Decimal to Fraction

What this tool is used for

• Turning a decimal measurement into the fractional form a tape measure or a drawing uses.
• Expressing a repeating decimal exactly rather than as a rounded value.
• Converting a decimal ratio into a fraction for a recipe or a mixing proportion.
• Checking whether a decimal is exactly representable as a simple fraction.
• Producing a mixed number for a value greater than one.

Frequently Asked Questions

How is a decimal converted to a fraction?

A terminating decimal becomes the digits over the appropriate power of ten, then reduced by the greatest common divisor: 0.75 is 75/100, which reduces to 3/4. Repeating decimals need algebra instead.

How are repeating decimals handled?

By the standard algebraic trick: for 0.333…, let x = 0.333…, then 10x − x = 3, so x = 1/3. A single repeating digit gives ninths, two digits give ninety-ninths — which is why 0.121212… is 12/99, reducing to 4/33.

Which decimals terminate?

Only those whose reduced denominator has no prime factors besides 2 and 5 — because ten is 2 × 5. So 1/8 terminates and 1/3 does not, and 1/7 produces a six-digit repeating cycle.

Why does 0.999… equal exactly 1?

Because they are two notations for the same real number, not two numbers that are close. The algebra is the same as above: 10x − x = 9, so x = 1. It is an artefact of decimal notation, not a paradox.

How do floating-point values complicate this?

Considerably — 0.1 stored as a double is not exactly 0.1, so a naive conversion can yield an enormous exact fraction like 3602879701896397/36028797018963968. Practical converters round to a tolerance first.

What is a continued fraction good for here?

Finding the simplest fraction within a tolerance. It is how 3.14159 becomes 355/113 rather than 314159/100000 — the successive convergents give the best approximation for each denominator size.

How should an improper fraction be presented?

As both forms, since they suit different uses — 7/4 for arithmetic and 1¾ for reading. Converting silently to a mixed number makes further calculation harder, which is why both are shown.

Common errors and gotchas

• Feeding in a rounded decimal and expecting the original fraction back. 0.33 is not one third.
• Assuming every decimal has a tidy fraction. An irrational value only ever gets an approximation.
• Losing the repeating notation, where 0.1666 and one sixth are close but not equal.
• Expecting the denominator you had in mind rather than the lowest terms, which is what reduction produces.
• Converting a decimal that came from floating-point arithmetic, where the stored value is already slightly off.