Decimal to Fraction Converter

How to convert a decimal to a fraction

A decimal is a fraction whose denominator is a power of ten, so the conversion is mostly a matter of writing that fraction down and reducing it.

1. Count the digits after the decimal point. Call that number n.
2. Write all the digits as a whole number and put it over 10ⁿ, which is a 1 followed by n zeros.
3. Divide the numerator and denominator by their greatest common divisor to reach lowest terms.
4. For a mixed number, divide the numerator by the denominator. The quotient is the whole part and the remainder becomes the new numerator.

Example: 0.375 has three decimal places, so it is 375/1000. The greatest common divisor of 375 and 1000 is 125, which reduces the fraction to 3/8.

Example with a whole part: 2.25 is 225/100. Dividing both by 25 gives 9/4, the simple fraction. Since 9 = 2 × 4 + 1, the mixed fraction is 2 1/4.

Repeating decimals need a small trick. Let x = 0.666… Then 10x = 6.666…, and subtracting the first line from the second leaves 9x = 6, so x = 6/9 = 2/3.

How the converter finds the fraction

The tool uses a continued-fraction search. It builds closer and closer fractions and stops as soon as one matches your number to within about 1e-16 of its size, which is the limit of double-precision arithmetic. The first box shows the simple fraction, which can be improper, and the second box shows the same value as a mixed number: a whole number alone (3), a proper fraction alone (3/4), or a whole part followed by a fraction (2 1/2).

The converter treats what you type as an exact decimal. That is why 0.3333 turns into 3333/10000 and not 1/3: those are different numbers. To recover a repeating decimal, enter all the digits a double-precision number can hold, which is about 16 or 17 significant digits.

Shorter truncations, such as 15 threes, are still valid decimals in their own right and give long fractions that look unfamiliar. If you only see an odd-looking fraction, check whether you typed a truncated repeating decimal.

Limits to keep in mind

• Use positive decimals. For a negative number the answer boxes fill with Infinity or NaN text. Convert the absolute value instead and put the minus sign in front of the result.
• Very large numbers, from roughly 1e17 upward, can overflow the search in the same way and print Infinity or NaN text.
• An empty box or 0 gives 0/1 and 0.
• Whole numbers show as n/1 in the simple box and as n alone in the mixed box.

Decimal to fraction conversion table

Related tools

Go back from a fraction to a decimal with the fraction to decimal converter. To reduce a fraction you already have, use the simplify fractions calculator, and to add, subtract, multiply or divide fractions try the fractions calculator. The decimal to percent converter covers the percentage form of the same number.