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Python mathematicsMoon Yong Joon1

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/() /() /() /() /() /() /() /() /() /() /() /() /() /() /() /() /() /() /() /() /() /( ) /() /()7

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, , , ,

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(, integer) (1, 2, 3, ...) (1, 2, 3, ...) 0 . , , , . 14

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(, rational number) ( 0 ) . (0 ) (, , , ) '' 16

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(, irrational number) . , .18

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[, real number] , . . , (x2 ) . .20

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(, complex number) .a+bi a, b i i2=-1 . a , b . . 0 . 22

(, complex number) 23

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() , . , 1 .

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: () 0 . 7 -7 7 0 7 -7. n ., , , . 26

: (,reciprocal) (-, : multiplicative inverse) 1, . x 1/x x -1 . 1 27

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0 , a , a (-1)

absolute value a , a . |a| .

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(, ratio) . (, ratio) . . 100 .31

Pythonsympy, , ,

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Symbolic mathematics33

sympy SymPy symbolic mathematics python . SymPy a full-featured computer algebra system (CAS) 34

sympy SymPy symbolic mathematics 35

Sympy pi (evalf)

sympy Sympy 36

Symbol name symbol

Symbol 37

sympy SymPy symbolic mathematics 38

Sympy pi (evalf)

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: sympy(, integer) (1, 2, 3, ...) (1, 2, 3, ...) 0 . 40

: sympy(, rational number) (p) (q) 41

1 : sympy [, real number] 42

2 : sympySympify sympy 43

: sympySympify sympy 44

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/pi sympy oo , pi 46

Sympy symbol 47

symbol Sympy x = Symbol(x) a, b, c = symbols(a, b, c) 48

symbol : symbols 49

symbol : symbols cls Function 50

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expand : expand 52

expand : expand 53

force=True

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Solve : solve 55

Pythonnumberdata type

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Numeric Type57

Numberic Typesintfloat long(2.x)complex 1 context id

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, operatorOperationResultNotesx+ysum ofxandyx-ydifference ofxandyx*yproduct ofxandyx/yquotient ofxandyx//y(floored) quotient ofxandyx%yremainder ofx/y-xxnegated+xxunchangedabs(x)absolute value or magnitude ofxint(x)xconverted to integerlong(x)xconverted to long integerfloat(x)xconverted to floating pointcomplex(re,im)a complex number with real partre, imaginary partim.imdefaults to zero.c.conjugate()conjugate of the complex numbercdivmod(x,y)the pair(x//y,x%y)pow(x,y)xto the poweryx**yxto the powery

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Numeric Type(int)60

- int int operator

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int int

real : int bit_length() : bit denominator : numerator :

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long python3 NotesPython 2Python 3x=1000000000000Lx=1000000000000x=0xFFFFFFFFFFFFLx=0xFFFFFFFFFFFFlong(x)int(x)type(x)islongtype(x)isintisinstance(x,long)isinstance(x,int)

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Numeric Type(float)64

- float float operator

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float float real : float hex() : 16 fromhex() : hex() float is_integer() : 0 true as_integer_ratio() : denominator : , numerator :

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Numeric Type(complex)67

- complex float operator

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complex complex real : float imag: conjugate () :

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fractions fractions . Float float

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fractions fractions . Float float

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, . an , a , n .

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x () x , sqrt(x) " x"( x) .

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x x1/2

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x +/-

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x

x x x 82

x,y

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a, b math.pow ** 86

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a b math.sqrt root2 **/pow 88

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(, numeral system) , .91

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(, binary) . 0 1 .93

(octal number system , ) 8 . 0 7 . 3 94

(, ) 10 . 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 .95

(, hexadecimal) 16 . 0 9 A F , . 4 , 2 2 .96

Python

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bin, oct, hex , , 16 99

Bit 100

: and, or, xorOperatorDescription& Binary AND | Binary OR ^ Binary XOR

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: ~, OperatorDescription~ Binary Ones Complement1 > Binary Right Shift .

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PEMDAS 105

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,S* ,S a,b a*b=b*a , (, commutative law)

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, , ., , 107

:

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( , associated law) , .

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, . . . 110

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S S * , S a, b, c a * (b + c) = (a * b) + (a * c) , (b + c) * a = (b * a) + (c * a) S * .

