One’s and Two’s Complement Numbers and
Arithmetic Operations
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Digital electronics13IDP14
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OutlineOutline Number of binary digits (bits) used in
computers for arithmetic operations Positive and negative number
representation using bits 1’s complement and 2’s complement Arithmetic operations in 2’s complement Carry and overflow concepts See module 2a notes for more examples
Bits Used in Computers for Numbers and Implications 4, 8, 16, 32, 64 Using 4 bits we can only represent 24 = 16
numbers -8, -7, -6, -5, …-1, 0, 1, … 7 Using 8 bits we can only represent 28 = 256
numbers -128, -127, …-1, 0, 1, … 127 Since we can represent only a fixed number of
positive and negative numbers, we may get wrong results when we compute arithmetic operations
We must determine when the results are wrong
One’s and Two’s Complement
0111
0110
0101
0100
0011
0010
0001
0000
1111
1110
1101
1100
1011 1010 1001 1000
+7 +6 +5 +4 +3 +2 +1 +0 -0 -1 -2 -3 -4 -5 -6 -7
Cryptography -Part -I 4
0111
0110
0101
0100
0011
0010
0001
0000
1111
1110
1101
1100
1011
1010
1001
1000
+7 +6 +5 +4 +3 +2 +1 +0 -1 -2 -3 -4 -5 -6 -7 -8
One’s Complement
Two’s Complement
Two’s Complement Number Representation and Value (1) MSB represents negative number with a
value. Example: 8-bit number say 1101 0011 Value in decimal = -1*27 + 1*26 + 0*25 + 1*24 + 0*23 +0*22 +
1*21 + 1 =-128 +64+0+16+0+0+2+1 =-128+83 =-45
Cryptography -Part -I 5
Two’s Complement Number Representation and Value (2) Example: 1111 1111 Decimal Value = -1*27 + 1*26 + 1*25 + 1*24 + 1*23 +1*22 +
1*21 + 1 = -128+64+32+16+8+4+2+1 =-128+127 = -
1 Example: 1000 0000 Value = -128 Example: 0111 1111 Value = 127
Cryptography -Part -I 6
Finding a Negative of a number Assume 1’s complement Number 8-bit: 0000 1111; value =15 Given 15 determine -15 Change all 0 to 1 and 1 to zero Result: 1111 0000 = -127+64+32+16 =-127+112 = -
15 Assume 2’s complement Number 8-bit: 0000 1111; value =15 Given 15 determine -15 Change all 0 to 1 and 1 to zero then add 1 Result: 1111 0001 = -128+64+32+16+1 =-128+113 =
-157
Arithmetic Operations in Two’s Complement (1) Example:Find 7 – 5 assuming 4-bit
numbers 7 = 0111 ; -5 =1011 in 2’s complement
0111 1011 -----10010
Carry in MSB 1, Carry out in MSB 1Result ok value 2
Arithmetic Operations in Two’s Complement (2) Example:Find 7 + 5 assuming 4-bit
numbers 7 = 0111 ; 5 =0101 in 2’s complement
0111 0101 -----1100
Carry in MSB 1, Carry out in MSB 0Result wrong; value -4. We can not represent
real value 12. This is overflow.
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