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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 0011 (3) from 1001 (9)

1. Given numbers

2. Two’s complement

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

• Invert: 1100

• Add 1: 1100 + 0001 = 1101

3. Addition operation

• Now add 1001 and 1101

• Binary sum: 1001 + 1101 = 1 0110

• That’s 5 bits: the leftmost 1 is overflow (carry out of MSB)

4. Remove overflow

• The result is 10110 with overflow

• Drop the leading 1 (overflow): 0110 = 6 (in decimal)

• 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

• Final answer = 6

• 9 – 3 = 6