Fraction Calculator
Add, subtract, multiply and divide fractions, simplify them, and convert between improper fractions, mixed numbers and decimals.
| Simplified fraction | — |
|---|---|
| Decimal | — |
| Mixed number | — |
| Percentage | — |
A third survives floating point. A tenth does not.
Fractions are usually treated as the schoolroom version of decimals — the thing you convert away from as soon as you have a calculator. The case for keeping them exact is stronger than that, and it is not the case people usually make. Floating point is not unreliable in a way you can predict and route around. It is unreliable arbitrarily.
| Calculation | In floating point | As fractions |
|---|---|---|
| 0.1 + 0.2 | 0.30000000000000004 ✗ | 3/10 ✓ |
| a tenth, ten times | 0.9999999999999999 ✗ | 1 ✓ |
| a third, three times | 1 ✓ | 1 ✓ |
Look at the last two rows. Adding a tenth ten times misses 1. Adding a third three times hits it exactly. Both fractions repeat forever in binary, neither is stored exactly, and nothing about the two problems distinguishes them — one is simply lucky with how the rounding errors land.
Trying every value from 2 to 100 makes the shape clear. Adding 1/n to itself n times reaches
exactly 1 for only 11 of the 99:
2, 3, 4, 5, 8, 12, 16, 17, 32, 34, 64. Six of those are powers of two, where success is
guaranteed because 1/2, 1/4 and 1/8 are stored perfectly. The other five —
3, 5, 12, 17, 34 — succeed purely by luck. No power of two ever
fails, and everything else is a coin flip that usually loses.
That is what makes it dangerous rather than merely imprecise. "It worked when I tested it" is not evidence about anything, because whether it worked was not a property of the calculation being easy.
Exactness is not free — the denominators grow
Fractions have none of that behaviour. 1/10 + 1/5 is 3/10 exactly, and — unlike floating point
— regrouping a sum cannot change the answer. In floating point it can:
(10¹⁶ + −10¹⁶) + 1 is 1, while 10¹⁶ + (−10¹⁶ + 1) is 0. Same
three numbers, same operations, different answer depending on the brackets.
The price of that reliability is size. Every addition multiplies the denominators before reducing, so a long chain of unrelated fractions gets heavy fast. Adding 1/1 + 1/2 + 1/3 + … and watching the denominator:
| Terms added | Digits in the denominator |
|---|---|
| 10 | 4 |
| 25 | 10 |
| 50 | 22 |
| 100 | 40 |
| 200 | 89 |
By two hundred terms the denominator is 89 digits long. A double-precision float would still be eight bytes, and still approximately right.
So the honest trade is not "fractions are better". Floats are fixed-size and approximately
correct; fractions are exactly correct and grow without bound. For money, for measurements
recorded as fractions, and for anything you will compare with ===, exactness is
worth the growth. For physics, graphics and statistics — where the inputs were approximations
to begin with — carrying an 89-digit denominator buys you nothing at all.
How to use
- Enter your fractions and pick an operation.
- Read the result in lowest terms.
- See the mixed number and decimal equivalents.
- Check the working if the answer surprises you.
Frequently asked questions
Why do fractions need a common denominator to add?
Because the denominator states what size the pieces are, and you cannot add quantities of different-sized pieces directly. Converting to a common denominator restates both in the same units, after which the numerators add. Multiplication needs no such step, which is why it feels easier than addition.
How are fractions simplified?
By dividing top and bottom by their greatest common divisor, found efficiently with Euclid's algorithm — repeatedly replacing the larger number with its remainder on division by the smaller. It is over two thousand years old and still the method computers use.
Why is dividing by a fraction the same as multiplying by its reciprocal?
Because dividing asks how many of the divisor fit into the dividend. Dividing by a half asks how many halves fit, and the answer is twice as many — which is multiplying by two, the reciprocal. The rule is not arbitrary; it follows from what division means.
Which fractions give repeating decimals?
Those whose denominator, in lowest terms, has a prime factor other than 2 or 5. One third repeats because 3 divides no power of ten; one eighth terminates because 8 divides 1000. The length of the repeating block relates to the denominator in a surprisingly intricate way.
Is 0.999 recurring really equal to 1?
Yes, exactly — not approximately. One third is 0.333 recurring, and multiplying both sides by three gives the result directly. It is a genuine equality between two ways of writing the same number, and one of the most reliably contested facts in mathematics.
Why do fractions matter if decimals exist?
Because fractions are exact and many decimals are not. One third has no finite decimal representation, so any decimal you write is an approximation that accumulates error through a calculation. Exact fraction arithmetic avoids that entirely, which is why algebra keeps them.
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