&6?-&�?(2+.456z*?0�FG6 45()B5C&'O7<()-&0k+.-&*?0�4<+k+!757<CH+.')( +.')*?C&-&6 X�C&6?Bf7<()0�-&(/&(V*,+.')*?0�')0�-�C&FG6?45()*?0>/&6?')'=a +.-U7<()*Pv&(O7![+ � j 'Y$&450 } ')6?Fk+�/&6?'Y*,+.')*?0�')0�/&6?')')6o*?(OST456o/&(�� [6�0�=�=�68757<0q/&(Bf7<C&/&()0�B5()-q/H+.')'=a +.-U7<()*Pv&(O7![+ � j '`@.+.')0�456A/&(���vH+�+!757<()4<+!7<0o'=a +!757<6?-&�?()0�-&6A/&(�FG0�'O7<(�Fk+!7<6?Fk+!7<()*?( N /H+.(7<6?FG$&(�/&(D#%45*Pv&()FG6?/&6Q+.(D-&0�Bf7<45(D=�()0�45-&(�6QB50�-&0EBf7P+!7<6QCD7<()')()�?�,+!7<6R/&(O@�6?45B56QFG687<0�/&0�')0�=�()6W*Pv&6 vH+.-&-&0*?0�-&/&0.757<0G+./>+.$&$&450�B5B5()Fk+.�?()0�-&( B56?FG$&456 $&(.9Cq+.*?*?C&4<+!7<6t/&6?'�-�C&FG6?450G/&(`*?(OST456E/&(�� � � �: -&0o/&6?(�$&45()FG(�45()B5C&'O7P+!7<( B5()=�-&(O�H*,+!7<(O@�( [6E/&0h@�CD7<0G+./q#%45*Pv&()FG6?/&6E*Pv&6t45()C&B5*P[ � +o/&()FG0�Bf7<4<+.456A*Pv&6.]��� 2M � �

2

� � 6M �� � � 2

� �M ����M+.$&$&450�B5B5()Fk+.-&/&0\'2+z')C&-&=�v&6?�?�,+o/&6?')'2+ *?()45*?0�-DST6?456?-&�,+o*?0�-q4<+.=�=�()0zC&-&(O7P+.45()0z*?0�-q()'�$�6?45()FG687<450\/&6?($�0�')()=�0�-&(�()-&B5*?45(O757<( 6t*?()45*?0�B5*?45(O757<( N +.4545(O@.+.-&/&0�+./>CD7<()')()�?�,+.456A$�0�')()=�0�-&(:�H-&0k+ 1 N '2+!7<( � ()$�0�4f7<(2+.FG0 C&-H+ } 4568@�6t7P+ } 6?')'2+z*Pv&6t()-&/&()*,+ �� � ��� � ����� � � � � � � � � /&6?'�*,+.')*?0�')0\/&6?')')6A*?(OST456/&(��Y]

��� Z � ���� �E� GB�B� F�N� �%F�F!%� ý � �6���X�M�� ��X�M���DF�"�������DF�N��� � ý �� � ����!"�" �E������M�� ��"F!%� þ E�!%�������" ��"F����DF!%� !9�" ��"�" GF� �X��� ��"��� � H � � �M��!'B��D�6��� ��Z ý B�!'F�N� �� N ������! þ �M����!'� F� L]� � � H�6�� ���F!%E�B�!'�" ���B� GF!%� � ; J �%B>���%�< �� B���E� F�N� � � H� ��F!�B>��E ý ����!%E��� � � $��1B>����J�� � � E�!%��!K��Z � � F� � � ��$�� !%�� ��" �� � �� ��E� ��� �" �� �D�#��� E������D�" ��"��� N� �!%�< T� �C��!"�"!K����� XF�N� T� � E������<!%E���!#��E����Z ��� �� �$�$G� [�D�@�D��E � N� �$�$G�@�" �� �D�@J �%B� [F!%E�E��DB��<!%� �" GB�B� 4�� ý �� � !T GB��DB>�� ����� 4��E��@� ��E�� ��� � !@ 4�D��F� E�F!%� þ �E� ���$G�@�"� ý �F �E�F�N�� ! ? ��F�N� 6�� ý �� � ![F�N� 6B� �� þ � �M��� ý ��� �� ����� � �T� ý �� �E�!�B��"F����������M��E��4�� ý ��� ���%�%�@ GB� ������ ! ���D�[FE�� ����! � E����<����YZ ��� ��E��I� ��E��� %�U� !RB>��� ��!%� !R�K�" �� �E���� � ��E� �B� �����E� T��� Hý F� þ E� ��"�� ý �� � !RB�L]� � ý ���<!R� ��� ��� ��! F�N� TE�������E� GB� ������I� !B>�� GB�B�!�� � H 6� � �M��! ��� F�N� '��� � 6��E�!%J�� ��X�M���DF�N� '��� F!%E��[���< �E��� %��F�N� 'E�� �¡ý ��E�������!4��Z GB���E� GB�B�� !%�� 6 6�D� B� � ý ���$G� �" ���" GF� �X��� �]�" ��]� ý �� �E�! _�� �]B��"F�N�� G�" %���%� GB� ������ ! ��B� QF�D�%B�F ý � �9�" �� � QF� þ E� ���� V � S ���������,.��MB��]E�� �< �� � ��� ��� ������ ����� B��< GB�B�! �G!9B� �� H ��� "!$#�* )&%' "!( . ,) .�*�+,!(-M* )+.#&'-.�� $/�0M�M�� ���B� �� B�!9F�N� ! � ����B ý �'F� þ E��6����� ��E� ý��¡ý ��� �� ����� B��< GB�B�! �� �CB� �� B�!R�%B�� ����!����DF! ���� �� B ý !KB�L]� � ý ���<! �" GF� �X��� %�Y!%��� ý E� �B� H #.�*�+,!(-M* I)+. &'-.�� 1/�0M� F� ! H TB� T! � ���1J�� !"F�F!R�"�F� þ E� 4�"� ý � �2�<B�B��M��� � ý � � N� �$�$G�@����� ��E� ý��¡ý ��� �� ����� [B��< GB�B�! ��� �KB� �� B�!T�%B�� ����!����DF! ���� ���B ý !TB�L]� � ý ���<!��" GF� �X��� %�! ��� F�N� XB� 3� H R.�*�+,!(-M* � � ���� �� �" �� �"! F!%�/F� H! � F�N� X! � ���1J�� !"F�F! �"� F� þ E� I�"� ý � �4�<B�B��M���#� ý � � N� �$�$G�R����� ��E� ý��¡ý ��� �� ����� B��< GB�B�! �]� � B� �� B�!X�%B�� ����!����DF! �]�� �� �D�XB ý �TE�������E� GB� �������$�� !%�� 9� � ý � �3 ý ���DB��D�%B��YJ �%B� %_

Page 135: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# �, &�� � � � �� � � ��� � � ��� � ����� � � � � �� � � ���� ���� 6 / +.')0�FG0�-&6 1 M � + � * 2 � 6 �#%45*Pv&()FG6?/&6 � � 6 + � * � 2 �O�, 2#%4 � + } vH+!7P+ 1�1 / � * � 2 �O�, &� N 4�( } 0�-H+.*?*?( � ���%6 2 �O�, &� P � P � ()687<6 ��� 1�2 2 �O�, &��� 1 � N � 2 1� 6�3 7<0�- � N�N � 2 �O�, &��� 1 � N � 2 � P 1 M 1�2 � � N/ vH+.45$ � N 1�1 2 � �

M � *?(OST456 " M �� +.*Pv&()- � M 6 N 2 � � �6�6 *?(OST456 " � 6�6�YC&')6?450 � M ��� 2 � ��%6 *?(OST456 " �%6

� 6?=U+ � M%P 1 2 � � �, 6 *?(OST456 " � � N� 6?=U+ � M 1 2 � � �

2 M *?(OST456 " � 2 N4H6?45=�C&B50�- 6 M Z � 1 N 2 � �M 6�6 *?(OST456 " N �%6

� 4<+I=�')(Q+.CD7<0�45( ()-&/&()*,+!7<(R()-37P+ } 6?')'2+�[6�*?C&45()0�B50 45()*?0�45/H+.456k*Pv&6 �YC&')6?450 N *Pv&6>+./&0.757![0 '2+�')68757<6?4<+=�456?*,+ � N 0.757<6?-&-&6t()'`B5C&0G45()B5C&'O7P+!7<0o()-qB50�')0oC&-Ba 0�4<+ �j 'D@.+.')0�456Q0.757<6?-�CD7<0t/H+ 4H6?45=�C&B50�-�-&6?' � C&=�')()0%/&6?' � 1 N 4<+.$&$&456?B56?-U7P+t'=a C&'O7<()FG0%*,+.')*?0�')0A/&6?')')6Q*?(OST456/&(���6?B56?=�C&(O7<0k+oFk+.-&0 �� 6?')'2+;7P+ } 6?')'2+�B56?=�C&6?-U7<6�B5(E45()$�0�4f7P+.-&0 N ()-U@�6?*?6 N +.')*?C&-&(%/&6?(E45()B5C&'O7P+!7<( 0.757<6?-�CD7<(EFG6?/&(2+.-U7<6*,+.')*?0�'2+!7<0�456 �� � � � �� � � ��� � � � �� � � ���� ���� � �� ����� ���4H6?45=�C&B50�- 6 � Z � 1 M M � 6 \ 6?B < *,+.')*?C&'2+!7<0�44H6?45=�C&B50�->6��;456?-&*Pv 6 1 Z � 1 M P 6 P \ 6?B < *,+.')*?C&'2+!7<0�4/ vH+.-U7<B 6��;456?-&*Pv��.4 6 M Z � 1 N � � 6�6G� N � j� � M 6 1 6� C&()')')0�C&/�6 4�()')')(2+!7<456 6U� Z � 1 N�N � � 6�6�6�6 j� � M 6 2 6� C&()')')0�C&/�6 \ ()*PvH+.FG$D7 6U� Z � 1 NOM � 6�6�6�6�6 3 \ 3 N�N 6�6� +.-H+./H+o6 : B5v&()450 � 6 Z � 1 P 2 � 6�6 � 2�2�1 � ; (O7P+.*Pv&( / Z P � 6 � �%6� +.-H+./H+o687%+.' � 6 � Z � 1 POM � 2 � �, M 6�6 � � 3 / � Z �� +.-H+./H+o6 � + < +.vH+.B5v&( 6 1 Z � 1�1�1 �%6 N ��� P 2 6�6�6�6 ; (O7P+.*Pv&( / P 6�6�6� +.-H+./H+o687%+.' � 6 1 Z �%6�6U� � � &��� 6�6�6�6�6�6�6�6 ; (O7P+.*Pv&( / P 6�6�6 � ��y

j -;$H+.4f7<()*?0�'2+.456o()'Y45()B5C&'O7P+!7<0q4<+.=�=�()C&-U7<0q/H+�� � � +.-H+./H+q6� � : B5v&()450>-&6?' � 1 P 2 [6G()'Y$&45()FG00.757<6?-�CD7<03FG6?/&(2+.-U7<6 ����6LJKCEGEH � G�D�� H%AD CEJ � \ +.' � 1 P 6 � +.B5C&Fk+.B<+ � +.-H+./H+ [6�C&-&0;/&6?(%Fk+.=�=�()0�45(6?B5$�0�-&6?-U7<( N 7<4<+�=�')(`()-U7<6?456?B5B<+!7<( +.'�*,+.')*?0�')0�/&(��;*?0�-�C&->6?')68@.+!7<0G-�C&FG6?450G/&( *?(OST456 � � +�Fk+.=�=�()0�4$H+.4f7<6u/&6?(HB5C&0�(H*,+.')*?0�')( [6uBf7P+!7P+E456,+.')()�?�,+!7P+AB5CGB5C&$�6?45*,+.')*?0�'2+!7<0�45(&6 } +.B<+!7P+EB5CGFG0�/&6?45-&(H+.')=�0�45(O7<FG((O7<6?4<+!7<(O@�( �� 6%$&45()FG6 ����� *?(OST456%/&6?'�-�C&FG6?450 �I*,+.')*?0�'2+!7<6uB5C�C&-�6?'2+ } 0�4<+!7<0�456 j� � M 6 B50�-&0\45()$�0�4f7P+!7<6uX�C&(

