Binary Addition & Subtraction (OCR A Level Computer Science)

Binary Addition

What is Binary Addition?

• Binary addition involves summing numbers in base-2, which uses only the digits 0 and 1

• Like denary addition, start from the rightmost digit and move towards the left

• Carrying over occurs when the sum of a column is greater than 1, passing the excess to the next left column

Example addition

Binary addition example

Overflow Errors

• Overflow occurs when the sum of two binary numbers exceeds the given number of bits

• In signed number representations, the leftmost bit often serves as the sign bit; overflow can flip this, incorrectly changing the sign of the result

• Overflow generally leads to incorrect or unpredictable results as the extra bits are truncated or wrapped around

An overflow occurring after a binary addition

Binary Subtraction

• As well as adding binary numbers, we can also subtract binary numbers

• One method of doing this is to use two's complement

Example 1

Subtract 1001 (9) from 0011 (3)

1. Given numbers

2. Two's complement

• Convert the number to subtract (0011) to its two's complement

• Invert bits to get 1100

• Add 1 to get 1101

3. Addition Operation

• Now add 1001 and 1101

• 1001 + 1101 = (1) 0110 - including an overflow

4. Remove Overflow

• The result is 10110 with overflow

• Remove the overflow bit to get 0110

• In two's complement arithmetic, the overflow bit does not contribute to the actual value of the operation but is more of a by-product of the method.