Decimal To Fraction Calculator

What this calculator does

So, you have a decimal like 0.75 and you want it written as a fraction, 3/4. This tool takes the decimal, writes it as a fraction over a power of 10, and reduces it to lowest terms. If there is a whole-number part, it keeps it, so 2.5 comes back as 2 1/2.

There is a single box: you type the decimal, and it does the rest.

How to use it

1. Type your decimal number.
2. Press Calculate.

It is built for positive decimals, the everyday case of turning a number like 0.375 into a clean fraction.

A decimal is already a fraction

Here is the idea the whole tool rests on. The digits after the point already name a fraction. 0.7 is seven tenths. 0.75 is seventy-five hundredths. 0.123 is one hundred twenty-three thousandths. The place of the last digit tells you the denominator: one decimal place means tenths, two means hundredths, three means thousandths, and so on down the line.

So a decimal is simply a fraction whose bottom is a power of 10. All the tool has to do is write that fraction out and then tidy it.

The method it uses

Count the decimal places. That many zeros gives the denominator: one place is 10, two places is 100, three is 1000. Put the digits over that denominator, then reduce the fraction by dividing top and bottom by their greatest common divisor. The tool finds that divisor with the Euclidean algorithm, the old repeat-the-remainder method for the greatest common divisor.

For 0.75, there are two decimal places, so the denominator is 100, giving 75/100. The greatest common divisor of 75 and 100 is 25, so both are divided by 25 to reach 3/4. A whole-number part is split off first, so 2.5 becomes the whole number 2 alongside 0.5, which reduces to 1/2, giving 2 1/2.

A worked example, step by step

Take 0.64.

• Two decimal places, so the denominator is 100: that gives 64/100.
• The greatest common divisor of 64 and 100 is 4.
• Divide both by 4: the answer is 16/25.

And 1.2 comes back as a mixed number: the whole part is 1, the 0.2 is 2/10, which reduces to 1/5, so the result is 1 1/5.

What it does with a rounded decimal

The tool converts exactly the number you type, digit for digit, and that is worth knowing. If you enter a rounded decimal like 0.33, it reads it faithfully as 33/100, not as 1/3. From the digits alone it has no way to know you meant the endless 0.333..., since a box can only hold a finite string of digits. So the fraction you get is a true and exact match for the decimal in front of it. If you are after 1/3, that value comes from the fraction itself, not from a shortened decimal of it.

The reverse trip: fraction to decimal

This tool goes from a decimal to a fraction. To travel the other way, you divide the top of the fraction by the bottom, which is what the fraction to decimal calculator does. It helps to remember that a fraction, a decimal, and a percent are three ways of writing the same amount, and each conversion is just a move between two of those forms.

Questions people ask

How does it choose the denominator?

By counting the decimal places. Each place adds a zero to the denominator, so two places give 100, three give 1000, and the digits sit on top of that.

Does it simplify the fraction?

Yes. After writing the fraction over a power of 10, it reduces to lowest terms by dividing through by the greatest common divisor.

Can it handle a number bigger than 1?

Yes. It splits off the whole-number part and converts the decimal part, so 2.5 comes back as 2 1/2.

Will 0.33 give me 1/3?

No, it gives 33/100, because it converts the exact digits you typed. The value 1/3 is an endless decimal, which a finite entry cannot represent.

Does it take negative decimals?

It is built for positive decimals, the usual case of turning a value like 0.375 into a fraction.

References

On decimal place value and the reducing step. A terminating decimal is a fraction whose denominator is a power of 10, read straight from the number of decimal places, and putting that fraction in lowest terms is done with Euclid's greatest common divisor.

1. Eric W. Weisstein, "Decimal Expansion," from MathWorld, a Wolfram resource, on how a number is represented in decimal places.
2. Euclid, Elements, Book VII, Proposition 2 (c. 300 BC), the Euclidean algorithm for the greatest common divisor, in David E. Joyce's online edition: Euclid's Elements, Book VII, Proposition 2.