If a, b, and c are integers, which of the following expressions always results in a negative integer?

a × b × c when all are negative

Assertion (A): For any integer n, n + 0 equals n. Reason (R): Zero is the identity element for addition.

Both A and R are true and R is the correct explanation of A.

Both A and R are true but R is not the correct explanation of A.

A submarine is at a depth of 450 meters below sea level (represented as -450 m). It ascends 120 meters and then descends 200 meters. What is its final depth?

-530 meters, or 530 meters below sea level

Is every integer a rational number? Justify your answer with an example.

Yes. Every integer can be expressed in the form p/q with q = 1. For example, 5 = 5/1.

Describe the Ishango bone and explain its mathematical significance.

The Ishango bone is a 20,000-year-old artifact found in Congo with carved notches. Its mathematical significance is that the notches are grouped into prime numbers (11, 13, 17, 19) and possibly show doubling, indicating early humans had knowledge of counting, primality, and multiplication.

Add the rational numbers −5/6 and 3/4, and express the result in simplest form. Then write an equivalent rational number with denominator 24.

Sum = −10/12 + 9/12 = −1/12. Equivalent with denominator 24: −2/24.

Using Brahmagupta's rules for zero and integers, compute the following and mention the rule used in each case: (i) 56 + 0 (ii) 0 – 34 (iii) (–9) × 8 (iv) (–7) × (–6) (v) 0 ÷ 15

(i) 56, a+0=a; (ii) -34, 0–a = –a; (iii) -72, negative × positive = negative; (iv) 42, negative × negative = positive; (v) 0, 0 ÷ non‑zero = 0.

Simplify the following expression and write the result in its lowest terms: [(–3/4) + (5/6)] × [(–2/3) ÷ (4/5)]. Then, find (i) the additive inverse of the result, and (ii) the multiplicative inverse (reciprocal) of the result.

Result = –5/72; additive inverse = 5/72; multiplicative inverse = –72/5.

In a mathematics class, students are exploring the closure property of operations on sets. They consider the set S = { -2, -1, 0, 1, 2 } and perform addition and multiplication between any two elements. They also recall the sets of natural numbers (ℕ) and integers (ℤ). (1) Is the set S closed under addition? Give a reason. (2) Is the set S closed under multiplication? Give a reason. (3) Is the set of integers closed under subtraction? Explain. (4) Give a counterexample to prove that natural numbers are not closed under subtraction.

S is not closed under addition (e.g., 2+1=3 ∉ S) nor multiplication (e.g., 2×2=4 ∉ S). Integers are closed under subtraction. Natural numbers are not closed under subtraction (e.g., 3-5 = -2).

Class 9 Maths Chapter 3: The World of Numbers — Important Questions & Sample Paper

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Yes — this page has 44+ original Class 9 Mathematics Chapter 3 (“The World of Numbers”) important questions with answers (Multiple Choice (MCQ), Assertion–Reason, Short Answer, Short Answer, Long Answer, Case Study). Practise them free, or generate a full CBSE board-pattern sample paper (80 marks) and export it to PDF or Word — in English & Hindi, for 2026-27.

Chapter 3, 'The World of Numbers' from the Class 9 NCERT textbook Ganita Manjari, delves into the foundational concepts of integers and rational numbers, tracing their historical development and formalizing their operations. Students explore Brahmagupta's pioneering rules for handling negative numbers and zero, learning to perform addition, subtraction, multiplication, and division with integers. The chapter connects these abstract ideas to real-life situations, such as tracking profit/loss or counting with pebbles, making the concepts tangible. It then extends to rational numbers—numbers expressed as p/q—covering their properties, closure under operations, and the computation of additive and multiplicative inverses. Key topics include the Ishango bone as evidence of early tallying systems, the formal rules for multiplying negatives (e.g., 'product of two debts is a fortune'), and simplifying complex expressions involving integers and fractions. Exam questions often require students to simplify multi-step expressions, justify rules using properties, and apply rational number operations in context. Mastering this chapter builds a strong arithmetic foundation for algebra and beyond.

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Mathematics — The World of Numbers

Class 9Time: 3 hrsMax Marks: 80

SECTION A

1.
A spice trader takes a loan of ₹850, makes a profit of ₹1,200 the next day, and then incurs a loss of ₹450 the following week. What is his final financial standing?
(a) ₹100 profit(b) ₹100 debt(c) ₹2,500 profit(d) ₹2,500 debt
1
2.
If a, b, and c are integers, which of the following expressions always results in a negative integer?
(a) a + b + c(b) a - b - c(c) a × b × c when all are negative(d) a + b - c
1
3.
Which of the following is equivalent to the fraction -1/2?
(a) -2/4(b) 1/-2(c) Both -2/4 and 1/-2(d) None of these
1

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In a CBSE exam, this chapter typically contributes questions across the following types. The last column shows how many original questions of each type we have ready in our bank for this chapter:

Question type	Marks each	In our bank
Multiple Choice (MCQ)	1 mark	13
Assertion–Reason	1 mark	6
Short Answer	2 marks	8
Short Answer	3 marks	6
Long Answer	5 marks	5
Case Study	4 marks	6

44 original, exam-style questions in our bank for this chapter — with answers.

