Two's Complement: Signed Binary Numbers

Two's complement is the standard method computers use to represent signed (positive and negative) integers. It elegantly handles negative numbers using the same binary addition hardware that works for positive numbers.

How two's complement works:

In an n-bit two's complement system, the most significant bit (MSB) is the sign bit: 0 for positive, 1 for negative. For an 8-bit number:

• Positive range: 00000000 to 01111111 (0 to +127)
• Negative range: 10000000 to 11111111 (-128 to -1)

Converting a positive number to its negative (finding two's complement):

To negate a number, invert all bits then add 1:

Example — finding -42 in 8-bit two's complement:

1. Start with +42: 00101010
2. Invert all bits: 11010101
3. Add 1: 11010110

So -42 is represented as 11010110₂.

Converting two's complement back to decimal:

If the MSB is 1 (negative number), invert all bits and add 1 to get the magnitude:

1. Given: 11010110
2. MSB is 1, so it is negative
3. Invert: 00101001
4. Add 1: 00101010 = 42
5. Result: -42

Why two's complement is brilliant:

The key advantage is that addition works identically for signed and unsigned numbers. The CPU does not need separate circuits for signed vs. unsigned addition. For example, 5 + (-3):

  00000101  (+5)
+ 11111101  (-3)
----------
 100000010

Discard the carry beyond 8 bits: 00000010 = 2. Correct.

Common ranges:

Bits	Signed Range	Unsigned Range
8	-128 to +127	0 to 255
16	-32,768 to +32,767	0 to 65,535
32	-2,147,483,648 to +2,147,483,647	0 to 4,294,967,295