Math
Rational or Irrational Calculator
Type any number the way you would write it, such as 0.75, 3/4, 0.(3), 0.333..., √50, ∛27, or 2π, and the calculator says whether it is rational or irrational and why.
Rational inputs are converted to a fraction in lowest terms with their exact decimal expansion, including the repeating block. The second mode takes a fraction p/q and decides whether its decimal terminates by checking that the reduced denominator has only 2s and 5s as prime factors, then shows the decimal.
Decimals, fractions, percents, repeating decimals with parentheses or ..., roots (√, ∛, sqrt(), cbrt(), root(4, 81)), and π, e, τ, φ expressions.
Try an example
Result
Classification
—
- As a fraction
- —
- Decimal expansion
- —
- Decimal type
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- Why
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Student quick launch
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Study path
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Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
What the rational or irrational calculator solves
A rational number is any number that can be written as a fraction p/q of two integers with q ≠ 0. Every integer, every terminating decimal, and every repeating decimal is rational. An A number that cannot be written as a fraction of two whole numbers, like π or √2. number cannot be written that way; its decimal goes on forever without ever settling into a repeating block. Famous examples are √2, π, and e. This calculator reads what you type, applies the right test, and explains the verdict.
| Input | Verdict | Reason |
|---|---|---|
| 0.75 | rational | = 3/4, terminating decimal |
| 0.(3) or 0.333... | rational | = 1/3, repeating decimal |
| √49 | rational | = 7, perfect square |
| √50 | irrational | 50 is not a perfect square |
| ∛27 | rational | = 3, perfect cube |
| 2π, π/4, e + 1 | irrational | non-zero rational multiple or shift of a transcendental |
| 0.101001000… | cannot decide from digits | if the pattern never repeats it is irrational |
Terminating vs repeating decimals
Reduce the fraction. If the The bottom number of a fraction; it says how many equal parts make one whole.'s only A whole number above 1 whose only factors are 1 and itself. factors are 2 and 5, the decimal terminates, because the denominator can be scaled up to a power of ten: 7/40 = 7/(2^3 × 5) = 175/1000 = 0.175. Any other prime in the denominator forces a repeating decimal: 1/12 = 1/(2^2 × 3) = 0.08333… The number of decimal places of a terminating decimal is the larger of the The small raised number saying how many times to multiply the base by itself. of 2 and 5, and the period of a repeating decimal divides φ(q') where q' is the denominator with its 2s and 5s removed.
Why √50 is irrational
Suppose √50 = p/q in lowest terms. Squaring gives p^2 = 50 q^2, so p^2 is even, so p is even, so p^2 is divisible by 4, so 50 q^2 is divisible by 4, so q^2 is even, so q is even, contradicting lowest terms. The same argument works for any integer that is not a perfect square, and for cube roots of non-cubes. You can still simplify: √50 = 5√2.
How to use it
- Choose the classify mode and type the number as you would write it: decimal, fraction, repeating decimal, A root expression written with the √ symbol, such as a square root., or π/e expression.
- Read the verdict, the fraction in lowest terms (when rational), the decimal expansion, and the reason.
- Choose the terminate mode and enter p and q to see the 2s-and-5s test with the decimal expansion.
- Open Show the work for the repeating-decimal shift, the perfect-square check, or the denominator factorization.
How to read the answer
Rational means the number is a fraction of integers and its decimal either stops or repeats; the page shows both forms and the repeating block in parentheses. Irrational means no such fraction exists; the decimal shown is only an approximation. A result of 'cannot be determined' appears for decimals ending in … whose digits show no repeating block: the finite digits do not settle the question.
Common mistakes and edge cases
- Assuming a long decimal on a calculator screen is irrational. 1/7 = 0.142857142857… repeats with period 6; the screen just runs out of room.
- Calling √49 or ∛-8 irrational; roots of perfect powers are integers.
- Treating 22/7 or 3.14 as π. They are rational approximations; π itself is irrational.
- Forgetting to reduce before the 2s-and-5s test: 6/12 = 1/2 terminates even though 12 has a factor of 3.
- Thinking irrational means 'not a real number'. Irrational numbers are real; √-1 is imaginary, which is a different category.
Worked examples
√50
50 is not a perfect square.
Classification
Irrational
0.75
= 3/4.
Classification
Rational
0.(3)
Repeating decimal = 1/3.
Classification
Rational
0.142857142857...
The block 142857 repeats: 1/7.
Classification
Rational
∛27
Perfect cube: 3.
Classification
Rational
2π
Rational multiple of π.
Classification
Irrational
Negative fraction
-7/2 is rational.
Classification
Rational
Non-repeating pattern
0.101001000… cannot be decided from finite digits.
Classification
Cannot be determined
Does 7/40 terminate?
40 = 2^3 × 5 → 0.175.
Classification
Terminating decimal
Does 1/12 terminate?
12 has a factor of 3 → 0.08(3).
Classification
Repeating decimal
Unreadable input
Words are rejected.
Classification
Error
Frequently asked questions
What is a rational number?+
A number that can be written as p/q with integers p and q ≠ 0. Examples: 5, -3/4, 0.125, 0.(6) = 2/3.
What is an irrational number?+
A real number that is not a fraction of integers. Its decimal never terminates or repeats. Examples: √2, √50, π, e, φ.
Is 0.333… rational?+
Yes. 0.333… = 1/3. Every repeating decimal is rational; shift and subtract to find the fraction.
Is √50 rational or irrational?+
Irrational, because 50 is not a perfect square. It simplifies to 5√2 ≈ 7.0711.
How do I know if a fraction terminates?+
Reduce it and factor the denominator. If only 2s and 5s appear, it terminates: 7/40 = 0.175. Otherwise it repeats: 1/12 = 0.0833…
Is π rational?+
No. π is irrational (in fact transcendental). 22/7 and 3.14159 are only approximations.
Are all square roots irrational?+
No. √49 = 7 and √(9/4) = 3/2 are rational. A square root is irrational exactly when the radicand in lowest terms is not a perfect square over a perfect square.
Can a decimal that looks random be rational?+
Yes, if it eventually repeats, even with a very long period; 1/97 repeats every 96 digits. Finite digits can never prove irrationality.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026