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Number System Worksheets for CBSE Class 9 Maths – Free PDF Solutions

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Practice Essential Number System Questions for Class 9 Maths (2025-26)

When it comes to Maths as a whole, not many people excel in this subject as it is a subject that solely relies on the logical reasoning functions of the brain. That is why the number system can be intimidating to most students. In Chapter 1, Number System for Class 9, students will learn the number system and their types and how to solve the equations. 


So, what is the number system, and what does the number system syllabus contain? A number system can be defined as an arithmetic system or practice of writing numbers to express them. It is the mathematical notation for continuously representing numbers of any given set by using a certain set of digits, symbols, or other characters. It offers a unique representation of every number. It signifies the arithmetic and algebraic structure of the given figures, permitting us to carry out mathematical calculations such as addition, subtraction, and division. 


All these figures carry their values, which can be determined by looking at the digit, the position in the number, and the base of the number. A number is a mathematical value used to count, measure, or label objects. Regarding the number system, these numbers are used as digits. 


With the help of worksheets such as the Number System Class 9 worksheet, Class 9 Maths Chapter 1 worksheet pdf, and worksheet for Class 9 Maths Chapter 1 with solutions and the operations on Real Numbers Class 9 worksheet, students will have a better understanding of what number systems are and how to solve them accurately.

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Access Worksheet for Class 9 Maths Number System

1. It is impossible to represent a rational number in decimal form.

  1. Terminating

  2. Non- terminating

  3. Repeating or Non- Terminating

  4. Non-repeating or Non- terminating


2. Between two rational numbers

  1. There is no rational number.

  2. There is exactly one rational number.

  3. There are infinitely many rational numbers.

  4. There are only rational numbers and no irrational numbers.


3. The product of any two irrational numbers,

  1. is always an irrational number.

  2. is always a rational number.

  3. is always an integer.

  4. can be rational or irrational.


4. Which of the following is irrational?

  1. $\sqrt{7}$

  2. $\sqrt{81}$

  3. $\dfrac{\sqrt{12}}{\sqrt{3}}$

  4. $\dfrac{\sqrt{4}}{9}$


5. What is the value is $\sqrt{4} \times \sqrt{81}$?

  1. 36

  2. 18

  3. 16

  4. 42


6. Fill in the blanks;

  1. Any two integers are separated by a finite number of others …..

  2. There are an ….. amount of rational numbers between 15 and 18.

  3. X+Y is a rational number if x and y are both ……

  4. Value of $\sqrt[3]{8}$ …….


7. Match the Column:


Column I

Column II

Value of 1.9999…..

$\dfrac{3}{7}$

The Simplest form of a rational number $\dfrac{177}{413}$

Recurring decimal and Non- Terminating

0.36

2

0.18181818………

Terminating Decimal


8. Using two irrational numbers as an example:

  1. Product is an irrational number.

  2. Difference is an irrational number.

  3. Division is an irrational number.


9. Simplify; $(\sqrt{5}+\sqrt{6})(\sqrt{5}-\sqrt{6})$.


10. Simplify; $\sqrt[3]{1331}-\sqrt{100}+\sqrt{81}$.


11. Calculate the value of $\dfrac{11^{\dfrac{1}{2}}}{11^{\dfrac{1}{4}}}$.


12. Calculate the $\dfrac{x}{y}$ form of $0.777 . . . . .$, where $\mathbf{x}$ and $\mathbf{y}$ are integers and $\mathbf{y}$ does not equal to zero.


13. Find three rational number between $\dfrac{9}{11}$ and $\dfrac{5}{11}$.


14. The value of $\dfrac{\sqrt{8}+\sqrt{12}}{\sqrt{32}+\sqrt{48}}$.


15. The value of $a^b+b^a$, if $\mathbf{a}=2$ and $\mathbf{b}=3$


16. Simplify; $2^{\dfrac{2}{3}} \cdot 2^{\dfrac{1}{5}}$


17. Find the value of $\dfrac{1}{a^b+b^a}$, where $a=5, \mathbf{b}=2$


18. Arrange in ascending order $\sqrt[3]{2}, \sqrt{3}, \sqrt[6]{5} \text {. }$


19. Simplify $(4 \sqrt{5}+3 \sqrt{7})^2$


20. Find the value of a, If $\left(\dfrac{y}{x}\right)^{2a-8}=\left(\dfrac{x}{y}\right)^{a-1}$.


21. Rationalize the denominators of $\dfrac{1}{\sqrt{7}}$.


22. Recall, $\pi$ is defined as the ratio of circumference (say c) to its diameter (say d). That is $\pi=\dfrac{c}{d}$. This seems to contradict the fact that $\pi$ is irrational. How will you resolve this contradiction?


