Base Converter
Convert a number from any base to any other base from 2 to 36, including fractions, with the working shown.
Change the values and press Calculate to work out your own figures.
Enter a number and choose its base and the base to convert to.
Show the working
- 10 ÷ 3 = 3 remainder 1
- 3 ÷ 3 = 1 remainder 0
- 1 ÷ 3 = 0 remainder 1
- Read the remainders from the bottom up for the whole part
- 0.5 × 3 = 1.5 → digit 1
- 0.5 has come up before, so the digits from there repeat
- Read the fraction digits from the top down
- 10.5₁₀ = 101.(1)₃
How to use it
Enter a number and choose its base and the base to convert to.
Letters stand for digits above 9 (A = 10 … Z = 35) and can be typed in either case; prefixes such as 0b and 0x, spaces and underscores are ignored. Show the working gives each step.
Key facts
From base b: value = Σ dᵢ × bⁱ To base b: divide the whole part by b repeatedly; multiply the fraction by b repeatedly
Repeating fractions
A fraction ends in base b only if its denominator's prime factors all divide b. So 0.5 ends in binary (0.1) but not in base 3 (0.(1)), and 0.1 ends in decimal but repeats in binary.
Questions
What is 10.5 in base 3?
101.(1): the 1 after the point repeats forever.
What is the largest base?
36, using 0–9 and A–Z.
Formulas
From base b: value = Σ dᵢ × bⁱ To base b: divide the whole part by b repeatedly; multiply the fraction by b repeatedly
Sources
Limitations
- Numbers have at most 1,000 characters; Roman numerals cover whole numbers from 1 to 3,999,999.