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8/18/2019 DDCA_Ch1 - Class1 (1)
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C 1
, 2
C 1
D H . H
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C 1
• •
•
• •
•
•
• •
C 1 ::
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C 1
•
– C , I, , .
• $21 1985 $300 2011
B
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C 1
H
f o c u s
o f t
h i s
c o u r s e
programs
device drivers
instructionsregisters
datapathscontrollers
addersmemories
AND gatesNOT gates
amplifiers
filters
transistorsdiodes
electrons
A
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C 1
•
– – F
–
• D
D A
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C 1
• – 1 0
– 1, E, HIGH, H – 0, FAE, ,
• 1 0 , , , .
• D 1 0
•
D D: B
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C 1
537410
=
1 0 ' s c ol umn
1 0 0 ' s
c ol umn
1 0 0 0
' s c ol umn
1 ' s c o
l umn
11012 =
2 ' s c ol u
mn
4 ' s c ol u
mn
8 ' s c ol u
mn
1 ' s c ol u
mn
• Decimal numbers
• Binary numbers
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C 1
537410
= 5 × 103 + 3 × 102 + 7 × 101 + 4 × 100five
thousands
1 0 ' s c ol umn
1 0 0 ' s
c ol umn
1 0 0 0 ' s c ol umn
threehundreds
seventens
fourones
1 ' s c o
l umn
11012 = 1 × 23 + 1 × 22 + 0 × 21 + 1 × 20 = 13
10oneeight
2 ' s c ol u
mn
4 ' s c ol u
mn
8 ' s c ol u
mn
onefour
notwo
oneone
1 ' s c ol u
mn
• Decimal numbers
• Binary numbers
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C 1
• 20 =
• 21 =
• 22 =
• 23 =
• 24 =
• 25 =
• 26
=• 27 =
• 28 =
• 29 =
• 210 =
• 211 =
• 212 =
• 213 =
• 214
=• 215 =
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C 1
• 20 = 1
• 21 = 2
• 22 = 4
• 23 = 8
• 24 = 16
• 25 = 32
• 26
= 64• 27 = 128
• Handy to memorize up to 210
• 28 = 256
• 29 = 512
• 210 = 1024
• 211 = 2048
• 212 = 4096
• 213 = 8192
• 214
= 16384• 215 = 32768
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C 1
• 224?
• H 32
?
E
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C 1
• 224?
24 × 220 ≈ 16
• H 32
?22 × 230 ≈ 4
E
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C 1
• Decimal to binary conversion:– Convert 100112 to decimal
• Decimal to binary conversion:– Convert 4710 to binary
C
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C 1
• Decimal to binary conversion:– Convert 100112 to decimal
– 16×1 + 8×0 + 4×0 + 2×1 + 1×1 = 1910
• Decimal to binary conversion:– Convert 4710 to binary
– 32×1 + 16×0 + 8×1 + 4×1 + 2×1 + 1×1 = 1011112
C
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C 1
• – H ?
– ? – E: 3 :
• – H ?
– : – E: 3 :
B
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C 1
• – H ? 10
– ? 0, 10
1 – E: 3 :
• 103 1000 • 0,
• – H ? 2
– : 0, 2 1
– E: 3 :• 23 • 0, 0002 1112
B
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C 1
Hex Digit Decimal Equivalent Binary Equivalent
0 0
1 1
2 2
3 3
4 4
5 5
6 6
7 7
8 8
9 9
A 10
B 11C 12
D 13
E 14
F 15
H
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C 1
Hex Digit Decimal Equivalent Binary Equivalent
0 0 0000
1 1 0001
2 2 0010
3 3 0011
4 4 0100
5 5 0101
6 6 0110
7 7 0111
8 8 1000
9 9 1001
A 10 1010
B 11 1011
C 12 1100
D 13 1101
E 14 1110
F 15 1111
H
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C 1
• B
• B &
• B
10010110nibble
byte
CEBF9AD7least
significantbyte
mostsignificant
byte
10010110least
significantbit
most
significantbit
B, B,
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C 1
37345168+
8902
carries11
10110011+
1110
11 carries
• Decimal
• Binary
A
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C 1
1001
0101+
10110110+
• Add the following
4-bit binary
numbers
• Add the following
4-bit binary
numbers
B A E
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C 1
1001
0101+1110
1
10110110+
10001
111
• Add the following
4-bit binary
numbers
• Add the following
4-bit binary
numbers
Overflow!
