1. Computer Science 210 Computer Organization
Number Systems
Signed Integers

2. Negatives: Sign-Magnitude
With a fixed number of bits, say N
The leftmost bit is used to give the sign
0 for positive number
1 for negative number
The other N-1 bits are for the magnitude
Example: -25 with 8 bit numbers
Sign: 1 since negative
Magnitude: for 25
8-bit result:
Note: This would be 153 as a positive.

3. Ranges for N-bit numbers
Unsigned (positive)
0000… or 0
1111… which is 2N-1
For N=8,
Sign-magnitude
1111…11 which is -(2N-1-1)
0111…11 which is 2N-1-1
For N=8, -127 to 127
2’s Complement
1000…00 which is -2N-1
0111…11 which is 2N
For N=8, to 127

4. Sign-Magnitude: Pros and Cons
Easy to comprehend
Easy to convert
Con:
Addition complicated (expensive) If signs same then … else if positive part larger …
Two representations of 0

5. Negatives: Two’s complement
With N bit numbers, to compute negative
Invert all the bits
Add 1
Example: -25 in 8-bit two’s complement
25 
Invert bits:
Add 1:

6. 2’s Complement: Pros and Cons
Not so easy to comprehend
Human must convert negative to identify
Pro:
Addition is exactly same as for positives No additional hardware for negatives, and subtraction.
One representation of 0

7. 2’s Complement: Examples
Compute negative of -25 (8-bits)
We found -25 to be
Invert bits:
Add 1:
Recognize this as 25 in binary
Add -25 and 37 (8-bits)
(-25) ( 37) (1)
Recognize as 12

8. Facts about 2’s Complement
Leftmost bit still tells whether number is positive or negative as with sign-magnitude
2’s complement is same as sign magnitude for positives

9. 2’s complement to decimal (examples)
Assume 8-bit 2’s complement:
X = X = = = = 39 (decimal) So, X = -39
X = Since X is positive, we have X = = 89