How to solve it
- Divide the numerator by the denominator.
- If the division ends, the decimal is exact.
- If a remainder repeats, the decimal is periodic: mark the repeating part.
- Round only if an approximation is requested.
3 ÷ 8 = 0.375
Result: 0.375
Common mistake:Wrong: 1/4 = 0.14.Right: 1/4 = 0.25.Writing the numerator and denominator on either side of a decimal point is not a division. 1/4 is 1 ÷ 4 = 0.25.
Type the fraction or mixed number, choose how many places you want to see, and select “Convert”. The answer uses the sign = if the whole decimal fits in those places and ≈ if it had to be rounded.
Dividing 5 by 12 with remainders
A board is 5/12 of a foot long, and you want to write that length as a decimal. How much is it?
- Divide the numerator by the denominator. 5 ÷ 12 gives 0 wholes with remainder 5, so you continue with decimals.
- Bring down a zero each time: 50 ÷ 12 = 4, remainder 2; then 20 ÷ 12 = 1, remainder 8; then 80 ÷ 12 = 6, remainder 8.
- The remainder 8 has already appeared, so the 6 repeats forever. The digits 4 and 1 come first and the 6 goes in parentheses: 0.41(6).
Result: 5/12 = 0.41(6), that is, 0.41666… feet.
Check: at 2 places, 0.42 × 12 = 5.04, very close to 5; the difference is 0.04.
Three kinds of decimal
When you divide, one of three things happens: the division ends, the digits repeat from the decimal point, or they repeat after a few fixed digits. You can tell which by looking at the denominator, always with the fraction in lowest terms. That is why 12/30 is reduced first to 2/5 and gives 0.4, even though 30 has a factor 3.
| Kind | Reduced denominator | Example |
|---|---|---|
| Terminating | Only factors 2 and 5 | 3/8 = 0.375, because 8 = 2³ |
| Purely repeating | No factor 2 or 5 | 1/3 = 0.(3) |
| Mixed repeating | Factors 2 or 5 and at least one other prime | 5/12 = 0.41(6), because 12 = 2² × 3 |
If you see digits that keep going, looking at the screen is not enough: not having ended among the visible digits does not prove that a decimal repeats. The denominator test gives the answer without dividing, and the guide to fractions with terminating decimals explains why only 2 and 5 count. The difference between 0.333… and decimals that never repeat is in exact and repeating decimals.
How the result is written
For 1/3 there are three ways of writing it that should not be mixed up. 0.(3) and 0.333… are the same number, written with parentheses or with an ellipsis; on paper it is written with a bar over the 3 (0.3̅). ≈ 0.33 is a rounded approximation, and the calculator marks it with ≈ so it is not mistaken for the value. Here 0.75 is written with a decimal point; in much of Europe it is written 0,75. The calculator accepts both.
The display rounds to the nearest value and, in a tie, to the one farther from zero. The limits are practical: the denominator cannot be 0, each number takes up to 100 digits, and the search for the repeating block examines 2,000 remainders.
Common fractions as decimals
The table lets you read the most-used conversions without calculating. The right-hand column shows the kind of decimal, and any repeating digits are in parentheses.
| Fraction | Decimal | Kind |
|---|---|---|
| 1/2 | 0.5 | terminating |
| 1/4 | 0.25 | terminating |
| 1/5 | 0.2 | terminating |
| 1/6 | 0.1(6) | mixed |
| 1/7 | 0.(142857) | pure |
| 1/9 | 0.(1) | pure |
| 1/11 | 0.(09) | pure |
| 1/12 | 0.08(3) | mixed |
| 1/16 | 0.0625 | terminating |
| 1/20 | 0.05 | terminating |
For more conversions, percents included, there is the fraction, decimal and percent table.
If you need the reverse or another form
To turn a terminating decimal such as 0.375 into a fraction, use decimal to fraction. If the decimal repeats, as in 0.41(6), repeating decimal to fraction returns the fraction that produces it. To express the value per hundred, there is fraction to percent. And if you would rather have a simple fraction that is close to a decimal with many digits, approximate a decimal as a fraction accepts a maximum denominator. OpenStax explains converting between fractions and decimals with practice problems.
Frequently asked questions
How many digits can the repeating block have?
At most one fewer than the denominator, because long division by b can only leave the remainders 1 to b − 1. That is why 1/7 has 6 digits and 1/97 reaches 96. The calculator looks for the block among the first 2,000 remainders and tells you if it does not find one.
How many digits come before the repeating block?
As many as the largest exponent of 2 or 5 in the reduced denominator. In 1/60 = 0.01(6), the denominator is 2² × 3 × 5, the largest exponent is 2, and two digits come before the block. If there is no 2 or 5, as in 1/11, the block starts right after the decimal point.
How do I convert a mixed number or a negative fraction?
Type the mixed number with a space, such as “2 3/4”, and the calculator turns it into 11/4 before dividing: you get 2.75. For a negative, type “-9/20” or “9/-20”: it divides the absolute value and keeps the sign, so the result is −0.45.
What does it mean when the full repeating block was not found?
It means the block is longer than 2,000 digits. That happens with 1/2017, whose block has 2,016 digits. The decimal still repeats and the fraction is still exact; only the display is rounded, with ≈. To see more digits, raise the places, up to 200.