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Every fraction, when converted to a decimal, ends up in exactly one of two categories: it either terminates after a finite number of digits, or it settles permanently into an endlessly repeating block of digits. Long division reveals which category applies, simply by tracking whether the running remainder ever hits exactly zero (terminating) or instead starts producing a remainder it has already seen before (repeating). This conversion matters whenever a fraction needs to be compared, entered into a calculator, or plugged into a formula that expects decimal input — spreadsheets, measuring instruments, and most consumer calculators all expect or display decimals rather than fractions, so being able to convert reliably, with proper repeating-decimal notation, avoids silently rounding off part of the true value.
How it works
Enter a numerator and denominator. The calculator performs long division of the numerator by the denominator, one digit at a time, and keeps a running log of every remainder it has produced along the way. If a remainder ever comes out to exactly 0, the division is complete and the decimal terminates there. If instead a remainder repeats a value it produced earlier in the process, everything between the first and second occurrence of that remainder must repeat forever after — since long division is deterministic, seeing the same remainder twice guarantees the exact same sequence of digits will follow both times.
- Enter numerator.
- Enter denominator.
- Click Calculate to see your results.
Examples
1/3
1 ÷ 3 gives a remainder of 1 repeatedly at every step, producing 0.(3), meaning 0.333... forever.
1/4
1 ÷ 4 = 0.25 exactly — the division terminates because the remainder reaches exactly 0 after two decimal digits.
5/6
5 ÷ 6 = 0.8(3), meaning 0.83333... — the first digit (8) doesn't repeat, but every digit after it is a repeating 3, since the remainder settles into a repeating cycle of just one value.
Common mistakes to avoid
- Rounding a repeating decimal and treating the rounded value as if it were exact, losing precision in any further calculation.
- Stopping long division too early and missing that a remainder has actually started repeating a previous value.
- Assuming every fraction terminates, and being surprised when a seemingly simple fraction like 1/7 produces a long repeating block instead.
- Confusing a genuinely repeating decimal with one that merely looks like it might repeat over just a few visible digits — always verify by checking the actual remainder cycle.