Rational or Irrational Calculator
Enter any number — a decimal, a fraction like 3/4, a repeating decimal like 0.(3), or a constant like √2, π, or e — and get an exact classification with the p/q form, the decimal expansion, and a proof sketch. Everything runs 100% in your browser.
Classify a number
Type a number or expression, then press Classify.
- As a fraction (p/q)
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- Decimal expansion
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How the classification works
- Integers and fractions
a/b→ rational, by definition. - Terminating decimals (
0.75) → rational — they equal a fraction (3/4). - Repeating decimals (
0.(3),0.333…) → rational — every repeating decimal converts to an exact fraction (1/3). - √n and ∛n → rational only for perfect squares/cubes (
√4 = 2); otherwise irrational. - π, e, φ, ln 2 → irrational (each proven mathematically).
- A non-zero rational multiple of an irrational, or a rational ± an irrational, is still irrational.
Classic examples
Frequently asked questions
What is a rational number?
A rational number is any number that can be written as p/q, where p and q are integers and q ≠ 0. Every integer, every fraction, every terminating decimal (like 0.75 = 3/4), and every repeating decimal (like 0.333… = 1/3) is rational.
What is an irrational number?
An irrational number cannot be written as p/q for any integers p and q. Its decimal expansion never terminates and never settles into a repeating pattern. Famous examples are √2, π, e, and the golden ratio φ.
Is 0.333… (0.3 repeating) rational or irrational?
Rational. The repeating decimal 0.333… equals exactly 1/3. In general, every repeating decimal can be converted to a fraction — for example, 0.142857 repeating equals 1/7, and 0.1666… equals 1/6.
Is the square root of 2 rational?
No — √2 is irrational; it cannot be written as a fraction p/q. This was proved by the ancient Greeks: if √2 = p/q in lowest terms, then p² = 2q² forces both p and q to be even, contradicting “lowest terms”.
Is pi (π) rational or irrational?
Pi is irrational. Johann Lambert proved in 1761 that no fraction p/q equals π, and its decimal expansion 3.14159265358979… never terminates or repeats. Note that 22/7 = 3.142857… is only an approximation of π — and 22/7 itself is rational.
How can you tell if a number is rational from its decimal expansion?
Look at the decimal digits: if they terminate (0.75) or eventually repeat the same block forever (0.1666… = 1/6), the number is rational. If the digits continue forever with no repeating pattern — like π = 3.14159265… — the number is irrational.
Free tool by NRTechStudio