. , , , , . , . , . * 113

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/, , . .12 ={1,2,3,6,12}16 ={1,2,4,8,16}18 ={1,2,3,6,9,18} 12 16 1,2,4 412, 16, 18 1,2 2 1

() 8 ={1,2,4,8}, 9 ={1,3,9} 8 9 1 8 9

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1, (relatively prime) . 1 .

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(, substitution) (, change of variables) 119

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(Rationalization) , 121

python operator Moon Yong Joon122

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special methodpython type class special method

SpecialMethod

special method 124

OperatorDescription0( )Parentheses (grouping)1f(args...)Function call2x[index:index]Slicing3x[index]Subscription4x.attributeAttribute reference5**Exponentiation()6~xBitwise not7+x, -xPositive, negative8*, /, %Multiplication, division, remainder9+, -Addition, subtraction10Bitwise shifts11&Bitwise AND12^Bitwise XOR13|Bitwise OR14in, not in, is, is not, =,, !=, ==Comparisons, membership, identity15notxBoolean NOT16andBoolean AND17orBoolean OR18lambdaLambda expression

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OperationFunctionMethodNegation (Arithmetic)-aneg(a)x.__neg__()Positive+apos(a)x.__pos__()

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: OperationFunctionMethodadditionx+yadd(a,b)x.__add__(y)subtractionx-ysub(a,b)x.__sub__(y)multiplicationx*ymul(a,b)x.__mul__(y)divisionx/ydiv(a,b)x.__div__(y)divisionx/ytruediv(a,b)x.__truediv__(y)floor divisionx // yfloordiv(a,b)x.__floordiv__(y)modulo (remainder)x%ymod(a,b)x.__mod__(y)floor division&modulodivmod(x,y)N/Ax.__divmod__(y)raise to powerx**ypow(a,b)x.__pow__(y)

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: operator Operator import

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: int methodInt class special method

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: python

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: OperationFunctionMethodadditionx+yadd(a,b)y.__radd__(x)subtractionx-ysub(a,b)y.__rsub__(x)multiplicationx*ymul(a,b)y.__rmul__(x)divisionx/ydiv(a,b)y.__rdiv__(x)divisionx/ytruediv(a,b)y.__rtruediv__(x)floor divisionx // yfloordiv(a,b)y.__rfloordiv__(x)modulo (remainder)x%ymod(a,b)y.__rmod__(x)floor division&modulodivmod(x,y)N/Ay.__rdivmod__(x)raise to powerx**ypow(a,b)y.__rpow__(x)

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Operationleft bit-shiftxyy bit : (x // (2** y) )bitwiseandx&yx y (1 0) 0 bitwisexorx^yx y bitwiseorx|yx y 1 1 0 0 Bitwise Inversion~a a

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python ,operator int OperationFunctionMethodleft bit-shiftxyrshift(a,b)x.__rshift__(y)bitwiseandx&yand_(a,b)x.__and__(y)bitwisexorx^yxor(a,b)x.__xor__(y)bitwiseorx|yor_(a,b)x.__or__(y)Bitwise Inversion~ainvert(a)x.__invert__()

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augmented operator140

augmented operatorpython

X += YX = X+ Y

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OperationFunctionMethodadditionx+=yiadd(a,b)x.__iadd__(y)subtractionx-=yisub(a,b)x.__isub__(y)multiplicationx*=yimul(a,b)x.__imul__(y)divisionx/=yidiv(a,b)x.__idiv__(y)divisionx/=yitruediv(a,b)x.__itruediv__(y)floor divisionx //= yifloordiv(a,b)x.__ifloordiv__(y)modulo (remainder)x%=yimod(a,b)x.__imod__(y)raise to powerx**=yipow(a,b)x.__ipow__(y)

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OperationFunctionMethodleft bit-shiftx=yirshift(a,b)x.__irshift__(y)bitwiseandx&=yiand_(a,b)x.__iand__(y)bitwisexorx^=yixor(a,b)x.__ixor__(y)bitwiseorx|=yior_(a,b)x.__ior__(y)

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boolean OperationSyntaxFunctionMethodand Logical ANDa and b NA NAor Logical ORaor b NA NANegation (Logical)notanot_(a) NA

