Q1. Rationalise the denominator of 1/√5.
Multiply numerator and denominator by √5
1/√5 × √5/√5 = √5/5
Number Systems — Class 9
Every number you know lives on the real line. Sort out rationals from irrationals and master surds and exponents.
Rational numberA number of the form p/q (q ≠ 0). Its decimal is terminating or repeating.
Irrational numberA number that cannot be written as p/q — its decimal is non-terminating and non-repeating (e.g. √2, π).
Real numbersRationals and irrationals together make up all the real numbers, each a point on the number line.
Surds & rationalisingA root like √3 is a surd. Rationalise a denominator by multiplying by the conjugate.
Solved examples
Q2. Is 0.101001000100001… rational or irrational?
The decimal never terminates and never repeats a fixed block.
So it is IRRATIONAL.
Formula sheet
| Quantity | Formula |
|---|---|
| Laws of exponents | aᵐ · aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ |
| Rationalising | 1/√a = √a / a |
| Product of surds | √a · √b = √(ab) |
Common mistakes & exam wins
- A terminating or repeating decimal is rational; a non-terminating non-repeating one is irrational.
- Between any two rational numbers there are infinitely many rationals (and irrationals).
- To rationalise a + √b in the denominator, multiply by the conjugate a − √b.
Frequently asked questions
What is the difference between rational and irrational numbers?
Rational numbers can be written as p/q with q ≠ 0 (terminating/repeating decimals); irrational numbers cannot, and have non-terminating non-repeating decimals.
Is π rational or irrational?
π is irrational — its decimal expansion never terminates or repeats.
What does it mean to rationalise a denominator?
To remove the surd (root) from the denominator by multiplying by a suitable factor, e.g. by the conjugate.