Class 9 · Chapter 3

The World of Numbers

Rationals, irrationals, real numbers, and decimal expansions.

Watch this chapter

Top CBSE tutorials for The World of Numbers

Curated video coming soon. For now, open the freshest CBSE-focused YouTube results — Magnet Brains, LearnoHub, Physics Wallah & more.

Read the full chapter

Ganita Manjari · Class 9 (2026-27)

Official NCERT PDF

Latest 2026-27 edition on ncert.nic.in

NCERT Book Portal

Browse every chapter of the latest book

CBSE Academic Portal

Official CBSE curriculum

Tip: NCERT updated the syllabus for 2026-27. The links above open the latest edition on the official portal — chapter titles and numbering may differ from the notes shown here.

Detailed theory

p/q

in lowest form: terminates iff denominator has only 2s and 5s as prime factors. Irrationals include

\sqrt 2, \sqrt 3, \pi, e

Key concepts & formulas

Rational

= p/q

q\ne 0

; terminating or non-terminating recurring decimal.

Irrational: non-terminating, non-recurring (e.g.,

\sqrt 2, \pi

Reals = rationals ∪ irrationals.

Laws of exponents extend to real numbers:

a^m a^n = a^{m+n}

, etc.

Rationalise denominator by multiplying by conjugate.