Page 136: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

�, �/&(`B56?=�C&(O7<0 4[ ]�

�2 wO�, &��� 1 � N � 2 � P 1 M 1�2 � 2 P � N � N 2�2 P 2 �.� M 1

� 1 � 2 6 M%P � N 6 N � P�N �%6 P 1�1 P�N � P 6 2 P � �2 � ��� M 6 NOM 1 P � �, P 6 P�N ��� 2 � P � 2 6 N�N M6 1�2 P � N 6 1 ��� 6 � P ��� 2 � M � � 2 � 1 6 P � � P �,�,�

�����1 �E ý � ��J�E� �L% [B>��!%E��D��B ý �UF����DF!%� !��" �� � F� þ E� �"� �C�� G�"�D������ F����DF!%�D�M��!%E� 4B�� B ý���� �E��DB�F 9� � B�� ��!������� ������������������������������� �!"�$#$%&��'������

G�Y�"� �R� ��E����DF!%�D��E� %�]�D�T� � � � � �������� ������������������������������� �!"�$#$%&��'������)(�'*�+����� ,

� Z ý � ��� ��!RE��DB ý � ���M��!KE��DB���� ����.-] ���� ��@J�E� S ��� S � ý ��� �"!K� �.� E�! þ _0/ �%B ý �X�%B��21 ��� �%���#�" �� ��Z 3'��� L% �E�B�� � H�K�"� 5 !�\4�5 ! �76 .�8"*�+,!(-M,�)&*M.:9 '%�;].�*M* * <)=?>� . ,) "+ � � � B�� ��� K�%� ��� ��E��Q��!ML% #F!%� �D��J<!%E��M��!%E���� E� G��� � $�$ H! � �6F����DF!%� ! �"��� F!%�V _ S�@ V _ V ����_ ������_ ����� F� þ E� 9�" GF� �X��� ����B ý �< �E���� �"! �"� ��� ý Z]�"�<B� ���L%!%� �� 6� � � ý �� �E�! �" �� � 9F� þ E� 9�" �� � !%E�! B>�� GB�B�!BA (")+. -.� ,�'�2C *�+ * D0E '%'*�+ DM�"F!�B>��� � ý � ��!���� S ��F�_ V�G)H _ @I ��_ �����9F� þ E� [�" GF� �X��� �� 4E�� ��� � ý ����! �� �� V .*.*.�_ ��YF����DF!%� !@E��DF�N�� GB� F� E�F��0F�� S !%E� ' þÿý E� G��� � $�$G�M��!@F!%� ý �KB ý �< �E�F!%��� ý �� �E�J'� ���%F�N�� ; J'� ���%F�N���-"3 H ����� ? ��F!%�#�%F�F GB�B�! ��� �D� �� ���!%E��D��"�UF� E�F�� ý �K�� �E���J 5 �� %_ 5 ��� �Y� � þ !%E��X��$�� !%���UB�!%��!TE� ��< �E�� J�� � � ��� �D�T� � � � � �LKQ �J

������� ������������%���')�)�M�)�)�����M%*���)N�O���P�N�M'��,��%��Q�R�RQM��QM�&S$����P������N���T.��P$%��

Page 137: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# �, 2�����)B ��� �0K�P ; ��;���� � Q 8��(6+FTK ��K����5N� ��� / (�*?0�-&B5()/&6?45(�C&-H+o*?()45*?0�-DST6?456?-&�,+�/&(�4<+.=�=�()0�� � �YB5B56?-&/&0

� � � � � �= �

� � � � M � 8 � � � � � � � � � �= �

j '�FG687<0�/&0 $&450�$�0�Bf7<0 /H+ #%45*Pv&()FG6?/&6 [6RX�C&6?')')0 /&(�+.$&$&450�B5B5()Fk+.456W()'�@.+.')0�456W/&( ��+!757<4<+,@�6?45B50A'=a +.456,+ N� � N /&6?'�$�0�')()=�0�-&0\456?=�0�'2+.456E/&( � �o'2+!7<( N ()-&B5*?45(O757<0o-&6?')'2+o*?()45*?0�-DST6?456?-&�,+�/&(�4<+.=�=�()0oC&-&(O7P+.45()0 � y (<[C$&456?*?()B<+.FG6?-U7<6.]�

�')()F�s�� �

/ (2+R<

�� N � � [6t'=a +.456,+\/&6?' X�CH+./&4<+!7<0�()-&B5*?45(O757<0o-&6?')'2+\*?()45*?0�-DST6?456?-&�,+ �

B

Aβ 2

O=H

� � ������92 ] � CH+./&4<+!7<0�()-&B5*?45(O757<0�-&6?')'2+\*?()45*?0�-DST6?456?-&�,+� B5B56?4f@.+.-&/&0k'2+ 4�()=�C&4<+ ��N '=a +.456,+o/&6?'`7<45(2+.-&=�0�')0\# � � $&C [0�6?B5B56?456z0.757<6?-�CD7P+�*?0�FG6.]

� � � �������

�� �

� � $�6?45*Pv 96 4<+.=�=�( /&(�C&-H+�*?()45*?0�-DST6?456?-&�,+�C&-&(O7P+.45(2+ � j -&0�'O7<456.]�4 � ����c �

� � M$�6?4f7P+.-U7<0 � 8 <

�4 � " � ��� ()B5C&'O7P+\X�C&()-&/&(L]

� � � ���� � � � � 8 < �4 � "� � � � �

� � � � � � 8 < �4 � "� w/&0h@�6z+ }&} (2+.FG0\$�0�Bf7<0 ���

� � 8 <

�4 � "��/ 6 /&(O@�()/&(2+.FG0o*?(2+.B5*?C&-I'2+!7<0�/&6?' X�CH+./&4<+!7<0k()->/&C&6 N B5( 0.757<()6?-&6 ()'�$�0�')()=�0�-&0o*?0�- � T '2+!7<(`$�6?4%*?C&('2+\B5(O7<CH+.�?()0�-&6o[6t0�4<+oX�C&6?')'2+\()')')C&Bf7<4<+!7P+z()- 4�()= � � �

Page 138: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

�, �

3 A

O H

B

β

� � ������ 8 ] � 687<0�/&0�/&(�#%45*Pv&()FG6?/&6t$�6?4u()' *,+.')*?0�')0o/&(�� +.=�()0�-H+.-&/&0\*?0�FG6t$&45()Fk+ N '=a +.456,+\/&6?'`7<45(2+.-&=�0�')0\# � � [6t/H+!7P+o/H+D]� � � ���� � � � � 8 < �

4 T "� � � T �� T �`� � � 8 < �

4 T "�6?B5B56?-&/&0�4 T ����� �j - =�6?-&6?4<+.')6 N $�6?4EC&- �HB5B<+!7<0 F J '=a +.456,+�/&6?'Y$�0�')()=�0�-&0�*?0�- � ��'2+!7<( N ()-&/&()*,+!7P+k*?0�- � � N [6 � �@�0�'O7<6E'=a +.456,+o/&6?(`7<45(2+.-&=�0�')(�*Pv&6 ')0o*?0�FG$�0�-&=�0�-&0 N 0h@�@�6?450 N *?0�FG6t()-&/&()*,+!7<0o/H+.')'2+ 4�()= � � N B5( vH+D]

� ��"� � �

� � � � � ���� �� � c � 8

<

�4 � "/&0h@�6^4 � �

���� � �y 0�Bf7<0&]

�� " { � �

� � c'2+

� ��"/&(O@�6?-U7P+D]

�2 " � � � { � � � 8

<

�4 � "j 'U$&450 } ')6?Fk+ /&6?'�*,+.')*?0�')0u/&( � � [6W45()*?0�-&/&0.757<0%+.'U*,+.')*?0�')0u/&(UC&- @.+.')0�456R+.$&$&450�B5B5()Fk+!7<0u/&( � 8 < �

4 � "��� 6?' FG687<0�/&0\$&450�$�0�Bf7<0\/H+\#%45*Pv&()FG6?/&6 N ()':@.+.')0�456%/&( � 8 < �4 � " @�()6?-&6A*,+.')*?0�'2+!7<0z+\$H+.4f7<()456%/H+X�C&6?')')0o/&( � 8 <

�� 4 � " +!757<4<+,@�6?45B50k'2+zST0�45F\C&'2+\/&( } ()B56?�?()0�-&6E/&6?=�')(�+.-&=�0�')(L]

� �"� 8 <

�4 � " �

�� > � �

�� 4 � "� �

�� >�� � > � 8 < �

�� 4 � "�/&0h@�6 6 a 4 � � � � � �

Page 139: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# �, ��y 0�Bf7<0 � � � � 8 <

�4 � " B5(�vH+q]

�,� � � 8 <

�4 � " � �6 N ()-q=�6?-&6?4<+.')6.]

� 8 <

�� 4 � " � � 8 <

� � � �� � � � � 8 <

�4 � c " �

� � c\ +.')'2+

� �"B5(�0.757<()6?-&6E'2+zST0�45F\C&'2+\45()*?0�45456?-U7<6.]

�,� � � M � � ��� > � � > � ��

c� < � �/ (�vH+�+.-&*Pv&6.]

{� �

� M{�� Q

{ c��

: � �$�6?4f7P+.-U7<0G'2+�2 "�N /&(O@�6?-U7P+D]

� � � { � � � � �� � { � c �

�� > � � > � ��

c�j -�*?0�-&*?')C&B5()0�-&6 C&-H+�Bf7<()Fk+�/&( �;B5( 0.757<()6?-&6z+!757<4<+,@�6?45B50�+.$&$&450�B5B5()Fk+.�?()0�-&(�B5C&*?*?6?B5B5(O@�6\0.757<6?-�CD7<6/H+.')')0\B5*Pv&6?Fk+G(O7<6?4<+!7<(O@�0oB56?=�C&6?-U7<6.]

� ��"����� ���� {

� �� M �,� � �{

� � { c�

� c � c

� i� e �� M : � ��� {

� � M : � �

� ������ � � � �� ��� � ���-/ � ��� � �� � �� � � �� �� � � � 0 �

� 0�')6?-&/&0q()FG$&')6?FG6?-U7P+.456\')0�B5*Pv&6?Fk+q(O7<6?4<+!7<(O@�0q/&6?B5*?45(O757<0q/H+.')'2+� ��"

()- C&-�B5()Bf7<6?Fk+q+.45(O7<FG687<()*?0+>$&456?*?()B5()0�-&6o�H-&(O7P+�� N } ()B50�=�-H+�7<6?-&6?456k*?0�-U7<0�/&6?')'2+q$&450�$H+.=U+.�?()0�-&6�/&6?')'=a 6?45450�456�/&(_CED���FB D � �3 0�-&B5()/&6?45(2+.FG0oX�C&()-&/&(�()' $H+.B5B50k=�6?-&6?45()*?0�/&6?')'=a +.')=�0�45(O7<FG0 /&(�#%45*Pv&()FG6?/&6.]

