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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