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(, equation) / . , . 149

X 2,3

. , a0xna1xn-1an-1xan x an, a0xna1xn-1an-1xan0 x an .150

, , , 151

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(Linear equation) 1 .153

a0x1+a1x0

a0 = a a1 = b n=1

x .154

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(, quadratic equation), 2 156

a0x2+a1x1+a2x0

a0 = a a1 = b a2 = c n=2

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(, inequality) . (, inequality sign) .159

a, b, c (,transitive relation) 160

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: +/- subs, evalf, replace 163

: *, / subs, evalf, replace 164

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2 expand solve 166

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python OperationFunctionMethodOrderingabgt(a,b)x.__gt__(y)

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(-, multiplication) . .172

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(, factorization)

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(,binomial theorem) x+y (x+y)n , xkyn-k

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: (a+b)3 (a+b)3=(a+b)(a+b)(a+b) =aaa+aab+aba+abb+baa+bab+bba+bbb =a3+3a2b+3ab2+b3 =a3+3C1a2b+3C2ab2+b3

3C1 = 3P1/1! = 3!/2!/1! = 3 3C2 = 3P2/2! = 3!/1!/2! = 3 181

: (a+b)3

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cancel : cancel 185

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factor factor expand solve 187

factor : modulus modulus 188

factor : (Gau, Gaussian integer) .(a+bi)(a-bi)=a2+b2 189

= (0+i)(0 i)= -i*I= -(i)**2= 1X**2+1

factor : extension 190

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(, : morphism) . map , (function) (morphism) .194

?(, function) f:XY X Y .

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(, codomain) () .196

/ (, domain) . (, range) "" 197

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?(injective) (one-to-one) (injection) (one-to-one function)

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? (, ) (surjective) X Y (onto) . (surjection) .

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?f:XY f (bijective) . (bijection) (one-to-one correspondence) . one-to-one & onto .

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() , . . , .202

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f:XY f -1: Y X.

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=

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function composition (, function composition) . (composite function) .

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Python mappingdata type

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mapping Type211

Mapping -dictionary Key/Value containerName 1Name 2container::Dictionary Type

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Mapping - dict dict

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Mapping dict type

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Hash 215

key hash mapping hash

tuple mutable hash 216

dict 217

Map - dict . key/value

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dict 219

Mapping - Accessing Elements Key/Value Key index

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Mapping - Updating Elements key key

key : key : 221

Mapping - Delete Elements , , dict instance

Dict

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dict 223

Mapping dict Dictionary , , dictionary instance Function Descriptioncmp(dict1, dict2)Compares elements of both dict.len(dict)Gives the total length of the dictionary. This would be equal to the number of items in the dictionary.str(dict)Produces a printable string representation of a dictionarytype(dict)Returns the type of the passed variable. If passed variable is dictionary, then it would return a dictionary type.dict(mapping)Converts a map into list.

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Mapping -dict class dict

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dict 226

Mapping -dictionary Method Descriptiondict.clear()Removes all elements of dictionarydictdict.copy()Returns a shallow copy of dictionarydictdict.fromkeys()Create a new dictionary with keys from seq and valuessettovalue.dict.get(key, default=None)Forkeykey, returns value or default if key not in dictionarydict.has_key(key)Returnstrueif key in dictionarydict,falseotherwisedict.items()Returns a list ofdict's (key, value) tuple pairsdict.keys()Returns list of dictionary dict's keysdict.setdefault(key, default=None)Similar to get(), but will set dict[key]=default ifkeyis not already in dictdict.update(dict2)Adds dictionarydict2's key-values pairs todictdict.values()Returns list of dictionarydict's values dict.iteritems()Iterable items

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dict.get() dict KeyError get()

Key default

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dict.setdefault() dict default

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dict.get/setdefault() dict [] get

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dict.update() dict dict

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iteritems/iterkeys/itervalues dict iterable

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keys,items NotesPython 2Python 3a_dictionary.keys()list(a_dictionary.keys())a_dictionary.items()list(a_dictionary.items())a_dictionary.iterkeys()iter(a_dictionary.keys())[iforiina_dictionary.iterkeys()][iforiina_dictionary.keys()]min(a_dictionary.keys())no change

