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CBSE Class 9 Mathematics Chapter 3 The World of Numbers Worksheet with Answers

Welcome to the CBSE Class 9 Mathematics Chapter 3 worksheet on 'The World of Numbers.' This chapter traces humanity's mathematical journey from ancient counting notches on bones to the sophisticated real number system. You will explore natural numbers, integers, rational and irrational numbers, and their decimal representations. This printable worksheet is designed for a 90-minute practice session and mirrors the CBSE exam pattern with varied question types and a detailed answer key.

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Key takeaways

✓Natural numbers originated from counting needs 35,000 years ago; integers include zero and negative numbers introduced by Brahmagupta in 628 CE.
✓Rational numbers are expressible as p/q where q ≠ 0; they have terminating or repeating decimal expansions.
✓Irrational numbers like √2 and π cannot be written as fractions and have non-terminating, non-repeating decimals.
✓Real numbers unite all rational and irrational numbers, forming a complete, continuous number line.
✓A denominator in lowest terms with only factors of 2 or 5 produces a terminating decimal; other primes create repeating decimals.
✓Between any two rational numbers exist infinitely many other rationals (density property), yet irrationals fill the gaps rationals cannot reach.
✓The worksheet covers all question types found in CBSE Class 9 term exams: MCQs, short answers, long answers, HOTS, and case studies.

Worksheet Instructions and Difficulty Level

This worksheet is designed for CBSE Class 9 students preparing for term examinations based on the NCERT Ganita Manjari textbook. The difficulty level is Medium to High, incorporating both foundational recall and higher-order thinking questions. Total marks: 80. Suggested time: 90 minutes. Students should attempt all sections sequentially. Use graph paper for number-line constructions in Section D. Each section tests a different cognitive skill — recognition (MCQs), recall (fill-in-the-blanks), application (short answers), analysis and synthesis (long answers), and real-world application (case study). Keep a calculator handy for verification but show all working steps. The answer key at the end provides brief explanations to aid self-assessment and concept reinforcement.

Difficulty Level: Medium to High
Total Marks: 80
Suggested Time: 90 minutes
Materials needed: pen, graph paper, calculator (for verification only)
Sections: A (MCQs), B (Fill-in-the-blanks), C (Match-the-following), D (Short answer), E (Long answer & HOTS), F (Case study)

Quick Chapter Recap: The World of Numbers

Chapter 3 chronicles the evolution of the number system. Natural numbers (ℕ = {1, 2, 3, …}) emerged from counting — evidence includes the 35,000-year-old Lebombo Bone with 29 notches. Zero and negative integers were formalized by Brahmagupta in 628 CE, creating the integers (ℤ = {..., −2, −1, 0, 1, 2,...}). He introduced the concepts of dhana (fortune, positive) and ṛiṇa (debt, negative). Rational numbers (ℚ) are numbers expressible as p/q with integers p and q (q ≠ 0); their decimals either terminate or repeat. Irrational numbers cannot be written as fractions; examples include √2, π, and e, with non-terminating, non-repeating decimals. The union of rationals and irrationals forms the Real Numbers (ℝ), a complete continuum on the number line. Key skills include identifying number types, converting repeating decimals to fractions, proving irrationality, and locating numbers on the number line using geometric constructions.

Natural numbers: counting numbers starting from 1, not closed under subtraction
Integers: include zero and negatives; Brahmagupta's dhana (positive) and ṛiṇa (negative)
Rational numbers: p/q form; decimals terminate or repeat based on prime factors of denominator
Irrational numbers: non-terminating, non-repeating decimals; discovered via geometry (√2 in Baudhāyana's Śhulbasūtra)
Real numbers: complete set uniting ℚ and irrational numbers; continuous number line
Decimal test: terminating/repeating → rational; non-terminating non-repeating → irrational

Section A: Multiple Choice Questions (1 mark each, 8 questions)

Choose the correct option for each question. Each MCQ carries 1 mark. These questions test your conceptual understanding and ability to distinguish between natural numbers, integers, rationals, and irrationals. Pay careful attention to the definitions and properties of each number set. Remember: a rational number has a decimal that either stops or repeats; an irrational decimal goes on forever without any repeating block. The set of real numbers is the union of rationals and irrationals, forming a continuous line with no gaps. Properties like closure, commutativity, and associativity vary across number sets. For example, natural numbers are NOT closed under subtraction (3 − 5 is not natural), but integers are. Work through each option carefully and eliminate clearly wrong answers first.

