NCERT Solutions for Class 9th Maths Chapter 3 The World of Numbers
Updated on 2026-09-19
About this chapter
Natural numbers grew out of one-to-one correspondence; integers ℤ appear once Brahmagupta (628 CE) turns śhūnya into a number and reads negatives as debts (ṛiṇa) against fortunes (dhana). A rational number is any number of the form p/q with p, q integers and q ≠ 0. ℚ is closed under +, −, × and under ÷ except by zero. ℚ is dense: the average (a + b)/2 of any two rationals is a rational lying strictly between them, so between any two points there are infinitely many rationals. √2 is irrational. Hippasus' proof by contradiction assumes √2 = p/q in lowest terms, forces both p and q to be even, and contradicts the assumption. A rational number in lowest terms p/q has a terminating decimal exactly when q has no prime factor other than 2 and 5; otherwise a remainder must repeat, so the decimal r
- .1 The Dawn of Mathematics: The Human Need to Count
- .3.1 The Arithmetic of Integers
- .3 Integers: Expanding the Horizon
- .4 Filling the Spaces: Fractions and Rational Numbers
- .4.1 Representation of Rational Numbers on the Number Line
- .4.2 The Density of Rational Numbers
- .5 Irrational Numbers
- .5.1 The Proof of Irrationality of √2
- .5.2 Construction of Length √n
- .6.1 Rational Decimals: Terminating and Repeating
Quick revision
| Idea | Symbol / form | Test or rule | Example from the chapter |
|---|---|---|---|
| Natural numbers | ℕ = {1, 2, 3, …} | closed under + and ×, not under − | 3 − 5 is not a natural number |
| Integers | ℤ = {…, −1, 0, 1, …} | fortunes and debts; (−) × (−) = (+) | (−3) × (−4) = 12 |
| Rational number | p/q, p, q ∈ ℤ, q ≠ 0 | q ≠ 0; take p, q co-prime | 12/30 = 2/5 |
| Equality of rationals | a/b = c/d | true exactly when ad = bc | 2/3 = 4/6 since 2·6 = 3·4 |
| Absolute value | |x| | distance from 0; |a − b| = distance a to b | |−4 − 3| = 7 |
| Density | (a + b)/2 | always rational, always between a and b | between 1 and 3/2 lies 5/4 |
| Terminating decimal | p/q in lowest terms | q = 2^m · 5^n only | 3/20 = 15/100 = 0.15 |
| Repeating decimal | p/q in lowest terms | q has a prime other than 2, 5 | 5/11 = 0.4545… repeating |
| Number of decimal places | q = 2^m · 5^n | larger of m and n | 1/40 = 0.025, 3 places |
| Irrational number | cannot be written p/q | decimal never ends, never repeats | √2, √3, π |
| Proof by contradiction | assume the opposite | derive an impossibility | √2 = p/q forces p, q both even |
| Real numbers | ℝ = ℚ ∪ irrationals | every point of the line | the number line has no gaps |
Exercises
- .1 The Dawn of Mathematics: The Human Need to Count — Exercise Set 3.1 Page 433
- .3.1 The Arithmetic of Integers — Think and Reflect Page 463
- .3 Integers: Expanding the Horizon — Exercise Set 3.2 Page 463
- .4 Filling the Spaces: Fractions and Rational Numbers — Think and Reflect Page 473
- .4 Filling the Spaces: Fractions and Rational Numbers — Think and Reflect Page 493
- .4 Filling the Spaces: Fractions and Rational Numbers — Exercise Set 3.3 Page 493
- .4.1 Representation of Rational Numbers on the Number Line — Think and Reflect Page 513
- .4.2 The Density of Rational Numbers — In-text Questions Page 523
- .4.2 The Density of Rational Numbers — Exercise Set 3.4 Page 523
- .5 Irrational Numbers — Think and Reflect Page 533
- .5.1 The Proof of Irrationality of √2 — Think and Reflect Page 553
- .5.2 Construction of Length √n — Think and Reflect Page 553
- .5.2 Construction of Length √n — Think and Reflect Page 563
- .6.1 Rational Decimals: Terminating and Repeating — In-text Questions Page 573
- .6.1 Rational Decimals: Terminating and Repeating — Think and Reflect Page 573
- .6.1 Rational Decimals: Terminating and Repeating — Think and Reflect Page 583
- .6.3 Irrational Decimals: Chaos and Infinity — Exercise Set 3.5 Page 613
- .7 Conclusion: The Never-Ending Journey — Think and Reflect Page 643
- The World of Numbers — End-of-Chapter Exercises Page 64