23. Express $0 . \overline{001}$ in the form of $\dfrac{p}{q}$, where $\mathrm{p}$ and $\mathrm{q}$ are integers and $\mathrm{q} \neq 0$.


24. Find five rational numbers between $\dfrac{3}{4}$ and $\dfrac{4}{5}$


25. Find six rational numbers between 3 and 4.


Answers to the Worksheet:

1. (d)

A rational number cannot have a non-terminating or non-repeating decimal form.


2. (c) 

Between two rational numbers, there are infinitely many rational numbers. 

E.g. $\dfrac{3}{5}$ and $\dfrac{4}{5}$ are two rational numbers, then $\dfrac{31}{50} \dfrac{32}{50} \dfrac{33}{50} \dfrac{34}{50} \dfrac{35}{50} \ldots$ are infinite rational number between them.


3. (d) 

The product of two irrational numbers can be rational or irrational depending on the two numbers.

For example, $\sqrt{3} \times \sqrt{3}$ is 3 which is a rational number whereas $\sqrt{2} \times \sqrt{4}$ is $\sqrt{8}$ which is an irrational number. As $\sqrt{3}, \sqrt{2}, \sqrt{4}$ are irrational.

Hence, option D is correct.


4. (a) $\sqrt{7}$ is an irrational number.


5. (b) 

$\sqrt{4} \times \sqrt{81}$ $= \sqrt{2^2} \times \sqrt{9^2}$ $= 2 \times 9$ = 18


6. Fill in the blanks.

  1. Any two integers are separated by a finite number of other integers.

  2. There are an endless amount of rational numbers between 15 and 18 .

  3. $\mathrm{X}+\mathrm{Y}$ is a rational number if $\mathrm{x}$ and $\mathrm{y}$ are both rational numbers.

  4. Value of $\sqrt[3]{8}$ is $\underline{2}$


7. Match The Column:


Column I

Column II

Value of 1.9999…..

2

The Simplest form of a rational number $\dfrac{177}{413}$

$\dfrac{3}{7}$

0.36

Terminating Decimal

0.18181818………

Recurring decimal and Non- Terminating


Explanation:



Explanation

Value of 1.9999…..

Let, $x=1.999$      …(1)

Since only 1 digit is repeating.

So, by multiplying $x$ by 10 , we get

$10 x=19.99$      …(2)

Subtracting equation (1) from (2), we get

$9 x=18$

$\Rightarrow x=\dfrac{18}{9}$

$\Rightarrow x=2$

The value of $1.999 \ldots$ in the form $\dfrac{p}{q}$, where $p$ and $q$ are integers an $q \neq 0$, is 2 .

The Simplest form of a rational number $\dfrac{177}{413}$

$\dfrac{177}{413}$ = $\dfrac{3 \times 59}{7 \times 59}$

59 will cancel out, therefore, we get

= $\dfrac{3}{7}$

0.36

A terminating decimal is a decimal, that has an end digit. It is a decimal, which has a finite number of digits(or terms). Hence, 0.36 is terminating decimal.

0.18181818………

Non-terminating decimals are the one that does not have an end term. Hence, 0.18181818……… is non-terminating decimal.


