Decimal to Fraction Calculator

Decimal places n → numerator = digits, denominator = 10ⁿ Then divide both by their greatest common divisor

Decimal

The number to convert, read to twelve decimal places. Trailing zeros do not change it, and a value below 0.000000000001 rounds away to nothing before the conversion begins

Decimal places

How many digits sit to the right of the point: three for 0.375, four for 0.0625, none for 2. This count is what sets the denominator, as that many zeros after a one

Numerator before reducing

The digits of the decimal read as a whole number, with the point removed: 375 for 0.375 and 625 for 0.0625. It is the top of the fraction that still shows where the number came from

Denominator before reducing

A power of ten — 1000 for three places, 10000 for four. It is the denominator that makes the fraction an exact decimal fraction rather than an approximation

Greatest common divisor

The largest number that divides both parts, and the only thing standing between the un-reduced fraction and the answer: 375 and 1000 share 125, so both are divided by it to reach 3 and 8

Four situations bring people here, and they call for slightly different levels of trust. Reading a decimal back as a fraction for a recipe, a tape measure or a drill size, where the decimals that matter are the ones with a terminating expansion — 0.375 is three eighths and 0.0625 is a sixteenth, and those conversions are exact. Checking a fraction you already have by working backwards, which is where the un-reduced output earns its place: entering 0.75 and seeing 75/100 confirms that the decimal is three quarters once 25 comes out of both parts. Turning a decimal into a ratio for display, where the reduced form is what goes on the page and the percent is a bonus. And the case where this page is the wrong tool, which is worth recognizing before you lose time to it: converting a repeating decimal. If what you have is 0.3333 rather than 1/3, the answer is 3333/10000 — a true statement about the digits in front of you and a poor stand-in for the number you had in mind. Going from a repeating decimal to its exact fraction is a different problem with a different method, and a decimal that stops after twelve places has already thrown away the evidence that would solve it.

1. 0.375 as a fraction

   1. Count the decimal places: 0.375 has three
   2. Denominator: three places means 1000, so the number is 375/1000
   3. Find the greatest common divisor of 375 and 1000: 125
   4. Divide both parts by 125: 375 ÷ 125 = 3 and 1000 ÷ 125 = 8
   5. The reduced fraction is 3/8, and the same decimal is 37.5 percent

   The default case, and the one that shows both outputs doing different jobs. 375/1000 is where the number came from and states plainly that three decimal places mean three zeros; 3/8 is the answer, and it is the form that keeps working — three eighths of a cup, three eighths of an inch on a ruler. Note that 3/8 is exact: 0.375 terminates after three places, which is the mark of a decimal that has an exact fraction behind it.

2. 1/16 as a decimal and back again

   1. Count the decimal places: 0.0625 has four
   2. Denominator: four places means 10000, so the number is 625/10000
   3. Divide both parts by 625: 10000 ÷ 625 = 16
   4. The reduced fraction is 1/16, and the decimal is 6.25 percent

   A sixteenth, which is where the un-reduced output becomes interesting: 625/10000 reduces all the way to 1/16 because 10000 is 16 × 625. This is the shape of every terminating decimal — the denominator of the reduced fraction divides a power of ten, which is the same statement as the denominator having no prime factors other than two and five. That is why 1/16 and 3/8 convert exactly and 1/3 and 1/7 never will.

3. 2.5 as an improper fraction

   1. Drop the point and read the digits: 25
   2. Count the places: one, so the denominator is 10
   3. Divide both parts by 5: 25 ÷ 5 = 5 and 10 ÷ 5 = 2
   4. The reduced fraction is 5/2, which is 2.5, and 250 percent

   A value above one, which is reported as an improper fraction rather than as a mixed number: 5/2 and not 2 1/2. The two forms are the same number and the improper one is the better input for further arithmetic — multiplying fractions does not want a whole number sitting beside them. Reading it back, 5 ÷ 2 = 2.5, is the arithmetic the next page in this group performs.

4. What 0.333333333333333 becomes

   1. Count the places: fifteen digits are typed, twelve are read
   2. Denominator: twelve places means 1000000000000
   3. Numerator: the twelve digits that survived, 333333333333
   4. Nothing cancels — no factor is shared with a power of ten — so both forms are the same fraction
   5. As a percentage it is 33.3333

   The case that separates this page from what a reader often hopes it does. Fifteen threes is 1/3 rounded to fifteen places, and the page converts the digits it was given rather than the number they approximate, so the answer is 333333333333/1000000000000 and not 1/3. That fraction is honest — it is exactly the decimal that was typed — and it is not what someone converting a repeating decimal is looking for. The un-reduced output equals the reduced one here because no cancellation is possible: a denominator that is a power of ten shares no factor with a numerator made of threes.

