Binary Subtraction Explained for KS3

You already know decimal subtraction: when the top digit is too small, borrow ten from the column to the left. Binary subtraction uses the same idea — except borrowing gives you two instead of ten, because binary only has two digits. Master that swap and the whole method falls into place.

What are the basic subtraction rules in binary?

Before attempting a full calculation, memorise these four single-bit subtraction facts — they are the only combinations that exist:

The only tricky case is the last one: 0 − 1. In decimal, 0 − 9 would require borrowing 10 from the next column. In binary, borrowing from the next column gives you 2 (because one group of the next-higher bit is worth two of the current bit). So 0 − 1 becomes (0 + 2) − 1 = 1, with a borrow of 1 carried to the left.

How does borrowing work column by column?

The process mirrors decimal long subtraction. Always work from the rightmost column (the least-significant bit) to the left.

Worked example: 1010 − 0110

Let us subtract 6 (0110) from 10 (1010) and confirm the answer is 4 (0100).

  1 0 1 0   (= 10 in decimal)
- 0 1 1 0   (=  6 in decimal)
---------

Column by column, right to left:

Column 1 (ones): 0 − 0 = 0. No borrow.
Column 2 (twos): 1 − 1 = 0. No borrow.
Column 3 (fours): 0 − 1. Cannot do this! Borrow from column 4.
→ Column 4 lends 1, so column 3 becomes 0 + 2 = 2 (in this column, borrowing gives two). Column 4 is reduced by 1.
→ 2 − 1 = 1.
Column 4 (eights): Was 1, reduced to 0 by the borrow. 0 − 0 = 0.

  1 0 1 0
- 0 1 1 0
---------
  0 1 0 0   (= 4 ✓)

How do you handle a borrow chain?

Sometimes you need to borrow from a column that is also 0, so you must borrow further left. Think of it like needing change from a shop that has no coins, so the cashier has to open a new roll.

Worked example: 1000 − 0001

Subtract 1 (0001) from 8 (1000). The answer should be 7 (0111).

Column 1: 0 − 1. Need to borrow, but column 2 is 0.
→ Column 2 must borrow from column 3 (also 0).
→ Column 3 must borrow from column 4.

Working the borrow chain:

• Column 4 (value 1) lends to column 3: column 4 → 0, column 3 → 2.
• Column 3 lends one of its 2 to column 2: column 3 → 1, column 2 → 2.
• Column 2 lends one of its 2 to column 1: column 2 → 1, column 1 → 2.
• Now column 1: 2 − 1 = 1.

  1 0 0 0
- 0 0 0 1
---------
  0 1 1 1   (= 7 ✓)

How can you check your answer?

Always verify by converting both your answer and the original numbers to decimal:

1000 in binary = 8
0001 in binary = 1
0111 in binary = 7

Check: 8 − 1 = 7  ✓

Alternatively, check by addition: if A − B = C, then B + C should equal A.

0001 + 0111 = 1000  ✓

This addition check takes only a few seconds and will catch most arithmetic slips before they cost you marks.

What mistakes do students most commonly make?

Three errors come up again and again in KS3 work:

1. Forgetting that borrowing gives 2, not 10. When you borrow in binary, the current column receives the value of one group from the column to its left — which is always exactly 2 in binary.

2. Not reducing the donor column after borrowing. When column 3 lends to column 2, column 3 must lose 1. Students often forget this step, especially in borrow chains.

3. Stopping the borrow chain too early. If you need to borrow and the next column is 0, you cannot stop — you must keep reaching further left until you find a 1.

How does this relate to two's complement subtraction?

At GCSE level you will also meet two's complement, which is how real computers handle negative numbers and subtraction internally. Instead of borrowing, the computer converts B into its two's complement (flip all bits, add 1) and then adds it to A. The borrowing method taught at KS3 is mathematically identical in its results — but it is easier to do by hand, which is why it appears in KS3 assessments.

Frequently asked questions

Why does borrowing in binary give 2 rather than 10?

In decimal, each column is worth ten times the one to its right, so borrowing one from the left gives ten on the right. In binary, each column is worth two times the one to its right — so borrowing one from the left gives exactly two.

Do I need to subtract binary numbers in my GCSE exam?

Binary subtraction is assessed at KS3, and the underlying concept feeds directly into two's complement, which is a GCSE topic. Practise the borrowing method until it is automatic — it will also strengthen your binary arithmetic generally.

Can I subtract a larger number from a smaller one in binary?

If the result would be negative, the answer requires either a negative sign or two's complement notation (a GCSE topic). At KS3, questions are set so that the top number is always larger.

Is there a quicker method?

Some students find it easier to convert both binary numbers to decimal, subtract in decimal, then convert the answer back to binary. This is perfectly valid for checking your work, but exam questions usually expect you to show the binary column-subtraction method with any borrows shown explicitly.

Want step-by-step binary practice with instant marking? Head to aitutors.me and let Professor Turing set you a sequence of worked problems at exactly your level.