Courses
By Karan Singh Bisht
|
Updated on 18 Jun 2026, 15:52 IST
CBSE Class 9 Maths Chapter 3, The World of Numbers, introduces students to the development and use of different types of numbers. The chapter explains natural numbers, zero, integers, rational numbers, decimal expansion, irrational numbers, and real numbers in a simple and concept-based manner.
To help students practice better, Infinity Learn provides The world of numbers class 9 worksheet with answers pdf for revision, homework support, and exam preparation. These class 9 maths worksheets chapter 3 include MCQs, fill in the blanks, very short answer questions, short answer questions, long answer questions, and chapter-based extra questions with answers.
Class 9 Chapter 3 the world of numbers explains how numbers developed from the basic human need to count. It begins with natural numbers and one-to-one correspondence, then explains the importance of zero, integers, rational numbers, decimal expansions, irrational numbers, and real numbers. The chapter also connects mathematics with history by discussing examples such as the Lebombo Bone, Ishango Bone, Indian numeral system, Śhūnya, and Brahmagupta’s contribution to zero and integers.
Also Read: Class 9 Maths The world of Numbers NCERT Solutions
Students looking for The world of numbers class 9 worksheet with answers pdf download can use this page to access chapter-wise practice material in a simple format. The worksheet is useful for quick revision before class tests, periodic assessments, and annual exams. The questions are arranged from basic to advanced level so that students can first revise definitions and then move to application-based questions.
The world of numbers class 9 worksheet with answers pdf is designed for students who want structured practice after reading the chapter. It helps learners revise important concepts such as one-to-one correspondence, the origin of natural numbers, Brahmagupta’s rules for zero, integers, rational numbers, and irrational numbers. Students can use this worksheet to check their understanding and improve accuracy. Each question is prepared to support conceptual clarity and exam-style practice.
Loading PDF...
1. What was the earliest human need that led to the development of numbers?
a) To draw shapes
b) To keep count
c) To write poetry
d) To measure temperature
Answer: b) To keep count
JEE
NEET
Foundation JEE
Foundation NEET
CBSE
2. One-to-one correspondence means:
a) Dividing one number by another
b) Matching one object with one other object
c) Writing numbers in decimal form
d) Multiplying two numbers
Answer: b) Matching one object with one other object
3. Which of the following represents Natural Numbers?
a) {0, 1, 2, 3, …}
b) {1, 2, 3, 4, …}
c) {…, -2, -1, 0, 1, 2, …}
d) {1/2, 2/3, 3/4}
Answer: b) {1, 2, 3, 4, …}
4. The Ishango bone is famous because one of its columns shows:
a) Only even numbers
b) Prime number groupings
c) Roman numerals
d) Decimal numbers
Answer: b) Prime number groupings
5. Who formally gave rules for zero as a number?
a) Aryabhata
b) Brahmagupta
c) Euclid
d) Newton
Answer: b) Brahmagupta
6. In Brahmagupta’s idea, positive numbers were linked with:
a) Debt
b) Loss
c) Fortune
d) Emptiness
Answer: c) Fortune
7. Which of the following is an integer?
a) 2/5
b) -8
c) √3
d) 0.75
Answer: b) -8
8. A rational number is written in the form:
a) p + q
b) p/q, where p and q are integers and q ≠ 0
c) √p
d) p × q only
Answer: b) p/q, where p and q are integers and q ≠ 0
9. Which of the following is an irrational number?
a) 5/2
b) 0.25
c) √2
d) -4
Answer: c) √2
10. Real numbers include:
a) Only natural numbers
b) Only whole numbers
c) Rational and irrational numbers
d) Only negative numbers
Answer: c) Rational and irrational numbers
1. The basic counting numbers are called __________ numbers.
Answer: Natural
2. The method of matching one object with another is called __________ correspondence.
Answer: One-to-one
3. The Lebombo Bone contains __________ distinct notches.
Answer: 29
4. The numbers 11, 13, 17 and 19 are examples of __________ numbers.
Answer: Prime
5. The Sanskrit word Śhūnya means __________.
Answer: Zero
6. Brahmagupta defined zero as the result of subtracting a number from __________.
Answer: Itself
7. Negative numbers were linked with __________ in Brahmagupta’s explanation.
Answer: Debts
8. The symbol for integers is __________.
Answer: Z
9. Rational numbers are denoted by the symbol __________.
Answer: Q
10. Irrational numbers have decimal expansions that are non-terminating and __________.
Answer: Non-repeating
1. What are Natural Numbers?
Answer: Natural numbers are counting numbers such as 1, 2, 3, 4, and so on.
2. What is one-to-one correspondence?
Answer: It is the method of matching each object in one group with exactly one object in another group.