It converts the decimal you typed, not the fraction you meant. Any decimal you can write down has an exact fraction, and this page will find it, which means a truncated approximation of a repeating decimal converts to a fraction of a power of ten rather than to the simple fraction it came from — 0.3333 becomes 3333/10000, and getting 1/3 back requires knowing it was 1/3 to begin with. The page reads at most twelve decimal places: a value below 0.000000000001 rounds to zero before the conversion starts and comes back as 0, with its percent form reduced to a correspondingly tiny number, so a number that small loses its fraction entirely rather than being approximated. Inputs with many digits before the decimal point are limited by the precision of ordinary floating-point arithmetic — past roughly fifteen significant digits the numerator stops being exact, and the fraction returned is as precise as the number that was entered and no more; it will not silently overstate precision, but neither can it recover digits that were never there. Fractions are reported as improper fractions rather than as mixed numbers, so 2.5 gives 5/2 and not 2 1/2, and a whole number gives a fraction over one that is then dropped to the integer itself — 1 comes back as 1. Negative inputs keep their sign on the numerator: minus 0.375 is minus 3/8. The page does no arithmetic beyond the conversion — it will not add, compare or simplify a fraction you already have, and it does not check whether the fraction it produced is in the form the destination wants. And it is not a rounding tool: nothing here decides how many decimal places your number should have had.

How do I convert a decimal to a fraction?

Count the decimal places, drop the point, and put the digits over a one followed by that many zeros: 0.375 has three places, so it is 375/1000. Then divide both parts by their greatest common divisor — 125 in this case — to get 3/8. The counting step is the whole method; everything after it is cancellation that a page can do faster than a person.

Why does 0.3333 give 3333/10000 instead of 1/3?

Because 0.3333 is not 1/3 — it is 1/3 cut off after four places, and 3333/10000 is exactly what those four digits say. The page converts the number in front of it and has no way to know that a repeating decimal was intended; the difference between the two is 0.0000333, which is small but real. To get 1/3 from a repeating decimal you need the repeating part, and that is a different calculation with a different method.

What does the unreduced fraction tell me?

Where the fraction came from. 375/1000 says the number had three decimal places and the digits were 375, which makes it possible to check the conversion by eye; 3/8 says what the answer is. The pair also shows the cancellation step explicitly, so the same page serves as a worked example of reducing a fraction rather than only as a converter. When no cancellation is possible — with a numerator made of threes, for instance — the two forms are identical and the page prints the same fraction twice.

Which decimals convert into exact fractions?

Those that stop. A decimal with finitely many places is a fraction over a power of ten, so the conversion is exact by construction, and the reduced denominator can only be built from twos and fives — 2, 4, 5, 8, 10, 16, 20, 25 and so on. Anything else, such as a third or a seventh, has a decimal expansion that never ends, and no finite number of digits can represent it exactly. This page will still convert the digits you give it, and the fraction it returns will be a fraction of a power of ten rather than the simple fraction behind it.

What happens to a very small decimal?

Below twelve decimal places it rounds away to zero. The page reads at most twelve places, so 0.0000000000001 — thirteen places — becomes 0 and comes back as the fraction 0 rather than as a fraction with a thirteen-digit denominator. Twelve places is a deliberate ceiling: it covers every decimal anyone writes down in ordinary work, and it keeps the numerator and denominator inside the range where whole numbers are exact.

Why is 2.5 shown as 5/2 instead of 2 1/2?

Because an improper fraction is the form that keeps working. The value is unchanged either way, and 5/2 is what you want if the next step is multiplying or dividing by another fraction, since a mixed number has to be turned back into an improper fraction before it can be used. Reading it back is a division — 5 ÷ 2 = 2.5 — and the page that performs that division is the next one in this group.

• Decimal fraction — an arithmetical fraction with an integral power of ten as its denominator, which is the definition this page converts from — Encyclopedia of Mathematics, European Mathematical Society (international)
• Reduced Fraction — a fraction written in lowest terms, obtained by dividing numerator and denominator through by their greatest common divisor, with 2/3 given as the reduced fraction of 8/12 — Wolfram MathWorld (United States)
• Regular Number — also called a finite decimal: a number with a finite decimal expansion, which is exactly the class of decimals whose reduced fraction has a denominator built only from twos and fives — Wolfram MathWorld (United States)