Page 140: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

�, N� � � {

� � �,� �� � � �

�� T � {

T � � T ��{� � � c

� c � ii� �

� � � � c� �� � ��� �� �

� � � �� c � {c � c �

�{T � c �

� T � � c � c

� i�� �� T � � c

� c �i� �

� � � � � > � �� � � { � � � �

�{c � � � � c � � c

� c � i�� �

� T �� c

� c i e�� i ��� �

� � � � � > � � > � ������ {

� ���{ c �

�� c �

� c � c

� i� e �� �� � � � � > � � > wOwOw > � � M : � �� � �: > � 4<+./&()*,+.')( ()-&-&6?Bf7P+!7<(

/ (`0�B5B56?4f@.+G*Pv&6 +.'�$H+.B5B50G( Z 6?B5()FG0\B5(�FG0�'O7<()$&')()*,+t()' 7<6?45FG()-&6E$&456?*?6?/&6?-U7<6z$�6?4?] �� �

*Pv&6E[6 N X�C&()-&/&( N +.-&*Pv&6 ()'�S|+!757<0�456Q/&(&+.FG$&')(O�H*,+.�?()0�-&6W/&6?')'=a 6?45450�456 /&(D450�C&-&/�Z 0 � � � 0EBf7<6?B5B50ES|+!757<0�456 N()- } +.B56 � 6 N [6.] � � 6 [��T [ c�� >

� @

B56?=�C&6 +.')')0�4<+z*Pv&6 '=a +.')=�0�45(O7<FG0 /&6?B5*?45(O757<0>[6 � � � ���-/ � �� �j ' @.+.')0�456E/&(�� ST0�45-&(O7<0o/H+.')'2+\$&450�*?6?/&C&4<+�/&6?B5*?45(O757P+o$�0.7<456 }&} 6t6?B5B56?456 N X�C&()-&/&( N/&6?'`7<CD757<0�6?454<+!7<0 �3 +.')*?0�')(2+.FG0�X�CH+.-&/&0~()' S|+!757<0�456I/&(E+.FG$&')(O�H*,+.�?()0�-&6>/&6?')'=a 6?45450�456�/&(A450�C&-&/�Z 0 � N *?0�FG$�0�4f7P+;C&-45()B5C&'O7P+!7<0\6?454<+!7<0�/&6?' � 6�6�� �y 0�()*Pv 96.]� p 2 wO�, &����wOwOwOwOwOwOw'2+\$�6?4f7<C&4 } +.�?()0�-&6E()-U7<450�/&0.757P+o/&68@�6 6?B5B56?456 /&6?')'=a 0�45/&()-&6A/&( � 6 c ]

� 6 [��T [ c�� >

� @� � 6 c�� 6 w 2 6 � � �

: > � " � � �

� : p � 62 �� � : p N

\ C&-&X�C&6o'=a +.')=�0�45(O7<FG0�$&450�/&C&*?6oC&- 45()B5C&'O7P+!7<0�B } +.=�')(2+!7<0�/&0�$�0�*?()45*,+ N $H+.B5B5( � � +.')6\45()B5C&'O7P+!7<0I[6*?0�-DST6?45Fk+!7<0o/H+.')')6A6?B5$�6?45()6?-&�?6 -�C&FG6?45()*Pv&6t45()$�0�4f7P+!7<6A-&6?' $H+.4<+.=�4<+!ST0 ���O� � � P �

Page 141: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# �, M� ������ � � � �� � � ���� �� � � ������ ����0 ���� � ������� �� � �y 6?4Q/H+.456AC&-H+zBf7<()Fk+ /&6?')'=a�6?45450�456A/&(:7<450�-&*,+.FG6?-U7<0o*Pv&6AB5( *?0�FGFG68757<6A+.45456?Bf7P+.-&/&0o+.'�$H+.B5B50;F�()'$&450�*?6?B5B50q(O7<6?4<+!7<(O@�0�/&6?B5*?45(O757<0 N B5(VCD7<()')()�?�,+G')0�Bf@�()')C&$&$�0�()-IB56?45()6\/&( � *�Z � +.C&45()- /&6?')'2+�STC&-&�?()0�-&6� 8 <

� "��� 0�Bf@�()')C&$&$�0�()-qB56?45()6E/&( � *�Z � +.C&45()-q/&6?')'2+zSTC&-&�?()0�-&6 � 8 <

� " [6t/H+!7<0�/H+�]

�N " � 8 <

� " �

> T2 � �

�� �

> �M � � w,w,w ()*?0�45/H+.-&/&0�'2+

� ��"B5(�vH+D]

� � � � � c � 8

< �� � c6?/ N CD7<()')()�?�,+.-&/&0o'2+ �

N "�N *,+.')*?0�'2+!7P+\()- �� � e � N B56?=�C&6 *Pv&6.]� � �� � c��

�� � c

> �N � TP � c � �� �%6

� �2 � �

c> w,w,w��

� � � > �N � T

� c � �� �%6� �� N � c

> w,w,w �X�C&()-&/&(�'=a 6?45450�456E/&(`7<450�-&*,+.FG6?-U7<0G+.-H+.')(O7<()*?0&]8 � �

� > � � �� �N � T

� c> w,w,w6t45()B5C&'O7P+D]

� 8 � � � ����� TN � � � c

����� � � Q � �0h@�@�6?450

8 � � �� � � � S+.'`$H+.B5B50kB5C&*?*?6?B5B5(O@�0&]

� 8 � _ c � � � � Q � � _ c �� � Q � � � �

�� � � _ c �*?()0&[6.]

8 � _ c � � �� � � � � �

8 �� C&()-&/&( N +./k0�=�-&(�$H+.B5B50 < N '=a 6?45450�456%/&(H7<450�-&*,+.FG6?-U7<0o+.-H+.')(O7<()*?0EB5( 45()/&C&*?6 /&(�C&-�X�CH+.4f7<0 �

0�45/&()-&6/&(�*?0�-U@�6?45=�6?-&�,+ ��"��

Page 142: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

�, P�����)B �CB �0K�P ; ��; � ����� 69K�P K��������/ (H*?0�-&B5()/&6?45(HC&-H+t*?()45*?0�-DST6?456?-&�,+ 3 /&(H*?6?-U7<450z'=a 0�45()=�()-&6 6u/&(:4<+.=�=�()0E4

� ��� j 'HFG687<0�/&0t$&450�$�0�Bf7<0/H+ � ()687<6I[6�X�C&6?')')0I/&( +.$&$&450�B5B5()Fk+.456k()'W@.+.')0�456k/&( � *?0�-1()'RB56?FG()$�6?45()FG687<450� � /&6?' $�0�')()=�0�-&0456?=�0�'2+.456o*?0�- � � '2+!7<(Y()-&B5*?45(O757<0q-&6?')'2+�*?()45*?0�-DST6?456?-&�,+ � j -&/&()*,+!7P+q*?0�- ��J " '2+�')C&-&=�v&6?�?�,+q/&6?')'2+*?()45*?0�-DST6?456?-&�,+ 3

��J " �

� � � � �B56 �� � +.')')0�4<+k]

���J "�6 �

�J " p � � �y 6?4 0�=�-&(H�HB5B<+!7<0z@.+.')0�456u/&( F N B5(�()-&/&()*Pv&(:*?0�-X? � '2+t')C&-&=�v&6?�?�,+ /&6?'�'2+!7<0 /&6?'�$�0�')()=�0�-&0E/&( � � '2+!7<(6E*?0�-� � ()' B5C&0GB56?FG()$�6?45()FG687<450 N B5(�0.757<()6?-&6o]

� ��"� � �

� � ? �� �� � c? � M

� +z456?'2+.�?()0�-&6%*Pv&6E')6?=U+z()' '2+!7<0 ? � /&6?'`$�0�')()=�0�-&0z/&( � � '2+!7<( +.'�'2+!7<0R? � _ c /&6?'`$�0�')()=�0�-&0z*?0�- � � _ c'2+!7<(�[6t'2+\B56?=�C&6?-U7<6.]

�� " ? � _ c � � �� � _ c � �

? �� M/H+o*?C&(�B5(�0.757<()6?-&6o]

�� � " ? � � � � ? � _ c � � �� � _ c � Mj -DS|+!757<(�*?0�FG6AB5(`0�B5B56?4f@.+o/H+.')'2+ 4�()=�C&4<+ 2 N B56 < �

� N ? � [6A()' '2+!7<0z/&6?'`X�CH+./&4<+!7<0o()-&B5*?45(O757<0\6 ? T [6()' '2+!7<0o/&6?')'=a 0.757P+.=�0�-&0o()-&B5*?45(O757<0 �

Page 143: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# �, 1

O

KA

C

B a2

a3

� � ������ A ] � 687<0�/&0G/&( � ()687<6E$�6?4u()'`*,+.')*?0�')0\/&(��/ 6kB5( *?0�-&B5()/&6?4<+.-&0I(W7<45(2+.-&=�0�')( ��� J 6 � � � N =�')(R+.-&=�0�')( ���� J 6?/����� � B50�-&0I/&C&6�+.-&=�0�')(45()B5$�68757<(O@.+.FG6?-U7<6E+.')'2+\*?()45*?0�-DST6?456?-&�,+�6t+.'�*?6?-U7<450G*Pv&6 ()-&B5()Bf7<0�-&0oB5C&')')0�Bf7<6?B5B50�+.45*?0

��� J " 4 c $�6?4*?C&(L]

���� J����� �� M ���� � �

� ���� J

��P

� a +.-&=�0�')0 J �� ��� � [6 C&=�CH+.')6z+.')'=aH+.-&=�0�')0 ���� J N 6?B5B56?-&/&0q+.-&=�0�')( +.')'2+ } +.B56z/&( C&-�7<45(2+.-&=�0�')0()B50�B5*?6?')6

�� J � " S $�6?4o*?C&( B56

Y i� [6�C&-~*,+!7<687<0 /&6?'Q7<45(2+.-&=�0�')0�� � J ? T -&6 [6�'=a ()$�0.7<6?-�C&B<+ N B5(0.757<()6?-&6E*Pv&6.]

? T � � � � P � �? T � � � �� T � �

? �� wy 0�-&6?-&/&0�4��� � 6# � � �

�4" 6?/�CD7<()')()�?�,+.-&/&0\'2+zST0�45F\C&'2+\/&( } ()B56?�?()0�-&6E$�6?4u()'`*?0�B56?-&0&]

� ��4" ��� �

� ��� 4 "� �

�� �

� ��4 c "�$�0�()*Pv 96.]

c � � � � � � �

6 MB5(�vH+D] ��� �

c� M : � �

��� 36�DF!%E��"�D����!�� �UB� �¡ý ����� [E��DB ý � ���M��!X�"� � �!%�� ��E��D�@ �� ��� �������E� ���� ���������������� 6 .R(".�-3%) +%'*M.�8" "+ .���-�� *�� -M.�< *M* * -M*+* - %) +%'*M.�8" "+ .���- � [* -�!( ,��- D� * %'*�+,+ ) � # *M.MD� . ,) -M.�< *M* * -M*%' . , +* %�;� @)+.�� ) � ,) 2� ("*+* *3� ,) ,�'��*1-�+%'*!

Page 144: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

��� 6y 6?4 :�� N ()'�7<45(2+.-&=�0�')0 � � � [6A()B50�B5*?6?')6E6E4568757<0o()- � 6E'=a +.-&=�0�')0 � �� �

�� c N $�6?4 *?C&(`6?B5B56?-&/&0� �

�� N 4<+.=�=�()0o/&6?')'2+\*?()45*?0�-DST6?456?-&�,+ N B5(�vH+o*Pv&6.]? �� �

� � � � �

� ? � �� � � �

� �� �� �

$�6?4 *?C&(L]� � �

� � ? � �� � � � ()*,+,@�(2+.FG0\C&-H+o456?'2+.�?()0�-&6E$�6?4_: � � N CD7<()')()�?�,+.-&/&0\'2+ � ��"

6t'2+�� " ]

��� c?�� c �

� � � � �4 _ c " � ?

