Chapter 3: The World of NumbersClass 9 Mathematics — summary, notes, extra questions & NCERT solutions

Summary

This chapter traces the expanding universe of numbers, from counting by one-to-one correspondence to the real number line. Natural numbers \(\mathbb{N}=\{1,2,3,\dots\}\) grew from the human need to count. Brahmagupta formalised zero (shunya) and the negative numbers, giving the integers \(\mathbb{Z}\) and rules such as \((-)\times(-)=(+)\). Combining integers with fractions gives the rational numbers \(\mathbb{Q}\), any number expressible as \(\dfrac{p}{q}\) with \(p,q\) integers and \(q\ne 0\). Rationals are dense — between any two there is always another, found by averaging. Some lengths, like the diagonal of a unit square \(\sqrt{2}\), cannot be written as a ratio; these are irrational, proved using proof by contradiction. Uniting rationals and irrationals forms the continuous real number line \(\mathbb{R}\). A number is rational exactly when its decimal expansion terminates or repeats, and irrational when it neither terminates nor repeats. The chapter shows how to convert repeating decimals into fractions and to predict, from prime factors of the denominator, whether a decimal terminates.

Key terms

Extra questions & answers

Exercise Set 3.1

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Exercise Set 3.2

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Exercise Set 3.3

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Exercise Set 3.4

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Think and Reflect

Exercise Set 3.5

Think and Reflect

End-of-Chapter Exercises

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