Adding binary numbers

Computer Science · Binary addition

Watch the carry ripple

Add A = 01011011 and B = 00111011. It's the column addition you already know from decimal, right to left, except you carry as soon as a column reaches two, not ten.

Before you press Next: the 1s column is 1 + 1. What do you write, and what do you carry?

_ _ _ _ _ _ _ 0

Predict, then check

The computer stores the result in 8 bits. There are only 8 boxes.

A computer adds 10110100 + 01100100 and stores the result in 8 bits. What does it store?

What you need to know

• Binary is base 2: it only uses the digits 0 and 1, called bits.
• In an 8-bit number the place values, from right to left, are 1, 2, 4, 8, 16, 32, 64 and 128.
• Add column by column from right to left, carrying into the next column to the left, just like decimal.
• The four column rules: 0 + 0 = 0; 0 + 1 = 1; 1 + 1 = 10 (write 0, carry 1); 1 + 1 + 1 = 11 (write 1, carry 1).
• Check an answer by converting both numbers and the result to decimal.
• Overflow happens when the result needs more bits than are available, so the stored result is wrong.

The big picture

Binary addition is column addition with a smaller limit. Work right to left, use the four column rules, carry as soon as a column reaches two, check in decimal, and watch for a carry that falls off the end.

⚠ Watch out

Writing 2 in a column. There is no 2 in binary: 1 + 1 is 10, so write 0 and carry 1. The other easy slip is forgetting the carry when a column already has two 1s. 1 + 1 + 1 is 11, so write 1 and carry 1.

Memory hook

Two in a column? Write 0, carry 1. Three in a column? Write 1, carry 1. Decimal carries at ten; binary carries at two.

Check yourself

Cover the page and add 00100111 + 00010101, writing in every carry. Then check it by converting all three numbers to decimal. You should get 00111100, because 39 + 21 = 60.