Download Representing Signed Integers

Computer Science 210
Computer Organization
Number Systems
Signed Integers
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: 11001 for 25
– 8-bit result: 10011001
• Note: This would be 153 as a positive.
Ranges for N-bit numbers
• Unsigned (positive)
– 0000…00 or 0
– 1111…11 which is 2N-1
– For N=8, 0 - 255
• 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-1 - 1
– For N=8, -128 to 127
Sign-Magnitude: Pros and Cons
• Pro:
– Easy to comprehend
– Easy to convert
• Con:
– Addition complicated (expensive)
If signs same then
…
else if positive part larger …
– Two representations of 0
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  00011001
– Invert bits: 11100110
– Add 1:
1
11100111
2’s Complement: Pros and Cons
• Con:
– 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
2’s Complement: Examples
• Compute negative of -25 (8-bits)
–
–
–
–
We found -25 to be 11100111
Invert bits: 00011000
Add 1:
00011001
Recognize this as 25 in binary
• Add -25 and 37 (8-bits)
–
11100111 (-25)
+ 00100101 ( 37)
(1)00001100
– Recognize as 12
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
2’s complement to decimal (examples)
• Assume 8-bit 2’s complement:
– X = 11011001
-X = 00100110 + 1 = 00100111
= 32+4+2+1 = 39 (decimal)
So, X = -39
– X = 01011001
Since X is positive, we have
X = 64+16+8+1 = 89