Decimal to Fraction Calculator

About the Decimal to Fraction Calculator

Going from a fraction to a decimal is division, and it is always possible. Going the other way is a question about what kind of number you are holding.

A decimal that stops converts immediately: the digits are already a count, and the number of places tells you what they are counting. A decimal that repeats needs a small piece of algebra to pin down, because you cannot write out an infinite tail and then divide it. And a decimal that neither stops nor repeats — pi, the square root of two — cannot be written as a fraction at all. Those are precisely the irrational numbers, and that is their definition.

This calculator handles the first two cases exactly, showing the working in each. The repeating method in particular is worth seeing once, because it is short, completely rigorous, and settles an argument people have been having about 0.999… for a long time.

How to Use the Decimal to Fraction Calculator

Choose whether your decimal ends or repeats.

For a decimal that ends, just enter it. 0.75, 2.4, −0.125 all work.

For a repeating decimal, the two parts go in separately. Enter the digits before the repeat begins as the non-repeating part, and the block that repeats as the repeating digits.

For 0.1666… that is 0.1 and 6. For 0.8333… it is 0.8 and 3. For 0.333… the non-repeating part is 0 and the repeating digit is 3.

Getting that split right matters. Entering 0.1666… as non-repeating 0 and repeating 1666 would describe 0.166616661666… which is a different number entirely.

Press Calculate for the fraction in lowest terms, the same value as a mixed number, and the unreduced form so you can see what was cancelled.

How Decimal to Fraction Conversion Is Calculated

For a decimal that ends, place value does the work. Every decimal digit sits over a power of ten: the first is tenths, the second hundredths, the third thousandths. So the digits after the point, read as a whole number, go on top, and the matching power of ten goes underneath. Then reduce.

0.75 has two places, so it is 75/100, which reduces to 3/4. That is the entire method.

For a decimal that repeats, you cannot write the digits down, so you remove them instead. Call the number x. Multiply it by two different powers of ten, chosen so that both results have exactly the same infinite tail. Subtract one from the other, and because the tails are identical, they cancel completely. What is left is a finite equation you can solve.

The trick works because the tail is infinite and identical in both lines. There is no approximation anywhere: the subtraction is exact, so the fraction is exact.

Decimal to Fraction Formula

Terminating, with k decimal places:

  value = digits / 10^k,  then reduce


Repeating, with n fixed places and a block of m digits:

  x = (high - low) / (10^(n+m) - 10^n)

  where high = all digits through one full repeat, read as a whole number
        low  = the digits before the repeat, read as a whole number

The denominator of a repeating decimal, once reduced, always turns out to be built from primes other than 2 and 5 — which is the same rule seen from the other side. A denominator made only of 2s and 5s gives a terminating decimal; anything else gives a repeating one.

Step-by-Step Example

Convert 0.1666… to a fraction.

Step 1 — name it. Let x = 0.1666…

Step 2 — shift it so the tails align. The repeat starts after one fixed digit and the block is one digit long, so multiply by 10¹ and by 10².

 10x =  1.666...
100x = 16.666...

Both lines now have the identical tail .666…

Step 3 — subtract.

100x - 10x = 16.666... - 1.666...
       90x = 15

The infinite tails cancelled exactly. Nothing was rounded.

Step 4 — solve and reduce.

x = 15/90 = 1/6

Check it by dividing: 1 ÷ 6 = 0.1666… It round-trips.

Now the famous one. Convert 0.999…

  x = 0.999...
10x = 9.999...
10x - x = 9.999... - 0.999...
     9x = 9
      x = 1

Not approximately 1. Exactly 1. The same subtraction that gave 1/6 gives 1 here, and there is nowhere in the working for an error to hide. 0.999… and 1 are two notations for one number, in the same way 1/2 and 2/4 are.

Understanding Your Result

The headline is the fraction in lowest terms — the standard form.

As a mixed number appears when the value is greater than one, showing the whole part separately.

Before reducing shows the raw fraction the method produced. For terminating decimals this is the digits over a power of ten, which makes the place-value logic visible. For repeating decimals it is the result of the subtraction, before cancelling.

The unreduced form is often the more instructive of the two. Seeing 0.1666… arrive as 15/90 before becoming 1/6 shows where the answer came from, which the final fraction alone does not.

If the fraction that comes back has a denominator containing 3, 7, 9, 11 or 13, that is consistent: those are exactly the denominators that produce repeating decimals.

When Should You Use This Calculator?

Exam questions. "Express 0.36 recurring as a fraction in its simplest form" is a standard question, and the method matters as much as the answer.

Keeping arithmetic exact. Fractions do not accumulate rounding error the way decimals do. Converting back to a fraction before a long calculation keeps the result exact.

Reading measurements. A caliper reading of 0.375 inches is 3/8, which is the number actually marked on the tool.

Recipes and imperial units. 0.25 cups is a quarter cup. Converting makes the figure usable with real measuring equipment.

Understanding a repeating result. If a division produced 0.428571428571…, converting it back reveals the fraction — here 3/7 — that generated it.

Common Mistakes

Splitting a repeating decimal incorrectly. The most common error by far. In 0.8333…, the 8 appears once and the 3 repeats. Entering the wrong split describes a different number and gives a wrong fraction.

Adding the percentage of decimal places wrongly. 0.075 has three decimal places, not two. Miscounting gives 75/100 instead of 75/1000 — an answer ten times too big.

Forgetting to reduce. 75/100 is correct but not finished. Almost every context expects lowest terms.

Trying to convert an irrational number. No fraction equals pi or √2. Any fraction you get for them — 22/7, for instance — is an approximation, sometimes a good one, never exact.

Rounding before converting. Converting a rounded decimal gives the fraction of the rounded value, not the original. If the underlying number was 1/3 and you convert 0.333, you get 333/1000, which is close and wrong.

Assuming a long decimal must be irrational. 1/17 has a sixteen-digit repeating block and is perfectly rational. Length is not the test; whether it eventually repeats is.