1. Which of the following is NOT a rational number? (a) 0.75 (b) 0.333... (c) √4 (d) √5
2. The decimal expansion of 22/7 is: (a) Terminating (b) Non-terminating repeating (c) Non-terminating non-repeating (d) Undefined
3. Zero was formally introduced as a number by: (a) Pythagoras (b) Euclid (c) Brahmagupta (d) Archimedes
4. Between any two rational numbers, there exist: (a) Exactly one rational (b) Exactly two rationals (c) Finitely many rationals (d) Infinitely many rationals
5. Which set is closed under subtraction? (a) ℕ (b) ℤ (c) Both ℕ and ℤ (d) Neither ℕ nor ℤ
6. The rational number 3/20 in decimal form is: (a) 0.15 (b) 0.3 (c) 0.6 (d) 0.03
7. (−3) × (−4) equals: (a) −12 (b) 12 (c) −7 (d) 7
8. An irrational number between 2 and 3 is: (a) 2.5 (b) √5 (c) 7/3 (d) 2.75

Section B: Fill in the Blanks (1 mark each, 6 questions)

Complete each statement with the appropriate term, number, or expression. These questions focus on recall of key definitions, historical facts, and fundamental properties from Chapter 3. Write your answers clearly. Remember that Brahmagupta called positive numbers 'dhana' (fortunes) and negative numbers 'ṛiṇa' (debts). The Lebombo Bone, dating to around 35,000 BCE, is the oldest known tally stick with 29 notches. A rational number in lowest terms has a terminating decimal if and only if the denominator has no prime factors other than 2 or 5. The set of real numbers includes every rational and every irrational number, forming a continuous number line without any gaps. Review the chapter summary if you are unsure about any term.

9. The set of natural numbers is denoted by the symbol _______.
10. Brahmagupta referred to negative numbers as _______ (debts).
11. A number that cannot be expressed in the form p/q is called an _______ number.
12. The decimal 0.142857142857... is an example of a _______ decimal.
13. The value of √2 was first encountered in the ancient Indian text called _______.
14. The union of all rational and irrational numbers is called the set of _______ numbers.

Section C: Match the Following (1 mark each, 5 questions)

Match the items in Column A with the correct descriptions or values in Column B. Write the pairs as (number, letter) in your answer sheet. This section tests your ability to connect concepts, historical milestones, decimal forms, and number types. For example, the Ishango Bone is famous for containing notches that correspond to prime numbers. Brahmagupta lived in the 7th century CE and formalized arithmetic rules for zero and negative numbers. A terminating decimal has a finite number of digits after the decimal point, while a repeating decimal has a block of digits that cycles forever. Every integer can be written as a rational number by placing it over 1 (e.g., 5 = 5/1). Review your notes on historical milestones and decimal classification before attempting this section.

Column A:
15. Ishango Bone
16. 7/8
17. π
18. Brahmagupta
19. 0.16̄
Column B:
(a) Irrational number, non-repeating decimal
(b) Contains prime number notches (11, 13, 17, 19)
(c) Indian mathematician, 628 CE, formalized zero
(d) Terminating decimal 0.875
(e) Repeating decimal, rational number 1/6

Section D: Short Answer Questions (3 marks each, 5 questions)

Answer the following questions in 3-4 sentences or show the necessary working steps. Each question carries 3 marks. Marks are awarded for correct method, accurate calculation, and clear presentation. Use proper mathematical notation. For questions involving proofs, state your assumptions and justify each step. For decimal conversions, show the algebraic manipulation clearly. When locating irrational numbers on the number line, describe the geometric construction step-by-step or provide a labelled diagram. These questions assess your ability to apply concepts from NCERT Class 9 Mathematics Chapter 3 in problem-solving contexts. Answers should be concise yet complete, demonstrating both understanding and computational skill.

