NCERT Solutions for Class 9 Maths Ganita Manjari Chapter 3 The World of Numbers

Chapter 3 contains five exercise sets plus end-of-chapter exercises for Session 2026-27:

• Exercise Set 3.1 – Natural Numbers, one-to-one correspondence, prime numbers
• Exercise Set 3.2 – Integers, Brahmagupta’s laws for signed numbers (real-life debt/fortune problems)
• Exercise Set 3.3 – Rational Numbers: equality, addition, subtraction, multiplication, division
• Exercise Set 3.4 – Representing rational numbers on the number line, density property
• Exercise Set 3.5 – Decimal expansions, cyclic numbers, classifying rational vs. irrational numbers
• End-of-Chapter Exercises – 16 problems including proof of irrationality of √5, converting decimals to p/q, locating numbers on the number line and two starred (*) challenge problems for advanced learners.

Start with a visual: draw a unit square and ask your child to measure its diagonal with a ruler – they’ll get approximately 1.41 cm, but the exact value (√2) cannot be written as a fraction. Chapter 3 uses exactly this example. Help your child understand the key rule: if a decimal terminates (stops) or repeats in a pattern, it is rational; if it goes on forever with no repeating pattern (like √2 = 1.41421356…), it is irrational. Practising with the “Think and Reflect” boxes in the textbook is also very effective.

The new Ganita Manjari (Session 2026-27) takes a strongly historical and cultural approach. It grounds the development of number systems in Indian mathematical heritage – highlighting the Indus Valley Civilisation, Vedic texts (Ṛigveda, Lalitavistara), the concept of Śhūnyatā from the Upanishads, Brahmagupta’s Brāhmasphuṭasiddhānta, Baudhāyana’s Śhulbasūtra and Mādhava of Sangamagrama’s infinite series for π. The chapter also introduces Cyclic Numbers and Imaginary Numbers as conceptual teasers and features rich “Think and Reflect” boxes and context-based word problems set in ancient Indian trade scenarios.

Chapter 3 provides a clear rule: Write the fraction p/q in its lowest terms. Then check the prime factorisation of the denominator q. If q has only 2s, only 5s, or both 2s and 5s as prime factors, the decimal terminates. If q has any other prime factor (like 3, 7, 11, etc.), the decimal repeats. For example, 7/20 terminates (20 = 2² × 5), but 4/15 repeats (15 = 3 × 5).

The chapter 3 provides an excellent 8-step scaffold. Teachers should:

1. First establish what “co-prime” means
2. Walk through the assumption step carefully
3. Emphasise that “if p² is even then p is even” is a key lemma (which can be proven separately)
4. Show the contradiction clearly in Step 8
5. Ask students to replicate the method for √3 independently.

Connecting it to a visual illustration (as shown in the textbook’s chalkboard figure) helps students internalize the logical structure.

The chapter references Mādhava of Sangamagrama’s infinite series (14th century CE): π = 4 × (1 − 1/3 + 1/5 − 1/7 + …). At Class 9 level, teachers should introduce this as a conceptual idea – that irrational numbers like π cannot be expressed as a single fraction but can be approached through an infinite sum of fractions. Students are not expected to work with infinite series formally; they will study convergence in higher grades. The goal is to build conceptual appreciation for why π is irrational.

Yes, it is fully relevant. The new NCERT curriculum (Session 2026-27) has been designed to give students a deeper appreciation of India’s mathematical contributions while teaching core mathematical concepts.
Students may be asked questions that reference Brahmagupta’s rules, the Bakhśhālī Manuscript or the Ishango Bone in context-based or short-answer questions. Understanding the historical context also helps students remember and apply the mathematical concepts more meaningfully.

These are ancient artefacts introduced in Section 3.1 to show that mathematics is at least tens of thousands of years old. The Lebombo Bone (~35,000 years old, discovered in southern Africa) has 29 carved notches, believed to be a lunar calendar. The Ishango Bone (~20,000 BCE, found near the Nile) contains tally groupings of 11, 13, 17, and 19 — the prime numbers between 10 and 20 – suggesting early humans had an intuitive grasp of prime numbers.

For a general repeating decimal (with non-repeating digits followed by a repeating block), the chapter teaches a two-step multiplication method. For 0.1̄6̄: first multiply by 10 (to shift the non-repeating part) to get 10x = 1.̄6̄, then multiply by 10 again (for the 1 repeating digit) to get 100x = 16.̄6̄. Subtracting gives 90x = 15, so x = 15/90 = 1/6.

A Cyclic Number is a special number whose digits rotate cyclically when multiplied by certain integers. Chapter 3 gives the classic example of 142857 – the repeating block of the decimal for 1/7 (= 0.̄142857̄). When multiplied by 1 through 6, the same six digits appear in a shifted cyclic order each time (e.g., 142857 × 2 = 285714, 142857 × 3 = 428571, and so on).

Yes! Chapter 3 explains this through the concept of non-uniqueness of decimal representations. Let x = 0.9̄ (= 0.999…). Multiplying by 10 gives 10x = 9.9̄. Subtracting: 9x = 9, so x = 1. Therefore 0.999… = 1 exactly, not “almost 1.” This is a beautiful consequence of how repeating decimals work.

Yes, the Proof by Contradiction for the irrationality of √2 is explicitly included in Chapter 3 for Session 2026-27. The proof assumes √2 = p/q in lowest terms (p, q co-prime), then shows that this forces both p and q to be even – contradicting the assumption that they share no common factors. This contradiction proves √2 cannot be rational.
Students are also asked to use the same method to prove √3, √5, and √7 are irrational.

A Rational Number can be expressed as p/q, where p and q are integers and q ≠ 0. Its decimal expansion is either terminating (e.g., 3/8 = 0.375) or repeating (e.g., 5/11 = 0.̄4̄5̄).
An Irrational Number cannot be written as p/q; its decimal expansion is non-terminating and non-repeating (e.g., √2 = 1.4142135…, π = 3.14159…).

Together, rational and irrational numbers form the Real Numbers (ℝ).

Brahmagupta (628 CE) was an Indian mathematician who wrote the Brāhmasphuṭasiddhānta. In Chapter 3, he is celebrated for three landmark contributions:

1. Formally defining zero as the result of subtracting a number from itself (a – a = 0)
2. Introducing negative numbers (debts/ṛiṇa) and rules for arithmetic with signed numbers, and
3. Establishing rules for operations on rational numbers. His laws are still used exactly as he wrote them over 1,300 years ago.

Chapter 3, The World of Numbers, covers the complete evolution of numbers – from Natural Numbers and Zero to Integers, Rational Numbers, Irrational Numbers and Real Numbers — along with their properties, arithmetic rules, decimal expansions and historical origins. It is part of the new NCERT textbook Ganita Manjari for Class 9, Session 2026-27.