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Irrational numbers

An irrational number cannot be written in the form p/q, where p and q are integers and q != 0.

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Student-friendly explanation

These numbers have decimal expansions that never end and never repeat. In Class 10, square roots like sqrt(2), sqrt(3), and sqrt(5) become important examples. They are not fractions, even if they look neat on the number line.

How to write this in exams

  1. 1

    Start with the exact idea

    An irrational number cannot be written in the form p/q, where p and q are integers and q != 0.

  2. 2

    Then show how to use it

    1. Try to write the number as a fraction. 2. Check its decimal pattern. 3. If it neither terminates nor repeats, classify it as irrational. 4. Use proof questions when the number is a square root.

  3. 3

    Add one concrete example

    sqrt(2) and pi are irrational numbers.

  4. 4

    Avoid this incomplete answer

    Thinking 0.333... is irrational is wrong because it repeats and equals 1/3.

Definition

An irrational number cannot be written in the form p/q, where p and q are integers and q != 0.

Example

sqrt(2) and pi are irrational numbers.

Rule to remember

An irrational number cannot be expressed as p/q with integers p and q, where q != 0.

Memory hook

Rational repeats or ends; irrational does neither.

Examples and method

Worked example

sqrt(5) is irrational because it cannot be written exactly as a fraction and its decimal continues without repetition.

Method to apply

1. Try to write the number as a fraction. 2. Check its decimal pattern. 3. If it neither terminates nor repeats, classify it as irrational. 4. Use proof questions when the number is a square root.

Diagram support

A number line can show irrational numbers lying between two nearby rational numbers.

How CBSE asks it

Identify irrational numbers, compare rational and irrational decimals, or justify a number classification.

Avoid common mistakes

Common confusion

Students call every non-terminating decimal irrational and forget that repeating decimals are rational.

Common wrong answer

Thinking 0.333... is irrational is wrong because it repeats and equals 1/3.

Exam tip

Check both conditions: the decimal should not end and should not repeat.

Quick check

A decimal never ends and never repeats. What conclusion should you draw if it comes from sqrt(2)?

The number is irrational, because it cannot be written as p/q and its decimal does not repeat.

Answer writing and exam use

1-mark answer

An irrational number cannot be written in the form p/q, where p and q are integers and q != 0.

2-mark answer

An irrational number cannot be written in the form p/q, where p and q are integers and q != 0. An irrational number cannot be expressed as p/q with integers p and q, where q != 0. sqrt(2) and pi are irrational numbers.

3-mark answer

These numbers have decimal expansions that never end and never repeat. In Class 10, square roots like sqrt(2), sqrt(3), and sqrt(5) become important examples. They are not fractions, even if they look neat on the number line. An irrational number cannot be expressed as p/q with integers p and q, where q != 0. sqrt(5) is irrational because it cannot be written exactly as a fraction and its decimal continues without repetition. Identify irrational numbers, compare rational and irrational decimals, or justify a number classification. Thinking 0.333... is irrational is wrong because it repeats and equals 1/3.
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