medium difficulty5 min

Irrational Numbers

An irrational number is a real number that cannot be written as p/q, where p and q are integers and q is not zero.

In one line: Perfect square root is rational; non-perfect square root stays irrational.

Understand it

Irrational numbers have decimal expansions that neither terminate nor repeat. Common examples include √2, √3, √5, and π. A square root of a non-perfect square is irrational.

Example

√2 is irrational because it cannot be written exactly as a fraction of two integers, and its decimal form continues without repeating.

Formula and rule

Square root rule: √n is irrational when n is a positive integer that is not a perfect square.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

1Classify √18.
2Since √18 = √(9 × 2) = 3√2 and √2 is irrational, √18 is irrational.

Method you can reuse

1For square roots, first check whether the radicand is a perfect square.
2If yes, simplify to an integer.
3If not, treat it as irrational unless it simplifies fully to a rational number.

Visual tip

A comparison table can separate perfect-square roots like √9 from non-perfect-square roots like √2.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Is √16 irrational? Give the reason.

Think first, then show answer

No, √16 is not irrational because √16 = 4, and 4 can be written as 4/1, so it is rational.

Common wrong answer

Calling √25 irrational.

Why it loses marks: √25 = 5, which is rational.

Don't mix this up

Students sometimes say every square root is irrational, but √4 = 2 and √9 = 3 are rational numbers.

Exam tip

Check whether the number under the square root is a perfect square before deciding whether the square root is rational or irrational.