To convert a base ten number to base b, divide by b repeatedly and read the remainders from the bottom up. The remainders are the digits of the new number.
This lesson starts SPM Mathematics number bases. The next lesson, converting between non-decimal bases, builds on it.
Why does repeated division work?
A number such as 213₅ means 2 × 25 + 1 × 5 + 3 × 1. Dividing it by 5 leaves the units digit 3 as the remainder, and the quotient 2 × 5 + 1 = 11 still contains the other digits.
Dividing 11 by 5 leaves 1 (the fives digit), and the quotient 2 is the leftmost digit. Each division peels off one digit, starting at the right.
Worked example: 94 to base 5
Convert 94 to base 5.
| Division | Quotient | Remainder |
|---|---|---|
| 94 ÷ 5 | 18 | 4 |
| 18 ÷ 5 | 3 | 3 |
| 3 ÷ 5 | 0 | 3 |
Read the remainders upwards: 94 = 334₅.
Check by expanding: 3(25) + 3(5) + 4(1) = 75 + 15 + 4 = 94.
The place-value method as a second route
List the powers of 5 up to 94: 1, 5, 25, 125. The last one is too big, so the answer has 3 digits.
Three 25s fit into 94, leaving 94 − 75 = 19. Three 5s fit into 19, leaving 4.
Four ones remain. The digits are 3, 3 and 4, so again 334₅.
Use whichever method feels natural, then use the other to check when time allows.
The mistake that costs marks
The usual slip is reading the remainders from the top. For 94 that gives 433₅, which expands to 4(25) + 3(5) + 3 = 118, not 94.
A second slip is stopping once the quotient is smaller than the base, for example after 18 ÷ 5 = 3 remainder 3, and writing 34₅. The final quotient 3 is itself a digit and belongs at the left. Keep dividing until the quotient reaches 0.
| Step | Wrong | Right |
|---|---|---|
| Reading remainders | Top to bottom: 433₅ | Bottom to top: 334₅ |
| When to stop | Quotient below base: 34₅ | Quotient reaches 0 |
| Expand check | 118 | 94 |
A number with a zero digit
Convert 65 to base 2. The divisions give remainders 1, 0, 0, 0, 0, 0, 1 in that order (65 ÷ 2 = 32 r 1, 32 ÷ 2 = 16 r 0, then 8, 4, 2 and 1 r 0, then 1 ÷ 2 = 0 r 1).
Reading upwards gives 1000001₂. Every zero must stay, since 64 + 1 = 65 and the five zeros hold the places for 32, 16, 8, 4 and 2.
Check yourself
Convert 200 to base 8. Then expand your answer to confirm.
Answer
200 ÷ 8 = 25 remainder 0. 25 ÷ 8 = 3 remainder 1. 3 ÷ 8 = 0 remainder 3.
Reading upwards gives 310₈.
Check: 3(64) + 1(8) + 0(1) = 192 + 8 = 200.
What to study next
When a question asks you to go from base 2 to base 5, you will convert through base ten, so this lesson is the first half of that route. Continue with converting between non-decimal bases, then learn to audit your answers in checking place-value errors in base conversions.
If a teacher watching your conversions would help, see online one-to-one Mathematics tuition.