Class 9 Mathematics Chapter 3 The World of Numbers – Revision Notes

Real numbers split neatly into rational and irrational types, and this chapter shows how to place a number like root 2 on the number line using a simple Pythagorean construction.

Last Updated: September 23, 2026

Number Types

• Rational: p/q form, terminating or repeating decimal.
• Irrational: non-terminating, non-repeating decimal (√2, π).
• Real numbers = rational + irrational.

Operations

• rational ±×÷ rational = rational.
• rational ± irrational = irrational.
• irrational × irrational = rational OR irrational (depends).

Number Line Representation

• √2 via Pythagoras construction (legs=1,1 → hypotenuse=√2).

Laws of Exponents

• a^m×aⁿ=a^m+n; a^m/aⁿ=a^m-n; (a^m)ⁿ=a^mn; a^mb^m=(ab)^m.

Rationalisation

• Convert irrational denominator to rational by multiplying by suitable term.

One-Line Summary

Real numbers combine rationals (p/q form) and irrationals (non-terminating, non-repeating decimals), both representable on the number line, governed by exponent laws and rationalisation techniques for simplification.

Quick visual: a worked diagram from the full Solutions page, for reference.

Frequently Asked Questions

What is the key difference between rational and irrational numbers covered in this chapter?
A rational number can be expressed as a ratio of two integers with a non-zero denominator and has a terminating or repeating decimal expansion, while an irrational number cannot be expressed this way and has a non-terminating, non-repeating decimal expansion.

How are irrational numbers represented on the number line in this chapter?
Irrational numbers like the square root of 2 are represented on the number line using geometric constructions based on the Pythagoras theorem, since they cannot be plotted directly using simple ratio-based methods.

Written by Satish

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