20. Convert the repeating decimal 0.363636... into a rational number in the form p/q. Show all steps.
21. State whether 5 − √3 is rational or irrational. Justify your answer with reasoning.
22. Simplify and express as a single rational number: (2/3) + (−5/6) − (1/4).
23. Insert two rational numbers between 1/3 and 1/2. Show your method.
24. Describe the method to locate √3 on the number line using a geometric construction.

Section E: Long Answer and HOTS Questions (5 marks each, 3 questions)

Attempt the following higher-order thinking and long-answer questions. Each carries 5 marks. Full marks require a detailed explanation, logical flow, and correct conclusions. For proof-based questions, use the contradiction method as demonstrated in the NCERT textbook for proving √2 is irrational. For questions on decimal expansions, explain why certain fractions terminate and others repeat, referencing the prime factorization of the denominator in lowest terms. HOTS questions test your ability to synthesize information, apply multiple concepts, and reason abstractly. Write clearly, define all terms, and connect each step in your argument. These questions often appear in CBSE board exams and require thorough understanding, not rote learning. Use examples to support your explanations where appropriate.

25. Prove that √5 is an irrational number using the method of contradiction. Clearly state each step and assumption.
26. Explain with examples why the decimal expansion of a rational number is either terminating or non-terminating repeating. Under what condition on the denominator (in lowest terms) does a rational number have a terminating decimal?
27. (HOTS) Show that the product of a non-zero rational number and an irrational number is always irrational. Illustrate with an example and provide reasoning.

Section F: Case Study Question (4 marks)

Read the case study carefully and answer the sub-questions that follow. This type of question integrates real-world scenarios with mathematical concepts from Chapter 3 'The World of Numbers.' Case-study questions have become a standard feature in recent CBSE Class 9 term exams. They test your ability to extract mathematical information from a narrative, apply number theory, and perform multi-step problem solving. Marks are distributed across sub-questions of varying difficulty. Show all working and reasoning for full credit. This question may involve calculating values, classifying numbers, or justifying mathematical statements in context. Read the passage twice before attempting the sub-questions to ensure you understand the scenario fully.

Complete Answer Key with Explanations

Below are the answers to all sections of the worksheet. Each answer includes a brief explanation to help you understand the reasoning. Use this key for self-assessment and to identify areas needing further revision. If you scored below 60%, revisit the NCERT Class 9 Mathematics textbook Chapter 3 and work through the solved examples. For conceptual doubts, CBSETUTOR.ai offers 24×7 AI-powered doubt solving with photo upload at a flat ₹999/month for all classes (6-12) — one subscription covers every subject, and you get a 3-day free trial to experience instant step-by-step explanations. Practice regularly to build speed and accuracy. Mark incorrect answers and rework those questions after reviewing the relevant chapter sections. Note that partial marks are awarded in board exams for correct method even if the final answer is wrong, so always show working steps.

Section A Answers: 1(d) √5 is irrational; 2(b) 22/7 = 3.142857... repeating; 3(c) Brahmagupta, 628 CE; 4(d) Infinitely many (density property); 5(b) ℤ (integers); 6(a) 0.15; 7(b) 12 (debt × debt = fortune); 8(b) √5 ≈ 2.236, irrational
Section B Answers: 9. ℕ; 10. ṛiṇa; 11. irrational; 12. repeating (or non-terminating repeating); 13. Śhulbasūtra (by Baudhāyana); 14. real
Section C Answers: 15–b; 16–d; 17–a; 18–c; 19–e
Section D Answers: 20. x=4/11; 21. Irrational (assume rational leads to √3 rational, contradiction); 22. −1/4; 23. e.g. 5/12 and 7/18 (find LCD and pick fractions between); 24. Draw unit square on number line, diagonal = √2, extend by 1 unit, new diagonal = √3 by Pythagoras; use compass to mark on line
Section E Answers: 25. Proof by contradiction (as outlined); 26. Terminating if denominator in lowest terms has only 2 and/or 5 as factors; else repeating; 27. Let r ≠ 0 rational, x irrational. If rx rational, then x = (rx)/r = rational/rational = rational, contradiction. Example: 2 × √2 = 2√2 irrational
Section F Answers: (i) Integers ℤ; (ii) Rational, 22/7 = 3.142857... repeating; (iii) 45/6 = 15/2; (iv) 22/7 is rational (repeating), π is irrational (non-repeating), so they cannot be equal; π = 3.1415926535... never repeats