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keys, values, items list dict

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dict.key/values/items() python list dict_keys, values,items list()

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dict.key/values/items: for python 3. for

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dict.key: set python 3. set

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viewitems/keys/values 239

dict.viewitems() dict view 3

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viewitems/viewkeys() dict viewitems/viewkeys set (3 keys()/items() )

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dict comprehension 242

Dict Comprehension

A = { for (k,v) in sequence if }

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Dict Comprehension : dict comprehension

A = { for i in sequence if }244

dict Comprehension : dict for

A = { for i in sequence for j in sequence if }

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plotting247

1 1 plot 248

2 2 plot 249

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1,2 plot 251

plot line_color 252

x 253

1 : x 1 plot (x,-5,5) x 254

Plot 1 2 x 255

Plot 1 2 x x 256

y 257

1 : y 1 plot ylim=(y,-5,5) y 258

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:legend legend260

1 (title, xlabel, ylabel) 261

show 262

sympy plot . show=False 263

Show 264

show sympy plot . show=False 265

Multi plot 266

plot 1 plot plot 267

plot : 2 plot 268

plot plot 269

Multi plot 270

append plot append plot 271

extend plot extend plot 272

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Numpy classMoon Yong Joon274

ndarray matrix 275

ndarray matrix matrix MATLAB ndarraymatrix 2 * numpy.multiply() numpy.dot()

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vector : ndarray Array vector

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ndarray matrix Matrix dot/* , ndarry */multiply

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// number, vector list(row or column), matrix array( rows, columns) vector Matrix

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vertor ndarray 1xN, Nx1, N 1 ,

scalarvector, , , , ()1N N , ,

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// //

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(magnitude) (direction)

tail head

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||v|| = sqrt(v0^2 + v1^2 + v2^2... + vn^2)

b= (6,8) |b| = ( 62+ 82) = ( 36+64) = 100 = 10

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Vector (Magnitude) x y

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(unit vector) 1

1 (hat)

sclae 289

0 ~ 1

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: +The vector (8,13) and the vector (26,7) add up to the vector (34,20)Example: add the vectorsa= (8,13) andb= (26,7)c=a+bc= (8,13) + (26,7) = (8+26,13+7) = (34,20)

ababc292

Vector : + edfed

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: -

Example: subtractk= (4,5) fromv= (12,2)a=v+ ka= (12,2) + (4,5) = (12,2) + (4,5) = (124,25) = (8,3)294

Vector : - edg-e-e

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:

m = [7,3]

A = 3m= [21,9]296

Vector : d3d

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vs Inner product, dot product, scalar productOuter product, vector product, cross product.(Dot)X(cross) n 3 a1 b1 + a2 b2 + . + an bn (a2 b3 a3 b2, a3 b1 a1 b3, a1 b2 a2 b1)|a||b| cos |a||b| sin n scalar vector

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(Inner Product) cos

a b= |a| |b| cos()Where:|a| : vectora |b| : vectorb : aandb

a b= ax bx+ ay by

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: 2 a b= |a| |b| cos()a b= 10 13 cos(59.5)a b= 10 13 0.5075...a b= 65.98... = 66 (rounded)

a b= ax bx+ ay bya b= -6 5 + 8 12a b= -30 + 96a b= 66

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3 1Dot

a b= ax bx+ ay by+ az bza b= 9 4 + 2 8 + 7 10a b= 36 + 16 + 70a b= 122304

3 2

a |a| = (42+ 82+ 102) = (16 + 64 + 100) = 180b |b| = (92+ 22+ 72) = (81 + 4 + 49) = 134 a b= 9*4+ 2*8+ 7*10 = 36+16+70 = 122

a b= |a| |b| cos()

122 = 180 134 cos()cos() = 122 / (180 134)cos() = 0.7855... = cos-1(0.7855...) = 38.2...305

(dot) (dot)

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vdot: vector (2) (dot)

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Vector product() 308

a b a b . c a b

Vector productCross product

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: 2 cross v = [a1,a2] u = [b1,b2] a1 a2 b1 b2a1*b2 a2*b1Example: The cross product ofa= (2,3) andb= (5,6)c= a1b2 a2b1= 26 35 = 3

