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Two’s Complement Explained – Representing Negative Binary Numbers (O Level Computer Science 2210 / IGCSE 0478)

Why Do Computers Need Two’s Complement?

Computers store numbers in binary, but they also need to represent negative numbers.

Example:

  • −5

  • −10

  • −127

To store negative numbers, computers use a system called two’s complement.

Two’s complement allows computers to perform binary subtraction and addition efficiently.

What Is Two’s Complement?

Two’s complement is a method used to represent negative binary numbers.

It is commonly used with 8-bit numbers.

In an 8-bit system, the range of numbers is:

TypeRangePositive numbers0 to 127Negative numbers−1 to −128

The most significant bit (MSB) shows whether the number is positive or negative.

MSBMeaning0Positive1Negative

Steps to Find Two’s Complement

To convert a positive binary number into its negative form, follow these steps.

Step 1: Write the binary number (8-bit)

Example: 5

00000101

Step 2: Invert all bits (find the one's complement)

11111010

Step 3: Add 1

11111011

So:

−5 = 11111011

Example 2

Find −12 in two’s complement.

Step 1: Convert 12 to binary

00001100

Step 2: Invert the bits

11110011

Step 3: Add 1

11110100

Therefore:

−12 = 11110100

Converting Back to Denary

To convert a negative binary number back to denary:

1️⃣ Invert the bits

2️⃣ Add 1

3️⃣ Convert to denary

4️⃣ Add the negative sign

Example:

11111011

Invert:

00000100

Add 1:

00000101

Result:

−5

Why Two’s Complement Is Important

Two’s complement helps computers to:

  • Represent negative numbers

  • Perform binary subtraction easily

  • Use the same circuits for addition and subtraction

This makes computer hardware simpler and more efficient.

Exam Tip (2210 / 0478)

Students are often asked to:

  • Convert positive numbers to two’s complement

  • Convert two’s complement numbers to denary

  • Identify negative binary numbers

Always remember:

Invert bits → Add 1

Practice Question

Find the two’s complement of 9.

Step 1:

00001001

Step 2:

11110110

Step 3:

11110111

Answer:

−9 = 11110111

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