Decimal to Fraction Calculator

How it works

To write a decimal as a fraction, put its digits over a power of 10 with one zero for each decimal place, then divide the top and the bottom by their greatest common factor. 0.375 has three places, so it is 375/1000, and both numbers divide by 125, which leaves 3/8. 3.66 is 366/100 = 183/50, or 3 33/50 as a mixed number, because 183 = 3 × 50 + 33. 0.2 is 2/10 = 1/5, and a negative decimal keeps its sign: -0.75 = -3/4. The calculator uses exact whole-number arithmetic, so nothing is rounded.

Some decimals come from fractions that never end: 1/3 is 0.3333... and 1/7 is 0.142857142857.... If you type only the digits you can see, the exact fraction is huge: 0.3333333 is exactly 3,333,333/10,000,000. Closest fraction finds the best fraction below a denominator limit from the continued fraction of your number. 0.3333333 gives 1/3, 0.142857 gives 1/7 and 0.11428571428 gives 4/35. It also shows the difference, so you can judge the fit. With limits of 10, 100 and 1,000, 3.14159 gives 22/7, 311/99 and 355/113.

Rulers and tape measures stop at halves, quarters, eighths, sixteenths, thirty-seconds and sixty-fourths. Nearest 1/n multiplies the decimal by n, rounds to a whole number of steps (halves round up) and reduces: 0.4 in is 6.4 sixteenths, which rounds to 6/16 = 3/8 in, off by 0.025 in, and to the nearest 1/32 it is 13/32 in. The chart lists every 1/64 of an inch with its decimal and its millimeters; for inches to centimeters see the inches to cm converter.

To go from a fraction to a decimal, divide the top by the bottom: 1/8 = 0.125 and 7/8 = 0.875. A mixed number first becomes one fraction (the whole number times the denominator, plus the numerator): 2 4/8 = 20/8 = 2.5. In a negative mixed number the minus sign applies to the whole number and the fraction together, so -2 1/2 = -(2 × 2 + 1)/2 = -5/2 = -2.5. The decimal ends when the denominator, in lowest terms, has only 2 and 5 as prime factors (8 = 2 × 2 × 2, 625 = 5 × 5 × 5 × 5). Every other denominator repeats: 1/3 repeats one digit, 1/7 six digits and 1/97 ninety-six. The calculator reports the length of the repeating block. To add, subtract, multiply or divide fractions use the fraction calculator, and for a part of a total as a percent the percentage calculator.

The formula

The exact and closest modes read the decimal as typed (its shortest digit string) and work with whole numbers of any size, so 0.11428571428 becomes 11428571428/100000000000 before it is reduced to 2857142857/25000000000. Up to 15 decimal places are accepted. The closest fraction uses the convergents of the continued fraction and the last allowed semiconvergent, whichever is nearer, which is the best fraction with a bounded denominator.

Nearest 1/n rounds half up: 0.03125 is exactly halfway between 0 and 1/16 and gives 1/16. Fraction to decimal divides in floating point and shows 10 places. For the repeating block, take the denominator in lowest terms and remove every factor 2 and 5: if 1 is left, the decimal ends and the length is 0. Otherwise the length is the smallest number of digits k for which 10 to the power k leaves a remainder of 1 when divided by what is left.