1's and 2's complements

1. One’s and Two’s Complement
Numbers and Arithmetic
Operations
1
Digital electronics
13IDP14

2. 2
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

3. 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

4. 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

5. 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

6. 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

7. 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 = -15
7

8. 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 1
Result ok value 2

9. 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 0
Result wrong; value -4. We can not represent
real value 12. This is overflow.
9

10. 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 0
Result wrong; value -4. We can not represent
real value 12. This is overflow.
9