_ c � � � ��4 _ c " � ?

_ c ��� _ c � � � �

4 _ c " � �

_ c � _ c/H+o*?C&(L]� _ c �

_ cy 6?4uX�CH+.-U7<0G/&68757<0G456?Bf7P+�()-&/&(O@�()/&CH+!7P+z'2+oB56?=�C&6?-U7<6tST0�45F\C&'2+\/&(�45()*?0�45456?-&�,+D]

� _ c �

� c _ � �� M : � �*?0�-

� � �� � � � � �

��� �

� ��� ������ � � � �� ��� � ���-/ � ��� � �� � �� � � �� �� � � � 0 �y 6?4uX�CH+.-U7<0G45()=�CH+.45/H+\'2+o$&450�$H+.=U+.�?()0�-&6A/&6?')'=a 6?45450�456E/&(�CED���FB D � N 6?B5B56?-&/&0&]

6 � � � � �� � � � M : � 2

Page 145: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# �����+.'`*?456?B5*?6?456z/&( :W')6EX�CH+.-U7<(O7![+o*,+.')*?0�'2+!7<6� 7<6?-&/&0�-&0k+./ � ] � � � � � �y 6?4f7P+.-U7<0G-&0�->*%aO[6 +.FG$&')(O�H*,+.�?()0�-&6%/&6?')'=a 6?45450�456E/&(�450�C&-&/�Z 0 � -&6?'�*,+.')*?0�')0o/&(

� _ c � �

� � 6t()'`FG687<0�/&0o/&( � ()687<6E$�6?4u()'`*,+.')*?0�')0\/&(�� [6 � ���-/ � �� �

� ������ � � � �� � � ���� �� � � ������ ����0 ���� � ������� �� � �y 6?4 /&687<6?45FG()-H+.456 C&-H+zBf7<()Fk+t/&6?')'=a 6?45450�456 /&(H7<450�-&*,+.FG6?-U7<0o+.-H+.')(O7<()*?0E*?0�FGFG6?B5B50o+./k0�=�-&(�$H+.B5B50F B5(�0�B5B56?4f@.+�*Pv&6.]

? � � � � � 8 < � �� � _ c � � � � � � c? � �

� � � 8 < �� �: 7<()')()�?�,+.-&/&0\')0�Bf@�()')C&$&$�0�()-�B56?45()6t/&( � + � ')0�4 /&6?')'2+zSTC&-&�?()0�-&6tB56?-&0 N *,+.')*?0�'2+!7<0o()- �� � N B5(�vH+D]

� � �� � � �� � > �N � TP � � �� �%6

� �� N � > w,w,w�� �

�� > �N � T� T � �� �%6

� �� � > w,w,wX�C&()-&/&(L]8 � � � > � �/H+o*?C&(L]

� 8 � � � ���� �N� T

� � �� �%6� �� N � > w,w,w ����

� ��w � Q � �

Page 146: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

��� ������)B ��� �0K�P ; ��; � � � ���/N=< <C6+I ��� � �� 6?' B5C&0o')( } 450

�C � A���I�JKA � GEH �FXW5F � AD C � ��I H �����KI J �.0�v&- �3+.')')()BW$&456?B56?-U7![0�'=a ()/&6?-U7<(O7![+ B56?=�C&6?-U7<6.]

� � ����2 2 � w,w,w

� <� < > �� <� <� � w,w,w()'�@.+.')0�456E/&(�� 96t*,+.')*?0�'2+!7<0\FG6?/&(2+.-U7<6E()'`$&450�/&0.757<0�/&(�()-D�H-&(O7<6?B5()FG( �� a +.$&$&450�B5B5()Fk+.�?()0�-&6k/&6?'Q@.+.')0�456�/&(�� @�()6?-&6q0.757<6?-�CD7P+ *?0�-&B5()/&6?4<+.-&/&0 ()' $&450�/&0.757<0 $H+.45�?(2+.')6k( Z6?B5()FG0 � /&6?(D$&45()FG(/: S|+!757<0�45( � � a.$�0�B5B5( } ()')6R+.')')0�4<+u*?0�-&B5()/&6?4<+.456Q'2+uST0�45F\C&'2+u45()*?0�45456?-U7<6QB56?=�C&6?-U7<6.]

� [ �� M �

�� c : � : � > �/&0h@�6t(

� ��N� N`w,w,w`N - �

� ������ � � � �� ��� � ���-/ � ��� � �� � �� � � �� �� � � � 0 �

\ (2+.FG0zC&-H+�Bf7<()Fk+\/&6?')'=a 6?45450�456E/&(�450�C&-&/�Z 0 � � / (`0�B5B56?4f@�( *Pv&6.] : � : � > � � � � � : � > � � � �

/&0h@�6 cc

i c �

�#%'`*?456?B5*?6?456z/&6?')'=a ()-&/&()*?6%( N $�6?4 : � �

� N (`@.+.')0�45( � 7<6?-&/&0�-&0G+��?6?450

��

� 6 "%� � a +.')=�0�45(O7<FG045()B5C&'O7P+\/&C&-&X�C&6 Bf7P+ } ()')6 �

� ������ � � � �� � � ���� �� � � ������ ����0 ���� � ������� �� � �

/ (`$&C [0G/&()FG0�Bf7<4<+.456A*Pv&6 '=a 6?45450�456E/&(`7<450�-&*,+.FG6?-U7<0G+.-H+.')(O7<()*?0;96�]� 8 � � � ���

� � < c ���� ��w N < c� +.')6 6?45450�456\[6%Bf7P+!7<0\45()*,+,@.+!7<0z6?B5$&45()FG6?-&/&0\'2+tST0�45F\C&'2+ /&( �3+.')')()B N FG6?/&(2+.-U7<6 '=a ()/&6?-U7<(O7![+t/&( � D CZ�_CED���F�G�LJKC�]

� � �� Q cc _ � ii�� � ii���� ii��� ii���

Page 147: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# ��� 2�����)B ��� �0K�P ; ��;�� � ����KX69VXE76����������j '`@.+.')0�456A/&(�� [6E0.757<6?-�CD7<0G/H+.')')0oBf@�()')C&$&$�0o()-qB56?45()6t/&( � * � +.C&45()- 4 � /&6?')'2+zSTC&-&�?()0�-&6 ? � 7 �

� " ]? � 7 �

� " �

> T2 �

��

> �M � w,w,wy 6?4 �

� ��NB5(�vH+�()-DS|+!757<(L]

� � > �2 � ��> �M � w,w,w()-&/&()*,+!7P+�*?0�-��

'2+IB50�FGFk+I$H+.45�?(2+.')6k/&6?( $&45()FG(9: +./&/&6?-&/&(E[6�$�0�B5B5( } ()')6�45()*,+,@.+.456�'2+�ST0�45F\C&'2+45()*?0�45456?-U7<6tB56?=�C&6?-U7<6.]� ��"

� c � � M � _ c � �

� �

> ��" c� : > �/&0h@�6 :

� � M � M wOwOw < > ���� ������ � � � �� ��� � ���-/ � ��� � �� � �� � � �� �� � � � 0 �j -3C&-1B5()Bf7<6?Fk+�+.45(O7<FG687<()*?0�+�$&456?*?()B5()0�-&6��H-&(O7P+ � N '=a 6?45450�456G/&(R450�C&-&/�Z 0 � ] � N -&6?' *,+.')*?0�')0>/&(C&-H+�B50�FGFk+\/&( < -�C&FG6?45( N B5(�$&C [0GBf7<()Fk+.456t+!757<4<+,@�6?45B50G'2+zST0�45F\C&'2+\B56?=�C&6?-U7<6.]

� ] � � a �< > ��" �

� ��c �

� ��

� ��" M *?0�- � I H*�� � I H�H/G�G���C0H%AJ?�� H

6 N X�C&()-&/&( N *?456?B5*?6 ')()-&6,+.45FG6?-U7<6E*?0�-q()'`-�C&FG6?450G/&6?=�')(�+./&/&6?-&/&( N < � / (�0�B5B56?4f@.+�*Pv&6�> ��"

c� : > �� 6 M X�CH+.-&/&0 : � �$�6?4u*?C&(�'=a +.')=�0�45(O7<FG0 $�6?4u()' *,+.')*?0�')0o/&6?')'2+� ��"

45()B5C&'O7P+\Bf7P+ } ()')6 �� ������ � � � �� � � ���� �� � � ������ ����0 ���� � ������� �� � �

� a 6?45450�456t/&(`7<450�-&*,+.FG6?-U7<0G+.-H+.')(O7<()*?0�[6E/H+!7<0G/H+D]8 � � � >

�� > �2 � ��

> �M � �h�h� � �> ��" � _ c� <� � �$�6?4u*?C&(L]

� 8 � � � � <� �

� � < c��7'�]��!%�� �����!9��!%�TB��D����!9� ���� �E� GB�B��M����������!"�"!4� �@F ý � H Q������E�!�B�B�� �X�M��� �D� þÿý ��$�� !%�� ��� ������ ;>= ? �� ��]F!%����� �D�M��!%E��

�" �� F����DF!%�D�M��!%E���_ �1!�B�B��D����!IB�!%� !K�%F�F ���� ��E� TF�N� @�D�KB ý �#LM��� ý ����$�� !%�� @��!%��L]� ��� @ GB� �¡ý � ���#�� G�"�D������ TB�L]� � ý ���<!� � B� �E�� �"��� F2� � ý E�� �U_

Page 148: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

���! �����)B ��� �0K�P ; ��; � � �0N�8��(6+E ��� � �j '�FG687<0�/&0�B56?=�C&6?-U7<6q[6oC&-H+�@.+.45(2+.-U7<6\/&(�X�C&6?')')0�/&( � 6?( } -&()�.] ()' @.+.')0�456z/&( � [6\*,+.')*?0�'2+!7<0�6�� 687fZ7<CH+.-&/&0�'2+\/&(�� 6?456?-&�,+o7<4<+o/&C&6 $�0�')()-&0�FG(:/&( � + � ')0�4 /&6?')'2+zSTC&-&�?()0�-&6 ? � 7 �

� " F/ (2+ 4;'=a +.45*?0.7P+.-&=�6?-U7<6t/&( c� ]

7 ��4 " � ��/H+.')'2+ ST0�45F\C&'2+\/&(`/&C&$&')()*,+.�?()0�-&6E/&6?=�')(�+.-&=�0�')( B5(�0.757<()6?-&6.]

7 ��� 4 " �

� 7 ��4 "� > 7 � ��4 " � �� �45(2+.$&$&')()*,+.-&/&0�'2+ -�C&0h@.+.FG6?-U7<6�]

7 �� 4 " �

� 7 ��� 4 "� > 7 � ��� 4 " � � �%6��� 1 w� a ()/&6,+3/&( � +.*Pv&()- STC X�C&6?')'2+3/&(E+.$&$&450�B5B5()Fk+.456 � �@.+.')0�456I/&6?')'=a +.45*?0.7P+.-&=�6?-U7<6 + � c "�N *?0�- '2+ST4<+.�?()0�-&6 c � [c c � N 6E*,+.')*?0�'2+.456%'2+\/&(�� 6?456?-&�,+ � c

� [c c � > ��" ()-�7<6?45FG()-&( /&( +.-&=�0�')( N FG6?/&(2+.-U7<6E'2+zST0�45F\C&'2+/&(`B50.757<4<+.�?()0�-&6E/&6?')'2+z7P+.-&=�6?-U7<6.]7 � 4 >

��

7 �� 4 " > 7 �

�� c "� � 7 �

� 4 " � 7 � �

� c " �7 �� 4 " > �� � 7 �� 4 "B5(�0.757<()6?-&6.]

