Fractions to Decimals and Back

A fraction and its decimal are the same number in two languages — but translation is easier in one direction. Fraction to decimal is plain long division, always possible, sometimes endless. Decimal back to fraction is either exact or an act of approximation, and telling those cases apart is the whole skill.

Fraction to decimal is long division

Divide the numerator by the denominator, track remainders, append digits. Two endings are possible. Either a remainder hits zero and the decimal terminates:

7 ÷ 8 = 0.875        (remainders 7, 6, 4 → then 0: terminates)

or a remainder repeats one you have seen before — and from that point the digits must repeat too, because the same remainder produces the same next digit forever:

1 ÷ 7 = 0.142857 142857 …
  remainders: 1, 3, 2, 6, 4, 5 → then 1 again: the cycle restarts

That remainder-watching is a genuine algorithm, not a trick: it is how the Fraction to Decimal Converter detects repeating cycles and annotates them as 0.(142857), with the parentheses marking the block that repeats forever.

Terminates or repeats — nothing in between

Which fate awaits a fraction is decided by its denominator, after simplifying. Factor it: if only 2s and 5s appear, the decimal terminates; if any other prime appears, it repeats. The reason is that terminating decimals are fractions whose denominator is a power of ten, and powers of ten are built from 2 × 5 — so 1/8 (2³) and 1/10 terminate while 1/3, 1/7, and 1/11 repeat. The cycle can be long but never longer than denominator − 1 digits: sevenths produce the six-digit cycle 142857, the maximum possible for 7. Every fraction lands in one camp or the other — that is what makes a number rational, literally "a ratio."

Rounding without lying

Repeating decimals force a decision about precision. The converter rounds to a chosen number of places: 22/7 to four places is 3.1429. A rounded value is a promise — the true value is within half a unit of the last digit — and the promise compounds if you chain rounded values through further calculations, so the honest habit is to keep full precision while computing and round once, at the end, for display.

Back again: exact when the decimal stops

Converting a terminating decimal to a fraction is mechanical: the digits after the point go over a power of ten, then simplify by the greatest common divisor:

0.625 = 625/1000 → divide by 125 → 5/8
2.375 = 2375/1000 → divide by 125 → 19/8 = 2 3/8   (mixed number)

Every terminating decimal converts exactly this way — the fraction and the decimal are perfect twins. The Decimal to Fraction Converter returns both the improper fraction and the mixed form, and the Fraction to Decimal Calculator shows the simplified form alongside the decimal so you can check the round trip.

When the decimal never stops: best-fit fractions

Non-terminating decimals have no exact fraction with small parts — their "true" fraction is the one they came from, with big numbers in it. So the converter approximates using continued fractions, which produce convergents: for a given cap on the denominator, the best possible fraction. The results reward attention:

0.3333333333 → 1/3        (the pattern is recognized — near-miss of 1/3)
0.333        → 333/1000   (honest: this really is not 1/3)
3.14159      → 9563/3044  (π truncated; 22/7 = 3.14286 is too far off)

The second line is the teaching moment: 0.333 and 0.3333333333 look like siblings, but one deserves 333/1000 and the other earns 1/3, because approximation is only worth doing when the error is tiny relative to the denominator you allow (up to 10,000 here). The famous 22/7 approximation of π survives for the same reason — it is the best fraction with a denominator under 60, not because π is a fraction. (π is not rational; no fraction is exact.) For the fraction-side fundamentals — anatomy, simplifying by the GCD, the four operations — see how fractions work.

Try it yourself

Convert 1/7 and 22/7 with the Fraction to Decimal Converter and watch the repeating block get its parentheses. Send 0.625, then 0.333, then 0.3333333333 through the Decimal to Fraction Converter and see exactly where exactness ends and best-fit begins. Use the Fraction to Decimal Calculator to check any round trip: fraction → decimal → simplified fraction should return where you started.