Decimal to Fraction / Fraction to Decimal

Decimal to Fraction

Turn a decimal into a fraction, or a fraction into a decimal. The interesting part is that a decimal has two honest fractional answers, and this page shows you both instead of guessing which one you meant.

Two fractions, two different questions

Type 0.333333 and the panel shows:

• Best rational approximation: 1/3, with an approximation error of 3.333e-7
• Exact fraction from the digits entered: 333333/1000000

Both are correct. The first answers "what simple fraction is this decimal probably standing in for?", which is what you want to know when the decimal came from rounding something. The second answers "what fraction is this decimal, exactly?", and that's the question to ask when the digits themselves are the measurement.

The two rows come from different methods. The exact row builds the fraction over a power of ten (333333 over 1,000,000) and reduces it with Euclid's greatest-common-divisor algorithm, which in this case divides by 1 and leaves it alone. The headline row comes from a continued-fraction expansion: it walks the convergents and stops at the last one whose denominator fits your budget. Euclid still runs, but only on the exact-from-digits row.

The denominator budget decides the answer

The budget is a dropdown, and changing it changes the actual result, not just how it's displayed.

Leave it at 10,000 and 3.14159265 returns 355/113 with an error of 2.7e-7. That's the famous approximation Chinese mathematicians had by the fifth century. Drop the budget to 16 and the same input returns 22/7, error 1.3e-3: cruder, but you can express it in sixteenths.

So set it to 16 when the answer has to be readable off a tape measure, 64 for fine woodworking, and leave it high when you want the tidiest fraction available.

Raising it too far undoes the trick. At a budget of 1,000,000, 0.333333 stops being 1/3 and becomes 333333/1000000 with zero error, because that fraction now fits and it's a closer match to the digits you typed. The approximation is only useful while it's constrained, so a larger budget can actually make the answer worse.

Repeating decimals get bracket notation

The result is echoed back as a decimal with the repeating block in brackets. 1/7 is written 0.(142857), 5/6 is 0.8(3), and 7/12 is 0.58(3), with the non-repeating part sitting outside the brackets.

Under the hood it's long division with a memory. The remainder at each step is recorded, and when a remainder comes up again, the digits between the two sightings are the cycle. Brackets are the only notation that tells the truth about these numbers, since 0.142857142857 is a different value from 1/7.

The expansion stops after 40 digits. A fraction with a longer period gets an ellipsis instead of brackets. 355/113 repeats with a period of 112, for example, so it's shown truncated, though the value behind it is still exact.

Signs, mixed numbers and one warning

Negatives are handled properly. -0.75 gives −3/4 and the mixed number −3/4. The previous build returned −1/1, which is a different number altogether, well beyond any rounding error. Values above one produce a mixed number alongside the improper fraction, so 2.5 shows as 5/2 and 2 1/2.

The warning is about how strong the approximation is. Continued fractions are very good at finding the fraction a decimal stands in for, which means they'll find one whether or not it exists. Feed in the golden ratio 1.6180339887 with a budget of 10,000 and you get 10946/6765. Those are two consecutive Fibonacci numbers, correct to nine decimal places, and it's still an approximation of a number with no fractional form at all. So read the error row first. When it says 0 (exact), it is; otherwise you're holding a very good guess.

Nearby number tools

• Number Base Converter: the same integers in base 2 through 36.
• Percentage Calculator: for the same ratio expressed as a percentage.
• Length Converter: pairs with the sixteenths budget for imperial measuring.