Answer:a b=-3

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: 3 cross v = [a1,a2,a3] u = [b1,b2,b3] a2 a3 a1 a2

b2 b3 b1 b2

x : a2*b3 a3*b2y : a3*b1 a1*b2z : a1*b2 a2*b1

Example: The cross product ofa= (2,3,4) andb= (5,6,7)cx= aybz azby= 37 46 = 3cy= azbx axbz= 45 27 = 6cz= axby aybx= 26 35 = 3Answer:a b=(3,6,3)311

(cross) 2 3

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Inner/outer 313

inner A = [[a1,b1] B = [[a2,b2]]numpy.inner(A,B) array([[a1*a2 + b1*b2]])[[1*4+0*1]]104110414

=.314

Inner

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dot/inner: (2) 2

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outer A = [[a1,b1]] B = [[a2,b2]]numpy.outer(A,B) array([[a1*a2 , a1*b2][ b1*a2, b1*b2]]) [[1*4,1*1] [0*4+0*1]]104110414100

=

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outer Dot

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Vector : + edfed

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Vector : - edg-e-e

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Vector : d3d

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Vector (Magnitude) x y

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vector 326

Vector : (dot) (dot)

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vector 328

Vector : (cross) (cross) 2 array 3 matrix

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,

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Diagonal matrix333

A(aij)(i, j1, 2, 3,, n) aij aij=0(ij) A () aij(ij) aij(ij) 0

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Identity matrix335

import numpy as np

a = np.array([[1,0],[0,1]])b = np.array([[4,1],[3,2]])print(np.dot(b,a))print(np.dot(a,b))[[4 1] [3 2]][[4 1] [3 2]]336

Triangular matrix337

(Upper triangular matrix) (lower triangular matrix) .

Upper triangular matrixlower triangular matrix338

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(+/-/*)

+-*/=+-*/+-*/

+-*/+-*/

+-*/+-*/1 2 34 5 61 2 34 5 6+=1+1 2+2 3+34+4 5+5 6+6=2 4 68 10 12340

+/-/*

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(transpose)342

: . T A .

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python T numpy transpose

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dot 345

dot vs inner (2)Dot inner dotinner N*M M*N , N*M N*M N*M . M*N N*NN*M N*M N*N

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dot 1*p, p*1 A B A B

1*pP111347

dot A B A B

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dot : 2A = [[a1,b1],[c1,d1]] B = [[a2,b2],[c2,d2]] numpy.dot(A,B) array([[a1*a2 + b1*c2, a1*b2 + b1*d2], [c1*a2 + d1*c2, c1*b2 + d1*d2]) [[1*4+ 0*2, 1*1+0*2],[0*4+1*2, 0*1+1*2]]10014122100141224122

=.349

dot Numpy.dot

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(det) , - , -

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(det) : 2

3 12 2

det= 3*2 1*2= 4

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(det) : 3

3 1 3 3 12 2 3 2 21 1 1 1 1= 3*2*1 3*2*1 + 1*3*1 3*3*1 + 3*2*1 1*2*1= 6 6 + 3 -9 + 6 -2= 15 17= -2

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(det) : 3n

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minor determinant356

2i ,j : Mij 2 11 2M112M12M21-1 -121 1

+

+M22 22 357

3i ,j : Mij

3 1 3 2 2 3 1 1 1 M112*1 -3*1-1M12M132*1 -3*1-12*1 -2*1 02 3 1 1 2 3 1 1 2 2 1 1

= 3M11+(-1)* 1M12 + 3M13 = -3+1+0 = -2

+

+358

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(cofactor)

3 1 3 2 2 3 1 1 1 m112 31 12-3-1+-1m122 31 12-3-1-1m132 21 12-20+0m211 31 11-3-2-2m223 31 13-30+0m233 11 13-12--2m311 32 33-6-3+-3m323 32 39-63--3m333 12 26-24+4

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(adj)

-1 2 -3 1 0 -3 0 -2 4-1 1 0 2 0 -2-3 -3 4

T 362

(inv) 2

[[ 0.66666667 -0.33333333] [-0.33333333 0.66666667]]

1/3 * 2 -1-1 22 11 2

-1A1=1/det(A) * CT363

(inv) 3

[[ 0.5 -1. 1.5] [-0.5 0. 1.5] [ 0. 1. -2. ]]