How CBSETUTOR.ai Supports Chapter 3 Mastery

CBSETUTOR.ai offers round-the-clock AI tutoring designed specifically for CBSE students from Class 6 to 12. For Chapter 3 'The World of Numbers,' students can photograph any problem — whether from this worksheet, NCERT exercises, or sample papers — and receive instant step-by-step solutions with conceptual explanations. The AI tutor covers proofs (like irrationality of √2), decimal conversions, number-line constructions, and HOTS questions. Unlike traditional coaching that costs thousands per subject, CBSETUTOR.ai charges a flat ₹999 per month for unlimited access across all subjects and classes. Parents appreciate the single subscription model: one child, one price, every subject. A 3-day free trial lets students test the platform before committing. The AI identifies weak areas — say, repeated mistakes in converting repeating decimals to fractions — and generates additional targeted practice. For busy families in metro cities juggling school, commute, and tuitions, CBSETUOR.ai brings the tutor home, available anytime on mobile or desktop. Students gain confidence tackling board exam–style questions and build a deep understanding of the real number system that is foundational for higher mathematics.

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Tips for Scoring Full Marks in Chapter 3 Questions

To excel in Chapter 3 'The World of Numbers,' focus on clarity and logical structure in your answers. For proof questions (e.g., proving √2 irrational), always state your initial assumption explicitly, show every algebraic step, and clearly identify the contradiction. CBSE examiners award marks for method, so even if you make an arithmetic slip, you earn partial credit for correct approach. When converting repeating decimals to fractions, write out the equation for x, the multiplication step, the subtraction, and the simplification — this demonstrates understanding beyond rote formula. For number-line constructions, draw neat diagrams with labels: mark the origin, unit length, and use a compass and ruler for geometric accuracy. In MCQs, eliminate obviously wrong options first and use properties (e.g., closure, density) to narrow choices. Memorize key historical milestones: Brahmagupta (628 CE, zero and negatives), Baudhāyana (Śhulbasūtra, √2), Lebombo and Ishango bones (ancient tallies). Practice mental conversion: know that denominators 2, 4, 5, 8, 10, 16, 20, 25, etc. (products of 2 and 5 only) yield terminating decimals; all others repeat. Finally, manage time in exams: spend ~1 minute per MCQ, 3-4 minutes per short answer, and 8-10 minutes per long answer or proof. Regular worksheet practice builds both speed and confidence.

For proofs: state assumption, show each step, highlight contradiction, conclude clearly
Decimal to fraction: write x = 0.abc..., multiply by 10^n, subtract, solve for x, simplify
Number-line construction: use Pythagoras theorem, draw diagrams, label all elements
Memorize historical facts: Brahmagupta, Baudhāyana, Lebombo and Ishango bones
Know terminating vs repeating: denominator factors {2,5} only → terminates; else repeats
Time management: 1 min/MCQ, 3-4 min/short answer, 8-10 min/long answer
Show all working steps for partial marks even if final answer is incorrect

Frequently asked questions

What is the difference between rational and irrational numbers?+

Rational numbers can be expressed as p/q where p and q are integers and q ≠ 0; their decimal expansions either terminate (like 0.75) or repeat (like 0.333...). Irrational numbers cannot be written as fractions; their decimals never terminate and never repeat (e.g., √2 = 1.414213..., π = 3.141592...).