B A E
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C 1
1001
0101+1110
1
10110110+
10001
111
• Substract the
following 4-bit
binary numbers
• Substract the
following 4-bit
binary numbers
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C 1
• Sign/Magnitude Numbers
• Two’s Complement Numbers
B
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C 1
• 1 sign bit, N -1 magnitude bits
• Sign bit is the most significant (left-most) bit
– Positive number: sign bit = 0
– Negative number: sign bit = 1
• Example, 4-bit sign/mag representations of ± 6:+6 =
- 6 =
• Range of an N -bit sign/magnitude number:
{ }
1
1 2 2 1 0
2
0
: , , , ,
( 1) 2n
N N
n
a i
i
i
a a a a a
A a−
− −
−
=
= − ∑
L
/
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C 1
• 1 sign bit, N -1 magnitude bits
• Sign bit is the most significant (left-most) bit
– Positive number: sign bit = 0
– Negative number: sign bit = 1
• Example, 4-bit sign/mag representations of ± 6:+6 = 0110
- 6 = 1110
• Range of an N -bit sign/magnitude number:[-(2N-1-1), 2N-1-1]
{ }
1
1 2 2 1 0
2
0
: , , , ,
( 1) 2n
N N
n
a i
i
i
a a a a a
A a−
− −
−
=
= − ∑
L
/
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C 1
• Problems:
– Addition doesn’t work, for example -6 + 6:
1110
+ 0110
10100 (wrong!)
– Two representations of 0 (± 0):
1000
0000
/
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C 1
• Don’t have same problems as sign/magnitude
numbers:– Addition works
– Single representation for 0
C
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C 1
( )
21
1
02 2
n
n i
n i
i A a a
−
−
−
=
= − +∑
• MSB has value of -2 N -1
• Most positive 4-bit number:
• Most negative 4-bit number:
• The most significant bit still indicates the sign
(1 = negative, 0 = positive)• Range of an N -bit two’s comp number:
C
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C 1
( )
21
1
02 2
n
n i
n i
i A a a
−
−
−
=
= − +∑
• Msb has value of -2 N -1
• Most positive 4-bit number: 0111
• Most negative 4-bit number: 1000
• The most significant bit still indicates the sign
(1 = negative, 0 = positive)• Range of an N -bit two’s comp number:
[-(2N-1), 2N-1-1]
C
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C 1
• Flip the sign of a two’s complement number
• Method:
1. Invert the bits
2. Add 1
• Example: Flip the sign of 310 = 00112
C
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C 1
• Flip the sign of a two’s complement number
• Method:
1. Invert the bits
2. Add 1
• Example: Flip the sign of 310 = 001121. 1100
2. + 1
1101 = -310
C
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C 1
• Take the two’s complement of 610 = 01102
• What is the decimal value of 10012?
C E
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C 1
• Take the two’s complement of 610 = 011021. 1001
2. + 1
10102 = -610
• What is the decimal value of the two’s
complement number 10012?
1. 01102. + 1
01112 = 710, so 10012 = -710
C E
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C 1
+01101010
+11100011
• Add 6 + (-6) using two’s complement
numbers
• Add -2 + 3 using two’s complement numbers
C A
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C 1
+01101010
10000
111
+11100011
10001
111
• Add 6 + (-6) using two’s complement
numbers
• Add -2 + 3 using two’s complement numbers
C A
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C 1 Copyright © 2012 Elsevier
• Extend number from N to M bits ( M > N ) :
– Sign-extension
– Zero-extension
I B
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C 1
• Sign bit copied to msb’s
• Number value is same
• Example 1:
– 4-bit representation of 3 = 0011
– 8-bit sign-extended value: 00000011
• Example 2:
– 4-bit representation of -5 = 1011
– 8-bit sign-extended value: 11111011
E
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C 1
• Zeros copied to msb’s
• Value changes for negative numbers
• Example 1:
– 4-bit value = 00112 = 310
– 8-bit zero-extended value: 00000011 = 310
• Example 2:
– 4-bit value = 1011 = -510
– 8-bit zero-extended value: 00001011 = 1110
E
C
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C 1
-8
1000 1001
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1010 1011 1100 1101 1110 1111 0000 0001 0010 0011 0100 0101 0110 0111 Two's Complement
10001001101010111100110111101111
00000001 0010 0011 0100 0101 0110 0111
1000 1001 1010 1011 1100 1101 1110 11110000 0001 0010 0011 0100 0101 0110 0111
Sign/Magnitude
Unsigned
Number System Range
Unsigned [0, 2 N -1]
Sign/Magnitude [-(2
N -1
-1), 2
N -1
-1]Two’s Complement [-2 N -1, 2 N -1-1]
For example, 4-bit representation:
C
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E: 4 ( )
Value Sign-and-Magnitude
1sComp.
2sComp.
+7 0111 0111 0111
+6 0110 0110 0110+5 0101 0101 0101+4 0100 0100 0100+3 0011 0011 0011+2 0010 0010 0010
+1 0001 0001 0001+0 0000 0000 0000
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E: 4 ( )
Value Sign-and-Magnitude
1sComp.
2sComp.
-0 1000 1111 --1 1001 1110 1111-2 1010 1101 1110-3 1011 1100 1101-4 1100 1011 1100-5 1101 1010 1011
-6 1110 1001 1010-7 1111 1000 1001-8 - - 1000
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E:1. For 2’s complement binary numbers, the range of
values for 5-bit numbers is
a. 0 to 31 b. -8 to +7 c. -8 to +8
d. -15 to +15 e. -16 to +15
2. In a 6-bit 2’s complement binary number system,what is the decimal value represented by (100100)2s?
a. -4 b. 36 c. -36 d. -27 e. -28
2