Number Bases
How to Add and subtract in a given base
Use this to add or subtract two numbers written in the same base without converting first.
Before you start
- Place value in a given base
- Carrying and borrowing in ordinary column arithmetic
- The digit range of a base (base five uses 0–4)
- Converting a number to base ten to check
When to use it
Use this to add or subtract two numbers written in the same base without converting first.
The steps
- Line up the digits by place value, as in ordinary arithmetic.
- Add or subtract column by column from the right.
- When a column reaches the base, carry one to the next column.
- When subtracting, borrow the base (not ten) from the next column.
- Check by converting both numbers and the answer to base ten.
Worked example
Add 24₅ + 13₅, giving your answer in base five.
- Line up the digits by place value, 24₅ above 13₅.
- Add the right column: 4 + 3 = 7.
- 7 reaches the base five (7 = 5 + 2), so write 2 and carry 1 to the next column.
- Add the left column: 2 + 1 + 1 (carried) = 4; write 4.
- Check: 24₅ = 14, 13₅ = 8, 14 + 8 = 22; and 42₅ = 4×5 + 2 = 22, they match.
A second example, with a twist
This is a subtraction in base two, so you borrow the base (2, not 10), and the borrow has to cascade through two columns. Subtract 1101₂ − 111₂, giving your answer in base two.
- Line up by place value, padding 111₂ to 0111₂ under 1101₂.
- Right column (2⁰): 1 − 1 = 0.
- Next column (2¹): 0 − 1 needs a borrow; borrow the base 2 from the next column, giving 2 − 1 = 1, and the next column drops by 1.
- The 2² column is now 0, and 0 − 1 needs another borrow: 2 − 1 = 1, and the 2³ column drops by 1.
- The 2³ column is now 0, and 0 − 0 = 0, giving 110₂. Check: 1101₂ = 13, 111₂ = 7, 13 − 7 = 6; and 110₂ = 6, they match.
Practise this in a KBAT problem
Frequently asked questions
When a column is too big, do I still carry 1?
Yes, you carry 1 just like normal addition, but you subtract the base from the column instead of subtracting 10. In base five, if a column makes 7, you write 7 − 5 = 2 and carry 1.
In base two, why do I borrow 2 and not 10?
Because each place is worth twice the place to its right, not ten times. So one unit borrowed from the next column is worth 2 in the column you are working on.
The size of a borrow always equals the base.
Do I have to convert to base ten before adding?
No. You can add or subtract directly in the given base, column by column, using the base for carrying and borrowing.
Converting to base ten is only a way to check the answer at the end, not a required step.