Binary addition

What you need to know

• A binary column holds only 0 or 1. When it runs out, it resets to 0 and the next column goes up.
• Place values double each step left (1, 2, 4, 8, …). Four binary digits reach 15, and 1111 is the biggest.
• Have a goYour mate says the next binary number after 0111 is 0112, “because you just add one”. Prove them wrong: what really comes next?

  1000. The three 1s reset to 0 and the next column goes up.

  A binary column can only hold 0 or 1, so a 2 can never appear. Full columns reset to 0 and the next one goes up, like 99 becoming 100 in decimal.

• The four addition facts: 0 + 0 = 0, 0 + 1 = 1, 1 + 1 = 10 and 1 + 1 + 1 = 11.
• There is no digit 2 in binary, so 1 + 1 is written 10: write 0, carry 1.
• To add, write the numbers in place value columns, work from the right, and treat an empty space as 0.
• Have a goTry 100 + 10 in columns, working from the right. Does any column ever reach 10?

  110. No column reaches 10, so nothing is carried.

  It goes 0 + 0 = 0, then 0 + 1 = 1, then 1 + 0 = 1, because the empty space counts as 0. Nothing hit 10, so there was nothing to carry.

• When a column totals 10 or 11, write the right-hand digit underneath and carry the 1 to the next column.
• Have a goNow work out 11 + 1 in columns. What's the answer, and how many columns end up carrying?

  100. Both columns total 10, so both write 0 and carry 1.

  Column 1 is 1 + 1 = 10. The carry then meets 1 in column 2, which is 1 + 1 = 10 again. The last carry becomes the leading 1.

• 1 + 1 + 1 = 11 is needed because a carry can land on a column that already holds two 1s.
• Three numbers? Same rules. A column of four 1s totals 100: write 0, carry the 1 two columns left.
• Check any answer by converting both numbers and the result to decimal. The sums should match.

The big picture

Binary addition is ordinary column addition in a number system that runs out of digits at 1. Add from the right using the four facts (0 + 0 = 0, 0 + 1 = 1, 1 + 1 = 10, 1 + 1 + 1 = 11). When a column totals 10 or 11, write the right-hand digit and carry the 1. Then check by converting to decimal.

⚠ Watch out

Writing 1 + 1 = 2: binary has no digit 2, so it's 10. Close second: writing the 0 but forgetting to carry the 1, so the answer comes out too small. And don't miss a carry landing on two 1s: that's 1 + 1 + 1 = 11.

Memory hook

Run out of digits? Write the zero, carry the one. And 1 + 1 isn't 2 here. It's one-zero.

Check yourself

Cover the page and add 1010 and 11. Say out loud what you write and carry in every column, then check it in decimal. You're aiming for 1101 (10 + 3 = 13).