Important & sample questions (with answers)

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Q1. A spice trader takes a loan of ₹850, makes a profit of ₹1,200 the next day, and then incurs a loss of ₹450 the following week. What is his final financial standing?
1 mark
Multiple Choice (MCQ)
(A) ₹100 profit(B) ₹100 debt(C) ₹2,500 profit(D) ₹2,500 debt
▸ Answer▾ Answer
₹100 debt
Q2. If a, b, and c are integers, which of the following expressions always results in a negative integer?
1 mark
Multiple Choice (MCQ)
(A) a + b + c(B) a - b - c(C) a × b × c when all are negative(D) a + b - c
▸ Answer▾ Answer
a × b × c when all are negative
Q3. Which of the following is equivalent to the fraction -1/2?
1 mark
Multiple Choice (MCQ)
(A) -2/4(B) 1/-2(C) Both -2/4 and 1/-2(D) None of these
▸ Answer▾ Answer
Both -2/4 and 1/-2
Q4. Brahmagupta's rules for zero in the Brāhmasphuṭasiddhānta include operations of addition, subtraction, and multiplication. Which of the following operations is NOT among the rules he defined for zero?
1 mark
Multiple Choice (MCQ)
(A) Adding zero to a number(B) Subtracting zero from a number(C) Multiplying a number by zero(D) Dividing a number by zero
▸ Answer▾ Answer
Dividing a number by zero
Q5. Assertion (A): For any integer n, n + 0 equals n. Reason (R): Zero is the identity element for addition.
1 mark
Assertion–Reason
(A) Both A and R are true and R is the correct explanation of A.(B) Both A and R are true but R is not the correct explanation of A.(C) A is true but R is false.(D) A is false but R is true.
▸ Answer▾ Answer
Both A and R are true and R is the correct explanation of A.
Q6. A submarine is at a depth of 450 meters below sea level (represented as -450 m). It ascends 120 meters and then descends 200 meters. What is its final depth?
2 marks
Short Answer
▸ Answer▾ Answer
-530 meters, or 530 meters below sea level
Q7. Is every integer a rational number? Justify your answer with an example.
2 marks
Short Answer
▸ Answer▾ Answer
Yes. Every integer can be expressed in the form p/q with q = 1. For example, 5 = 5/1.
Q8. Describe the Ishango bone and explain its mathematical significance.
3 marks
Short Answer
▸ Answer▾ Answer
The Ishango bone is a 20,000-year-old artifact found in Congo with carved notches. Its mathematical significance is that the notches are grouped into prime numbers (11, 13, 17, 19) and possibly show doubling, indicating early humans had knowledge of counting, primality, and multiplication.
Q9. Add the rational numbers −5/6 and 3/4, and express the result in simplest form. Then write an equivalent rational number with denominator 24.
3 marks
Short Answer
▸ Answer▾ Answer
Sum = −10/12 + 9/12 = −1/12. Equivalent with denominator 24: −2/24.
Q10. Using Brahmagupta's rules for zero and integers, compute the following and mention the rule used in each case: (i) 56 + 0 (ii) 0 – 34 (iii) (–9) × 8 (iv) (–7) × (–6) (v) 0 ÷ 15
5 marks
Long Answer
▸ Answer▾ Answer
(i) 56, a+0=a; (ii) -34, 0–a = –a; (iii) -72, negative × positive = negative; (iv) 42, negative × negative = positive; (v) 0, 0 ÷ non‑zero = 0.
Q11. Simplify the following expression and write the result in its lowest terms: [(–3/4) + (5/6)] × [(–2/3) ÷ (4/5)]. Then, find (i) the additive inverse of the result, and (ii) the multiplicative inverse (reciprocal) of the result.
5 marks
Long Answer
▸ Answer▾ Answer
Result = –5/72; additive inverse = 5/72; multiplicative inverse = –72/5.
Q12. In a mathematics class, students are exploring the closure property of operations on sets. They consider the set S = { -2, -1, 0, 1, 2 } and perform addition and multiplication between any two elements. They also recall the sets of natural numbers (ℕ) and integers (ℤ).
4 marks
Case Study
1. (i) Is the set S closed under addition? Give a reason.1 mark
2. (ii) Is the set S closed under multiplication? Give a reason.1 mark
3. (iii) Is the set of integers closed under subtraction? Explain.1 mark
4. (iv) Give a counterexample to prove that natural numbers are not closed under subtraction.1 mark
▸ Answer▾ Answer
S is not closed under addition (e.g., 2+1=3 ∉ S) nor multiplication (e.g., 2×2=4 ∉ S). Integers are closed under subtraction. Natural numbers are not closed under subtraction (e.g., 3-5 = -2).

Frequently asked questions

What are Brahmagupta's rules for multiplying negative numbers?

Brahmagupta's rules state that a negative number multiplied by a positive number yields a negative result (e.g., (–a) × b = –ab), while the product of two negative numbers is positive (e.g., (–a) × (–b) = ab). He famously described the latter as 'the product of two debts is a fortune,' forming the foundation for modern integer multiplication.

How are rational numbers defined and what are their closure properties?

A rational number is any number expressed as p/q where p and q are integers and q ≠ 0. The set of rational numbers is closed under addition, subtraction, and multiplication—the result of these operations on any two rational numbers is always a rational number. However, it is not closed under division because division by zero is undefined, and division may yield a rational number only if the divisor is non-zero.

What is the mathematical significance of the Ishango bone?

The Ishango bone, a 20,000-year-old artifact found in Congo, features a series of notches believed to represent a tallying system or early arithmetic. Its significance lies in demonstrating that prehistoric humans had developed a concept of numbers and possibly even prime numbers, making it one of the oldest known mathematical tools.

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