3. What was the use of pebbles in the cattle-counting example?
Answer: Pebbles were used to match and count each cow that went out and returned.
4. Name one ancient bone related to early counting.
Answer: Lebombo Bone or Ishango Bone.
5. What is the value of a - a?
Answer: 0
6. What is the result of a + 0?
Answer: a
7. What is the product of any number and zero?
Answer: 0
8. What is the value of (-3) × (-4)?
Answer: 12
9. What is the absolute value of -5/3?
Answer: 5/3
10. Write one example of an irrational number.
Answer: √2
1. Are Natural Numbers closed under subtraction? Give an example.
Answer: No, Natural Numbers are not closed under subtraction. For example, 7 - 3 = 4 is a natural number, but 3 - 7 = -4 is not a natural number.
2. A merchant receives 15 copper ingots for every 2 bags of spices. How many ingots will he receive for 12 bags?
Answer:
2 bags = 15 ingots
12 bags = 6 groups of 2 bags
6 × 15 = 90
So, he will receive 90 copper ingots.
3. List the next three prime numbers after 11, 13, 17 and 19.
Answer: The next three prime numbers are 23, 29 and 31.
4. The temperature in Ladakh is 4°C at noon. It drops by 15°C by midnight. What is the midnight temperature?
Answer:
4 - 15 = -11
The midnight temperature is -11°C.
5. A trader has a debt of ₹850, earns a profit of ₹1200, and then has a loss of ₹450. Find his final standing.
Answer:
-850 + 1200 - 450 = -100
Final standing: ₹100 debt.
6. Calculate (-12) × 5.
Answer:
(-12) × 5 = -60
7. Calculate 0 - (-14).
Answer:
0 - (-14) = 0 + 14 = 14
8. Prove that 2/3 and 4/6 are equal rational numbers.
Answer:
Two rational numbers a/b and c/d are equal if ad = bc.
For 2/3 and 4/6:
2 × 6 = 12 and 3 × 4 = 12
Since both products are equal, 2/3 = 4/6.
9. Find the sum of 2/5 and 3/10.
Answer:
2/5 = 4/10
4/10 + 3/10 = 7/10
Answer: 7/10
10. Why can q not be zero in p/q?
Answer:
q cannot be zero because division by zero is not defined in mathematics.
1. Explain how the need to count led to the birth of Natural Numbers.
Answer:
Early humans needed to count objects such as animals, food, tools, and goods. Before written numbers existed, they used one-to-one correspondence. For example, a herder could place one pebble in a pot for every cow that went out to graze. When each cow returned, one pebble was removed. If no pebble remained, all cows had returned. This matching of one object with another led to the idea of counting and the birth of Natural Numbers: 1, 2, 3, 4, and so on.
2. Explain the importance of the Ishango Bone in the history of numbers.
Answer:
The Ishango Bone is important because it shows that early humans were not only counting but also arranging numbers in meaningful groups. One column of the bone contains the numbers 11, 13, 17 and 19, which are prime numbers between 10 and 20. Another column suggests doubling. This shows that humans were thinking about number patterns thousands of years ago.
3. Write a note on Brahmagupta’s rules for zero.
Answer:
Brahmagupta gave important arithmetic rules for zero. He explained that when zero is added to a number, the number remains unchanged. When zero is subtracted from a number, the number remains unchanged. When any number is multiplied by zero, the result is zero. He also defined zero as the result of subtracting a number from itself, such as a - a = 0.
4. Explain the concept of integers using fortune and debt.
Answer:
Brahmagupta explained positive and negative numbers using real-life ideas. Positive numbers were called fortunes, representing wealth or gain. Negative numbers were called debts, representing loss or money owed. When positive numbers, negative numbers and zero are combined, they form the set of integers. For example, ..., -3, -2, -1, 0, 1, 2, 3, ... are integers.
5. Explain why the product of two negative numbers is positive using a debt example.
Answer:
A negative number can represent a debt. Multiplying by a negative number can be understood as removing a debt. For example, if four debts of ₹3 each are removed, then the person becomes ₹12 richer. This is why (-3) × (-4) = +12. So, the product of two negative numbers is positive.
6. Define rational numbers and give examples.
Answer:
A rational number is any number that can be written in the form p/q, where p and q are integers and q ≠ 0. Rational numbers include natural numbers, whole numbers, integers, and fractions. For example, 5 can be written as 5/1, -10 can be written as -10/1, and 3/4 is already in rational form. Therefore, 5, -10 and 3/4 are rational numbers.
7. Explain the density property of rational numbers.
Answer:
Rational numbers are dense, which means that between any two rational numbers, there is always another rational number. For example, between 1 and 3/2, we can find a rational number by taking their average:
(1 + 3/2) ÷ 2 = 5/4
So, 5/4 lies between 1 and 3/2. This process can be repeated again and again, which shows that there are infinitely many rational numbers between any two rational numbers.