8. Given an example of two irrational numbers whose;

  1. Product is an irrational number $\sqrt{6} \times \sqrt{3}=\sqrt{6 \times 3}=\sqrt{18}=3 \sqrt{2}$

  2. Difference is a irrational number $\sqrt{6}-\sqrt{3}$ = $\sqrt{3}$

  3. Division is an irrational number $\dfrac{\sqrt{6}}{\sqrt{3}}=\sqrt{\dfrac{6}{3}}=\sqrt{2}$


9. Simplify; $(\sqrt{5}+\sqrt{6})(\sqrt{5}-\sqrt{6})$ 

We know that, $(a+b)(a-b)=a^2-b^2$

Then,

= $\left((\sqrt{5})^2-(\sqrt{6})^2\right)$

= $5-6$

= $-1$


10. $\sqrt[3]{1331}-\sqrt{100}+\sqrt{81}$

= $\sqrt[3]{11^3}-\sqrt{10^2}+\sqrt{9^2}$

= $11-10+9$

= $1+9$

= 10


11. $\dfrac{11^{\dfrac{1}{2}}}{11^{\dfrac{1}{4}}}$

$\dfrac{11^{\dfrac{1}{2}}}{11^{\dfrac{1}{4}}}=11^{\dfrac{1}{2}-\dfrac{1}{4}}$

$=11^{\dfrac{2-1}{4}}$

$=11^{\dfrac{1}{4}}$


12. Let, 

$p= 0.777…$ ....   (1)

Multiply both side in above equation 10

Then, 

$10p= 7.777…$ ….(2)


Subtracting equation (1) from (2), we get;

$10p-p= 7.777… - 0.777…$

$9p= 7$

$p= \dfrac{7}{9}$


13. Three rational number between $\dfrac{9}{11}$ and $\dfrac{5}{11}$

Rational number of $\dfrac{9}{11}$ and $\dfrac{5}{11}$ is denominator same

Then,

$= \dfrac{9}{11}, \dfrac{8}{11}, \dfrac{7}{11}, \dfrac{6}{11}, \dfrac{5}{11}$


14. $\dfrac{\sqrt{8}+\sqrt{12}}{\sqrt{32}+\sqrt{48}}$

$= \dfrac{\sqrt{2^3}+\sqrt{4 \times 3}}{\sqrt{8 \times 4}+\sqrt{8 \times 6}}$

$= \dfrac{2 \sqrt{2}+2 \sqrt{3}}{4 \sqrt{2}+4 \sqrt{3}}$

$= \dfrac{2(\sqrt{2}+\sqrt{3})}{4(\sqrt{2}+\sqrt{3})}$

$= \dfrac{(\sqrt{2}+\sqrt{3})}{2(\sqrt{2}+\sqrt{3})}$

$= \dfrac{1}{2}$


15. If $a=2$ and $b=3$

The value of $a^b+b^a$

$= 2^3+3^2$

$= 8+9$

$= 17$


16. $2^{\dfrac{2}{3}} \cdot 2^{\dfrac{1}{5}}$

$2^{\dfrac{2}{3}} \cdot 2^{\dfrac{1}{5}}=2^{\dfrac{2}{3}+\dfrac{1}{5}} \quad \because a^p \cdot a^q=a^{p+q}$

$=2^{\dfrac{10+3}{15}}$

$=2^{\dfrac{13}{15}}$


17. Value of $\dfrac{1}{a^b+b^a}$, where $a=5, b=2$

$= \dfrac{1}{5^2+2^5}$

$= \dfrac{1}{25+32}$

$= \dfrac{1}{57}$


18. Here we have : $\sqrt[3]{2}, \sqrt{3}, \sqrt[5]{5}$

We can also write the expression in simpler form as follows:

$2^{\dfrac{1}{3}}, 3^{\dfrac{1}{2}}, 5^{\dfrac{1}{6}}$

Now we can see that in the denominators of the exponents we have: $3,2,6$

We will now take the LCM of $3,2,6$, which is 6 .

Now we will make all the denominators equal to 6 , so we have to multiply by the multiples in both numerator and denominator.

We can write the numbers as:

$2^{\dfrac{1}{3}} \times \dfrac{2}{2}=2^{\dfrac{2}{6}}$

For the second number we can write:

$3 \dfrac{1}{2} \times \dfrac{3}{3}=3 \dfrac{3}{6}$

Since in the third number we already have the desired denominator, so the third number is

$5^{\dfrac{1}{6}}$

Now we will again write the numbers in the root under, but we have to keep in mind that the numerator will turn as the exponential powers inside the root.

