How this calculator works
A rational number can be written as p/q where p and q are integers and q is not zero. Every terminating decimal qualifies because it is a fraction with a power of ten in the denominator, and every fraction qualifies by definition. A square root of a whole number is rational only when that number is a perfect square; a cube root is rational only when the radicand is a perfect cube. All other whole-number roots are irrational, and their decimals never terminate or repeat.
Formula
A square root √n is rational exactly when n = k² for an integer k. A cube root ∛n is rational exactly when n = k³. The decimal expansion of a rational number either terminates or eventually repeats; an irrational decimal does neither.
Worked example
√9 = 3 is rational, while √2 ≈ 1.41421356 is irrational. The decimal 0.375 is rational because 0.375 = 3/8. The fraction 22/7 is rational, even though its decimal 3.142857... is not the exact value of π.
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Frequently asked questions
Is every decimal a rational number?
Every terminating decimal is rational. A repeating decimal such as 0.333... is also rational. Only decimals that go on forever without repeating are irrational, and a typed decimal always has finitely many digits.
How do I know a square root is irrational?
Check whether the radicand is a perfect square. If it is not, the square root is irrational. The same test works for cube roots with perfect cubes.
Why is π irrational?
π cannot be written as a ratio of two integers; Johann Lambert proved this in 1761. Any finite decimal such as 3.14159 is a rational approximation, not the value itself.
Can an irrational number be written as a fraction?
No, by definition. A fraction of integers is rational. Irrational numbers can be approximated by fractions, and 22/7 is a well-known approximation of π, but the approximation is not equal to the irrational value.