The Number System

Rational vs. Irrational Numbers

Distinguishing rational numbers from irrational numbers like √2 and π.

Intermediate15 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Numbers That Never Settle Into a Pattern

Every number you've worked with so far — whole numbers, fractions, decimals, even repeating decimals like 0.333... — can be written as one whole number divided by another. Before reading on, think about √2, the length of the diagonal of a 1-by-1 square. Do you think it's possible to write that exact length as a fraction too?

Definition — Rational and Irrational Numbers

A rational number can be written as a fraction of two integers, and its decimal form either terminates or repeats in a fixed pattern forever. An irrational number cannot be written as such a fraction — its decimal form goes on forever without ever settling into a repeating pattern.

It turns out √2 genuinely cannot be written as a fraction — a fact mathematicians in ancient Greece discovered and found deeply surprising, since it meant not every length could be measured using whole-number ratios. Its decimal expansion, 1.41421356..., continues forever with no repeating block, which is exactly the signature of an irrational number.

Worked Example — Classifying Numbers as Rational or Irrational

Classify 0.75, √9, and π. 0.75 is rational — it's exactly 3/4. √9 is rational too, since it equals exactly 3, a whole number, even though it involves a square root symbol. π is irrational — its decimal expansion, 3.14159265..., never terminates or repeats.

Worked Example — Recognizing a Perfect Square Root as Rational

Is √64 rational or irrational? Since 8 × 8 = 64, √64 equals exactly 8, a whole number — rational. Only square roots of numbers that aren't perfect squares, like √2 or √50, are irrational.

Tip

A square root symbol doesn't automatically mean irrational — check whether the number underneath is a perfect square (1, 4, 9, 16, 25, ...) first, since those square roots simplify to whole numbers.

Common Mistakes

Assuming every number written with a square root symbol must be irrational.
Check whether the number inside the radical is a perfect square — √25 = 5 is perfectly rational, even though it's written with a radical symbol.
Assuming a long decimal, like 0.121212..., must be irrational just because it has many digits.
Check whether the decimal eventually settles into a repeating block — 0.121212... repeats '12' forever, which makes it rational (equal to 12/99), regardless of how long it looks.

Key Takeaways

A rational number can be written as a fraction of two integers, with a decimal form that terminates or repeats.
An irrational number's decimal form continues forever without ever repeating, and cannot be written as such a fraction.
A square root is rational exactly when the number underneath is a perfect square.

Summary

Irrational numbers exist because not every length or ratio reduces to a fraction of whole numbers. The next lesson finds a practical way to work with these never-ending numbers — estimating their value closely enough to be useful.

Track your progress

Back to The Number System Estimating Irrational Numbers→