Two's Complement

Two's complement is a method used to represent negative numbers in binary and to perform binary subtraction efficiently.

Key Concept

When adding two binary numbers, an extra carry bit may appear, which is discarded:
$1001+0111=10000$

How to Calculate Two's Complement

To compute the two's complement of a binary number:

1. Take the 1's complement (invert all bits).
2. Add 1 to the result.

Example:
Find two's complement of 1011 (4-bit representation):
$\text{1’s complement: }1011\to 0100$
$\text{Add 1: }0100+0001=0101$

Signed Numbers

Signed numbers use the most significant bit (MSB) as a sign bit:

• 0 → Positive number
• 1 → Negative number

Examples:

• $0100=+4$
• $1100=-4$ (two's complement representation)

Finding Negative of a Number

To find the negative representation of a binary number:

1. Take its binary form.
2. Apply two's complement.

Example: Represent $-6$ in 4-bit:
$+6=0110$
Two's complement:
$0110\to 1001+1=1010$
So, $-6=1010$.

Binary Subtraction Using Two's Complement

Instead of direct subtraction, add the two's complement of the subtrahend to the minuend.

Example:
$0111-0101$
Step 1: Two's complement of $0101$:
$0101\to 1010+1=1011$
Step 2: Add to minuend:
$0111+1011=0010(\text{discard carry})$
Result: $0010=2$.