So we have the numbers as:

$\sqrt[6]{2^2}, \sqrt[5]{3^3}, \sqrt[5]{5}$

We will simplify the values inside the root, so we have:

$\sqrt[5]{4}, \sqrt[6]{27}, \sqrt[5]{5}$

From this we can write the smaller value in the front and then the larger value:

$\sqrt[5]{4}, \sqrt[6]{5}, \sqrt[5]{27}$

Hence the original numbers in ascending form are:

$\sqrt[3]{2}, \sqrt[6]{5}, \sqrt{3}$


19. $(4 \sqrt{5}+3 \sqrt{7})^2$

We know that,

$(a+b)^2=a^2+b^2+2 a b$

$=(4 \sqrt{5})^2+(3 \sqrt{7})^2+2 \times (4 \sqrt{5}) (3 \sqrt{7})$

$=80+63+24 \sqrt{5 \times 7}$

$=143+24 \sqrt{35}$


20. $\left(\dfrac{y}{x}\right)^{2 a-8}=\left(\dfrac{x}{y}\right)^{a-1}$

Rewrite,

$\left(\dfrac{y}{x}\right)^{2 a-8}=\left(\dfrac{x}{y}\right)^{8-2 a}$   $ \because (x)^{-a}=\dfrac{1}{x^a}$

Then,

$\left(\dfrac{x}{y}\right)^{8-2 a}=\left(\dfrac{x}{y}\right)^{a-1}$

When the bases of both sides of an equation are the same, then their exponents are also equal.

$\Rightarrow 8-2 a=a-1$

$\Rightarrow 2 a+a=8+1$

$\Rightarrow 3 a=9$

$\Rightarrow a=\dfrac{9}{3}$

$\Rightarrow a=3$


21. $\dfrac{1}{\sqrt{7}}=\dfrac{1}{\sqrt{7}} \times \dfrac{\sqrt{7}}{\sqrt{7}}$

(Dividing and multiplying by $\sqrt{7}$ )

$=\dfrac{\sqrt{7}}{7}$


22. Writing $\pi$ as $\dfrac{22}{7}$ is only an approximate value and so we can't conclude that it is in the form of a rational. In fact, the value of $\pi$ is calculating as non-terminating, non-recurring decimal as $\pi=3.14159$ Whereas

If we calculate the value of $\dfrac{22}{7}$ it gives $3.142857$ and hence $\pi \neq \dfrac{22}{7}$

In conclusion $\pi$ is an irrational number.


23. Let $x=0.001001 \ldots \ldots$ (1)

Since 3 digits are repeated multiply both the sides of (1) by 1000

$1000 x=1.001001 \ldots$

$1000 x=1+0.001001 \ldots$

$1000 x=1+x$

$1000 x-x=1$

$999 x=1$

$x=\dfrac{1}{999}$

$\therefore 0 . \overline{001}=\dfrac{1}{999}$


24. Since we make the denominator the same first, then

$\dfrac{3}{4}=\dfrac{3 \times 5}{4 \times 5}=\dfrac{15}{20}$

$\dfrac{4}{5}=\dfrac{4 \times 4}{5 \times 4}=\dfrac{16}{20}$

Now we need to find 5 rational no.

$\dfrac{15}{20}  =\dfrac{15 \times 6}{20 \times 6}=\dfrac{90}{120}$

$\dfrac{16}{20}=\dfrac{16 \times 6}{20 \times 6}=\dfrac{96}{120}$

$\therefore$ Five rational numbers between $\dfrac{3}{4}$ and $\dfrac{4}{5}$ are $\dfrac{91}{120}, \dfrac{92}{120}, \dfrac{93}{120}, \dfrac{94}{120}$ and $\dfrac{95}{120}$


25. We can find any number of rational numbers between two rational numbers. First of all, we make the denominators same by multiplying or dividing the given rational numbers by a suitable number. If denominator is already same then depending on number of rational no. we need to find in question, we add one and multiply the result by numerator and denominator.

$3=\dfrac{3 \times 7}{7} \text { and } \quad 4=\dfrac{4 \times 7}{7}$

$3=\dfrac{21}{7} \quad \text { and } \quad 4=\dfrac{28}{7}$

We can choose 6 rational numbers as: $\dfrac{22}{7}, \dfrac{23}{7}, \dfrac{24}{7}, \dfrac{25}{7}, \dfrac{26}{7}$ and $\dfrac{27}{7}$

Benefits of Learning Number System in Class 9 Chapter 1 Maths Worksheet The Class 9 Maths Chapter 1 worksheet pdf contains more than enough material to help students better understand what number systems are and how to solve them. The worksheets come with extensive questions, attempting to clear any doubts the students might have about the number system and their types.

