Binary addition

What you need to know

• ✓Add together up to three binary numbers, each up to 8 bits long.
• ✓The rules: 0 + 0 = 0. 0 + 1 = 1. 1 + 1 = 10 (write 0, carry 1). 1 + 1 + 1 = 11 (write 1, carry 1).
• ✓AQA's questions have at most three 1s in any column, so the biggest column total is 1 + 1 + 1 = 11.
• ✓If the total is more than 255, the answer needs a 9th bit. This is overflow: an 8-bit register has nowhere to store that extra bit, so it would be lost. When a question asks for the full answer, write all 9 bits.

What you don’t need

• Binary subtraction and multiplication — the specification's binary arithmetic is addition and shifts.

Worked examples

1. Add the 8-bit binary numbers 0011 0101 and 0001 1110.

   1. Write one number under the other and work from the right.
   2. 1st and 2nd columns: 1 + 0 = 1, then 0 + 1 = 1.
   3. 3rd column: 1 + 1 = 10. Write 0, carry 1.
   4. 4th column: 0 + 1 + the carried 1 = 10. Write 0, carry 1.
   5. 5th column: 1 + 1 + the carried 1 = 11. Write 1, carry 1.
   6. 6th column: 1 + 0 + the carried 1 = 10. Write 0, carry 1.
   7. 7th column: 0 + 0 + the carried 1 = 1. 8th column: 0 + 0 = 0.
   8. Check in denary: 53 + 30 = 83.

   Answer: 0101 0011, which is 64 + 16 + 2 + 1 = 83

2. Add the three binary numbers 0000 0111, 0000 0011 and 0000 0001.

   1. 1st column: 1 + 1 + 1 = 11. Write 1, carry 1.
   2. 2nd column: 1 + 1 + the carried 1 = 11. Write 1, carry 1.
   3. 3rd column: 1 + the carried 1 = 10. Write 0, carry 1.
   4. 4th column: the carried 1 = 1. The rest are 0.
   5. Check in denary: 7 + 3 + 1 = 11.

   Answer: 0000 1011, which is 8 + 2 + 1 = 11

3. Add the 8-bit binary numbers 1100 1000 and 0110 0100. Give the full answer.

   1. Work from the right as usual. The first six columns have no carries: 0, 0, 1, 1, 0, 1.
   2. 7th column: 1 + 1 = 10. Write 0, carry 1.
   3. 8th column: 1 + 0 + the carried 1 = 10. Write 0, carry 1.
   4. There's no 9th column to add, so the carried 1 becomes a 9th bit on the left.
   5. Check in denary: 200 + 100 = 300, which is more than 255 — too big for 8 bits.

   Answer: 1 0010 1100, which is 256 + 32 + 8 + 4 = 300. This is an overflow: kept to 8 bits, the 9th bit is lost and you're left with 0010 1100, which is only 44.

Common mistakes

• ✕Writing 2 in a column. 1 + 1 is 10 in binary: write 0 and carry 1.
• ✕Forgetting to add the carried 1 into the next column.
• ✕Writing 3 for 1 + 1 + 1. In binary that's 11: write 1 and carry 1.
• ✕Dropping the last carry. If the 8th column carries, write the extra 1 as a 9th bit — and if the question asks, say that it's an overflow.