Binary Addition

Binary addition follows the same rules as denary addition but carries at 2 instead of 10. There are four basic rules:

Example: Add 01101011 + 00110110:

  01101011
+ 00110110
----------
  10100001

Result: 10100001 = 161 in denary (107 + 54 = 161 ✓)

Overflow in Binary Addition

Overflow occurs when the result of an addition is too large to be stored in the available number of bits. In an 8-bit system, if the result exceeds 255 (unsigned) or goes outside the signed range, a carry out of the most significant bit indicates overflow.

Two's Complement

Computers represent negative numbers using two's complement notation. This allows the same addition circuits to handle both positive and negative numbers.

Representing Negative Numbers with Two's Complement

In n-bit two's complement:

• The most significant bit (MSB) has a negative place value: −2^(n−1)
• All other bits have their normal positive place values
• Range: −2^(n−1) to +2^(n−1)−1

For 8-bit two's complement:

Example: 11110100 in 8-bit two's complement:
= −128 + 64 + 32 + 16 + 0 + 4 + 0 + 0 = −128 + 116 = −12

Converting a Positive Denary to Two's Complement

For positive numbers, two's complement is the same as unsigned binary (as long as the MSB is 0).

Converting a Negative Denary to Two's Complement

Method 1 — Invert and add 1:

1. Write the positive value in binary
2. Invert all bits (flip 0→1 and 1→0) — this is the one's complement
3. Add 1 to the result

Example: Represent −37 in 8-bit two's complement:

• +37 in binary: 00100101
• Invert: 11011010 (one's complement)
• Add 1: 11011011
• Verify: −128+64+16+8+2+1 = −128+91 = −37 ✓

Subtraction Using Two's Complement

To subtract A − B, compute A + (−B), where −B is the two's complement of B. This means computers only need addition circuits — no separate subtraction hardware.

Example: 53 − 37 = 16
Two's complement of 37 = 11011011
00110101 + 11011011 = 100010000
The carry out of bit 7 is discarded → 00010000 = 16 ✓

Sign and Magnitude (for comparison)

An alternative method where the MSB is a sign bit (0=positive, 1=negative) and the remaining bits represent the magnitude. This is not two's complement. Cambridge 9618 focuses on two's complement as the standard method.