- 0.5 * -1 2 -3 1 0 -3 0 -2 43 1 3 2 2 3 1 1 1

-1

A1=1/det(A) * CT364

(inv)

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Dot 366

Dot A B A B

n*m n*m n*n 367

dot n*m 2

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dot : 2 a(2,2) b(2,2) n*m, m*n = n*n

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cross product370

cross A = [[a1,b1],[c1,d1]] B = [[a2,b2],[c2,d2]] numpy.cross(A,B) = A.T * Barray([[a1*b2 - c1*a2 , b1*d2 d1*c2])[[1*1- 0*4,0*2-1*2]]100142121-2=371

Cross n*m 2

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Inner 373

inner A = [[a1,b1],[c1,d1]] B = [[a2,b2],[c2,d2]] numpy.inner(A,B) array([[a1*a2 + b1*b2, a1*c2 + b1*d2], [c1*a2 + d1*b2, c1*c2 + d1*d2])[[1*4+0*1,1*2+0*2],[0*4+1*1, 0*2+1*2]]10014122100141224212

=.374

Inner : 2 a(2,2) b(2,2) out.shape = a.shape[:-1] + b.shape[:-1]

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Inner : 3 a(2,3,2) b(2,2) out.shape = a.shape[:-1] + b.shape[:-1]

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outer product377

outer 1 1 104110414100

=

222*2378

outer: 1 Out out[i,j]=a[i]*b[j]

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outer: 2 5*5

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outer: 3

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tensordot382

tensordotTensordot axes 0 tensor product

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tensordotTensordot axes 0 tensor product =10014122

10014122412241224122=42120000000042122,22,22,2,2,2384

tensordot: 1 2 2 4

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tensordot: 2 axes = 0 tensor product, axes = 1 tensor dot product, axes = 2 tenser double contraction

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Trace : 3 3(2,2,2) 1 0213465701

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trace

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Python matrix class Moon Yong Joon390

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n [2] "nn" (,

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Numpy matrix

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: dot()N*M M* N dot M*M ( xij )( yij )=(kxikykj)

a1a2a3a4

b1b2b3b4

a1*b1+a1*b3a2*b2+a2*b4a3*b1+a3*b3a4*b2+a4*b4

=.395

: dot dot

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: cross cross

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: +/-N*M N*M +/- N*M a1a2a3a4

b1b2b3b4

a1 +/- b1a2 +/- b2a3+/- b3a4 +/- b4

=+/-398

: +/- +/-

399

: (k) N*M a1a2a3a4

k* a1k * a2k* a3k* a4

=k400

: k

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: (transpose)N*M M*N a1a2a3a4

a1 a3a2 a4

=T402

: (transpose)N*M M*N T, transpose

403

matmulMatrix dot

404

Matmul: N*m, M*n 1

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matrix_power matrix_power dot

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matrix_power: dot

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pythonNumpy linalg Moon Yong Joon408

Matrix and vector products409

dot(a,b[,out])n n*m m*l production( n*l) vdot(a,b)Vector produtioninner(a,b)N Inner product ( ).outer(a,b[,out])2 .matmul(a,b[,out]) Matrix product (dot )tensordot(a,b[,axes])Compute tensor dot product along specified axes for arrays >= 1-D.linalg.matrix_power(M,n)Raise a square matrix to the (integer) powern.cross(a, b, axisa=-1, axisb=-1, axisc=-1, axis=None) einsum(subscripts,*operands[,out,dtype,...])Evaluates the Einstein summation convention on the operands.kron(a,b)Kronecker product of two arrays.

410

Decompositions411

linalg.cholesky(a)Cholesky decomposition.linalg.qr(a[,mode])Compute the qr factorization of a matrix.linalg.svd(a[,full_matrices,compute_uv])Singular Value Decomposition.

412

Matrix eigenvalues413

linalg.eig(a)Compute the eigenvalues and right eigenvectors of a square array.linalg.eigh(a[,UPLO])Return the eigenvalues and eigenvectors of a Hermitian or symmetric matrix.linalg.eigvals(a)Compute the eigenvalues of a general matrix.linalg.eigvalsh(a[,UPLO])Compute the eigenvalues of a Hermitian or real symmetric matrix.linalg.eig(a)Compute the eigenvalues and right eigenvectors of a square array.

