Key Stage 3 computing · Data and logic

KS3 binary addition: a guide for parents

Adding binary numbers works like column addition in decimal, but with only two digits. In Years 7 to 9, children learn the four simple rules of binary addition, how to carry a 1 into the next column, how to check an answer by converting to decimal, and what overflow means when an answer is too big for the space a computer has for it.

Binary addition, explained

Binary addition follows the same pattern as the column addition children learned in primary school. Numbers are lined up by place value, you start in the right-hand column and work left, and when a column adds up to more than the largest digit you can write, you exchange and pass the extra to the next column. In decimal the largest digit is 9, so 7 + 5 is 12 and you write 2 and pass on a 1 to the tens. In binary the largest digit is 1, so the passing on happens much more often. Only a handful of rules are needed, so they are quick to learn.

The rules are these. 0 + 0 = 0. 0 + 1 = 1. 1 + 1 = 10, which is read as “one zero” and is the number two, so you write 0 and carry 1 to the next column. And 1 + 1 + 1 = 11, which is the number three, so you write 1 and carry 1. That last rule is needed when a carried 1 meets two 1s in a column. Teachers often say “carry”, which is the same idea as the exchange used in primary maths.

The curriculum lists binary addition as an example of a simple operation on binary numbers, so schools choose how much further to go. Many teach adding two 8-bit numbers, and some go on to overflow. A computer sets aside a fixed number of bits for each number. If the answer needs one more bit than is available, the extra bit is lost, and the result is wrong. In an 8-bit store, 255 + 1 would need nine bits (100000000), so the computer would end up with 0. Overflow is a real cause of bugs in software and games.

Why does this matter? Inside a processor, adding is done by electronic circuits built from logic gates, which use exactly these rules. Learning to add by hand helps children see that arithmetic in a computer is not magic, but a pattern of simple on-off steps. It also reinforces place value in a new base, which is one of the most useful ideas in the topic. A reliable habit is to convert the numbers and the answer to decimal as a check, because that catches nearly every slip.

Words your child will hear

Binary addition: Adding numbers written in binary, column by column, using the rules for 0 and 1.
Carry: The 1 passed to the next column on the left when a column adds to 2 or more.
Bit: One binary digit, either 0 or 1.
Byte: A group of 8 bits, able to store numbers from 0 to 255.
Overflow: What happens when an answer needs more bits than are available to store it.
Place value: The value of a digit given by its column: 1, 2, 4, 8 and so on in binary.
Least significant bit: The right-hand bit, worth 1, where addition starts.

What the National Curriculum says

The programme of study for Key Stage 3 says:

Pupils should be taught to understand simple Boolean logic [for example, AND, OR and NOT] and some of its uses in circuits and programming; understand how numbers can be represented in binary, and be able to carry out simple operations on binary numbers [for example, binary addition, and conversion between binary and decimal]
Aim: pupils can understand and apply the fundamental principles and concepts of computer science, including abstraction, logic, algorithms and data representation.

Objectives from the National Curriculum in England, © Crown copyright, used under the Open Government Licence v3.0.

A worked example

Add the 8-bit binary numbers 01101011 and 00100110, then check the answer in decimal.

Estimate in decimal first. 01101011 is 64 + 32 + 8 + 2 + 1 = 107, and 00100110 is 32 + 4 + 2 = 38. So the answer should be 107 + 38 = 145.
Line up the numbers by place value and start on the right. Column 1 (ones): 1 + 0 = 1, no carry. Column 2 (twos): 1 + 1 = 10, so write 0 and carry 1. Column 3 (fours): 0 + 1 + the carried 1 = 10, so write 0 and carry 1.
Column 4 (eights): 1 + 0 + 1 carried = 10, so write 0 and carry 1. Column 5 (sixteens): 0 + 0 + 1 carried = 1, write 1, no carry. Column 6 (thirty-twos): 1 + 1 = 10, write 0 and carry 1.
Column 7 (sixty-fours): 1 + 0 + 1 carried = 10, write 0 and carry 1. Column 8 (one hundred and twenty-eights): 0 + 0 + 1 carried = 1, write 1. There is no carry out of the last column, so the answer fits in 8 bits.
```
  carries  1 1 0 1 1 1 0 0
           0 1 1 0 1 0 1 1
         + 0 0 1 0 0 1 1 0
         -----------------
           1 0 0 1 0 0 0 1
```
Check: 10010001 is 128 + 16 + 1 = 145, which matches the decimal estimate.

01101011 + 00100110 = 10010001, which is 107 + 38 = 145 in decimal.

Common mistakes and how to help

Writing 2 in a column, as in 1 + 1 = 2

The sum is true in everyday numbers, and the habit of writing the total in a column is strong. But binary has no digit 2.

What helps: Remind your child that two in binary is 10. In the column, write the 0 and carry the 1, just as ten in decimal is written as a 0 with a 1 carried into the tens.

Forgetting to add the carried 1 into the next column

The small carried digit is written outside the main numbers and it is easy to skip it when moving on.

What helps: Write each carry directly above the column it will be added to, and cross it off once used. Checking the final answer in decimal catches any missed carry.

Getting 1 + 1 + 1 wrong

Children have the rule for two 1s well learned. When a carry arrives to join two 1s, the sum is three, which is 11 in binary, and many write 10 or 100.

What helps: Count the 1s in the column: three 1s make 11, so write 1 and carry 1. Some children find it helps to add the first two as 10 and then add the third.

Thinking a 9-bit answer simply fits in an 8-bit store

On paper there is always room for another column. A computer, however, has only the bits it was given.

What helps: Show 255 + 1 on paper: it gives 100000000, nine bits. Explain that in an 8-bit store the leftmost 1 has nowhere to go, leaving 00000000, which is wrong. That is overflow.

How to help at home

Start with small numbers. Use 4 bits first. Ask your child to add 0101 and 0011 and then convert to decimal to check: 5 + 3 = 8, which is 1000. Once that feels easy, move up to 8 bits.
Compare with decimal column addition. Ask your child to show how 8 + 5 is worked out in columns, with the 1 passed on. Then compare it with 1 + 1 in binary. Both cases are the same idea: a column gets too full, so it is emptied and the extra goes next door.
Always check by converting. Make it a rule that every answer gets converted back to decimal. Getting the same answer both ways builds trust in the method, and a mismatch is the cue to look for a carry that was missed.
Use cards or coins. Binary place-value cards, or coins turned heads up for 1, make carrying physical. Lay out two numbers, then combine them column by column. Whenever a column ends up with two coins, take both away and put one coin in the next column to the left.
Ask your child to teach you. Pretend you do not know the method and ask them to teach you using a number you choose, such as 00010110 + 00001101 (the answer is 00100011, which is 22 + 13 = 35). Explaining each carry aloud shows quickly which step is shaky.