Class 9 Maths Chapter 1 Number Systems – Exercise 1.6 NCERT Solutions

Introduction

Exercise 1.4 teaches you how to represent real numbers on the number line. You’ll learn the magnification method to locate irrational numbers like $\sqrt{2}, \sqrt{3}, \sqrt{5}$ , and decimals such as $1.732...$ .

Key Concept

Magnification Method: To represent irrational numbers, divide the number line into smaller intervals repeatedly until the required decimal approximation is reached.

Solved Questions (Step by Step)

Q1. Show that $2$ is irrational.

Solution:

Assume $\sqrt{2} = \frac{p}{q}$ .
Then $2 q^{2} = p^{2}$ .
$p$ must be even, let $p = 2 k$ .
Substituting: $2 q^{2} = 4 k^{2} ⟹ q^{2} = 2 k^{2}$ .
So $q$ is also even.
Contradiction: both $p, q$ have common factor 2.
Hence, $\sqrt{2}$ is irrational.

Q2. Show that $3$ is irrational.

(Same method as Q1).

Q3. Show that $5$ is irrational.

(Same method as Q1).

Q4. Show that $7$ is irrational.

(Same method as Q1).

Q5. Show that $11$ is irrational.

(Same method as Q1).

Q6. Show that $13$ is irrational.

(Same method as Q1).

Q7. Show that $2 + \sqrt{3}$ is irrational.

Solution:

Suppose $2 + \sqrt{3}$ is rational.
Then $\sqrt{3} = (r a t i o n a l - 2)$ .
But $\sqrt{3}$ is irrational.
Contradiction.
Hence, $2 + \sqrt{3}$ is irrational.

Q8. Show that $5 \sqrt{2}$ is irrational.

Solution:

Suppose $5 \sqrt{2}$ is rational.
Then $\sqrt{2} = \frac{r a t i o n a l}{5}$ .
But $\sqrt{2}$ is irrational.
Contradiction.
Hence, $5 \sqrt{2}$ is irrational.

Q9. Show that $\frac{1}{\sqrt{3}}$ is irrational.

Solution:

Suppose $\frac{1}{\sqrt{3}}$ is rational.
Then $\sqrt{3} = \frac{1}{r a t i o n a l}$ .
But $\sqrt{3}$ is irrational.
Contradiction.
Hence, $\frac{1}{\sqrt{3}}$ is irrational.

Q10. Show that decimal expansion of $\frac{1}{3}$ is non‑terminating repeating.

Solution:

Divide 1 by 3: $1 \div 3 = 0.333...$ .
Decimal expansion is repeating.
Hence, rational.

Q11. Show that decimal expansion of $\frac{1}{7}$ is non‑terminating repeating.

Solution:

Divide 1 by 7: $0.142857142857...$ .
Repeats in block of 6 digits.
Hence, rational.

Q12. Show that decimal expansion of $\frac{2}{11}$ is non‑terminating repeating.

Solution:

Divide 2 by 11: $0.181818...$ .
Repeats in block of 2 digits.
Hence, rational.

Q13. Show that decimal expansion of $\frac{1}{2}$ is terminating.

Solution:

Divide 1 by 2: $0.5$ .
Terminates.
Hence, rational.

Q14. Show that decimal expansion of $\frac{1}{4}$ is terminating.

Solution:

Divide 1 by 4: $0.25$ .
Terminates.
Hence, rational.

Q15. Show that decimal expansion of $\frac{1}{8}$ is terminating.

Solution:

Divide 1 by 8: $0.125$ .
Terminates.
Hence, rational.

Conclusion

Exercise 1.2 has 15 questions that strengthen your understanding of irrational numbers and decimal expansions. These proofs and examples are essential for mastering number systems.

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