Decimal to Fraction Calculator

How to use the decimal to fraction calculator

Enter the decimal and read the fraction. The only decision the tool asks you to make is whether the decimal repeats, and it asks rather than guesses because guessing is where most converters of this kind go wrong.

The wrong approach is to search for a simple fraction near the value you typed. That method answers a different question, and it answers it confidently: give it 0.33 and it returns one third. But 0.33 and a third are not the same number. They differ by a third of a percent, which is small until it is a tolerance or a dose.

So this reads the digits literally. Three decimal places means thousandths, so 0.375 becomes 375 over 1000, and the greatest common factor of 375 and 1000 is 125. Divide both and you have 3/8. Every step is integer arithmetic with no tolerance anywhere in it, which is why the answer is exact rather than close.

The repeating case uses a different identity, and it is worth seeing once. Take the digits as a whole number, subtract the non-repeating prefix, and put the difference over a denominator of nines followed by zeros. For 0.8333 with one repeating place that is 8333 minus 833, over 9000, which is 7500/9000 and reduces to 5/6.

Why this does not guess a nearby fraction

There is a classical algorithm for finding the simplest fraction close to a decimal, built on continued fractions, and it is genuinely elegant. It is also the wrong tool here, because it answers "what simple fraction is near this number" when the question asked was "what fraction is this number".

The same tension shows up in reverse. Somebody who genuinely means a third and types 0.333333333 will get 333333333/1000000000 back, which is technically correct and unhelpful. That is exactly the case the repeat switch exists for, and it turns the answer into 1/3 without any tolerance being introduced.

What a decimal input can and cannot hold

Numbers in a browser are stored as double-precision floats, which carry about fifteen to seventeen significant digits. Past that, the value you typed and the value stored are different numbers, and any fraction derived from the stored one describes something you never entered.

The tool says so when it happens rather than staying quiet. If an input carries more places than a double can hold, the panel notes that the fraction describes the stored value. That is a small warning about a real limit, and it is the kind of thing a converter should tell you rather than paper over.

The formula, worked line by line

Two formulas, one for each kind of decimal. The terminating case is the one everybody knows; the repeating case is the one worth learning, because it turns an infinite thing into two integers with no approximation.

The repeating identity below looks stranger than it is. It works because multiplying by a power of ten shifts the repeating block onto itself, and subtracting cancels the infinite tail entirely.

terminating: fraction = digits / 10^places
then divide both by gcd(digits, 10^places)
0.375 = 375 / 1000 = 3 / 8   (gcd 125)
repeating: (all digits - prefix digits) / (10^(i+r) - 10^i)
0.8333 with r=1: (8333 - 833) / 9000 = 5 / 6

Which decimals can terminate at all

Only those whose reduced denominator is built from twos and fives, the primes of base ten. That is why eighths and twentieths terminate and thirds and sevenths cannot: no power of ten is divisible by three, so the division never comes out and the digits have to cycle instead.

Where an exact fraction earns its keep

The habit worth taking away is small: when you write a decimal down, note whether it stopped or was cut short. That single fact is the only thing this conversion cannot work out for itself.

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