Add and subtract in other bases using the same column method as in base ten, but carry and borrow the base instead of ten. Then check the answer by converting to base ten.
This lesson is part of the SPM Mathematics number bases chapter. It builds on converting base ten to another base.
How does carrying work in base five?
In base five, each column is worth five times the one on its right. When a column reaches 5, it becomes 1 in the next column.
A column total of 6 is 1 carry 1, because 6 = 1 × 5 + 1. A total of 8 is 1 carry 3.
Worked example: adding in base five
Find 234₅ + 142₅.
- Units: 4 + 2 = 6. Since 6 = 1 × 5 + 1, write 1 and carry 1.
- Fives: 3 + 4 + 1 = 8. Since 8 = 1 × 5 + 3, write 3 and carry 1.
- Twenty-fives: 2 + 1 + 1 = 4.
So 234₅ + 142₅ = 431₅.
Base-ten check. 234₅ = 2(25) + 3(5) + 4 = 69. 142₅ = 25 + 20 + 2 = 47.
Their sum is 116, and 431₅ = 4(25) + 3(5) + 1 = 116. The two agree.
Worked example: subtracting in base five
Find 402₅ − 143₅.
- Units: 2 − 3 cannot be done. The fives column is 0, so borrow from the 25s column first.
- The 25s column goes from 4 to 3. The fives column becomes 5, then lends 1 to the units, leaving 4.
- The units become 2 + 5 = 7, so 7 − 3 = 4.
- Fives: 4 − 4 = 0. Twenty-fives: 3 − 1 = 2.
So 402₅ − 143₅ = 204₅.
Base-ten check. 402₅ = 100 + 2 = 102. 143₅ = 25 + 20 + 3 = 48.
The difference is 54, and 204₅ = 2(25) + 0 + 4 = 54.
The mistake that costs marks
The common slip is to carry as in base ten. A student adds the units 4 + 2 = 6 and writes 6, even though 6 is not a digit in base five. The carry is skipped, so the whole answer is wrong.
| Wrong | Right | |
|---|---|---|
| Units of 4 + 2 | Write 6 | Write 1, carry 1 |
| Answer | 376₅ (the 7 and 6 are not digits in base five) | 431₅ |
| Check | Skipped | Base ten gives 116 |
Any digit equal to or larger than the base is an immediate warning sign.
Check yourself
Find 1101₂ + 1011₂ and check your answer in base ten.
Answer
Units: 1 + 1 = 2 = 10₂, so write 0 and carry 1.
Twos: 0 + 1 + 1 = 2, so write 0 and carry 1.
Fours: 1 + 0 + 1 = 2, so write 0 and carry 1.
Eights: 1 + 1 + 1 = 3 = 11₂, so write 1 and carry 1. The final carry gives another 1.
The answer is 11000₂. Check: 1101₂ = 13, 1011₂ = 11, the sum is 24, and 11000₂ = 16 + 8 = 24.
What to study next
Learn to catch errors in checking place-value errors in base conversions. Then try the number bases practice set. The algebra step-repair trainer builds the habit of checking every line, and the mistake log and paper-error review records slips.
If you want a teacher to go through your columns, see online one-to-one Mathematics tuition.