Simple repeating
- Input
- 1/3
- Expected output
- 0.3[3]
The multiplicative order of 10 modulo 3 is 1, so the period is a single digit.
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fraction to decimal with repeating
Every fraction a/b has either a terminating decimal or a repeating one, never both and never neither. The tool computes up to 40 digits after the decimal point and automatically identifies the repeating block, showing it inside brackets.
The multiplicative order of 10 modulo 3 is 1, so the period is a single digit.
A 6-digit period, the maximum possible for a denominator of 7: no digit is left outside the repeating block.
The 2¹ factor of 6 = 2×3 produces 1 digit before the repeating part; without it, as in 1/3, the period would start right at the first decimal place.
Simplifying means dividing both the numerator and denominator by their Greatest Common Divisor (GCD) until no integer greater than 1 divides both. The resulting fraction is equivalent but in its most reduced form.
Yes: every rational fraction has either a terminating decimal (when the reduced denominator has only factors of 2 and 5) or a repeating one (in every other case). There is no third possibility, and the tool automatically identifies which of the two applies.
For prime denominators, the period of 1/b can have up to b−1 digits; the tool computes at most 40 decimal places, and if the repeating block does not close by then, it shows an ellipsis (…) instead of the full bracket. This only affects denominators with a very long period.
Yes, using Euler's theorem: the period of 1/b (with b free of factors 2 or 5) always divides φ(b), the count of integers smaller than b that are coprime with it; for a prime b, that reduces to a divisor of b−1, like the 6 digits of 1/7.
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