Fraction to Decimal Converter

How to use the fraction to decimal converter

The tool works in both directions.

Fraction → decimal

1. Type a Fraction such as 5/12, a mixed number such as 1 3/4, or a whole number.
2. Read the Exact decimal (repeating part in brackets): 5/12 = 0.41(6), which means 0.41666… with the 6 repeating forever.
3. Next to it you get the value Rounded to 15 decimal places and the Period length, the number of digits in the repeating block.
4. The steps show the long division, one digit at a time, and explain why the decimal stops or repeats.

Decimal → fraction

1. Type a Decimal number. For a repeating decimal, put the repeating digits in brackets: 0.(3) for 0.333…, 0.1(6) for 0.1666…, 2.(142857) for 2.142857142857…
2. The ( and ) buttons insert brackets if your keyboard makes them hard to reach.
3. Read the Fraction in lowest terms and the algebra that produced it.

A decimal comma works as well as a decimal point: 0,1(6) and 0.1(6) are the same number.

Why some fractions repeat

A fraction in lowest terms has a terminating decimal only when its denominator has no prime factors other than 2 and 5, the prime factors of 10. So 3/8 = 0.375 (8 = 2³) and 7/20 = 0.35 (20 = 2² × 5) stop, while 1/3, 1/6 or 1/7 repeat forever. The tool factorises the denominator and tells you which case applies.

When you divide by hand, each step leaves a remainder smaller than the denominator. There are only so many possible remainders, so sooner or later one comes back, and from that moment the digits repeat. For 1/7 the remainders cycle through 1, 3, 2, 6, 4, 5 and the decimal is 0.(142857), a period of 6 digits. The period of 1/d is at most d − 1 digits long.

Converting a repeating decimal to a fraction

The classic method multiplies the number by powers of ten so that the repeating parts line up, then subtracts. For x = 0.1(6):

1. x = 0.1(6)
2. 10x = 1.(6)
3. 100x = 16.(6)
4. 100x − 10x = 16 − 1, so 90x = 15
5. x = 15/90 = 1/6

The tool performs exactly these steps, whatever the length of the repeating block, and simplifies the result. For a terminating decimal the method is simpler: 0.75 = 75/100 = 3/4.

Precision and limits

All calculations use whole numbers of unlimited size, so the results are exact, not approximations. The long division shows the first twelve steps in detail and summarises the rest. If a period is longer than 5,000 digits (which only happens with very large denominators), the expansion is cut and marked with “…”.