Fractional number base converter
Enter a number with a point and pick the two bases to see whether the fraction ends or repeats. It shows, for example, why 0.1 has no finite binary form.
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How to use it
- Enter number, from base, to base and maximum digits after the point.
- The result appears straight away and updates as you edit.
- Copy the result, or copy a link that keeps the same values to share it.
How it is calculated
The integer part uses repeated division. The fraction uses repeated multiplication: multiply by the output base and the integer part is the next digit. The fraction is kept as an exact ratio, and when a remainder recurs the cycle is shown in brackets: 0.0(0011) means 0.000110011 with 0011 repeating.
Example
Inputs: Number = 0.1, From base = Base 10 (decimal), To base = Base 2 (binary), Maximum digits after the point = 40. Result: Result in base 2: 0.0(0011); Fraction type in the new base: Repeating; Length of the repeating block: 4.
0.1
0.0(0011)
Limits and caveats
- If the cycle is longer than the chosen limit, the result is cut without brackets; raise the limit to see it.
- Input up to 400 digits; the point can be a dot or the Arabic decimal mark ٫.
Questions about this tool
Why can't 0.1 be written exactly in binary?
0.1 is 1/10, and the factor 5 in 10 does not divide base 2, so the fraction repeats forever: 0.0(0011). That is where small rounding errors in computers come from.
When does a fraction terminate in a given base?
A reduced fraction terminates when every prime factor of its denominator also divides the base. For example 1/4 ends in base 12 (0.3) because 2 divides 12, while 1/3 repeats in decimal.
What do the brackets in the result mean?
The digits inside the brackets repeat forever. 101.(1) in ternary means 101.111 with the 1 repeating, which equals 10.5 in decimal.