Chapter 3 — The World of Numbers

Chapter 3 of the Class 9 Maths NCERT textbook, "The World of Numbers", traces how number systems grew from natural numbers and zero to integers, rationals, irrationals and real numbers, covering Brahmagupta's rules, decimal expansions and the proof that sqrt(2) is irrational.

• How Numbers Evolved — Numbers grew to meet human needs, from counting by one-to-one correspondence to the Indian invention of zero (Shunya), which Brahmagupta formalised with arithmetic rules in 628 CE.
• Integers and Rationals — Integers extend counting numbers with zero and negatives, framed as fortunes and debts. Rational numbers, written p/q, are densely packed on the line, with one always sitting between any two.
• Irrational Numbers — Some quantities like sqrt(2) and pi cannot be written as p/q. A proof by contradiction establishes this irrationality, expanding the number line beyond fractions.
• Decimals and Real Numbers — Decimal expansions separate rationals (terminating or repeating) from irrationals (never repeating). Together these families unite into the single system of real numbers.

1. 01Natural numbers arose from counting using one-to-one correspondence
2. 02Brahmagupta formalised zero and its arithmetic rules in 628 CE
3. 03Integers (Z) include positive numbers, zero and negative numbers (debts)
4. 04Rational numbers are p/q with integers p, q and q not equal to 0
5. 05Rational numbers are dense; one exists between any two via their average
6. 06Proof by contradiction shows sqrt(2) is irrational; pi is also irrational
7. 07Rational decimals terminate or repeat; irrational decimals never repeat