? � 7 � � �� 2�1 � � 4 >�

� ? � 7 �� �� �

>�

45()B50�'O@�6?-&/&0�45()B5$�68757<0G+ � c ]�

� ? � 7 �� �� �

> ? � 7 � � �� 2�1 � wj -&/&()*,+.-&/&0\*?0�- � � � "()'�$�0�')()-&0�FG()0E/&( � + � ')0�4 /&( =�4<+./&0 <�/&6?')'=a ? � 7 �

� "�N $�0�B5B5(2+.FG0 45()B5*?45(O@�6?456'2+\456?'2+.�?()0�-&6A0.757<6?-�CD7P+ N -&6?' FG0�/&0oB56?=�C&6?-U7<6.]

� p � N � � �� �� �

> � � �� �� 2�1 �

� ������ � � � �� � � ���� �� � � ������ ����0 ���� � ������� �� � �

/ (`$&Cu90G*,+.')*?0�'2+.456A'2+\Bf7<()Fk+o/&6?')'=aD6?45450�456t/&(`7<450�-&*,+.FG6?-U7<0&]� 8 � � � � N�

� <� ��"5� � � _ c � ��w N < c

� � �

*?0�-&B5()/&6?4<+.-&/&0G'=a 6?45450�456E/&(`7<450�-&*,+.FG6?-U7<0G+.-H+.')(O7<()*?0\*Pv&6 B5(�*?0�FGFG68757<6t+.$&$&450�B5B5()Fk+.-&/&0 �;*?0�->()'$�0�')()-&0�FG()0 /&( � + � ')0�4 /&(�=�4<+./&0�< *Pv&6 +.$&$&450�B5B5()Fk+D]� N � ? � 7 �

� �� � w

Page 149: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# ����������)B ��� �0K�P ; ��; � �j '.@.+.')0�456V/&(��>[6Y*,+.')*?0�'2+!7<0QCD7<()')()�?�,+.-&/&0Q')0 Bf@�()')C&$&$�0 ()-EB56?45()6Y/&( � + � ')0�4`$�6?4 '2+RSTC&-&�?()0�-&6W+.45*?0�B56?-&0&]

? � � 8 < �� �� T2 � �� 2

�� � w,w,wy 6?4 �

� �� � N B5(�vH+o()-DS|+!757<(L] �

�N � �� � �� �2 �� T � �� 2 �� �� � � w,w,w �j -&/&()*,+!7P+�*?0�- �

6 7N 45()B5$�68757<(O@.+.FG6?-U7<6k'2+>B50�FGFk+>/&6?( $&45()FG( � 7<6?45FG()-&(R6k'*<�( Z�J� � I�D�S|+!757<0�456/&6?')')0oBf@�()')C&$&$�0�()-qB56?45()6EB5(�0.757<()6?-&6E'2+oB56?=�C&6?-U7<6tST0�45F\C&'2+\(O7<6?4<+!7<(O@.+D]

� ��"7 c � � � � M 7

_ c �7�� : > ��"P :

�� " � [ �

6 M � _ c � �

7

� : > �M6?/>()-&0�'O7<456.]

� � � N � �/&0h@�6 (� ��NO�O�

- �� ������ � � � �� ��� � ���-/ � ��� � �� � �� � � �� �� � � � 0 �j -3C&-1B5()Bf7<6?Fk+�+.45(O7<FG687<()*?0�+�$&456?*?()B5()0�-&6��H-&(O7P+ � N '=a 6?45450�456G/&(R450�C&-&/�Z 0 � ] � N -&6?' *,+.')*?0�')0>/&(C&-H+�B50�FGFk+\/&( < -�C&FG6?45( N B5(�$&C [0GBf7<()Fk+.456t+!757<4<+,@�6?45B50G'2+zST0�45F\C&'2+\B56?=�C&6?-U7<6.]

� ] � � a �< > ��" �

� ��c �

� ��

� ��" M *?0�- � I H*�� � I H�H/G�G���C0H%AJ?�� H

6 N X�C&()-&/&( N *?456?B5*?6 ')()-&6,+.45FG6?-U7<6E*?0�-q()'`-�C&FG6?450G/&6?=�')(�+./&/&6?-&/&( N < � / (�0�B5B56?4f@.+�*Pv&6�� : > ��"P : � �

M X�CH+.-&/&0 : � �$�6?4u*?C&(�'=a +.')=�0�45(O7<FG0 $�6?4u()' *,+.')*?0�')0o/&6?')'2+� ��"

45()B5C&'O7P+\Bf7P+ } ()')6 S+.-H+.')0�=U+.FG6?-U7<67

� : > �� 6 M X�CH+.-&/&0 : � �

$�6?4u*?C&(�'=a +.')=�0�45(O7<FG0 $�6?4u()' *,+.')*?0�')0o/&6?')'2+�� " 45()B5C&'O7P+\Bf7P+ } ()')6 �

Page 150: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

��� N� ������ � � � �� � � ���� �� � � ������ ����0 ���� � ������� �� � �

� +.')CD7<(2+.FG0I'=a 6?45450�456�/&( 7<450�-&*,+.FG6?-U7<0�+.-H+.')(O7<()*?0 *Pv&6>B5(u*?0�FGFG68757<6q+.$&$&450�B5B5()Fk+.-&/&0 � *?0�- ()'$�0�')()-&0�FG()0 /&(�=�4<+./&0�< /&( � + � ')0�4u/&6?')'2+zSTC&-&�?()0�-&6^? � � 8 <

� " ()- � c� ]

� 8 � � � ����� > N � �� � �� �2 �� T � �� 2 �� �� � ������$�6?4 *?C&(Y]� 8 � � � 6 w M � < c��

� � �

Page 151: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# ��� M�����)B ��� �0K�P ; ��; � � ��� � E�P K�Y Q N ��69;UE�K���� FTKXQR698:Nj ':@.+.')0�456 /&(��~[6%*,+.')*?0�'2+!7<0z+!757<4<+,@�6?45B50oC&-H+ ST0�45F\C&'2+t/&(�X�CH+./&4<+!7<C&4<+ 7<4<+.$�6?�?0�()/H+.')6 �

[�~$�0�B5B5( } ()')6+.$&$�0�B5B5()Fk+.4562� � *?0�-\'2+uB50�FGFk+ � � /&6?')')6R+.456?6 /&6?(�7<4<+.$�6?�?(�/&(D+.'O7<6?�?�,+ c� ()-&B5*?45(O757<(�()- C&-\X�CH+.4f7<0/&(�*?()45*?0�-DST6?456?-&�,+�/&(�4<+.=�=�()0\C&-&(O7P+.45()0 �j -DS|+!757<(`'2+\STC&-&�?()0�-&6E*Pv&6 /&68�H-&()B5*?6 C&-�+.45*?0�/&(�*?()45*?0�-DST6?456?-&�,+�C&-&(O7P+.45(2+z-&6?' j X�CH+./&4<+.-U7<6G[6.]KV� " �

� � > �y 6?4u*?C&( ()-U7<6?=�4<+.-&/&0�-�C&FG6?45()*,+.FG6?-U7<6KV� " ()- 6 M �

� B5(�0.757<()6?-&6t*Pv&6.]� c[� � > ��� qp

� �����N "

� � ������ � ] � 687<0�/&0�/&(�()-U7<6?=�4<+.�?()0�-&6: 7<()')()�?�,+.-&/&0�'2+oST0�45F\C&'2+o/&( X�CH+./&4<+!7<C&4<+�7<4<+.$�6?�?0�()/H+.')6t$�6?4 ()'�*,+.')*?0�')0�-�C&FG6?45()*?0k/&6?')'=a ()-U7<6?=�4<+.')6� �����

N " B5(�0.757<()6?-&6 N $�6?4 � � N '=a 6?B5$&456?B5B5()0�-&6 B56?=�C&6?-U7<6.]� � � �� � � � � { c �

�{� � �h�h� �

�{ � c

�/&0h@�6.]

{��� >

: �< �

� ������ � � � �� ��� � ���-/ � ��� � �� � �� � � �� �� � � � 0 �j -qC&-qB5()Bf7<6?Fk+o+.45(O7<FG687<()*?0o+\$&456?*?()B5()0�-&6A�H-&(O7P+�� N '=a 6?45450�456A/&(`450�C&-&/�Z 0 � ] � N -&6?'�*,+.')*?0�')0\/&6?')'2+B50�FGFk+ � � N B5(�$&C [0�Bf7<()Fk+.456E-&6?'�FG0�/&0oB56?=�C&6?-U7<6.]� ] � � a �� � c �

< > ��" �� c ��c � {

� ��

� ��"6 N X�C&()-&/&( N *?456?B5*?6z')()-&6,+.45FG6?-U7<6t*?0�- < N -�C&FG6?450k/&6?=�')( +./&/&6?-&/&( � y 6?4f7P+.-U7<0k'=a +.')=�0�45(O7<FG0\45()B5C&'O7P+Bf7P+ } ()')6 �

Page 152: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

��� P� ������ � � � �� � � ���� �� � � ������ ����0 ���� � ������� �� � �

� a 6?45450�456E/&(`7<450�-&*,+.FG6?-U7<0k+.-H+.')(O7<()*?0z*Pv&6 B5(�*?0�FGFG68757<6o[6.]� 8 � � � � <

T � �6?B5B56?-&/&0 C&-H+�*?0�Bf7P+.-U7<6 $�0�B5(O7<(O@.+z6?/ 8 � Fk+.=�=�()0�4<+!7<0\/H+.')'2+zBf7<()Fk+o+.B5()-U7<0.7<()*,+\/&6?( { � �

Page 153: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# ��� 1�����)B ��� �������"� +���� ������ ��� ����+- ,+�� ������� �� � �

\ ( B56?=�C&(O7<0k@�6?-&=�0�-&0�$&450�$�0�Bf7<(�+.')*?C&-&( $&450�=�4<+.FGFG(�()- 4H0�4f7<4<+.-�456?'2+!7<(O@�( +./I( FG687<0�/&( /&6?B5*?45(O757<(-&6?'�$H+.4<+.=�4<+!ST0o$&456?*?6?/&6?-U7<6 �. ��� ����0 0 � � � �������� � � � � ���� � ���� � � � � ��� � ��0 � � � � � � � � ����� 0 �

�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t��� ����� �������������������� ��! ��"��$#

%�&('*),+ -�./-�.�021,.�'*3 '*4,.�5267+$-�+,8�82+:9;'*3,.�'�<�.,82.>=

?A@ !��*���* @CBED !��2� ������F� BHG7?IBED�?IB �* � ��� B � ?IB �* � ��� 7J BK@ J

%:LNM�'AO�PQ'$'�97.�-�+�5R-�+,8SPI3;5�<782+TO('$-�'U3,.2);578V97+23,+

W ? !��NXVY B Y�ZW ? !��NXVY B Y�ZW ? !��NXVY B Y�ZW ? !��NXVY B Y�ZW ? !��NXVY B Y�ZW ? !��NXVY B Y�Z

aaaaaa

��������������" ?\[ #XAJ*Z � ?2]�G X ?�^ J*Z D X ?�^ J*ZX`_�Z D X ? Z ]�D�a �!bXAJ ^�D�a �!bXAJ ^�D X ?�^ J*Z�c�_�Z*d�_ Z

aaaaaa

W ? !��NXVY B Y�ZfeV# @,D �* ? �� ? � @�g �;�* ���" ? ����! ? "����h�;��� ? ��� @ �Ne ����*" Y BH@�� ?A@ !;Y BH@

