Number Bases
How to Convert between number bases
Use this to change a number from one base to another, for example base two to base ten or base ten to base five.
Before you start
- Place value (ones, twos or fives, and so on)
- Powers of a number such as 2², 5³
- Long division with remainders
- Reading the base subscript, e.g. 142₅
When to use it
Use this to change a number from one base to another, for example base two to base ten or base ten to base five.
The steps
- To convert TO base ten: multiply each digit by its place value and add the results.
- To convert FROM base ten: divide by the new base repeatedly, writing down each remainder.
- Keep dividing until the quotient is zero.
- Read the remainders from bottom to top to get the answer.
- Check by converting your answer back to base ten.
Worked example
Convert 234₅ to base ten.
- The number is in base five and we want base ten, so multiply each digit by its place value and add.
- 234₅ = 2×5² + 3×5¹ + 4×5⁰.
- = 2×25 + 3×5 + 4×1 = 50 + 15 + 4.
- = 69.
- Check by converting 69 back to base five: 69÷5 = 13 r4, 13÷5 = 2 r3, 2÷5 = 0 r2; reading bottom to top gives 234₅, which matches.
A second example, with a twist
This time we go the other way, from base ten to base two, using repeated division, and base two gives a long line of remainders. Convert 45 (base ten) to base two.
- We convert from base ten to base two, so divide repeatedly by the new base 2.
- 45÷2 = 22 r1; 22÷2 = 11 r0; 11÷2 = 5 r1.
- 5÷2 = 2 r1; 2÷2 = 1 r0; 1÷2 = 0 r1, keep dividing until the quotient is 0.
- Read the remainders from bottom to top: 1 0 1 1 0 1, giving 101101₂.
- Check: 1×2⁵ + 0×2⁴ + 1×2³ + 1×2² + 0×2¹ + 1×2⁰ = 32 + 8 + 4 + 1 = 45.
Practise this in a KBAT problem
Frequently asked questions
Do I read the remainders from the top or the bottom?
Read them from the bottom up. The last remainder you get is the first (left-most) digit of the answer.
Reading top to bottom reverses the number and is a very common mistake, so always start from the final division.
Can I convert straight from base two to base five?
Not directly. Convert to base ten first (multiply by the place values), then convert that base-ten number to base five by repeated division.
Base ten acts as a bridge between any two non-ten bases.
What does the small number like ₅ mean?
It is the base of the number. So 234₅ means the digits 2, 3, 4 are read in base five, not base ten.
Always write the subscript so your reader knows which base you are working in.