NCERT Solutions for Class 9 Maths Chapter 1 – Number Systems

Number Systems is the first and foundational chapter of Class 9 Mathematics that builds a strong base for higher classes. This chapter introduces students to natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers, along with their properties and representation on the number line. A clear understanding of this chapter is essential for school exams and future topics in algebra and real numbers.

NCERT Solutions for Class 9 Maths Chapter 1 – Number Systems are prepared strictly according to the latest CBSE syllabus and exam pattern. The solutions are written in simple, step-by-step language with clear explanations and worked examples, helping students understand concepts easily, avoid common mistakes, and score well in Class 9 examinations.

NCERT Solutions for Class 9 Maths Chapter 1 – Number Systems

Number Systems

Easy

Q.

$\mathrm{Find} \mathrm{five} \mathrm{rational} \mathrm{numbers} \mathrm{between} \frac{3}{5} \mathrm{and} \frac{4}{5}.$

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Number Systems

Easy

Q.

$\mathrm{Visualise} 4.\overline{26 }\mathrm{on} \mathrm{the} \mathrm{number} \mathrm{line}, \mathrm{upto} 4 \mathrm{decimal} \mathrm{places}.$

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Number Systems

Easy

Q.

$\text{Find:\hspace{0.33em}}\left(\text{i}\right){\text{9}}^{\frac{\text{3}}{\text{2}}}\left(\text{ii}\right){\text{32}}^{\frac{\text{2}}{\text{5}}}\left(\text{iii}\right){\text{16}}^{\frac{\text{3}}{\text{4}}}\left(\text{iv}\right){\text{125}}^{\frac{-\text{1}}{\text{3}}}$

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Number Systems

Easy

Q.

$\mathrm{Find}:\left(\text{i}\right) {64}^{\frac{1}{2}}\left(\mathrm{ii}\right) {32}^{\frac{1}{5}}\left(\mathrm{iii}\right) {125}^{\frac{1}{3}}$

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Number Systems

Easy

Q.

$\begin{array}{l}\mathrm{Rationalise} \mathrm{the} \mathrm{denominators} \mathrm{of} \mathrm{the} \mathrm{following}:\\ \left(i\right)\frac{1}{\sqrt{7}}\\ \left(ii\right)\frac{1}{\sqrt{7}-\sqrt{6}}\\ \left(iii\right)\frac{1}{\sqrt{5}+\sqrt{2}}\\ \left(iv\right)\frac{1}{\sqrt{7}-2}\end{array}$

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Number Systems

Medium

Q.

$\mathrm{Represent} \sqrt{9.3} \mathrm{on} \mathrm{the} \mathrm{number} \mathrm{line}.$

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Number Systems

Easy

Q.

$\begin{array}{l}\mathrm{Recall}, \mathrm{\pi } \mathrm{is} \mathrm{defined} \mathrm{as} \mathrm{the} \mathrm{ratio} \mathrm{of} \mathrm{the} \mathrm{circumference} (\mathrm{say} \mathrm{c})\\ \mathrm{of} \mathrm{a} \mathrm{circle} \mathrm{to} \mathrm{its} \mathrm{diameter} \left(\mathrm{say} \mathrm{d}\right).\mathrm{That} \mathrm{is}, \mathrm{\pi }=\frac{\mathrm{c}}{\mathrm{d}}.\mathrm{This} \mathrm{seems}\\ \mathrm{to} \mathrm{contradict} \mathrm{the} \mathrm{fact} \mathrm{that} \mathrm{\pi } \mathrm{is} \mathrm{irrational}.\mathrm{How} \mathrm{will} \mathrm{you}\\ \mathrm{resolve} \mathrm{this} \mathrm{contradiction}?\end{array}$

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Number Systems

Easy

Q.

$\begin{array}{l}\mathrm{Simplify} \mathrm{each} \mathrm{of} \mathrm{the} \mathrm{following} \mathrm{expressions}:\\ \left(\mathrm{i}\right)(3+\sqrt{3})(2+\sqrt{2}) \left(\mathrm{ii}\right) (3+\sqrt{3})(3-\sqrt{3})\\ \left(\mathrm{iii}\right) {(\sqrt{5}+\sqrt{2})}^2\left(\mathrm{iv}\right) (\sqrt{5}-\sqrt{2})(\sqrt{5}+\sqrt{2})\end{array}$

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Number Systems

Medium

Q.

