The Number System
GradesGrade 8The Number SystemRational vs. Irrational Numbers

Rational vs. Irrational Numbers

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Rational vs. Irrational Numbers

Every real number is either rational (expressible as p/q) or irrational — never both.

A rational number can be written as a fraction p/q where p and q are integers and q ≠ 0. Its decimal expansion either terminates or repeats. An irrational number cannot be expressed as such a fraction; its decimal expansion never terminates and never repeats.

Rational Numbers

1/2 = 0.5 (terminates)1/3 = 0.333… (repeats)−7 = −7/1 (integer)0.75 = 3/4

Irrational Numbers

√2 = 1.41421356… (never repeats)π = 3.14159265… (never repeats)√3 ≈ 1.7320508…√5 ≈ 2.2360679…

If a square root simplifies to a whole number (e.g., √9 = 3), it is rational. Otherwise it is irrational.

4=2∈Q2∉Q\sqrt{4} = 2 \in \mathbb{Q} \qquad \sqrt{2} \notin \mathbb{Q}
🔎Classifying numbers
Is √25 rational or irrational?
√25 = 5 → rational (it is a perfect square).

Is √7 rational or irrational?
√7 ≈ 2.6457513… → irrational (7 is not a perfect square).
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Remember This!

Perfect squares (1, 4, 9, 16, 25, 36, 49, 64, 81, 100, …) have rational square roots. All other positive integers under a square root are irrational.

✏️ Try It!

Which of the following is an irrational number?

Take Quiz 📝 — 25 Questions