Rational or Irrational Calculator (fractions, decimals, repeating decimals & roots)

Understanding the rational or irrational number calculator — concept and principle

This calculator decides whether a number is rational or irrational. It handles whole numbers, fractions and terminating decimals, repeating decimals, and roots of the form ᵏ√(p/q). For rational numbers it gives the exact fraction; for all it shows the decimal value and the type of decimal. Use it for algebra and number-sense homework.

A root of a whole number is rational only if the number is a perfect power; 50 is not.

How it is calculated

Whole numbers, fractions and terminating decimals are rational; the decimal is turned into a fraction in lowest terms.
Repeating decimals are rational: x = 0.1666… gives 100x − 10x = 15, so x = 15/90 = 1/6.
A root ᵏ√(p/q), with p/q in lowest terms, is rational only if p and q are both perfect k-th powers; otherwise it is irrational.
A fraction in lowest terms has a terminating decimal exactly when its denominator has no prime factors other than 2 and 5.

The principle

A rational number is any number that can be written as p/q with whole numbers p and q ≠ 0. Long division of p by q can produce only finitely many remainders, so the digits must either stop or start repeating. Conversely, any stopping or repeating decimal can be turned back into a fraction. Irrational numbers are the ones whose decimals never stop and never repeat. For roots, if ᵏ√n were a fraction a/b in lowest terms, then n = aᵏ/bᵏ, which forces n to be a perfect k-th power; that is why √2, √50 and ∛10 are irrational.

Worked example

Default inputs: root of a number, index k = 2, number under the root 50.

50 = 50/1 in lowest terms.
50 = 2 × 5², which is not a perfect square (the exponent of 2 is odd); 1 is a perfect square.
Since 50 is not a perfect square, √50 is Irrational.
√50 = 5√2 ≈ 7.071067812.
Its decimal never ends and never repeats, so no fraction p/q exists.

How to read the result

"Rational" comes with the exact fraction; use it instead of a rounded decimal. "Irrational" means any decimal you write is an approximation. Any decimal typed in full is rational, because it stops; irrational numbers only appear here through roots or special constants such as π, e and the golden ratio.

Common mistakes

Thinking a long decimal on a calculator screen is irrational; the screen just cuts it off.
Calling 22/7 equal to π; it is a rational approximation.
Assuming every root is irrational; √(9/4) = 3/2 is rational.
Assuming the sum of two irrational numbers is irrational; √2 + (−√2) = 0.

Keep in mind: The calculator decides from the exact form you enter. It cannot tell whether a decimal you copied from somewhere was rounded from an irrational number.

Frequently asked questions

Is 0 a rational number?

Yes. 0 = 0/1.

Is π rational if I write 3.14159?

3.14159 is rational, but it is only an approximation of π, which is irrational.

Is 0.999… equal to 1?

Yes. With x = 0.999…, 10x − x = 9, so x = 1.

Sources: OpenStax, Prealgebra 2e (2020), §7.1 Rational and Irrational Numbers

Checked 9 Nov 2026. OpenStax, Prealgebra 2e (2020), §7.1 Rational and Irrational Numbers (rational: p/q with integers p, q ≠ 0; every rational number has a decimal that stops or repeats; a decimal that neither stops nor repeats is irrational; square roots of whole numbers that are not perfect squares are irrational; π is irrational). Repeating-decimal-to-fraction method and the 2-and-5 rule for terminating decimals — standard results.

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