$\begin{array}{l}\text{Classify\hspace{0.33em}the\hspace{0.33em}following\hspace{0.33em}numbers\hspace{0.33em}as\hspace{0.33em}rational\hspace{0.33em}or\hspace{0.33em}irrational:}\\ \left(\text{i}\right)\text{\hspace{0.33em}2}-\sqrt{\text{5}}\left(\text{ii}\right)\left(\text{3+}\sqrt{\text{23}}\right)-\sqrt{\text{23}}\left(\text{iii}\right)\frac{\text{2}\sqrt{\text{7}}}{\text{7}\sqrt{\text{7}}}\\ \left(\text{iv}\right)\frac{\text{1}}{\sqrt{\text{2}}}\left(\text{v}\right)\text{\hspace{0.33em}2π}\end{array}$

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Number Systems

Easy

Q.

$\begin{array}{l}\text{Classify the following numbers as rational or irrational:}\\ \left(\text{i}\right)\sqrt{\text{23}}\left(\text{ii}\right)\sqrt{\text{225}}\left(\text{iii}\right)\text{\hspace{0.33em}0.3796\hspace{0.33em}}\\ \left(\text{iv}\right)\text{\hspace{0.33em}7.478478}...\left(\text{v}\right)\text{\hspace{0.33em}1.101001000100001}...\end{array}$

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Number Systems

Easy

Q.

State whether the following statements are true or false. Give reasons for your answers.
(i) Every natural number is a whole number.
(ii) Every integer is a whole number.
(iii) Every rational number is a whole number.

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Number Systems

Easy

Q.

$\begin{array}{l}\mathrm{Look} \mathrm{at} \mathrm{several} \mathrm{examples} \mathrm{of} \mathrm{rational} \mathrm{numbers} \mathrm{in} \mathrm{the} \mathrm{form}\\ \frac{\mathrm{p}}{\mathrm{q}} \left(\mathrm{q}\ne 0\right) , \mathrm{where} \mathrm{p} \mathrm{and} \mathrm{q} \mathrm{are} \mathrm{integers} \mathrm{with} \mathrm{no} \mathrm{common}\\ \mathrm{factors} \mathrm{other} \mathrm{than} 1 \mathrm{and} \mathrm{having} \mathrm{terminating} \mathrm{decimal}\\ \mathrm{representations} \left(\mathrm{expansions}\right).\mathrm{Can} \mathrm{you} \mathrm{guess} \mathrm{what} \mathrm{property}\\ \mathrm{q} \mathrm{must} \mathrm{satisfy}?\end{array}$

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Number Systems

Medium

Q.

$\begin{array}{l}\mathrm{What} \mathrm{can} \mathrm{the} \mathrm{maximum} \mathrm{number} \mathrm{of} \mathrm{digits} \mathrm{be} \mathrm{in} \mathrm{the} \mathrm{repeating}\\ \mathrm{block} \mathrm{of} \mathrm{digits} \mathrm{in} \mathrm{the} \mathrm{decimal} \mathrm{expansion} \mathrm{of} \frac{1}{17}? \mathrm{Perform} \mathrm{the}\\ \mathrm{division} \mathrm{to} \mathrm{check} \mathrm{your} \mathrm{answer}.\end{array}$

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Number Systems

Easy

Q.

$\begin{array}{l}\text{Express\hspace{0.33em}0.99999,\hspace{0.33em}in\hspace{0.33em}the\hspace{0.33em}form\hspace{0.33em}}\frac{\text{p}}{\text{q}}\text{.\hspace{0.33em}Are\hspace{0.33em}you\hspace{0.33em}surprised\hspace{0.33em}by\hspace{0.33em}your\hspace{0.33em}answer?}\\ \text{With\hspace{0.33em}your\hspace{0.33em}teacher\hspace{0.33em}and\hspace{0.33em}classmates\hspace{0.33em}discussway\hspace{0.33em}the\hspace{0.33em}answer\hspace{0.33em}makes\hspace{0.33em}sense.}\end{array}$

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Number Systems

Easy

Q.

$\begin{array}{l}\text{Express\hspace{0.33em}the\hspace{0.33em}following\hspace{0.33em}in\hspace{0.33em}the\hspace{0.33em}form\hspace{0.33em}}\frac{\text{p}}{\text{q}}\text{,\hspace{0.33em}where\hspace{0.33em}p\hspace{0.33em}and\hspace{0.33em}q\hspace{0.33em}are}\\ \text{integers\hspace{0.33em}and\hspace{0.33em}q¹0.}\\ \left(\text{i}\right)\text{\hspace{0.33em}0\hspace{0.33em}\hspace{0.33em}}\left(\text{ii}\right)\text{\hspace{0.33em}0.4}\overline{\text{7 }}\left(\text{iii}\right)\text{\hspace{0.33em}0.}\overline{\text{001}}\end{array}$

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Number Systems

Easy

Q.

