To convert between two non-decimal bases, go through base ten. Expand the original number into base ten, then convert that number to the new base by repeated division.
This lesson sits in SPM Mathematics number bases and relies on converting base ten to another base. Read that one first if division remainders still feel uncertain.
What is the two-step route?
Step 1 turns the source number into base ten by multiplying each digit by its place value. Step 2 turns the base ten number into the target base by dividing.
Convert 3142₅ to base 8.
Step 1, expand to base ten. The place values in base 5 are 125, 25, 5 and 1.
3(125) + 1(25) + 4(5) + 2(1) = 375 + 25 + 20 + 2 = 422
Step 2, divide by 8.
| Division | Quotient | Remainder |
|---|---|---|
| 422 ÷ 8 | 52 | 6 |
| 52 ÷ 8 | 6 | 4 |
| 6 ÷ 8 | 0 | 6 |
Reading upwards: 3142₅ = 646₈.
Check: 6(64) + 4(8) + 6(1) = 384 + 32 + 6 = 422, which matches step 1.
The mistake that costs marks
The common slip is using the wrong base in one of the two steps. A student divides 422 by 5 instead of 8, or expands 3142₅ with powers of 8.
Another slip is forgetting a place value. Writing 3(25) + 1(5) + 4(1) + 2 drops the 125 and gives a number far too small.
| Step | Wrong | Right |
|---|---|---|
| Expand 3142₅ | 3(25) + 1(5) + 4 + 2 = 86 | 3(125) + 1(25) + 4(5) + 2 = 422 |
| Target base | Divide by 5 | Divide by 8 |
| Final digit set | May contain 8 or 9 | Digits 0 to 7 only |
When base 2 and base 8 meet
Because 8 = 2³, each base 8 digit matches exactly three binary digits. Take 1101011₂ and group in threes from the right: 1 | 101 | 011. Pad the left group to 001.
Now convert each group: 001 = 1, 101 = 5, 011 = 3. So 1101011₂ = 153₈.
Check with the long route: 1101011₂ = 64 + 32 + 8 + 2 + 1 = 107. Then 1(64) + 5(8) + 3(1) = 107. The shortcut works only between bases that are powers of one another, so it does not apply from base 5 to base 8.
Check yourself
Convert 2031₄ to base 5. Expand to base ten first, then divide.
Answer
Base 4 place values: 64, 16, 4, 1.
2(64) + 0(16) + 3(4) + 1(1) = 128 + 12 + 1 = 141.
141 ÷ 5 = 28 remainder 1, then 28 ÷ 5 = 5 remainder 3.
5 ÷ 5 = 1 remainder 0, then 1 ÷ 5 = 0 remainder 1.
Reading upwards gives 1031₅.
Check: 1(125) + 0(25) + 3(5) + 1 = 141.
What to study next
With both conversions secure, you can add and subtract in other bases. Continue with adding and subtracting numbers in different bases, or test yourself on the number bases practice set.
A teacher can also review your layout and checking habits in online one-to-one Mathematics tuition.