Digital Logic & Design Lecture 02
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Transcript of Digital Logic & Design Lecture 02
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Digital Logic & Design
Lecture 02
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Recap
Last lecture discussion Decimal Number Systems Caveman Base 5 Number System Binary Number System Number System Conversion
Today’s lecture discussion Decimal-Binary Conversion
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Binary to Decimal Conversion
Sum-of-Weights Expression base number & weights Sum terms Paper and pencil method
Sum of non-zero terms Mental Arithmetic, quick method Sum of weights of non-zero terms
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Binary to Decimal Conversion
Sum-of-Weights 100112
(1 x 24) + (0 x 23) + (0 x 22) + (1 x 21)
+ (1 x 20) Terms 16, 0, 0, 2 and 1 19
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Binary to Decimal Conversion
Add weights of non-zero terms Weights increase/decrease by power of 2 100112 = 16 + 2 + 1 = 19
1011.1012 = 8 + 2 + 1 + 1/2 + 1/8
= 11 + 5/8
= 11.625
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Decimal to Binary Conversion
Sum-of-Weights method used in reverse Highest binary weight less than the decimal
number Subsequent smaller weights that add up to
decimal number Repeated division by 2
Paper and pencil method Number repeatedly divided by 2
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Decimal to Binary Conversion using Sum-of-Weights
number Weight Result after subtraction Binary Bit
392 256 392-256 = 136 1 b8
136 128 136-128 = 8 1 b7
8 64 0 b6
8 32 0 b5
8 16 0 b4
8 8 8-8 =0 1 b3
0 4 0 b2
0 2 0 b1
0 1 0 b0
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Decimal to Binary Conversion
number Quotient after division Remainder after division
392 196 0 (b0)
196 98 0 (b1)
98 49 0 (b2)
49 24 1 (b3)
24 12 0 (b4)
12 6 0 (b5)
6 3 0 (b6)
3 1 1 (b7)
1 0 1 (b8)
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Binary-Decimal fraction conversion
Binary to Decimal Conversion Sum-of-Weights method Weights decrease by a factor of 2 0.11012 weights ½, ¼, 1/16 Sum up to 0.8125
Decimal to Binary Conversion Repeated Multiplication by 2 example
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Decimal-Binary fraction conversion
Decimal to Binary Conversion Repeated multiplication by 2
Number Mult. By 2 Integer
0.8125 1.625 1 (b-1)
0.625 1.250 1 (b-2)
0.250 0.500 0 (b-3)
0.500 1.000 1 (b-4)
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Binary Arithmetic
Binary Addition Binary Subtraction Binary Multiplication Binary Division
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Binary Addition
Four Basic rules for binary addition
1st digit 2nd digit Sum Carry
0 0 0 0
0 1 1 0
1 0 1 0
1 1 0 1
Addition of multiple binary numbers
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Binary Addition
Carry 1 10 1
1st Number 1 0 1 1
2nd Number 1 1 0
3rd Number 1 0 0 0
4th Number 1 1
Result 1 1 1 0 0
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Binary Subtraction
Four Basic rules for binary subtraction
1st digit 2nd digit Difference Borrow
0 0 0 0
0 1 1 1
1 0 1 0
1 1 0 0
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Binary Subtraction
Borrow 1
1st Number 1 0 1 1
2nd Number 1 1 0
Result 1 0 1
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Binary Multiplication
Four Basic rules for binary multiplication
1st digit 2nd digit Product
0 0 0
0 1 0
1 0 0
1 1 1
Example of Binary Multiplication
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Binary Multiplication
1101 (13)
x 101 (5)
1st product term 1101
2nd product term 0000
3rd product term 1101
Product 1000001 (65)
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Multiplication by shifting left
Decimal 29 shifted left by one digit 290 Shift left 1 digit is multiply by 10
Binary 111012 (29) shifted left by one bit
1110102 (58) Shift left 1 bit is multiply by 2
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Binary Division
10
101 | 1101
101
011
000
11
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Division by shifting right
Decimal 29 shifted right by one digit 2.9 Shift left 1 digit is divide by 10
Binary 111012 (29) shifted left by one bit
1110.12 (14.5) Shift left 1 bit is divide by 2
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Signed and Unsigned Numbers
Unsigned Binary Numbers Signed Binary Numbers
Most significant bit represents sign 0 represents a positive number 1 represents a negative number
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2’s Complement form
1’s complement form 2’s complement form
Binary number 01101 (13)
1’s complement 10010
+ 1
2’s complement 10011 (-13)
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Addition and Subtraction with 2’s Complement
0101 +5 0101 +50010 +2 1110 -20111 +7 0011 +3
1011 -5 1011 -51110 -2 0010 +21001 -7 1101 -3
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Range of Numbers
Maximum Range Number of digits Decimal number example Binary number example Overflow
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Range of Binary Numbers
DecimalNumber
Sign Magnitude2n-1, 2n-1 -1
2’s Complement2n-1, 2n-1 -1
Unsigned2n, 2n - 1
0 0000 0000 000
1 0001 0001 001
2 0010 0010 010
3 0011 0011 011
4 0100 0100 100
5 0101 0101 101
6 0110 0110 110
7 0111 0111 111
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Range of Binary Numbers
DecimalNumber
Sign Magnitude2n-1-1, -(2n-1 -1)
2’s Complement2n-1, -2n-1
Unsigned
-8 1000
-7 1111 1001
-6 1110 1010
-5 1101 1011
-4 1100 1100
-3 1011 1101
-2 1010 1110
-1 1001 1111
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Summary
Binary to Decimal Conversion Sum-of-Weights Sum of non-zero terms
Decimal to Binary Conversion Sum-of-Weights (in reverse) Repeated Division by 2
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Summary
Binary to Decimal fraction conversion Sum-of-Weights
Binary to Decimal fraction conversion Repeated Multiplication by 2
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Summary
Binary Addition Binary Subtraction Binary Multiplication
Multiplication by shift left operation Binary Division
Division by shift right operation
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Summary
Unsigned Binary Signed Binary
Sign Bit 2’s Complement 1’s Complement Range of Binary Numbers
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Lecture No. 2
Number Systems