Fraction to Decimal Converter

Division is all it is

A fraction is a division waiting to happen: 3/8 is 3 divided by 8, which is 0.375. Every fraction converts by dividing the numerator by the denominator, and the only interesting question is what kind of decimal comes out.

Terminating or repeating is decided by the denominator

A fraction in lowest terms produces a terminating decimal if and only if its denominator's prime factors are just 2s and 5s — the factors of 10. So 1/8 terminates because 8 is 2³, and 1/20 terminates because 20 is 2²×5. Anything else repeats: 1/3 is 0.333…, 1/7 is 0.142857 repeating, 1/6 is 0.1666… . This is not a quirk of the decimal system so much as a consequence of it, and in base 12 the fractions that terminate would be a different set.

Repeating patterns have a predictable length

The repeating block of 1/7 is six digits long, and no repeating block for a denominator n can exceed n − 1 digits. Denominators achieving that maximum, such as 7, 17 and 19, are called full reptend primes. The notation for a repeat is a bar over the repeating digits, or parentheses around them in plain text: 0.1(6) for one sixth.

Rounding a repeating decimal loses exactness

0.333 is not one third, and three of them sum to 0.999 rather than 1. Wherever repeated arithmetic follows, keep the fraction as long as possible and convert at the end — this is the same reason financial and engineering systems hold rational values rather than decimals through intermediate steps.

Recurring decimals convert back exactly

Any repeating decimal is a fraction. The standard technique multiplies by a power of ten to shift one full period, subtracts the original, and solves. It also produces the result that 0.999… equals exactly 1 — not approximately, but identically, since 9x = 9 when x = 0.999… . This is a true statement that reliably provokes disagreement.

Binary makes different fractions exact

Computers store decimals in binary, where the terminating fractions are those with denominators that are powers of 2. One tenth is not one of them, so 0.1 has no exact binary representation and 0.1 + 0.2 gives 0.30000000000000004. A fraction that looks clean in decimal can be inexact in floating point, which is why money is stored as integer minor units or in a decimal type.