The Maths assignment for Class 9 Number System list of questions and answers provide thorough insights on the topic’s resources and offers easy tricks to identify quicker ways to solve the questions faster while also being more aware and making sure students don’t go wrong or commit any silly mistakes in their solutions.

All of these worksheets have been developed by the best mathematicians and experienced arithmetic representatives who are very aware of the needs and requirements of the students of Class 9.

Examples of Usage of Number System for Class 9

These are a few examples of Maths assignments for Class 9 Number System exercises’ examples :

  1. Answer the following.

  • Find two irrational numbers and two rational numbers between 0.7 and 0.77.

  • Every integer is not a whole number. True or false?

  • Find at least 7 rational numbers between 2 and 9.

  • Write down 4567 in the decimal and binary number systems.

  • Is 0 a rational number? State your reasons based on your answer.


Interesting Facts About Number System for Class 9

  • There are nine types of number systems in mathematics. They are :

    • Natural numbers

    • Whole numbers

    • Integers

    • Fractions

    • Rational numbers

    • Irrational numbers

    • Real numbers

    • Imaginary numbers

    • Prime and composite numbers

  • Natural numbers are the root forms of numbers between 0 to infinity. They are also named “positive numbers” or “counting numbers” and are represented by the symbol N. (1, 2, 3, 4, 5 and so on)

  • Whole numbers are natural numbers, with the only difference being the inclusion of 0. They are represented by the symbol W. (0, 1, 2, 3, 4, 5 and so on)

  • Integers contain whole numbers and the negative values of natural numbers and don’t include fractions, so their numbers can’t be written in the “a/b” format. It ranges from infinity at the negative end to infinity at the positive end, including 0 and is represented by Z. (...-3, -2, -1, 0, 1, 2, 3… and so on)

  • Fractions are numbers written in the “a/b” format, where “a” (numerator) is a whole number, and “b” (denominator) is a natural number. Hence, the denominator can never be 0. (2/4, 0/10, 5/7, etc.)

  • Rational numbers can be written in fractions where “a” and “b” are both integers and b ≠ 0. All fractions are rational numbers, but all rational numbers are not fractions.(-5/9, 3/9, -8/14, etc)

  • Irrational numbers are numbers that can’t be written in fractional forms. (√8, √.127, √3.209, etc)

  • Real numbers can be written in decimals, including whole numbers, integers, fractions, etc. All integers belong to real numbers, but not all real numbers belong to integers. (1.25, 0.467, 8.9, etc.)

  • Imaginary numbers are not real numbers, resulting in negative numbers when squared or put together. They are also named complex numbers and are represented by the symbol i. (√-3, √-16, √-1, etc.)

  • Numbers that don’t have other factors except 1 are called prime numbers, and the rest of the numbers - except 0 - are called composite numbers, as 0 is neither a prime nor a composite number. ( 2, 3, 5… are prime numbers whereas 4, 6, 8… are composite numbers)

  • Other definitions and elaborated explanations will be provided in the operations on real numbers class 9 worksheet pdf and more.


Important Topics for Class 9 Number System

The important topics students will have to learn in the number system syllabus for Class 9 are as follows :

  • What are number systems, and how to solve them?

  • What are the four types of number systems?

  • How to convert one number system to another number system.

  • Solving the problems and choosing the correct answer.

  • Various other exercises in the Number System Class 9 worksheet


What does the PDF Consist of?

  • Most schools have syllabuses that don’t include just spoon-feeding the information to the students.

  • It only means that the students must learn by themselves, with the teachers guiding and aiding them throughout their learning process.

  • With technology being part of most school curriculums, a huge part of their assignments, tests, and worksheets are online, favoured as pdfs.

  • Vedantu’s pdf format is highly sought-after as it is used for creating, editing, highlighting, saving, and sharing content.

  • The worksheet for Class 9 Maths Chapter 1 with Solutions pdfs is free for download at Vedantu’s website.

  • Rest assured, all the worksheets adhere to the CBSE guidelines' strict, updated, and revised rules.


