A rational number is any value that is the quotient of two integers, with a nonzero divisor. The fraction is the way of writing it and the rational number is the value: −6/3 and −2 are two writings of the same rational number, just as 1/2, 2/4 and 0.5 are. To know whether a decimal is rational, see if it ends or repeats a block of digits (the guide to terminating and repeating decimals details each type). √2 does neither, and that is why it is not a fraction.
| If you have… | Then | Example |
|---|---|---|
| A fraction of integers, nonzero denominator | It is rational | −6/3 = −2 |
| A decimal that ends | It is rational, with a power of 10 as the denominator | 0.375 = 375/1000 = 3/8 |
| A decimal with a block that repeats | It is rational, and a subtraction removes the tail | 0.(27) = 3/11 |
| A decimal that never ends and never repeats | It is irrational | √2 = 1.41421356… |
The table sums up the criterion: what decides is the shape of the complete decimal, not how many digits the screen shows. A number that is displayed with many digits can be rational, and one with few digits can be cut off. The definition matches that of the English Wikipedia article on rational numbers, which describes rationals as quotients of two integers and says their decimal expansion either terminates or eventually repeats (consulted October 17, 2026).
One number, many fractions
Multiplying the numerator and denominator by the same nonzero integer keeps the value, because you are multiplying by a fraction that equals one. So 1/2 becomes 2/4 with the factor 2/2, and dividing 2 and 4 by their greatest common factor brings back 1/2. None of those writings is a different number from the others.
The integers are in the set too: 5 = 5/1 and 0 = 0/1. The sign does not change the class, so −3/5 is rational. The only value ruled out is a zero denominator, and the reason is explained in the guide on why the denominator cannot be zero.
How to turn a repeating decimal into a fraction
Going from a repeating decimal to a fraction is done with a subtraction. For 0.(27), read “zero point twenty-seven repeating,” the repeating block has two digits.
Worked example. 0.(27) as a fraction
What fraction in lowest terms is x = 0.272727…?
- Shift one period. Since the block has two digits, multiply by 100: 100x = 27.2727…
- Subtract to erase the tail. 100x − x = 27, that is, 99x = 27.
- Solve. x = 27/99.
- Simplify. The greatest common factor of 27 and 99 is 9, and you get 3/11.
Result. 0.(27) = 3/11.
Check. Dividing 3 by 11 gives 0.2727…, the same decimal you started with.
If the decimal has digits before the repeating block, like 0.58(3), you first multiply by the power of 10 that puts them to the left of the point and then apply the same procedure. The result is 7/12. The repeating decimal to fraction calculator does the same calculation with any block, and the guide to terminating and repeating decimals explains when a decimal is of each type.
Which decimal each denominator gives
The decimal of a/b, with the fraction already in lowest terms, depends only on b. If b has only the factors 2 and 5, the decimal ends. Otherwise it repeats: remove the factors 2 and 5 from b, and the length of the repeating block is the smallest k such that 10^k − 1 is a multiple of what is left. It never exceeds b − 1.
| Fraction | Decimal | Repeating block |
|---|---|---|
| 3/8 | 0.375 | Ends, because 8 = 2³ |
| 1/3 | 0.(3) | 1 digit |
| 1/11 | 0.(09) | 2 digits |
| 1/7 | 0.(142857) | 6 digits |
| 1/13 | 0.(076923) | 6 digits |
| 7/12 | 0.58(3) | 1 digit after 2 that do not repeat |
In 7/12, the 12 = 4 × 3 contains two factors of 2, and that is why there are two digits before the 3 starts to repeat. With 1/7 and 1/13 the block is 6 digits long because 10⁶ − 1 is the first number of the form 10^k − 1 divisible by 7 and by 13. To see these steps on another fraction, use fraction to decimal or review the terminating case in fractions with a terminating decimal.
Where the irrationals begin
An irrational number cannot be written as an exact quotient of two integers, and its decimal neither ends nor repeats. √2 is the classic example: no fraction squared gives exactly 2. Writing 1.414 as an approximation does not make it rational, because 1.414 = 707/500 is a nearby, different rational, and the difference from √2 is less than 0.0003. The full proof is in the post why √2 is not a fraction.
A decimal having infinitely many digits is not enough for it to be irrational: 0.(3) has them and equals a third. The root symbol does not decide it by itself either, because √4 = 2 is rational and √2 is not. Classify the value once it is calculated, not the way it is written.
Where to go next
If you are looking for a short definition of a term, the fraction glossary gathers them in a table. To see a rational number as a quotient of integers in practice, the fraction calculator operates with them without going through decimals. If what interests you is going from a decimal to its fraction, the decimal to fraction calculator does it step by step.
Sources
- Wikipedia, Rational number, with the definition as a quotient of two integers with a nonzero denominator and the characterization by decimal expansions that terminate or eventually repeat (consulted October 17, 2026).