8. Differentiate between rational and irrational numbers.
Answer:
Rational numbers can be written in the form p/q, where p and q are integers and q ≠ 0. Their decimal expansion is either terminating or repeating. Examples include 1/2, 0.25 and 0.3333... Irrational numbers cannot be written in the form p/q. Their decimal expansion is non-terminating and non-repeating. Examples include √2 and π.
9. Classify the following numbers as rational or irrational: √81, √12, 0.3333..., 0.123123123..., 1.01001000100001...
Answer:
√81 = 9, so it is rational.
√12 is irrational because it cannot be written as a ratio of two integers.
0.3333... is rational because it is repeating.
0.123123123... is rational because the block 123 repeats.
1.01001000100001... is irrational because it does not have a fixed repeating pattern.
10. Explain the evolution of the number system from Natural Numbers to Real Numbers.
Answer: The number system developed gradually according to human needs. First came Natural Numbers, which were used for counting. Then zero was introduced to represent nothingness. Integers were formed by including positive numbers, negative numbers and zero. Fractions helped represent parts of a whole, and rational numbers included all numbers that can be written in the form p/q. Later, irrational numbers such as √2 and π were understood as numbers that cannot be written as fractions. Together, rational and irrational numbers form the set of Real Numbers.
11. Why does a negative times a negative equal a positive? Think of it in terms of action and debt. If a negative number represents a debt, then multiplying by a negative number represents the removal of that debt. (Hint: If someone takes away (-) four of your debts that are each worth ₹ 3 (that is, -3), you are effectively ₹ 12 richer! Therefore, (-3) × (-4) = +12.)
Solution: A negative number can represent a debt. Multiplying by a negative number can be understood as removing that debt.
For example, suppose each debt is worth Rs. 3. So, one debt is represented as -3.
If 4 such debts are removed, it means:
(-3) x (-4) = +12
This means the person becomes Rs. 12 richer because the debts have been taken away.
Therefore, a negative times a negative gives a positive.
Think and Reflect (NCERT Textbook Page No. 47)
12. Can you explain why we need q ≠ 0 in the definition of a rational number?
Solution: We need q ≠ 0 in the definition of a rational number because division by zero is not defined in mathematics.
A rational number is written as:
p/q
where p and q are integers, and q ≠ 0.
If q = 0, then the number becomes:
p/0
This has no meaningful value because no number can be multiplied by 0 to give p, when p is not zero.
For example:
5/0 is not defined.
So, q must not be equal to 0.
Therefore, in a rational number p/q, we need q ≠ 0 because division by zero is undefined.
Think and Reflect (NCERT Textbook Page No. 49)
13. While adding or subtracting two rational numbers having different denominators, how will you make the denominators equal?
Solution: To add or subtract two rational numbers with different denominators, first make their denominators equal by finding the LCM of the denominators.
Example:
2/5 + 3/10
The denominators are 5 and 10.
LCM of 5 and 10 = 10
Now convert 2/5 into an equivalent fraction with denominator 10:
2/5 = 4/10
So,
2/5 + 3/10 = 4/10 + 3/10 = 7/10
Therefore, while adding or subtracting rational numbers with different denominators, we make the denominators equal by converting them into equivalent fractions using the LCM of the denominators.
The World of Numbers Worksheet Class 9 Chapter 3 helps students revise the chapter in a structured and exam-focused way. Since this chapter introduces important concepts such as natural numbers, zero, integers, rational numbers, irrational numbers, and real numbers, worksheet practice makes it easier for students to understand and remember each topic clearly.
No courses found
The World of Numbers Class 9 worksheet by Infinity Learn includes multiple choice questions, fill in the blanks, very short answer questions, short answer questions, long answer questions, and extra practice questions with answers. It is designed to help students revise Chapter 3 in a structured way.
Yes, The World of Numbers Class 9 worksheet is useful for CBSE exams because it helps students practise important concepts from the chapter in different question formats. Infinity Learn’s worksheet supports revision, self-assessment, and exam preparation.
Yes, students can download The World of Numbers Class 9 worksheet with answers PDF from Infinity Learn for offline practice. The PDF format makes it easy to revise the chapter anytime and practise questions before school tests and exams.
The World of Numbers Class 9 covers important topics such as natural numbers, one-to-one correspondence, zero, integers, rational numbers, representation on the number line, decimal expansion, irrational numbers, and real numbers. Infinity Learn’s worksheet includes questions based on these key concepts.
Yes, MCQs are included in The World of Numbers Class 9 worksheet by Infinity Learn. These multiple choice questions help students quickly revise definitions, properties, examples, and important concepts from Chapter 3.