$\begin{array}{l}\text{You\hspace{0.33em}know\hspace{0.33em}that\hspace{0.33em}}\frac{\text{1}}{\text{7}}\text{\hspace{0.33em}=\hspace{0.33em}0}\overline{\text{.142857}}\text{.\hspace{0.33em}Can\hspace{0.33em}you\hspace{0.33em}predict\hspace{0.33em}what\hspace{0.33em}the}\\ \text{decimal\hspace{0.33em}expansions\hspace{0.33em}of\hspace{0.33em}}\frac{\text{2}}{\text{7}}\text{\hspace{0.33em},}\frac{\text{3}}{\text{7}}\text{\hspace{0.33em},}\frac{\text{4}}{\text{7}}\text{,\hspace{0.33em}}\frac{\text{5}}{\text{7}}\text{,\hspace{0.33em}}\frac{\text{6}}{\text{7}}\text{\hspace{0.33em}are,\hspace{0.33em}without\hspace{0.33em}actually}\\ \text{doing\hspace{0.33em}the\hspace{0.33em}long\hspace{0.33em}division?\hspace{0.33em}If\hspace{0.33em}so,\hspace{0.33em}how?}\end{array}$

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Number Systems

Easy

Q.

$\begin{array}{l}\text{Write the following in decimal form and say what kind of}\\ \text{decimal expansion each has:}\\ \left(\text{i}\right)\frac{\text{36}}{\text{100}}\left(\text{ii}\right)\frac{\text{1}}{\text{11}}\left(\text{iii}\right)\text{\hspace{0.33em}4}\frac{\text{1}}{\text{8}}\left(\text{iv}\right)\frac{\text{3}}{\text{13}}\left(\text{v}\right)\frac{\text{2}}{\text{11}}\left(\text{vi}\right)\frac{\text{329}}{\text{400}}\end{array}$

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Number Systems

Easy

Q.

$\mathrm{Show} \mathrm{how} \sqrt{5} \mathrm{can} \mathrm{be} \mathrm{represented} \mathrm{on} \mathrm{the} \mathrm{number} \mathrm{line}.$

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Number Systems

Easy

Q.

State whether the following statements are true or false. Justify your answers.
(i)  Every irrational number is a real number.
(ii) Every point on the number line is of the form $\sqrt{\text{m}}$ where m is a natural number.
(iii) Every real number is an irrational number.

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Number Systems

Easy

Q.

$\mathrm{Simplify}:\left(\mathrm{i}\right)2^{\frac{2}{3}}.2^{\frac{1}{5}}\left(\mathrm{ii}\right){\left(\frac{1}{3^3}\right)}^7\left(\mathrm{iii}\right)\frac{{11}^{\frac{1}{2}}}{{11}^{\frac{1}{4}}}\left(\mathrm{iv}\right)7^{\frac{1}{2}}.8^{\frac{1}{2}}$

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1) Q. Find five rational numbers between $\frac{3}{5}$ and $\frac{4}{5}$.

Answer:

Take a common denominator 50.

Five rational numbers between them are:

2) Q. State whether the following statements are True or False. Give reasons.

(i) Every natural number is a whole number.
(ii) Every integer is a whole number.
(iii) Every rational number is a whole number.

Answer:

(i) True — Natural numbers $1, 2, 3, \dots$ are part of whole numbers $0, 1, 2, 3, \dots$.

(ii) False — Integers include negative numbers such as $-1, -2$, which are not whole numbers.

(iii) False — Rational numbers like $\frac{1}{2}$ and $\frac{3}{4}$ are not whole numbers.

3) Q. State whether the following statements are True or False. Justify your answers.

(i) Every irrational number is a real number.
(ii) Every point on the number line is of the form $m$, where $m$ is a natural number.
(iii) Every real number is an irrational number.

Answer:

(i) True — Irrational numbers are a subset of real numbers.

(ii) False — The number line contains fractions, negative numbers and irrational numbers, not only natural numbers.

(iii) False — Real numbers include both rational and irrational numbers.

4) Q. How can the number 5 be represented on the number line?

Answer:

Start from 0 and move 5 units to the right on the number line. The point obtained represents the number 5.

