Why Signed Integers?

Unsigned binary can only represent non-negative numbers. To represent negative integers in hardware, computers use two's complement — the standard method used by virtually all modern processors.

Two's Complement Representation

In two's complement, the MSB (most significant bit) has a negative weight:

Range for n-bit two's complement: −2^(n−1) to 2^(n−1)−1. For 8 bits: −128 to +127.

Reading two's complement

• If MSB = 0 → positive number (read normally).
• If MSB = 1 → negative number. Multiply MSB by −128 (for 8-bit), add remaining bits normally.

Example: 11110010 = −128+64+32+16+2 = −14

Converting Positive → Negative (Two's Complement)

Method — invert all bits, then add 1:

Verify: −128+64+32+8+1 = −128+105 = −23 ✓

Binary Subtraction Using Two's Complement

Subtraction A − B is performed as A + (two's complement of B). This avoids separate subtraction circuitry.

Example: 45 − 28

Result: 00010001 = 16+1 = 17 ✓ (discard overflow carry)

Sign and Magnitude (Alternative)

An alternative representation: use the MSB as a sign bit (1=negative), remaining bits represent magnitude. Problem: two representations of zero (+0 and −0), and arithmetic is more complex. Not used in practice — two's complement is preferred.