Fraction to Decimal Calculator
Result
As a decimal
- Fraction in lowest terms
- 3/8
- Repeating digits (0 = terminating)
- 0
A fraction to decimal calculator divides the numerator by the denominator and reports three things: the quotient as a decimal, the fraction itself reduced to lowest terms, and the length of the repeating cycle in the decimal expansion. Three eighths is 0.375 and its decimal stops, so the cycle length is zero. One third is 0.333333 on the display and its decimal never stops, so the cycle length is one — a single digit repeats forever. That third number is what separates this page from a plain division. Every fraction has a decimal expansion, and that expansion falls into exactly two kinds: it stops after finitely many digits, or it goes on forever with a block of digits repeating. There is no third possibility, and which kind you get is decided entirely by the denominator before anything is divided. Reduce the fraction first — this page does that for you, so two quarters is reported as one half — and look at what the reduced denominator is built from. If its only prime factors are twos and fives, the decimal stops, and the number of places is however many factors of two or five the longer of the two runs to. If the denominator has any other prime factor, the decimal repeats, and it repeats forever. The cycle length is then a property of that denominator alone, which is why it is a useful thing to print: one seventh repeats every six digits, one ninety-seventh every ninety-six, and the fraction twenty-two sevenths — a very good approximation of pi — repeats every six as well, because seven is the denominator underneath. Reading the outputs in the other order is worth doing once. The reduced fraction tells you whether the decimal will stop; the decimal tells you what it is to six places; the cycle length tells you how much of it you are not seeing. When that third output is zero, the six digits are the whole answer and nothing was lost. When it is not, the six digits are a rounded view of an expansion that has no end — one third is displayed as 0.333333, which is not one third but one third rounded at the sixth place, and the difference is 0.000000333 and does not go away. Two inputs are refused. A denominator of zero has no answer, and a numerator that is not a whole number is rejected rather than converted, since a mixed number has to be turned into an improper fraction before it can be divided by anything.
One over each denominator from two to twelve
| Denominator | Decimal | Repeating period |
|---|---|---|
| 2 | 0.5 | 0 |
| 3 | 0.333333 | 1 |
| 4 | 0.25 | 0 |
| 5 | 0.2 | 0 |
| 6 | 0.166667 | 1 |
| 7 | 0.142857 | 6 |
| 8 | 0.125 | 0 |
| 9 | 0.111111 | 1 |
| 10 | 0.1 | 0 |
| 11 | 0.090909 | 2 |
| 12 | 0.083333 | 1 |
Eleven unit fractions, and the two columns that matter are the first and the last — the denominator and its cycle length, with the decimal in between as the thing you would otherwise be looking up. Read down the period column and the pattern is the whole theory of this page: 2, 4, 5, 8 and 10 give zero because those numbers are built from twos and fives alone; 3, 6, 9 and 12 give 1 because a single three sits inside each of them; 11 gives 2, its own short cycle; and 7 gives 6, the longest on the table. Read the decimal column next to a nonzero period and the point becomes concrete: thirds, sixths, twelfths and ninths all display as approximations — 0.333333, 0.166667, 0.083333 — while the terminating rows display exactly. The seventh row is worth a second look, because 0.142857 is the same six-digit block that appears in twenty-two sevenths, and it is the reason that fraction makes such a good approximation of pi.