5) Q. Write the following in decimal form and state the type of decimal expansion.

Answer:

6) Q. If $\frac{1}{7} = 0.\overline{142857}$, predict the decimal expansions of

without long division.

Answer:

The digits repeat in a cyclic pattern.

7) Q. Express the following in the form $\frac{p}{q}$, where $q \neq 0$.

Answer:

(i) $0 = \frac{0}{1}$

(ii) $0.\overline{47} = \frac{47}{99}$

(iii) $0.\overline{001} = \frac{1}{999}$

8) Q. Express $0.9999\ldots$ in the form $\frac{p}{q}$. Are you surprised?

Answer:

Let $x = 0.9999\ldots$

Subtracting,

Therefore,

This result may seem surprising, but it is mathematically exact.

9) Q. What is the maximum number of digits in the repeating block of the decimal expansion of $\frac{1}{17}$?

Answer:

The maximum length of the repeating block is:

Thus, the decimal expansion of $\frac{1}{17}$ can have at most 16 repeating digits.

10) Q. What condition must the denominator $q$ satisfy for a rational number $\frac{p}{q}$ (in lowest form) to have a terminating decimal expansion?

Answer:

The prime factorisation of $q$ must contain only 2 and/or 5.

That is,

11) Q. Classify the following numbers as rational or irrational.

Answer:

12) Q. Visualise $4.\overline{26}$ on the number line up to four decimal places.

Answer:

Up to four decimal places:

Mark the point 4.2626 on the number line.

13) Q. Classify the following as rational or irrational.

Answer:

(i) $2 - \sqrt{5}$ → Irrational
(ii) $3 + \sqrt{23} - \sqrt{23} = 3$ → Rational
(iii) $\frac{2}{7777}$ → Rational
(iv) $\sqrt{12} = 2\sqrt{3}$ → Irrational
(v) $2\pi$ → Irrational

14) Q. Simplify the following expressions.

Answer:

(i) $(\sqrt{3}+\sqrt{3})(\sqrt{2}+\sqrt{2}) = 4\sqrt{6}$
(ii) $(\sqrt{3}+\sqrt{3})(\sqrt{3}-\sqrt{3}) = 0$
(iii) $(\sqrt{5}+\sqrt{2})^2 = 7 + 2\sqrt{10}$
(iv) $(\sqrt{5}-\sqrt{2})(\sqrt{5}+\sqrt{2}) = 3$

15) Q. π is defined as $\frac{c}{d}$. Why is π irrational?

Answer:

The ratio $\frac{c}{d}$ represents the ratio of two real quantities: circumference and diameter.
π being irrational means it cannot be expressed as $\frac{p}{q}$ where p and q are integers.
Thus, there is no contradiction.

16) Q. Represent 9.3 on the number line.

Answer:

Divide the interval between 9 and 10 into 10 equal parts.
The third division from 9 represents 9.3.

17) Q. Rationalise the denominators.

Answer:

(i) $\frac{1}{\sqrt{7}} = \frac{\sqrt{7}}{7}$

(ii) $\frac{1}{\sqrt{7}-\sqrt{6}} = \sqrt{7}+\sqrt{6}$

(iii) $\frac{1}{\sqrt{5}+2} = \sqrt{5}-2$

(iv) $\frac{1}{\sqrt{7}-2} = \frac{\sqrt{7}+2}{3}$

18) Q. Find the following values.

Answer:

19) Q. Find the following values.

Answer:

20) Q. Simplify the following expressions.

Answer:

(i) $2^{2/3} \cdot 2^{1/5} = 2^{13/15}$
(ii) $\left(\frac{1}{3^3}\right)^7 = 3^{-21}$
(iii) $11^{1/2} \cdot 11^{1/4} = 11^{3/4}$
(iv) $7^{1/2} \cdot 8^{1/2} = 2\sqrt{14}$

FAQs: Class 9 Maths Chapter 1 – Number Systems

Q1. Is Number Systems an important chapter for Class 9 exams?
Yes, it is a core foundational chapter for Class 9 Mathematics.

Q2. Which topics are covered in this chapter?
Rational numbers, irrational numbers, real numbers, and number line representation.

Q3. Are proofs asked from this chapter?
Yes, proofs related to irrational numbers are commonly asked.

Q4. Is this chapter important for higher classes?
Yes, it forms the base for Class 10 Real Numbers and Algebra.

Q5. How do NCERT Solutions help?
They provide NCERT-aligned, exam-ready explanations with solved examples.