How do I know if a fraction will have a terminating or repeating decimal?+

Write the fraction in lowest terms. Check the prime factorization of the denominator. If it contains only the factors 2 and/or 5, the decimal terminates (e.g., 3/8 = 0.375 because 8=2³). If the denominator has any other prime factor (like 3, 7, 11), the decimal repeats (e.g., 5/11 = 0.454545...).

Who invented zero and when?+

The concept of zero as a number — not just a placeholder — was formalized by the Indian mathematician Brahmagupta in 628 CE. He defined its arithmetic rules and introduced negative numbers, calling positives 'dhana' (fortunes) and negatives 'ṛiṇa' (debts). This revolutionized mathematics globally.

What is the density property of rational numbers?+

Between any two rational numbers, no matter how close, there exist infinitely many other rational numbers. For example, between 1/2 and 1, you can find 3/4, then between 1/2 and 3/4 find 5/8, and so on forever. This means the rationals are 'dense' — yet they still leave gaps that irrational numbers fill.

How do I convert a repeating decimal like 0.636363... into a fraction?+

Let x = 0.636363... Multiply both sides by 100 (since the repeating block has 2 digits): 100x = 63.636363... Subtract the original: 100x − x = 63, so 99x = 63. Therefore x = 63/99 = 7/11 in lowest terms. This method works for any repeating decimal.

Why is √2 irrational? Can you prove it?+

Proof by contradiction: Assume √2 = a/b in lowest terms (a, b coprime). Squaring: 2 = a²/b², so 2b² = a², meaning a² is even, hence a is even. Let a = 2k. Then 2b² = 4k², so b² = 2k², meaning b is even. But if both a and b are even, they share a factor of 2, contradicting 'lowest terms.' Hence √2 cannot be rational.

How can I locate √3 on the number line?+

Start with a unit square (side 1) at the origin; its diagonal is √2 by Pythagoras. Extend this diagonal by 1 unit perpendicular to form a right triangle with sides √2 and 1. The new hypotenuse is √(2+1) = √3. Use a compass with this length to mark √3 on the number line from the origin.

What are the main number sets I need to know for CBSE Class 9?+

Natural numbers ℕ = {1,2,3,...}, Whole numbers W = {0,1,2,...}, Integers ℤ = {...,−2,−1,0,1,2,...}, Rational numbers ℚ (all fractions p/q), Irrational numbers (non-fractions like √2, π), and Real numbers ℝ (union of rationals and irrationals). Each set builds on the previous, expanding to solve new problems.

Is 22/7 equal to π?+

No. 22/7 is a rational approximation of π (≈ 3.142857, a repeating decimal), but π is irrational with a non-terminating, non-repeating decimal (3.141592653589793...). They are close in value, so 22/7 is often used for quick calculations, but they are fundamentally different types of numbers.

How does CBSETUOR.ai help with Chapter 3 practice?+

CBSETUOR.ai provides 24×7 AI tutoring for CBSE Class 9 Mathematics. You can photograph any problem from this worksheet or NCERT and get instant step-by-step solutions with explanations. At ₹999/month flat for all subjects and classes (6-12), it replaces expensive coaching. A 3-day free trial lets you test it risk-free before committing.

Related resources

Important Questions: CBSE Class 9 Mathematics Chapter 3 Coordinate Geometry CBSE Class 9 Mathematics Chapter 3 Coordinate Geometry Worksheet with Answers Class 9 Mathematics Chapter 3 Coordinate Geometry — Formulas & Key Points Class 9 Mathematics Chapter 3 The World of Numbers — Formulas & Key Points CBSE Class 9 Mathematics Chapter 1 Number Systems — Notes CBSE Class 9 Mathematics — Number Systems: complete chapter guide NCERT Solutions for Class 9 Mathematics Chapter 1: Number Systems – Complete Guide CBSE Class 9 Mathematics Chapter 1 Number Systems Worksheet with Answers

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