Download The World of Numbers – NCERT Class 9 Maths Chapter 3 PDF from the new NCERT Class 9 Maths book Ganita Manjari (2026-27). Read the chapter online or download the free PDF. 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).
The World of Numbers PDF – Chapter 3 of Ganita Manjari (Part I)
Read the chapter online or download the PDF.
| Chapter | Download | |
|---|---|---|
| Chapter 3 PDF | The World of Numbers | View PDF |
| Full book | NCERT Class 9 Maths – Ganita Manjari | View Book |
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What Is in The World of Numbers? (Class 9 Maths Chapter 3)
- 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 recurs.
- Rationals ∪ irrationals = the real numbers ℝ, an unbroken line. Terminating and repeating decimals are rational; non-terminating non-repeating decimals are irrational.
Quick revision – The World of Numbers
| 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 |
Old NCERT vs New – Class 9 Maths
This chapter is from Ganita Manjari, the new NCERT Class 9 Maths book (NCF-SE 2023, used from 2026-27). It replaced the old book “Mathematics”, so chapter numbers and names differ from older PDFs.
Solutions and All Chapters
Answers to the exercises and the full chapter list of Ganita Manjari:
The World of Numbers – Overview
| Class | Class 9 |
| Subject | Maths |
| Book | Ganita Manjari – new NCERT book |
| Part | Part I |
| Chapter | 3 of 14 – The World of Numbers |
| Book pages | 41 – 67 |
| Edition | Latest NCERT edition (2026-27) |
| Session | 2026-27 |
| Free – read online or download |
Class 9 Maths Chapter 3 – FAQs
Where can I download The World of Numbers Class 9 PDF?
From the download table on this page – the chapter PDF is free, and the full book is linked too.
What is Class 9 Maths Chapter 3 about?
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).
Is The World of Numbers from the new NCERT Class 9 Maths book?
Yes. It is chapter 3 of ganita manjari (part i), the new NCERT Class 9 Maths book for 2026-27.
Is this the latest NCERT edition for 2026-27?
Yes – the PDF is NCERT’s latest edition for the 2026-27 session.
Where are the NCERT Solutions for Class 9 Maths Chapter 3?
Open the NCERT Solutions link above for answers to the exercise questions.