Class 9 Maths Chapter 1 Number Systems – Exercise 1.1 Solutions

Introduction

Exercise 1.1 of NCERT Class 9 Maths Chapter 1 introduces the basics of rational and irrational numbers. You’ll learn how to represent numbers in the form $\frac{p}{q}$, find rational numbers between two integers, and understand decimal expansions.

Key Formula Used

• Rational Number Definition: A number is rational if it can be expressed as $\frac{p}{q}$, where $p,q\in \mathbb{Z}$ and $q\mathrm{\ne }0$.

• Irrational Number Definition: Numbers that cannot be expressed in the form $\frac{p}{q}$. Examples: $\sqrt{2},\pi$.

Solved Questions (Step by Step)

Q1. Show that zero is a rational number.

Solution:

• Rational numbers are of the form $\frac{p}{q}$, where $q\mathrm{\ne }0$.

• $0=\frac{0}{1}$.

• Hence, zero is rational.

Q2. Find six rational numbers between 3 and 4.

Solution:

• Divide the interval into tenths: $\frac{31}{10}=3.1,\frac{32}{10}=3.2,\frac{33}{10}=3.3,\frac{34}{10}=3.4,\frac{35}{10}=3.5,\frac{36}{10}=3.6$.

• These six lie between 3 and 4.

Q3. Express $0.\overline{333}$ as a fraction.

Solution:

• Let $x=0.\overline{333}$.

• $10x=3.\overline{333}$.

• Subtract: $10x-x=3$.

• $9x=3⟹x=\frac{1}{3}$.

Q4. Express $\mathrm{0.777...}$ as a fraction.

Solution:

• Let $x=0.\overline{777}$.

• $10x=7.\overline{777}$.

• Subtract: $9x=7$.

• $x=\frac{7}{9}$.

Q5. Express $\mathrm{0.999...}$ as a fraction.

Solution:

• Let $x=0.\overline{999}$.

• $10x=9.\overline{999}$.

• Subtract: $9x=9$.

• $x=1$.

Q6. Find rational numbers between $\frac{1}{2}$ and $\frac{3}{4}$.

Solution:

• Convert to decimals: $\frac{1}{2}=0.5,\frac{3}{4}=0.75$.

• Choose: $0.55,0.6,0.65,0.7$.

• These are rational numbers between them.

Q7. Express $\mathrm{0.142857...}$ as a fraction.

Solution:

• Let $x=0.\overline{142857}$.

• ${10}^{6}x=142857.\overline{142857}$.

• Subtract: $999999x=142857$.

• $x=\frac{142857}{999999}=\frac{1}{7}$.

Q8. Express $\mathrm{0.0833...}$ as a fraction.

Solution:

• Let $x=\mathrm{0.0833...}$.

• Multiply by 10: $10x=\mathrm{0.833...}$.

• Multiply by 100: $100x=\mathrm{8.333...}$.

• Subtract: $100x-10x=\mathrm{8.333...}-\mathrm{0.833...}$.

• $90x=7.5⟹x=\frac{1}{12}$.

Q9. Express $\mathrm{0.0016...}$ as a fraction.

Solution:

• Let $x=\mathrm{0.0016...}$.

• Multiply by 10000: $10000x=16.\overline{66}$.

• Simplify: $x=\frac{1}{600}$.

Q10. Show that $\sqrt{2}$ is irrational.

Solution:

• Assume $\sqrt{2}=\frac{p}{q}$.

• Then $2q^2=p^2$.

• $p^2$ is even, so $p$ is even.

• Let $p=2k$. Then $p^2=4k^2$.

• Substituting: $2q^2=4k^2⟹q^2=2k^2$.

• So $q$ is also even.

• Contradiction: both $p,q$ have common factor 2.

• Hence, $\sqrt{2}$ is irrational.

Q11. Show that $\sqrt{5}$ is irrational.

(Same proof method as Q10).

Q12. Show that $\sqrt{3}$ is irrational.

(Same proof method as Q10).

Q13. Show that $\sqrt{7}$ is irrational.

(Same proof method as Q10).

Q14. Show that $2+\sqrt{5}$ is irrational.

Solution:

• Suppose $2+\sqrt{5}$ is rational.

• Then $\sqrt{5}=(rational-2)$.

• But $\sqrt{5}$ is irrational.

• Contradiction.

• Hence, $2+\sqrt{5}$ is irrational.

Q15. Show that $3\sqrt{2}$ is irrational.

Solution:

• Suppose $3\sqrt{2}$ is rational.

• Then $\sqrt{2}=\frac{rational}{3}$.

• But $\sqrt{2}$ is irrational.

• Contradiction.

• Hence, $3\sqrt{2}$ is irrational.

Q16. Show that $\frac{1}{\sqrt{2}}$ is irrational.

Solution:

• Suppose $\frac{1}{\sqrt{2}}$ is rational.

• Then $\sqrt{2}=\frac{1}{rational}$.

• But $\sqrt{2}$ is irrational.

• Contradiction.

• Hence, $\frac{1}{\sqrt{2}}$ is irrational.

FAQs

• Q: How do you check if a number is rational? A: Try expressing it as $\frac{p}{q}$ with integers $p,q$. If possible, it’s rational.

• Q: Are terminating decimals always rational? A: Yes, because they can be expressed as fractions.

Conclusion

Exercise 1.1 builds the foundation of rational and irrational numbers. Mastering these basics will help you in later topics like algebra and real numbers.

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