%�&('*),+/-�. %�'>820�578�5

G7?2] _biD�?2] JIiD !��2� ]�j� ]�j

k ��� @ ! ?A@�g �

� ]�G7? Y D�?G7?2] _�Y G7?D�?2]�D�a �!bX�XAJ ^�D�a �!bXAJ ^�D�? Y�Y*_�Z�Z*d�_ ZD !��2� ]�D !��2��l,J

%nm�+232Oo.�6;'*4,.�5267+$-�+,8�8p'hq('*),+/-�.�%�'>820�578�5(=

?�r X D !��2�sit���bi @ Z:���2! � k

Page 154: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

� N 6@ J ]�@�* � ��� ]�� Y�Y @ J�* � ��� ] _;J�Y�J2d��* � ��� � ?2]�� Y*��!�� @ XAJIi`Z�* � ��� 7J ] XK� ?�^ �7Z*d�� ?�� ?A@ !;Y B e [ #U�*������������! � ] e B ��� ?A@ !;Y B e��� � ��� ��R� @ ��� ? ! ? ��� D ! ? �;��! ����" ? e B �* � ���

%R% 526�q,3;526(M*5 0�526 8p'\q���6�4,.�5267+R-�. 82.�<�3,+23,.�'

�� ?A@ !;Y B e [ # �;�*" ? � @ !�� r*g�@�;? � @7?\G�g7? �2! ^�?A@ ] e B � ?�� ?A@ !;Y B e��� � ��� ��U �������! ?� �Ne B �* � ��� 7JD ! ���� @ "

�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�. �� � � � � �� �� ���������

3 0�-&B5()/&6?45(2+.FG0�+.')*?C&-&6z$&450h@�6o/&(�6?B56?*?C&�?()0�-&6o/&6?')'=a +.')=�0�45(O7<FG0o/&6?B5*?45(O757<0�-&6?' � 687<0�/&0�/&(�#%45*Pv&( ZFG6?/&6 N '=a 0 } ()68757<(O@�0q[6EX�C&6?')')0�/&(�FG68757<6?456t()-q68@�()/&6?-&�,+�'=a � � � ���-/ � ��� � �� /&6?' $&450�*?6?/&()FG6?-U7<0 � 3 0�FG6@�6?/&456?FG0�$�6?4%-�

M N ()' 45()B5C&'O7P+!7<0I[6\+.*?*?C&4<+!7<0q+ � *?(OST456zB5()=�-&(O�H*,+!7<(O@�6 N FG6?-U7<456\$�6?4A- � ����N ()'45()B5C&'O7P+!7<0\0.757<6?-�CD7<0>[6t*?0�FG$&')687P+.FG6?-U7<6E6?454<+!7<0 ��t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t���������������" ?\[ #

XAJ*Z � ?2]�G X ?�^ J*Z D X ?�^ J*ZX`_�Z D X ? Z ]�D�a �!bXAJ ^�D�a �!bXAJ ^�D X ?�^ J�Z�c2_�Z�d�_�Z

# @,D �* ? �� ? � @�g �;�* ���" ? ����! ? "����h�;��� ? ��� @ ��

[ #��*������������! � ] iVJ � J�� j�j J _�� � ��� ��R� @ ��� ? ! ? ��� D ! ? �;��! ����" ? j i j�j J�_��,J �� ��_��[ # �;�*" ? � @ !�� r*g�@�;? � @ � G�g7? �2! ^�?A@�] iVJ � J k�� _ ����� � ��� ��U �������! ?� � j i�_ k _���_ j � � � ^�j ��t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�

�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�

Page 155: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# � N ���������������" ?\[ #XAJ*Z � ?2]�G X ?�^ J*Z D X ?�^ J*ZX`_�Z D X ? Z ]�D�a �!bXAJ ^�D�a �!bXAJ ^�D X ?�^ J*Z�c�_�Z*d�_ Z

# @,D �* ? �� ? � @�g �;�* ���" ? ����! ? "����h�;��� ? ��� @ �J [ #U�*������������! � ] _bi���_�� � _ ��j �QXVY�Z�� � ��� ��R� @ ��� ? ! ? ��� D ! ? �;��! ���:" ? iVJ�_ � _ � � k � ^�j��[ # �;�*" ? � @ !�� r*g�@�;? � @7?hG�g7? �2! ^�?A@�] iVJ � J k�� _ ����� � ��� ��U �������! ?� � j i j ��� ��� �� _ �

�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t��r " j 'U45()B5C&'O7P+!7<0Q/&6?')'2+QB56?*?0�-&/H+u6?B56?*?C&�?()0�-&6 [6W/&6?'�7<CD757<0u6?454<+!7<0 � j -DS|+!757<(U()'.B5()Bf7<6?Fk+u+.45(O7<FG687<()*?0/&6?':*,+.')*?0�'2+!7<0�456u6?B56?=�C&6%')6u0�$�6?4<+.�?()0�-&(H-&6?(�456?=�()Bf7<45(:+ /&0�$&$&(2+t$&456?*?()B5()0�-&6 6 $�6?49<

� � 2 N '=a 6?45450�456/&(�CED���FB;D � [6E/&6?' � 6�6�� N *?0�FG6E/&6?B5*?45(O757<0G-&6?'�$H+.4<+.=�4<+!ST0 ���O� � �O� �+.')=�0�45(O7<FG0z()-&Bf7P+ } ()')6 "��

Page 156: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

� N �. ��� ����0 0 � � � �������� � � � � ���� � ���� � � � � ��� � ��0 � � � � � � � �� ��

�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t��� ����� �������������������� ��! ��"��$#�#

%h&('*),+ -�./-�.�021,.�'*3 '*4,.�5267+$-�+,8�82+:9;'*3,.�'�<�.,82.>=

?A@ !��*���* @CBED !��2�"�� g�G ���:�� ���� ?�D�? � @ � BHG7?IBED�?IB �* � ��� B � ?IB �* � ��� 7J BH@ J

%�LNM�'AO�PQ' ':97.�-�+�5 -�+,8 PI3;5�<782+TO(' -�'U3,.2);578V97+23,+

W ? !��NXVY B Y�ZW ? !��NXVY B Y�ZW ? !��NXVY B Y�ZW ? !��NXVY B Y�ZW ? !��NXVY B Y�ZW ? !��NXVY B Y�Z

aaaaaa

��������������" ?\[ #XAJ*Z � ?2] � X ?�^ J*Z*d�� ?X`_�Z �IX ? Z ]�D�a �!bXAJ ^ �IX ?�^ J*Z�Z*d�_ Z

aaaaaa

W ? !��NXVY B Y�Z eV# @,D �* ? �� ? � @�g �;�* ��:" ? ����! ? "����h�;��� ? ��� @ �Ne ����*" Y BH@

%h&('*),+R-�. %�'>820�578�5

� ?2]�j i� �* ] _biD !��2� ] J� ] _bi

k ��� @ ! ?A@�g �

D !��2� ]�D !��2��l,J� ?2]�D�a �!bX�XAJpl�� ? Z*d�_�Z� ] � �* �d�� ?� �* ] �

% m�+232Oo.�6;'*4,.�5267+$-�+,8�8p'hq('*),+R-�.�%�'>820�578�5(=

?�r X D !��2�sit���bi @ Z����2! � k@ J ]�@�* � ��� ]�� Y�Y @ J�� ?A@ !;Y B e �* � ��� �� ] e B �* � ��� �* � ��� ] k i�_�Y�J2d��* � ��� � ?2]�� Y*��!�� @ XAJIi`Z�* � ��� 7J ] XK� ?�^ �7Z*d�� ?

Page 157: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# � N 2�� ?A@ !;Y B e [ #U�*������������! � ] e B ��� ?A@ !;Y B e��� � ��� ��R� @ ��� ? ! ? ��� D ! ? �;��! ���:" ? e B �* � ���

% % 526�q,3;526(M*5 0�526 8p'\q���6�4,.�5267+ -�. 82.�<�3,+23,.�'

�� ?A@ !;Y B e [ # �;�*" ? � @ !�� r*g�@�;? � @7?\G�g7? �2! ^�?A@ ] e B � ?�� ?A@ !;Y B e��� � ��� ��/ �������! ?� �Ne B �* � ��� 7JD ! ���� @ "

�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�. �� � � � � �� �� ���������

3 0�-&B5()/&6?45(2+.FG0�+.')*?C&-&6o$&450h@�6G/&(V6?B56?*?C&�?()0�-&6G/&6?')'=a +.')=�0�45(O7<FG0�/&6?B5*?45(O757<0q-&6?' � 687<0�/&0q/&( � ()687<6 N'=a 0 } ()68757<(O@�01[6>X�C&6?')')0 /&( FG68757<6?456q()-~68@�()/&6?-&�,+�'2+ � ���-/ � ��� � �� /&6?'u$&450�*?6?/&()FG687<0 � ()B5$�68757<0�+.(45()B5C&'O7P+!7<(`/&6?' $H+.4<+.=�4<+!ST0�$&456?*?6?/&6?-U7<6\$�6?4u@.+.')0�45(`$&(<[C>+.'O7<(`/&( < '=a 6?45450�456t/&( 450�C&-&/�Z 0 � -&0�->*?0�FoZ$&450�FG68757<6E()'`45()B5C&'O7P+!7<0 ��t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t���������������" ?\[ #

XAJ*Z � ?2] � X ?�^ J*Z*d�� ?X`_�Z �IX ? Z ]�D�a �!bXAJ ^ �IX ?�^ J*Z�Z*d�_ Z

# @,D �* ? �� ? � @�g �;�* ���" ? ����! ? "����h�;��� ? ��� @ ��

[ #U�*������������! � ] iVJ � J k J k � � j ��� J�_�� � ��� ��R� @ ��� ? ! ? ��� D ! ? �;��! ���:" ? iVJ �� ��_��,J�_ k j�j�j�j�j�j J � ^�j �[ # �;�*" ? � @ !�� r*g�@�;? � @7?hG�g7? �2! ^�?A@ ] iVJ � J k�� _ ��� J j J�_ k �� �� � ��� ��U �������! ?� � j i�_ k J ��k � j ��_ �� J _ j J � � ^�j �

�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t��t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t���������������" ?\[ #

XAJ*Z � ?2] � X ?�^ J*Z*d�� ?X`_�Z �IX ? Z ]�D�a �!bXAJ ^ �IX ?�^ J*Z�Z*d�_ Z

# @,D �* ? �� ? � @�g �;�* ���" ? ����! ? "����h�;��� ? ��� @ �J��

[ #U�*������������! � ] iVJ � J k�� _ k j _ k�� ��� ��� � ��� ��R� @ ��� ? ! ? ��� D ! ? �;��! ���:" ? JIi�_;J j�� J � �� ��j�j�j J�_�_;J � ^�j �

Page 158: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

� N [ # �;�*" ? � @ !�� r*g�@�;? � @7?\G�g7? �2! ^�?A@ ] iVJ � J k�� _ ��� J j J�_ k �� �� � ��� ��U �������! ?� � j i � k � � j � ��� ����� j � j _ � _ � ^�j��

�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t��t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t���������������" ?\[ #

XAJ*Z � ?2] � X ?�^ J*Z*d�� ?X`_�Z �IX ? Z ]�D�a �!bXAJ ^ �IX ?�^ J*Z�Z*d�_ Z

# @,D �* ? �� ? � @�g �;�* ���" ? ����! ? "����h�;��� ? ��� @ � �j

[ #��*������������! � ] iVJ � J k�� _ k j _ k�� ��� ��� � ��� ��R� @ ��� ? ! ? ��� D ! ? �;��! ����" ? � i k J j _��,J j� �� k � � � � � � ^ J��[ # �;�*" ? � @ !�� r*g�@�;? � @7?\G�g7? �2! ^�?A@ ] iVJ � J k�� _ ��� J j J�_ k �� �� � ��� ��U �������! ?� � � i k � � j � ��� ����� j � j _ � _ � ^�j �

�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�� B5B56?4f@.+.�?()0�-&6� a 6?45450�456G456?'2+!7<(O@�0q*Pv&6k@�()6?-&6G*,+.')*?0�'2+!7<0q-&6?'W*?0�45B50I/&6?')'=a +.')=�0�45(O7<FG0k*?0�FG$&456?-&/&6k+.-&*Pv&6k'=a 6?45450�456/&(`450�C&-&/�Z 0 � N $�6?4u*?C&(�45()B5C&'O7P+D]] � � � � 8 � 8 � ? 7 : �

��

] � � � � 8 � : 7 � � <G K?U� w ?O<B? � : 7 : � �] � � � � 8 � : M � � � < � > �

K`K� � 6 �

Page 159: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# � N �. ��� ����0 0 � � � �������� � � � � ���� � ���� � � � � ��� � ��0 � � � � � � � � / ����# " y 45()Fk+\Bf7<4<+!7<6?=�(2+�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t��� ����� ���� ��������������� ��! ��"��$#�� D !� ���!��*� ? � J

%�&('*),+ -�./-�.�021,.�'*3 '*4,.�5267+$-�+,8�82+:9;'*3,.�'�<�.,82.>=

?A@ !��*���* �

������ � BV?IB �* � ��� B � ?IB �* � ��� 7J BH@ J BED !���� BH@

%:LNM�'AO�PQ'$'�97.�-�+�5R-�+,8SPI3;5�<782+TO('$-�'U3,.2);578V97+23,+

W ? !��NXVY B Y�ZW ? !��NXVY B Y�ZW ? !��NXVY B Y�ZW ? !��NXVY B Y�ZW ? !��NXVY B Y�Z

aaaaa

��������������" ?\[ #� ,? � g ���;� ?A@nD �* ? ��" ?�� �������� R" ? �* ��!�� X��;Z

aaaaa

W ? !��NXVY B Y�Z eV# @,D �* ? ��$�� �" ?A@ ��"�������� D�,? � g ���;�Ne ����*" Y BH@

%�&('*),+/-�. %�'>820�578�5

a ?2] JIiD !��2� ] J

k ��� @ ! ?A@�g �

D !��2� ]�D !��2��l,J?2] X ^ J*Z�Y�YIX D !��2� ^ J*Z?2] ? doX`_�Y D !��2� ^ J*Za ?2]�a ? l ?

%nm�+232Oo.�6;'*4,.�5267+$-�+,8�8p'hq('*),+/-�.�%�'>820�578�5(=

?�r X D !��2�sit���bi @ ^ J*Z����2! � k� ]�� Y a ?@ J ]�@�* � ��� ] J2d @ J�* � ��� ] _�Y*�* � ��� � ?2]�� Y*��!�� @ XAJIi`Z�* � ��� 7J ] XK� ?�^ �7Z*d�� ?�� ?A@ !;Y B e [ #U�*������������! � ] e B �

Page 160: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

� N�N�� ?A@ !;Y B e��� � ��� ��R� @ ��� ? ! ? ��� D ! ? �;��! ����" ? e B �* � ���

%R% 526�q,3;526(M*5 0�526 8p'\q���6�4,.�5267+R-�. 82.�<�3,+23,.�'

�� ?A@ !;Y B e [ # �;�*" ? � @ !�� r*g�@�;? � @7?\G�g7? �2! ^�?A@ ] e B � ?�� ?A@ !;Y B e��� � ��� ��U �������! ?� �Ne B �* � ��� 7JD ! ���� @ "

�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�� " / 6?*?0�-&/H+GBf7<4<+!7<6?=�(2+�� ����� ���� ��������������� ��! ��"��$#�� D !� ���!��*� ? � _

%h&('*),+ -�./-�.�021,.�'*3 '*4,.�5267+$-�+,8�82+:9;'*3,.�'�<�.,82.>=

?A@ !��*���* �bBED !��2� BH@ �������� BV?IB �* � ��� B � ?IB �* � ��� 7J BH@ J ������ a ?IBta ? J

%�LNM�'AO�PQ' ':97.�-�+�5 -�+,8 PI3;5�<782+TO(' -�'U3,.2);578V97+23,+

W ? !��NXVY B Y�ZW ? !��NXVY B Y�ZW ? !��NXVY B Y�ZW ? !��NXVY B Y�ZW ? !��NXVY B Y�Z

aaaaa

��������������" ?\[ #� ,? � g ���;� ?A@nD �* ? �U" ?�� �������� /" ? �* ��!�� X��;Z

aaaaa

W ? !��NXVY B Y�Z eV# @,D �* ? �� �� �" ?A@ �U"�������� D�,? � g ���;�Ne ����*" Y BH@�� ?A@ !;Y BH@

%h&('*),+R-�. %�'>820�578�5

a ?2] JIia ? J ] JIi

"�� J j D !��2� ]�@CB J B ^ J

?�r X`_�YIX D !��2�;d�_�ZNi�� a i D !��2�7ZU!�� � @?2] J2doX`_�Y�������� � X D !��2�7Z ^ J*Za ?2]�a ? l ?��� D �

Page 161: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# � NOM?2] J2doX`_�Y�������� � X D !��2�7Z ^ J*Za ? J ]�a ? Jpl ?� @ " ?�r

J j ��� @ ! ?A@�g �

a ?2] � G,D X a ?�^�a ? J*Z

%nm�+232Oo.�6;'*4,.�5267+$-�+,8�8p'hq('*),+/-�.�%�'>820�578�5 i

� ]�� Y a ?@ J ]�@�* � ��� ] J2d @ J

�* � ��� ] _�Y*�* � ��� � ?2]�� Y*��!�� @ XAJIi`Z�* � ��� 7J ] XK� ?�^ �7Z*d�� ?�� ?A@ !;Y B e [ #U�*������������! � ] e B ��� ?A@ !;Y B e��� � ��� ��R� @ ��� ? ! ? ��� D ! ? �;��! � �:" ? e B �* � ���

% % 526�q,3;526(M*5 0�526 8p'\q���6�4,.�5267+ -�. 82.�<�3,+23,.�'

�� ?A@ !;Y B e [ # �;�*" ? � @ !�� r*g�@�;? � @7?hG�g7? �2! ^�?A@ ] e B � ?�� ?A@ !;Y B e��� � ��� ��U �������! ?� �Ne B �* � ��� 7JD ! ���� @ "

. �� � � � � �� �� ���������# " y 45()Fk+\Bf7<4<+!7<6?=�(2+�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t���������������" ?\[ #

� ,? � g ���;� ?A@nD �* ? ��" ?�� �������� R" ? �* ��!�� X��;Z

# @,D �* ? �� �� �" ?A@ �U"�������� D�,? � g ���;���j[ #U�*������������! � ] iVJ�_ �� �j�� J ����� � ��� ��R� @ ��� ? ! ? ��� D ! ? �;��! ���:" ?:j i�_�� k � J � _�� �� � ^�j J[ # �;�*" ? � @ !�� r*g�@�;? � @7?hG�g7? �2! ^�?A@ ]� iVJ � J k�� _ ��� J�� � ��� ��U �������! ?� � j i � k � � _ � � � � � ^�j _

�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�

Page 162: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

� N�P

�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t���������������" ?\[ #� ,? � g ���;� ?A@nD �* ? �U" ?�� �������� /" ? �* ��!�� X��;Z

# @,D �* ? ��$�� �" ?A@ ��"�������� D�,? � g ���;�_ j�j[ #��*������������! � ] iVJ � k�� _���_ ��� � ��� ��R� @ ��� ? ! ? ��� D ! ? �;��! ����" ?:j i ������������� ��� � � ^�j _[ # �;�*" ? � @ !�� r*g�@�;? � @7?\G�g7? �2! ^�?A@ ] iVJ � J k�� _ ��� J�� � ��� ��U �������! ?� � j iVJ k�� J k � k � j � � ^�j _

�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t��t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t���������������" ?\[ #

� ,? � g ���;� ?A@nD �* ? �U" ?�� �������� /" ? �* ��!�� X��;Z

# @,D �* ? ��$�� �" ?A@ ��"�������� D�,? � g ���;���j�j[ #��*������������! � ] iVJ ��j J�� k j � J�� � ��� ��R� @ ��� ? ! ? ��� D ! ? �;��! ����" ?:j i�_�� k � J � _���_ � � ^�j _[ # �;�*" ? � @ !�� r*g�@�;? � @7?\G�g7? �2! ^�?A@�] iVJ � J k�� _ ��� J�� � ��� ��U �������! ?� � j i � k ��� k _ � J � � ^�j�

�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t��t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t���������������" ?\[ #

� ,? � g ���;� ?A@nD �* ? �U" ?�� �������� /" ? �* ��!�� X��;Z

# @,D �* ? ��$�� �" ?A@ ��"�������� D�,? � g ���;�k j�j�j�j�j[ #��*������������! � ] iVJ � J k�� � � �� � ��� ��R� @ ��� ? ! ? ��� D ! ? �;��! ����" ?:j i ��������������� j � ^�j k[ # �;�*" ? � @ !�� r*g�@�;? � @7?\G�g7? �2! ^�?A@ ] iVJ � J k�� _ ��� J�� � ��� ��U �������! ?� � j i �� � � k � � � k � � ^�j �

�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�

Page 163: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# � N 1� " / 6?*?0�-&/H+�Bf7<4<+!7<6?=�(2+

�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t���������������" ?\[ #� ,? � g ���;� ?A@nD �* ? ��" ?�� �������� R" ? �* ��!�� X��;Z

# @,D �* ? �� �� �" ?A@ �U"�������� D�,? � g ���;���j[ #U�*������������! � ] iVJ�_ �� �j�� � � _�� � ��� ��R� @ ��� ? ! ? ��� D ! ? �;��! ���:" ?:j i�_�� k � J � _�� �� � ^�j J[ # �;�*" ? � @ !�� r*g�@�;? � @7?hG�g7? �2! ^�?A@ ]� iVJ � J k�� _ ��� J�� � ��� ��U �������! ?� � j i � k � ��j�j � ��� � � ^�j _

�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t��t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t���������������" ?\[ #

� ,? � g ���;� ?A@nD �* ? ��" ?�� �������� R" ? �* ��!�� X��;Z

# @,D �* ? �� �� �" ?A@ �U"�������� D�,? � g ���;�_ j�j[ #U�*������������! � ] iVJ � k�� J � J�J�� � ��� ��R� @ ��� ? ! ? ��� D ! ? �;��! ���:" ?:j i ������������� ��� � � ^�j _[ # �;�*" ? � @ !�� r*g�@�;? � @7?hG�g7? �2! ^�?A@ ]� iVJ � J k�� _ ��� J�� � ��� ��U �������! ?� � j iVJ k�� J��,J k J�� � ^�j _

�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t��t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t���������������" ?\[ #

� ,? � g ���;� ?A@nD �* ? ��" ?�� �������� R" ? �* ��!�� X��;Z

# @,D �* ? �� �� �" ?A@ �U"�������� D�,? � g ���;���j�j[ #U�*������������! � ] iVJ � J�J�� j k �;J�� � ��� ��R� @ ��� ? ! ? ��� D ! ? �;��! ���:" ?:j i�_�� k � J � _���_ � � ^ _[ # �;�*" ? � @ !�� r*g�@�;? � @7?hG�g7? �2! ?A@$]� iVJ � J k�� _ ��� J�� � ��� ��U �������! ?� � j i � k�k � ��� J��;J�� � ^�j �

�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�

Page 164: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

� M 6

�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t���������������" ?\[ #� ,? � g ���;� ?A@nD �* ? �U" ?�� �������� /" ? �* ��!�� X��;Z

# @,D �* ? ��$�� �" ?A@ ��"�������� D�,? � g ���;�k j�j�j�j�j[ #��*������������! � ] iVJ � J k�� _ j _���� � ��� ��R� @ ��� ? ! ? ��� D ! ? �;��! ����" ?:j i ��������������� j � ^�j k[ # �;�*" ? � @ !�� r*g�@�;? � @7?\G�g7? �2! ^�?A@ ] iVJ � J k�� _ ��� J�� � ��� ��U �������! ?� � j i�_�_ � � � _ � ��j� � ^�j��

�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�t�

Page 165: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

����������� ������������������� ������������� ��� �!"� � #�$�� � ���%!&#('���*)+#�!&�,�.-/�.#��0�����# � M ������)B ��� � ���������#-� �����# $H+.4f7<()456Q/H+ � 6?( } -&()� N ()'�*,+.')*?0�')0A/&6?')')6 *?(OST456Q/&( �qB5( } +.B<+EB5C&')'=a +.$&$&450�B5B5()Fk+.�?()0�-&6R/&6?')'2+ STC&-&�?()0�-&6+.45*?0.7P+.-&=�6?-U7<6 � #Q757<CH+.')FG6?-U7<6 N -&6?=�')(�+.')=�0�45(O7<FG(&-�C&FG6?45()*?( N ()':FG0�/&0z$&(<[Ck-H+!7<C&4<+.')6 $�6?4Q*,+.')*?0�'2+.456C&-Ba +.$&$&450�B5B5()Fk+.�?()0�-&6E/&(��;B56?*?0�-&/&0k'2+\$&456?*?()B5()0�-&6E/&6?'�B5()Bf7<6?Fk+q[6

� � ? � � 7 � � ��"*Pv&6 N 0h@�@�(2+.FG6?-U7<6 N S|+�C&B50 /&6?')'=a +.$&$&450�B5B5()Fk+.�?()0�-&6�/&6?')'2+qSTC&-&�?()0�-&6�+.45*?0.7P+.-&=�6?-U7<6 N *?0�-&B5()/&6?4<+!7P+*?0�FG6ASTC&-&�?()0�-&6QT@� � � A Y� FG/&6?'�$H+.4f7<()*?0�'2+.456A*?0�FG$&()'2+!7<0�456%CD7<()')()�?�,+!7<0 �/ (Y45()$�0�4f7P+�C&-H+�7P+ } 6?')'2+k*?0�-U7<6?-&6?-U7<6k'2+�Bf7<()Fk+�/&6?')'=a 6?45450�456o/&(V7<450�-&*,+.FG6?-U7<0>+.-H+.')(O7<()*?0k6�'2+Bf7P+ } ()')(O7![+o$�6?4 ( @.+.45( FG687<0�/&( /&6?B5*?45(O757<( S B56?=�C&6 N ()-&0�'O7<456 N ()' =�4<+!�H*?0k/&6?')')6t*?C&4f@�6o*?0�4545()B5$�0�-&/&6?-U7<( N0.757<6?-�CD7<6G+.'�@.+.45(2+.456\/&(�< � \ +�6?-U7<4<+.F } (YB5(Y/&6?/&C&*?6�*Pv&6G()'�FG687<0�/&0�/&( � +.*Pv&()- ST0�45-&()B5*?6oC&-H+Bf7<()Fk+G/&6?')'=a 6?45450�456z/&( 7<450�-&*,+.FG6?-U7<0�+.-H+.')(O7<()*?0G$&(<[C�$&()*?*?0�'2+k45()B5$�68757<0�+.=�')( +.'O7<45( FG687<0�/&(�()-U@�6?450X�C&6?Bf7<(�C&'O7<()FG(`B50�-&0�Bf7P+!7<( 6?')6?-&*,+!7<( N ()->7P+ } 6?')'2+ N ()-�0�45/&()-&6 /&6?*?456?B5*?6?-U7<6o/&( @�6?')0�*?(O7![+�/&( *?0�-U@�6?4mZ=�6?-&�,+ " 0h@�@�6?450�*?0�-U@�6?45=�6�+.B5()-U7<0.7<()*,+.FG6?-U7<6\+.' @.+.')0�456 �1$&(<[C>@�6?')0�*?6?FG6?-U7<6\45()B5$�68757<0�+.( FG687<0�/&($&450�$�0�Bf7<( � � � � � � � �� 0 � ��� ( ������ � ( � ���-/ � ��� � ��

� +.*Pv&()-� �M 6 N " ��w N Q <

cQ � �

� / j� + � ')0�4 6 w M Q <

c��� Q

� / j4H0�45F � � 4<+.$�6?�?0�()/H+.')6 � Q <

T � � / j

� ()687<6� ���

1�2 " ��w � Q � / j

�3+.')')()B� �N ����" ��w N Q <

c/ j

� 6?( } -&()�� �N�P�P " � Q <

c/ j#%45*Pv&()FG6?/&6

�� 6 + � * ��" � � Q

� � �

Page 166: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

� M �

1 2 3 4 5 6 7 8

10−10

10−8

10−6

10−4

10−2

100

Archimede Viete Wallis Leibniz Machin Taylor Trapezoidale

Stime dell’errore di troncamento analitico

Page 167: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli� ������������ ���

��

�� �� � � � � 7*��A����� � H � �:��CEJ J �_JY6LD C�A T � A��LJ� �FB>@� � C � � D H%CZ J � �LJ � ��H � C�I H%F DU� A��LJ

� D H%CZ�� CEDU�@F�� F � F �,� D F1�vU757<$`] � �!B5B5$&= ��� } -&B5* � 45' � +.* � C < � / vH+.456 � j / �uy �.+.45(2+.-&6 � 4 � vU7<F NvU757<$`] � �!4<+,@�6?' � 6?B545()- � 6?B<+ � (O7 �!/&0�*?B �!6?B<+�Z���Z � P � 1 6?-&= � $�/DS N y +.45()B N�� 1�1 N �.C&' � � 1 � � � 3 0�/ � � � � � 687�+.' � Z � CED6LD�� J? � H� � � H%FB�� D CZ1� JKF�7%A�� �FB/JY6LJKFB/JKFLA � A�H%FBOH%CZ � D C� � D H%A � F�7 _D � FLA�C � A���I�JKA � GRZ j � � � � ()*?450 N P &N�� 1 P &� 2 � \ 6�3 y � � �ON ��+.FG6?B � � � Z� �FLA,CEDX �LGKA � D FbAD � ��I�JKC � GEH � � D IC6B��A�H%A � D F � F H*� GEH �%Z

/ $&45()-&=�6?4mZ � 6?45'2+.= N � 1 P ��� � 4H0�45B � 7<v&6q# � j �ON � 6?6?-H+.- � � # �ON � 45=U+.-&()* < � � j �ON / 7<6?- } 6?45= � � Z � D IC6B��AJKC � G � JKF�G�JX� H�E� C?�KA � D���C?� JXFDZ �.0�v&- � ()')6 ��� / 0�-&B N�� 1 N 1 � �� 4H0�45B � 7<v&6 � � � � Z��Q��H%A � � H � H%A � � ��H/GKAD C � LN��H�%C0H%A � G � >@��H%A � D F � D��:9%JKC�S Z�()-��� @,D !� g ��! ?� � D � ����! D � r !�� � � g�@ "����;� @ !���� � � ���� ���� � r ������ G �� Z y 450�*?6?6?/&()-�Z=�BW0.S`+ / � FG$�0�B5()C&F *?0�-&/&C&*87<6?/�+!7Y7<v&6 j� � 6?B56,+.45*Pv � + } 0�4<+!7<0�4 � N ���� 45()*Pv�Z ��� B5*Pv&')( < 0�- N/ 3u(O7<�?6?45'2+.-&/ N �.C&-&6 � Z M N � 1 NOM � �Y/&(O7<6?/ } � � 45C&-&0 \ 6 � 0�-q+.-&/ y 687<6?4�; 6?-&45()*?( N $&$ �:� 2�� N ��� N � 4H0�45B � 7<v&6 � � � �ON � +.')*?0�F � � # �ON � 0�')6?4 3 � � � Z � D IC6B��AJKC�I�JKA��LDX*�C� D C�I H%A��LJKI H%A � GEH �G�D IC6B��A�H%A � D F1�RZ y 456?-U7<()*?6�Z ;%+.')' N`� 1 M�M � M � � 6,+.4 3 � � � Z � D IC6B��AJKC � C 7OH%F � � H%A � D F H%FB CED?7%C0H%IRI � F�7uZ � * � 4<+X3QZ ; ()')' N j � 6?/&(O7<()0�- N� 1 P ��� P � ; ()=�vH+.F � � � � Z � G�G���C0H/G � H%FBV�KA�H�T � � � A � DU� F ��I�JKC � GEH � H � 7/D C � A���IA�WZ j5j 6?/ �ON / j # � �%6�6U� � 1 � vU757<$`] � �*3 3 3 � +.Bf7<450�-H+.CD7<()*,+ � C&BvU757<$`] � �*3 6 } � 7<()B5*,+.')( � (O7 �!-&0!Z 456?/&()456?*87fZ 7<()B5*,+.')( �.+.Bf7<450�-H+.CD7<()*,+ � �.+.Bf7<450�-H+.CD7<(O*,+ @�68757<0�45( � vU7<F � 6

� � +.vH+.- � �ON 3 0�0�-&6?- � � � �ON / 7<0�-&6 ; � Z � CED6LD�� J? � A�H%FBOH%CZ � � D H%A � F�7 _D � FLA�C � A���I�JKA � GZR# 3 � / j � � : � � 6�3uB5')68757<6?4 N B5$�6?*?(2+.'�()B5B5C&6 N � *87 ��� 1 M 1 � ���

� � +.-&/H+.C � �ON � (OST*Pv&(O7<� � � Z E� � � � >@�LJ � ���J D C � >@�LJXF � ���J D C � J /J �Z< �� � H*�KA � G � A�J Zt7<0�FG6 �Aj5j N� ()4 N`� 1 NOM � � �� � +�� y �ON � C&45Bf7<6?()- / �ON � +��t# � Z � F&H � � � � I H%AJKI H%A � GEH G�D F�HK6�6B� � GEH�� � D F � J GEH � G�D�� D F ��I�JKC � G�DZ � +.-&()*Pv&6?')')( N�� 1 P�N �

� M 2

Page 168: A. Murli - unina.itwpage.unina.it/lapegna/educational/aritm_fl_point.pdfA. Murli Yº©µ ºx³n sÀ ¼> p· p·8¸ d ¾ ¾ ÿ º©º©µ»º© 6¼p½F½Hµ ¾ ¿À3µ Á ¢ Æq«p¢6Äp«

A. M

urli

� 2� / 7<6?4 } 6?-U7<� y � Z � � D H%A � F�7�6LD � FLA�G�D IC6B��A�H%A � D FoZ y 456?-U7<()*?6�Z ;%+.')' � �

1 M%P " �, � �3+.')')()B y � � � � � Z �IC6&CED�9 � F�7 � � D H%A � F�7 _D � FLA CED?7%C0H%IRI � F�7uZ �.0�v&-�� ()')6 � � / 0�-&B N�� 1�1 6 � ���� � ()' < ()-&B50�- � � ; � Z �_D���FB � F�7�� C�CED C?� � F

�� 7/JXT�C0H � G CED G�J�� J�WZ y 456?-U7<()*?6�Z ;%+.')' N`� 1 N 2 