What is an irrational number? Give examples.

An irrational number is a real number that cannot be expressed as p/q where p and q are integers and q ≠ 0. Its decimal expansion is non-terminating and non-repeating. Examples: √2 = 1.41421..., √3 = 1.73205..., π = 3.14159..., e = 2.71828...

How do I know if a square root is rational or irrational?

√n is rational if and only if n is a perfect square. Examples: √4 = 2 (rational, since 4 = 2²), √9 = 3 (rational), √16 = 4 (rational). But √2, √3, √5, √6, √7 are all irrational because 2, 3, 5, 6, 7 are not perfect squares.

Can the sum of two irrational numbers be rational?

Yes. For example, √3 + (−√3) = 0, which is rational. Similarly, (3 + √2) + (3 − √2) = 6, which is rational. However, √2 + √3 is irrational. So the sum of two irrationals may or may not be rational.

What Are Irrational Numbers?

Number Systems • Class 9 Mathematics • NCERT • CBSE

Irrational numbers are real numbers that cannot be expressed as a fraction p/q, where p and q are integers and q ≠ 0. Their decimal expansions are non-terminating and non-repeating. Examples include √2, √3, π, and e. They are denoted by Q' or ℝ\Q.

Key Formulas

Irrational numbers: cannot be expressed as p/q (q ≠ 0)
√n is irrational if n is not a perfect square
√2 × √2 = 2 (rational — product of two irrationals can be rational)
R = Q ∪ Q' (Real numbers = Rational ∪ Irrational)

Frequently Asked Questions

What is an irrational number? Give examples.: An irrational number is a real number that cannot be expressed as p/q where p and q are integers and q ≠ 0. Its decimal expansion is non-terminating and non-repeating. Examples: √2 = 1.41421..., √3 = 1.73205..., π = 3.14159..., e = 2.71828...
How do I know if a square root is rational or irrational?: √n is rational if and only if n is a perfect square. Examples: √4 = 2 (rational, since 4 = 2²), √9 = 3 (rational), √16 = 4 (rational). But √2, √3, √5, √6, √7 are all irrational because 2, 3, 5, 6, 7 are not perfect squares.
Can the sum of two irrational numbers be rational?: Yes. For example, √3 + (−√3) = 0, which is rational. Similarly, (3 + √2) + (3 − √2) = 6, which is rational. However, √2 + √3 is irrational. So the sum of two irrationals may or may not be rational.

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