Unit III — 1's & 2's Complement Methods

Complement methods are used to represent negative numbers in binary and to perform subtraction using addition circuits, simplifying CPU hardware design.

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Why Complements?

Computers represent negative numbers using complements rather than a separate sign. This allows the same addition circuits to handle both addition and subtraction.

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1's Complement

Definition: The 1's complement of a binary number is obtained by inverting every bit (0→1, 1→0).

Example:

• Number: (10110011)₂
• 1's Complement: (01001100)₂

Another Example:

• Number: (00001010)₂ = 10
• 1's Complement: (11110101)₂ = −10 in 1's complement notation

Properties:

• Two representations of zero: 00000000 (+0) and 11111111 (−0).
• Range for 8-bit: −127 to +127.

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2's Complement

Definition: The 2's complement is the 1's complement plus 1.

Method 1:

1. Find 1's complement.
2. Add 1 to the result.

Example: Find 2's complement of (10110011)₂

• 1's Complement: 01001100
• Add 1: 01001100 + 1 = (01001101)₂

Method 2 (Shortcut):

• Starting from the rightmost bit, copy all bits up to and including the first 1.
• Invert all remaining bits to the left.

Example: 2's complement of (10110000)₂

• Copy from right up to first 1: ...10000 → keep 10000
• Invert remaining: 101 → 010
• Result: (01010000)₂

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Subtraction Using 1's Complement

Rule: A − B = A + (1's complement of B); if there is an end-around carry, add it to the result.

Example: Subtract (1011)₂ − (0101)₂ = (0110)₂ ? (11 − 5 = 6)

1. 1's complement of 0101 = 1010
2. Add: 1011 + 1010 = 1 0101 (carry out = 1)
3. End-around carry: 0101 + 1 = (0110)₂ = 6 ✓

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Subtraction Using 2's Complement

Rule: A − B = A + (2's complement of B); discard any carry out.

Example: Subtract (1011)₂ − (0101)₂ (11 − 5 = 6)

1. 2's complement of 0101 = 1010 + 1 = 1011
2. Add: 1011 + 1011 = 1 0110 (carry out = 1, discard it)
3. Result: (0110)₂ = 6 ✓

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Example 2: Subtract (01000)₂ − (01011)₂ (8 − 11 = −3)

1. 2's complement of 01011 = 10100 + 1 = 10101
2. Add: 01000 + 10101 = 11101
3. No carry out → result is negative; find 2's complement of 11101: 00010+1 = 00011 = 3
4. Result: −3 ✓

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Comparison

Key Takeaway: 2's complement is the universal standard for representing negative integers in modern CPUs. It eliminates the double-zero problem of 1's complement and allows subtraction via addition with a simple carry discard. Master both methods with worked examples for the exam.