Signed Number Representation – Sign-Magnitude, 1's and 2's Complement (7 Marks)

For $n$ bits, MSB is sign bit: $0$=+ve, $1$=-ve.

1. Sign-Magnitude:
MSB = sign, remaining $(n-1)$ bits = magnitude.
Example $n=4$: $+3=0\,011$, $-3=1\,011$
Range: $-(2^{n-1}-1)$ to $+(2^{n-1}-1)$. Zero has two forms: $0\,000$ (+0) and $1\,000$ (-0).

• Advantages: Simple for humans, easy to find magnitude.
• Disadvantages: Two zeros, addition/subtraction needs separate logic checking signs and magnitudes (complex ALU), not used internally.

2. 1's Complement:
+ve numbers as in sign-magnitude; -ve numbers obtained by complementing all bits of +ve.
Example $n=8$: $+25=00011001$, $-25=11100110$
Range: $-(2^{n-1}-1)$ to $+(2^{n-1}-1)$, two zeros: $00000000$ and $11111111$.
Addition: binary add, then end-around carry (carry out added back to LSB).

• Advantages: Subtraction via addition, better than sign-magnitude.
• Disadvantages: Two zeros, end-around carry makes arithmetic slower, extra circuitry.

3. 2's Complement:
+ve as usual; -ve = 1's complement +1.
Example $n=8$: $+25=00011001$, $-25=11100110+1=11100111$
Range: $-2^{n-1}$ to $+(2^{n-1}-1)$ (one extra negative value, e.g., 8-bit $-128$ to $+127$). Single zero $00000000$.
Addition/Subtraction: simple binary addition, ignore carry out beyond $n$ bits. Subtraction $A-B = A + (-B)_{2's}$.
Example $5-3 = 0101 + 1101 = 0010$ (2) correct (carry discarded).

• Advantages: One zero, simplest arithmetic (single adder for add/sub), no end-around carry, widely used in modern ALUs (x86, ARM).
• Disadvantages: Asymmetric range, human interpretation slightly less intuitive, negation needs add.

Comparison Table:

Feature	Sign-Mag	1's Comp	2's Comp
Zeros	2	2	1
Range (8-bit)	-127 to +127	-127 to +127	-128 to +127
Arithmetic	complex	end-around carry	simple add

Conclusion: 2's complement dominates due to uniform add/sub hardware.