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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.

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Student-friendly explanation

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.

How to write this in exams

  1. 1

    Start with the exact idea

    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.

  2. 2

    Then show how to use it

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

  3. 3

    Add one concrete example

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

  4. 4

    Avoid this incomplete answer

    Calling √25 irrational is wrong because √25 = 5, which is rational.

Definition

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.

Example

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

Rule to remember

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

Memory hook

Perfect square root is rational; non-perfect square root stays irrational.

Examples and method

Worked example

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

Method to apply

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

Diagram support

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

How CBSE asks it

Students may be asked to identify irrational numbers from a list or justify why √2 is irrational at an introductory level.

Avoid common mistakes

Common confusion

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

Common wrong answer

Calling √25 irrational is wrong because √25 = 5, which is rational.

Exam tip

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

Quick check

Is √16 irrational? Give the reason.

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

Study the irrational numbers diagram carefully

Use the labelled diagram to keep irrational numbers clear in short answers and revision.

What this diagram makes clear

This diagram keeps the labels and direction of irrational numbers in the right order.

Where this helps in exams

Use this for labelled diagram work and short exam answers on irrational numbers.

Revision cue

Revise irrational numbers through the labels before writing the answer.

Answer writing and exam use

1-mark answer

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.

2-mark answer

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. Square root rule: √n is irrational when n is a positive integer that is not a perfect square. √2 is irrational because it cannot be written exactly as a fraction of two integers, and its decimal form continues without repeating.

3-mark answer

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. Square root rule: √n is irrational when n is a positive integer that is not a perfect square. Classify √18. Since √18 = √(9 × 2) = 3√2 and √2 is irrational, √18 is irrational. Students may be asked to identify irrational numbers from a list or justify why √2 is irrational at an introductory level. Calling √25 irrational is wrong because √25 = 5, which is rational.
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