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Norms and other numbers415

linalg.norm(x[,ord,axis,keepdims])Matrix or vector norm.linalg.cond(x[,p])Compute the condition number of a matrix.linalg.det(a)Compute the determinant of an array.linalg.matrix_rank(M[,tol])Return matrix rank of array using SVD method Rank of the array is the number of SVD singular values of the array that are greater thantol.linalg.slogdet(a)Compute the sign and (natural) logarithm of the determinant of an array.trace(a[,offset,axis1,axis2,dtype,out])Return the sum along diagonals of the array.

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Solving equations and inverting matrices417

linalg.solve(a,b)Solve a linear matrix equation, or system of linear scalar equations.linalg.tensorsolve(a,b[,axes])Solve the tensor equationax=bfor x.linalg.lstsq(a,b[,rcond])Return the least-squares solution to a linear matrix equation.linalg.inv(a)Compute the (multiplicative) inverse of a matrix.linalg.pinv(a[,rcond])Compute the (Moore-Penrose) pseudo-inverse of a matrix.linalg.tensorinv(a[,ind])Compute the inverse of an N-dimensional array.

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4.

Moon Yong Joon419

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() "a b " , n- . , "a, b, c " .422

a b R A, B A B (binary relation) AxB aA bB (a, b)R , a Rb (a, b)R

(domain) R : dom(R)(range) : ran(R)423

424

: (arrow diagram) A, B A a B b 425

: (coordinate diagram) A, B A a x , B b y 426

: (relation matrix) A, B . 0 1 (boolean matrix) 427

: (directed graph) A A (vertex) , (a, b) a b (edge)

428

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: 1 1) : R A a A (a,a) R R . ) A = {a, b, c} R (a,a), (b,b), (c,c) .2) : R X . x X (x,x) ! R R .

430

: 2 3) : R A a, b A (a,b) R (b,a) R R . ) A = {a, b, c} R (a,b) (b,a) .431

: 34) : R A a, b A (a,b) R, (b,a) R a b ) A = {a, b, c} R1 = { (a,b), (b,a) } . R2 = { (a,b), (c,a) } .

5) : R A a, b, c A (a,b) R (b,c) R (a,c) R R . ) A = {a, b, c} R (a,b), (b,c) (a,c) R .

432

: 46) : , , . , X ~ , a, b, c : a ~ a: a ~ b => b ~ a: a ~ b, b ~ c => a ~ c .433

Python relation/sequence(tuple) data type434

Sequence : Tuple Type435

Sequence - Tuple tuple immutable Slicing String

Python ExpressionResultsDescriptionT =(1,)(1,) (,) T = (1,2,3,4)(1, 2, 3, 4) len((1, 2, 3))3Length (1, 2, 3) + (4, 5, 6)(1, 2, 3, 4, 5, 6) Concatenation('Hi!) * 4'Hi!Hi!Hi!Hi!' string 3 in (1, 2, 3)True Membershipfor x in (1, 2, 3): print x,1 2 3 - Iteration

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Sequence- Tuple tuple Function Descriptioncmp(tuple1, tuple2)Compares elements of both tuples.len(tuple)Gives the total length of the tuple.max(tuple)Returns item from the tuple with max value.min(tuple)Returns item from the tuple with min value.tuple(seq)Converts a list into a tuple.str(tuple)Produces a printable string representation of a tupletype(tuple)Returns the type of the passed variable. If passed variable is tuple, then it would return a tuple type.

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Tuple Tuple (count), (index)

438

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Tuple : Tuple tuple copy

list 440

Tuple 441

Tuple mutable tuple immutable mutable . ,

442

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Moon Yong Joon443

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(Graph) . G (vertex or node) V (edge or line) E .

V : (vertex) E : (edge) 446

(Graph) : V, E 447

(adjacent) (adjacent) a, b (a,b) , a b (adjacent) 448

449

(Degree) v deg(v) loop 2 450

G n , e 451

452

(Path) : (length):

(closed) : (cycle): 453

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() (undirected graph)455

, , 456

() G . G .457

Moon Yong Joon458

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() digraph/ directed graph () . 460

(arc) V V A (V,A) . V (vertex) A (arc) . (a,b) a (initial vertex), b (terminal vertex) .461

, , 462

() H . H .463

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(complete) 465

3 3, 4 6.466

(subgraph) .467

G1, G2, G3 G .

