Decimal to fraction calculator

Type a decimal, like 0.375, 2.75 or −0.04. The fraction appears straight away in lowest terms, with a mixed number too when it's more than 1, and fraction bars to show its size. The steps show how it was found.

For a repeating decimal, put the repeating digits in brackets, as in 0.1(6) for 0.1666…, or type a few repeats and three dots: 0.1666... and 0.142857142857... both work, and the calculator finds the repeating block. A bar typed over the digits (0.16̅) and the word repeating after them work too. The answer is exact, however many digits you type, up to 30.

Count the decimal places, and write the digits over 1 with that many zeros. 0.375 has three places, so it's 375/1000. Then simplify by dividing the top and bottom by their greatest common factor, here 125, to get 3/8.

A whole-number part comes along for the ride. 2.75 is 275/100, which simplifies to 11/4, or 2 3/4 as a mixed number. Every decimal that ends gives a fraction whose bottom, in lowest terms, has no prime factors but 2 and 5, because 10 is 2 × 5.

A repeating decimal like 0.1666… goes on for ever, but it's still an exact fraction, and a little algebra finds it. Call the number x. Multiply it by 100 to get 16.666… and by 10 to get 1.666…. The two have the same endless tail of 6s, so subtracting one from the other leaves 90x = 15, and x = 15/90, which simplifies to 1/6.

The same works for any repeating block. For 0.142857142857…, multiply by 1,000,000 (one 0 for each repeating digit) and subtract x itself, leaving 999,999x = 142,857, and x = 1/7. That's why 0.999… equals exactly 1, since 10x − x = 9.999… − 0.999…, so 9x = 9.