Why Two's Complement?

Computers need to represent both positive and negative numbers. Two's complement is the standard method used by computers to store signed (positive and negative) integers in binary. It allows subtraction to be performed using the same addition circuits, making hardware design simpler.

Representing Negative Numbers in 8-bit Two's Complement

In 8-bit two's complement:

• The most significant bit (MSB) — the leftmost bit — is the sign bit
• If the sign bit is 0, the number is positive
• If the sign bit is 1, the number is negative
• Range: −128 to +127

Converting a Positive Number to Its Negative (Two's Complement Method)

To find the two's complement (negate a number):

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

Example: Represent −19 in 8-bit two's complement

1. +19 = 00010011
2. Invert bits: 11101100
3. Add 1: 11101100 + 00000001 = 11101101

So −19 = 11101101 in 8-bit two's complement

Verify: −128 + 64 + 32 + 8 + 4 + 1 = −128 + 109 = −19 ✓

Reading a Negative Two's Complement Number

To find the denary value of a negative binary number:

• If the MSB is 1, the number is negative
• The sign bit has value −128 (for 8-bit)
• Add the values of all remaining 1-bits to −128

Example: What is 11110001 in denary?

MSB=1 (negative). Values: −128 + 64 + 32 + 16 + 1 = −128 + 113 = −15