468

G1, G2, G3 G .G1 26 . , G2, G3 .G3 G2, .

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(adjacency matrix) G = (V,E), |V| = n(1) .adj_mat[i][j] =1 if (vi, vj) (adjacent)0 470

: 471

472

:

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Moon Yong Joon473

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, , ,

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477

180 A

A sin() b/c cos() a/c tan() b/a478

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: .

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: *sin() , *cos()

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A rcosB rsin

r 1 483

3, 4 5

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()

Moon Yong Joon485

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xy r P OP

1

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sqrt(3**2 + 4**2)=5

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x * , Y *

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Radian & degree

Moon Yong Joon491

degrees/radians 492

degrees radians degrees radians

Degrees Radians

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degree radian 494

degrees radians 2 360, 1radian 57.3

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degrees radians: 90, pi/2

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radians degrees : numpy np.deg2rad, np.rad2deg radians degree

180

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radians -> degrees : numpy np.degrees radians degree

np.degrees(radian, )

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degrees-> radians : numpy np.radians degree radian np.radians(radian, )

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Moon Yong Joon500

cosine501

cosine .

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cosine .

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cosine : numpy .

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cosine .

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sine506

sine sine radians

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sine .

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sine : numpy .

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sine .

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tangent511

tan tangent radians

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tangent tan .

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tangent : numpy tan .

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tangent tan .

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Moon Yong Joon516

517

n*/2 + 90n + 1. n*/2 + 90n + , n 0< < /2 , 0 < cos, cos-> sin, tan-> cot 3. (+,-) 1234 sin++-- cos+--+ tan+-+-

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519

: n*/2 + 90n + n n=0, , 4n=2, , 2 sin(-x) = -sin(x) sin( -x) = sin(x) cos(-x) = cos(x) cos(-x) = -cos(x) tan(-x) = - tan(x) tan(-x) = -tan(x)

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: n*/2 + 90n + n

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: n*/2 + 90n + n n=1, , 1n=3, , 3 sin(/2 -x) = cos(x) sin(3/2 -x) = -cos(x) cos(/2-x) = sin(x) cos(3/2-x) = -sin(x) tan(/2-x) = cot(x) tan(3/2-x) = cot (x)

1522

: n*/2 + 90n + n

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/2 :524

: /2 A 120 /2(90)+30 sin cos, cos sin

rabA=120B=30

525

: 90 90 , B sin A cos

AB

cos(A) = sin(B), sin(B) = cos(A) , tan(B) = cot(A) A + B = 90 B 526

/2

527

: 180 180 90 +

rabA=120B=30 sin(A) = cos(B) = b/r cos(A) = - sin(B) = - a/r tan(A) = - cot(B) = -b/a528

/2 +

529

530

Sin sin pi 2pi

531

cos cos pi 2pi

532

: - 180 B

rabB=30 sin(-B) = sin(B) = b/r cos(-B) = - cos(B) = - a/r tan(-B) = - tan(B) = -b/a

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: + A 120 (180)+30 sin sin, cos cos

rabA=210B=30

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2 535

: 2 - A 360 2(360)-30 sin -sin, cos cos

rabA=360B=30

536

: 2 + 2 .

rabA=360B=30537

Moon Yong Joon538

Sin 539

Sin sin sin

rabtssin(+ ) = sin() cos() + cos() sin() = a/r * b/s + b/r * t/s

sin() cos() = a/r * b/s cos() sin() = b/r * t/s

sin() = a/r , sin() = t/s540

Sin sin

541

Sin 542

Sin sin

Cos 90 0 543

cos /544

cos cos

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cos cos

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Moon Yong Joon547

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(, : reciprocal) (-, : multiplicative inverse) 1, . x 1/x x -1 . 1

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Python

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* = 1 * = 1

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Moon Yong Joon553

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555

y= arcsinxy= sin-1xx= siny1 +1/2 y /2y= arccosxy= cos-1xx= cosy1 +10 y y= arctanxy= tan-1xx= tany/2