All About Numbers
Rational vs Irrational Numbers
Having met rational and irrational numbers separately, it's worth putting them side by side. Every real number is one or the other – never both, and never neither – and the fastest way to tell them apart is to look at the decimal expansion.
Side-by-Side Comparison
| Feature | Rational | Irrational |
|---|---|---|
| Can be written as p/q? | Yes | No |
| Decimal expansion | Terminates or repeats | Never terminates, never repeats |
| Examples | 1/2, 5, 0.75, 0.333… | π, e, √2, √3 |
| Symbol | ℚ | (no single standard symbol; often ℚ′ or ℝ\ℚ) |
The Decimal Test
Look at the decimal expansion: if it stops (terminates) or eventually repeats a block of digits forever, the number is rational. If it goes on forever with no repeating pattern, it's irrational.
A repeating decimal is always rational – this one equals 5/11. It is Rational.
7 is not a perfect square, so √7's decimal never terminates or repeats. It is Irrational.
A Common Misconception
Not every square root is irrational! √16 = 4, a whole number – because 16 is a perfect square. Only square roots of non-perfect squares are irrational.
Key Takeaways
- Every real number is either rational or irrational, never both.
- A terminating or repeating decimal is rational; a non-terminating, non-repeating decimal is irrational.
- Square roots of perfect squares are rational; square roots of non-perfect squares are irrational.