Decimal to Fraction Converter

Terminating decimals are straightforward

Put the digits over the appropriate power of ten and reduce. 0.375 is 375/1000, which divides by 125 to give 3/8. The power of ten is set by the number of decimal places, so two places means hundredths, three means thousandths, and reduction by the greatest common divisor finishes the job.

Repeating decimals need the algebraic trick

Let x = 0.666…, multiply by 10 to get 10x = 6.666…, subtract to get 9x = 6, so x = 2/3. The multiplier is 10 raised to the length of the repeating block, which shifts exactly one period and makes the infinite tails cancel. For 0.142857 repeating, you multiply by 10⁶. Mixed decimals with a non-repeating prefix, such as 0.16666…, need two shifts to isolate the repeat.

Approximation is the harder problem

Converting an unrounded decimal like 0.3333333 gives 3333333/10000000, which is technically correct and useless. What people usually want is the simplest fraction close enough to the value — 1/3. This is a different computation, solved by continued fractions or the Stern-Brocot tree, which find the fraction with the smallest denominator within a stated tolerance. Any converter returning enormous denominators for a clearly simple value is answering the exact question rather than the useful one.

Irrational numbers have no fraction

π, e and √2 cannot be written as a ratio of integers, which is what irrational means. 22/7 is a famous approximation to π accurate to about two decimal places; 355/113 is far better, matching to six. Neither is π, and no fraction ever will be.

Why anyone needs the fraction back

Exactness through further calculation, since rationals do not accumulate rounding error. Presentation, because 3/4 communicates better than 0.75 in a recipe, a measurement or a music notation. Fitting a convention, since imperial measurements, drill bits and stock sizes are expressed in fractional inches. And reverse-engineering a ratio, since seeing 1.618 as roughly 1618/1000 tells you nothing while recognising the golden ratio does.

Floating point limits what you get back

A decimal read from a float is already an approximation, so converting 0.1 to a fraction from its stored binary value gives an exact but grotesque result rather than 1/10. Converting from the original text, before any float conversion, is what produces the answer people expect.