Class 9 Mathematics Chapter 3 The World of Numbers – Extra Questions with Answers

Not every number can be written as a terminating or repeating decimal, and this set of questions tests the difference between rational and irrational numbers along with the exponent rules that govern them.

Last Updated: September 23, 2026

Very Short Answer Questions (1 mark)

Q1. Is √4 rational or irrational?
Ans: Rational, since √4 = 2 = 2/1.

Q2. What type of decimal expansion does an irrational number have?
Ans: Non-terminating and non-repeating.

Q3. Simplify: 2³ × 2².
Ans: 2⁵ = 32.

Q4. Is every integer a rational number?
Ans: Yes, since every integer n can be written as n/1.

Q5. Name the set that includes both rational and irrational numbers.
Ans: Real numbers.

Short Answer Questions (2–3 marks)

Q6. Classify the following as rational or irrational: 0.333…, √9, √7, 22/7.
Ans: 0.333… is rational (repeating decimal, equals 1/3). √9 = 3 is rational. √7 is irrational (non-perfect square). 22/7 is rational (a fraction of integers — note this is only an approximation of π, not π itself).

Q7. Rationalise the denominator of 1/√2.
Ans: Multiply numerator and denominator by √2: (1×√2)/(√2×√2) = √2/2.

Q8. Describe, in words, how √2 can be represented accurately on the number line using a geometric construction.
Ans: Construct a right-angled triangle with both legs of length 1 unit on the number line; by the Pythagoras theorem, the hypotenuse has length √(1²+1²) = √2. Using a compass, this hypotenuse length can be transferred (swung as an arc) from the origin onto the number line, precisely marking the point √2.

Higher-Order Thinking / Application Questions

Q9. A student claims that the product of two irrational numbers is always irrational, citing √2 × √3 = √6 as an example. Show, using a different example, that this claim is not always true.
Ans: The claim is false. Consider √2 × √2 = 2, which is rational, even though both √2 factors are irrational. So the product of two irrational numbers can be either rational or irrational, depending on the specific numbers involved.

Q10. Simplify the expression (3²)³ ÷ 3⁴ using laws of exponents, showing each step, and explain which law is applied at each step.
Ans: Step 1: (3²)³ = 3²×³ = 3⁶ (using the law (a^m)ⁿ = a^mn). Step 2: 3⁶ ÷ 3⁴ = 3⁶⁻⁴ = 3² = 9 (using the law a^m/aⁿ = a^m−n).

Frequently Asked Questions

How can you show that a number like 0.333… repeating is a rational number even though its decimal expansion never ends?
A repeating decimal like 0.333… can be expressed exactly as the fraction one-third, and since it can be written as a ratio of two integers, it satisfies the definition of a rational number.

Why is the square root of 2 classified as an irrational number?
The square root of 2 cannot be expressed exactly as a ratio of two integers, and its decimal expansion continues infinitely without repeating in any pattern, the defining characteristic of an irrational number.

Written by Satish

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