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Mathematics · pattern A, Numeric fields

Fraction to decimal, terminating or recurring.

Enter a numerator and denominator for the decimal value, and find out whether it terminates — which depends entirely on the prime factors of the denominator.

Inputs

The formula used

divide the numerator by the denominator

A fraction in lowest terms terminates if and only if its denominator's only prime factors are 2 and 5. Anything else recurs forever.

Decimal value

0.3750000000

Percentage
37.500000 %
Does it terminate?
yes — only 2s and 5s in the denominator
Denominator prime factors
2 × 2 × 2
In lowest terms
3/8
Rounded to 3 places
0.375
Reciprocal
2.66666667

Only 2s and 5s terminate, because we count in tens.

Ten factorises as 2 × 5, so a denominator built only from those primes divides some power of ten exactly and the decimal stops. A factor of 3 or 7 never does, which is why a third recurs and an eighth does not. In base 3 the situation would be reversed.

Questions about fraction to decimal

Why does 1/7 recur with six digits?
Because the block length is the order of 10 modulo 7, which is six. The pattern 142857 repeats indefinitely.
Are all fractions terminating or recurring?
Yes. Every rational number is one or the other; irrationals such as π and √2 are neither.
Should I convert at all?
Only if asked. Fractions are exact and decimals often are not.

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Arithmetic runs in double-precision floating point, so results beyond about fifteen significant figures are not exact. Where a question wants an exact fraction or surd, keep the exact form rather than a decimal.