Convert numbers WITH a fractional part between any bases 2-36, using exact fraction arithmetic that detects repeating digits instead of silently rounding them.
Show calculation steps
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Radix points in any base
Digits do not stop at the point β they continue with negative powers: in base b, the first fractional place is worth 1/b, the second 1/bΒ². This converter applies that principle between any two bases from 2 to 36, holding the value as an exact fraction so that nothing is silently rounded.
The deep result it makes visible: whether a fraction terminates depends on the BASE. One tenth is finite in decimal, infinite in binary; one third is infinite in both but finite in base 3.
Worked example
Input: 3.14 (decimal) β binary, 20 digit limit
Integer part: 3 β 11
Fraction: 0.14 Γ 2 = 0.28 β 0
0.28 Γ 2 = 0.56 β 0
0.56 Γ 2 = 1.12 β 1 (keep 0.12)
β¦
Result: 11.00100011110101110000 (truncated at 20 β 3.14 has no finite binary form)Repeats are detected, not guessed
The converter tracks every remainder produced during digit generation. The first time a remainder recurs, the digits since its first appearance form a cycle that repeats forever β shown in parentheses, as 0.2 β 0.(1463) in octal. A value that neither terminates nor cycles within your digit budget is marked truncated rather than passed off as complete.
Why this matters practically
Floating-point surprises (0.1 + 0.2), fixed-point format selection for embedded work, DSP coefficient quantisation, and radix coursework all reduce to one question: what does this fraction look like in that base? An exact answer with honest repeat notation settles it.
Precision policy and privacy
Display precision is yours to set (10β64 digits); correctness is not negotiable β the underlying arithmetic is exact rational at every step. All of it runs in your browser; values never leave the page.
How to use the Fractional Base Converter
- Enter a number that may include a radix point, such as 3.14 or 101.101.
- Choose the source and target bases (2-36).
- Set the maximum number of fraction digits you want.
- Click "Convert fraction" – repeating digits are shown in parentheses, and the steps explain the two-part method.
Frequently asked questions
How are the two sides of the point converted differently?
The integer part uses repeated DIVISION by the target base; the fractional part uses repeated MULTIPLICATION, peeling off one digit each time. The steps panel shows both phases applied to your number.
What do the parentheses in a result mean?
Digits that repeat forever. The tool tracks remainders exactly, and the moment one recurs it knows the cycle: 0.1 in decimal becomes 0.0(0011) in binary, meaning 0011 repeats without end. No finite number of binary digits equals decimal 0.1.
When does a fraction terminate in the new base?
Exactly when the reduced denominator's prime factors all divide the target base. Tenths (denominator 5Γ2) terminate in base 10 but not base 2, because 5 does not divide 2. A half terminates in every even base.
Is the conversion really exact if it shows limited digits?
The arithmetic is exact – the value is held as an integer fraction throughout. What is limited is the DISPLAY: you choose the maximum digits, and if the expansion neither terminated nor repeated within that limit, the tool says so explicitly instead of pretending.
Can I convert a repeating input like 0.333… ?
Enter the digits you have – 0.333333 is read as exactly 333333/1000000, not as a third. Converting the true fraction 1/3 requires knowing it IS 1/3; a digit string can only ever represent the digits it contains.
Does this run on my device?
Entirely. The exact-fraction arithmetic is JavaScript BigInt in your browser; the value never leaves the page and analytics never see it.
Why would I need fractional base conversion at all?
Understanding floating-point behaviour is the big one – why 0.1 misbehaves in every language. It also appears in fixed-point DSP work, embedded scaling factors, and computer science coursework on radix representation.
