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 $\mathrm{1.732...}$.

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 $\sqrt{2}$ is irrational. Solution: Assume $\sqrt{2}=\frac{p}{q}$, where $p,q\in \mathbb{Z},q\mathrm{\ne }0$.

$$2q^2=p^2$$

So $p^2$ is even, hence $p$ must be even. Let $p=2k$.

Substituting:

$$2q^2=(2k{)}^2=4k^2\Rightarrow q^2=2k^2$$

Thus $q$ is also even.

Contradiction: both $p$ and $q$ have common factor 2. Hence, $\sqrt{2}$ is irrational.

Q2. Show that $\sqrt{3}$ is irrational. (Same method as Q1).

Q3. Show that $\sqrt{5}$ is irrational. (Same method as Q1).

Q4. Show that $\sqrt{7}$ is irrational. (Same method as Q1).

Q5. Show that $\sqrt{11}$ is irrational. (Same method as Q1).

Q6. Show that $\sqrt{13}$ is irrational. (Same method as Q1)..

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

Solution:

• Suppose $2+\sqrt{3}$ is rational.

• Then $\sqrt{3}=(rational-2)$.

• 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 $\sqrt{2}=\frac{rational}{5}$.

• 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 $\sqrt{3}=\frac{1}{rational}$.

• 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: $1÷3=\mathrm{0.333...}$.

• Decimal expansion is repeating.

• Hence, rational.

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

Solution:

• Divide 1 by 7: $\mathrm{0.142857142857...}$.

• 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: $\mathrm{0.181818...}$.

• 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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