Formula
Decimal = numerator ÷ denominator Reduced fraction = numerator/denominator with the greatest common divisor cancelled Repeating period = the length of the digit block that repeats (0 if the decimal terminates)
- Numerator
- The top of the fraction, a whole number. It may be negative or zero; a numerator of zero gives a decimal of zero, and a fraction whose numerator is larger than its denominator gives a decimal above one
- Denominator
- The bottom of the fraction, a whole number up to a hundred thousand and not zero. This is the field that decides whether the decimal stops or repeats, and if it repeats, how long the cycle is — a fact about the denominator alone
- Decimal
- The numerator divided by the denominator, shown to six places. When the decimal terminates these six places contain the whole answer with trailing zeros; when it repeats they are the expansion rounded at the sixth place, not a truncated or exact form of it
- Reduced fraction
- The fraction in lowest terms, which is the form its repeating behaviour has to be judged in: 2/4 is reported as 1/2, and it is the 2 in that denominator that decides the decimal stops, not the 4 you typed
- Repeating period
- How many digits long the repeating block is, or zero if the decimal terminates. One third gives 1, one seventh gives 6, one ninety-seventh gives 96 — and the number depends only on the reduced denominator, so every fraction with ninety-seven underneath has a cycle of ninety-six regardless of its numerator
Three uses, in increasing order of how much the third output helps. Converting a fraction to a decimal for measuring or for a spreadsheet is the plain one: three eighths is 0.375, one half is 0.5, and the four fractions with small denominators are worth knowing outright. Checking whether a fraction will convert exactly is the second, and it is the question that matters when the answer has to go into a fixed number of decimal places — a denominator built only from twos and fives converts cleanly, and one containing a seven or a three never will, no matter how many places you allow. And estimating how much precision you are giving up is the third: a fraction with a cycle of one or two digits is represented well by six decimal places, while one with a cycle of ninety-six is not — one ninety-seventh displays as 0.010309, and the next six digits of the true expansion are entirely different digits, which is worth knowing before that number is used in a calculation. Two cases are outside what the page does. Turning a repeating decimal back into its exact fraction is the reverse direction and a separate problem, and a decimal printed to six places has thrown away the repeating structure that would solve it. And mixed numbers are not accepted as input: two and a quarter goes in as 9/4, because the division needs a single numerator and denominator.
Worked examples
Three eighths — a decimal that stops
- Divide the numerator by the denominator: 3 ÷ 8 = 0.375
- The division comes out exactly, so the repeating period is 0
- The fraction is already in lowest terms, so it is reported as 3/8
The default case and the easy kind: eight is built from twos alone, so the decimal terminates and the six places on the display hold the whole answer with zeros after it. The period output of zero is what says so — a zero here means nothing was rounded and nothing was thrown away, which makes it the first thing to look at when the exactness of a decimal matters.
One third — a decimal that never stops
- Divide: 1 ÷ 3 = 0.333333…
- Three is not built from twos and fives, so the expansion never ends
- The repeating block is the single digit 3, so the period is 1
- The display shows the first six places, 0.333333
The case the period output exists for, and the one where the decimal is least trustworthy. 0.333333 is not one third — it is one third rounded at the sixth place, and multiplying it by three gives 0.999999 rather than 1. The period of 1 tells you the display is a view of an infinite expansion rather than the whole answer; a period of zero on the same page would tell you the opposite. Adding a third and two thirds is the classic demonstration: the decimals give 0.999999 while the fractions give exactly 1.
Twenty-two sevenths — a long period and an improper fraction
- Divide: 22 ÷ 7 = 3.142857142857…
- Seven contributes a prime factor other than two or five, so the decimal repeats
- The repeating block is 142857, six digits long
- The display shows 3.142857, and the fraction is already in lowest terms
A fraction larger than one, so the decimal has a whole part, and a denominator of seven, whose cycle is six digits — the longest any one-digit denominator produces. Twenty-two sevenths is the classical approximation of pi, correct to two decimal places, and the repeating block 142857 is itself famous as the cycle of one seventh. The period output is doing real work here: it says the six digits shown are the full repeating block and not a rounded fragment of a longer one.
Two quarters is reported as one half
- Reduce first: 2 and 4 share a factor of 2, so the fraction is 1/2
- Divide the reduced form: 1 ÷ 2 = 0.5
- Two is built from twos alone, so the period is 0
The page reduces before it divides, and this is what that looks like: the fraction output is 1/2 and not the 2/4 that was typed. The value is identical either way, and the reduced form is the one the repeating behaviour has to be read from — the denominator that decides whether a decimal stops is the reduced one. Seeing this once explains why a fraction typed in a form you did not expect comes back in a different one.
One ninety-seventh — a cycle longer than the display
- Divide: 1 ÷ 97 = 0.010309278350515…
- Ninety-seven is prime and is neither two nor five, so the decimal repeats
- The cycle is 96 digits long
- The display shows the first six places, 0.010309
The extreme case, and the reason the period is worth printing even when it is large. Ninety-six digits is far more than the six on display, so what you are seeing is a sixteenth of the first repetition — a number that would have to be carried to a great many places before its pattern appeared. A period of 96 next to a six-place decimal is a warning: it does not mean the decimal is wrong, it means the decimal is a snapshot. Compare it with the one-third case, where the period of 1 makes the same six places a much better representation of the number.