Many other Number Systems Class 9 worksheet pdfs are present at Vedantu’s platform, created by their own arithmetic subject matter experts, ensuring that the students receive the best training and exercises needed to test their skills and excel in their examinations.

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FAQs on Number System Worksheets for CBSE Class 9 Maths – Free PDF Solutions

1. What are the most frequently asked types of important questions from Chapter 1, Number System, for the Class 9 final exam?

For the CBSE Class 9 exam for the year 2025-26, the important questions from Number Systems are typically structured by marks as follows:

  • 2-Mark Questions: Finding a set of rational numbers between two given numbers, or simplifying expressions using the laws of exponents.

  • 3-Mark Questions: Representing irrational numbers like √3 or √5 on the number line, and converting recurring decimals into the p/q form.

  • 4/5-Mark Questions (HOTS): These often involve complex rationalisation or multi-step simplification problems that combine several laws of exponents.

2. How should I answer a question on representing an irrational number like √9.3 on the number line to get full marks?

To secure full marks for representing a decimal root, you must show the geometric construction steps clearly. For √9.3, the key steps are: draw a line segment of 9.3 units, extend it by 1 unit, find the midpoint to draw a semicircle, and then draw a perpendicular from the end of the 9.3 unit mark to the semicircle. The length of this perpendicular is exactly √9.3, which you then project onto the number line. Clearly labelling all points (A, B, C, etc.) and using a compass and ruler accurately is crucial.

3. Which types of rationalisation questions are considered most important for exams?

The most important and expected questions on rationalisation involve denominators of the form 1/(√a ± √b) or 1/(a ± √b). To solve these, you must multiply the numerator and denominator by the conjugate of the denominator. This process tests your ability to correctly apply the algebraic identity (x - y)(x + y) = x² - y², which is a fundamental skill for this chapter.

4. What are some expected MCQ-type questions from the Number System chapter?

Important multiple-choice questions (MCQs) for Chapter 1 often test core conceptual understanding. Expect questions such as:

  • Identifying an irrational number from a given set of options (e.g., √16, √8, 0.5).

  • A question on the nature of the decimal expansion of an irrational number, which is always non-terminating and non-recurring.

  • A simple calculation based on the laws of exponents, like finding the value of (64)^(1/2).

5. Are all integers rational numbers, but not all rational numbers are integers? Explain why this is a common trap in 1-mark questions.

This is a critical distinction for scoring well in objective questions. Yes, all integers are rational numbers because any integer 'z' can be expressed in the form p/q as z/1 (e.g., -5 = -5/1). However, not all rational numbers are integers. A number like 2/3 is a rational number because it is in the form p/q, but it is not an integer. Understanding this hierarchy helps avoid common mistakes in 'True/False' or 'Choose the correct statement' questions.

6. Why is rationalising the denominator considered such an important final step when solving problems with irrational numbers?

Rationalising the denominator is an essential skill in exams for two main reasons. First, it converts the number into a standard and simplified form, which is often required to be awarded full marks. Second, it makes further operations like addition or subtraction much easier by creating an integer in the denominator. An expression with an irrational root in the denominator is technically considered un-simplified in mathematics.

7. How do the laws of exponents apply to questions with rational powers, a concept frequently tested in exams?

Questions involving rational exponents are a staple of this chapter. For an expression like (81)^(3/4), you must apply the laws of exponents systematically. The key is to first express the base as a power that cancels the denominator of the exponent. For instance, 81 = 3⁴. The expression becomes (3⁴)^(3/4). Using the law (x^a)^b = x^(ab), the powers multiply to give 3³, which equals 27. Showing these steps is vital for securing marks.

8. What is the fundamental difference between a recurring decimal and a non-terminating, non-recurring decimal, and how does this impact exam questions?

The fundamental difference determines whether a number is rational or irrational, a key concept for Higher Order Thinking Skills (HOTS) questions. A number is rational if its decimal form is either terminating (e.g., 2.5) or non-terminating but recurring (e.g., 1.333...). In contrast, a number is irrational if its decimal form is both non-terminating and non-recurring (e.g., the value of π or √2). Exam questions can ask you to classify numbers or justify why a product of a rational and an irrational number is irrational, based on this principle.