Limitations
The decimal is shown to six places, so it is exact only when the repeating period is zero. When the period is greater than zero the number has no finite decimal form and the six digits are the expansion rounded at the sixth place — one third appears as 0.333333, which is not a third, and a third added to two thirds shows as 0.999999 rather than 1. A period larger than six means the display is showing only part of a single repetition. The page reduces before dividing, so a fraction entered in a form other than lowest terms comes back reduced: 2/4 is reported as 1/2 and 100/8 as 25/2, and the period, being a property of the reduced denominator, is computed from that reduced form. Whole results are printed as integers rather than as fractions over one, so 5/1 comes back as 5 and a numerator of zero as 0. Mixed numbers are not accepted: two and a quarter has to be entered as 9/4, because the division works on a single numerator and denominator, and a non-whole value in either field is refused rather than converted. Denominators are capped at a hundred thousand, which also bounds the period — the longest cycle any permitted denominator can produce is just under a hundred thousand digits. And the period is reported as a count of digits and not as the digits themselves: the page tells you that the expansion of one ninety-seventh repeats every ninety-six places without showing you which ninety-six digits they are.
Frequently asked questions
- How do I convert a fraction to a decimal?
- Divide the numerator by the denominator: three eighths is 3 ÷ 8 = 0.375, and one half is 1 ÷ 2 = 0.5. The division is the whole method — the only thing worth doing first is reducing the fraction to lowest terms, because the reduced denominator is what decides whether the decimal will stop or go on forever.
- How do I know whether the decimal will terminate?
- Look at the reduced denominator's prime factors. If it is built only from twos and fives, the decimal stops: 2, 4, 5, 8, 10, 16, 20, 25 and 100 all give terminating decimals. If it contains any other prime — a three, a seven, an eleven — the decimal repeats forever and cannot be written exactly in any finite number of places. The repeating period output tells you which case you are in: zero means the decimal terminates, and anything else is the length of the repeating block.
- Why does 1/3 show as 0.333333?
- Because the display carries six decimal places and a third has no finite decimal form, so the sixth place is where the expansion is rounded. 0.333333 is not exactly one third — multiply it by three and you get 0.999999 rather than 1, which is the arithmetic way of seeing the difference. The repeating period of 1 is the page telling you this: the decimal goes on, and what is shown is a view of it rather than the answer.
- Why is 2/4 shown as 1/2?
- Because the page reduces the fraction before dividing, and the reduced form is the one that matters for everything else it reports. The value is the same either way — 0.5 — but the repeating period is a property of the reduced denominator, so the reduction has to happen first: it is the 2 in 1/2 that says the decimal terminates, not the 4 you typed. Fractions entered in a higher form than necessary always come back reduced.
- What does a repeating period of 96 mean?
- That the decimal expansion of that fraction repeats every ninety-six digits — for one ninety-seventh, the block of ninety-six digits goes on forever. It is not a measure of how accurate the displayed decimal is: the six places shown are still correct as far as they go, they are simply a small part of a single repetition. What it does tell you is that six decimal places cannot represent the number well, since the digits that follow are not a continuation of the visible pattern but a much longer one.
- Can the answer be larger than one, or negative?
- Both are ordinary. A numerator larger than the denominator gives a decimal above one — twenty-two sevenths is 3.142857 — and the fraction output stays improper rather than becoming a mixed number. A negative numerator gives a negative decimal, and the sign stays on the numerator of the fraction: minus three quarters is minus 0.75 and prints as −3/4. The repeating period is unaffected by either, because it depends only on the denominator.
References
- Repeating Decimal — a decimal expansion in which a digit or block of digits repeats forever, the property the period output measures, together with the rule that a terminating decimal is one whose reduced denominator has no prime factor other than 2 or 5 — Wolfram MathWorld (United States)
- Decimal fraction — an arithmetical fraction with an integral power of ten as its denominator, which is exactly the class of fractions whose decimal expansion terminates — Encyclopedia of Mathematics, European Mathematical Society (international)
- Fraction — numerator, denominator and the reduction of a fraction to lowest terms, which is the step that must happen before the denominator's prime factors can be read — Wolfram MathWorld (United States)