NCERT Solutions for Class 9 Maths Chapter 1: Number System - Exercise 1.1

Exercise (1.1)

1.  Is zero a rational number? Can you write it in the form  $\dfrac{ {p}}{ {q}}$, where $ {p}$ and $ {q}$ are integers and $ {q}\ne  {0}$? Describe it.

Ans: Remember that, according to the definition of rational number,

a rational number is a number that can be expressed in the form of  $\dfrac{p}{q}$, where $p$ and $q$ are integers and  $q\ne \text{0}$. 

Now, notice that zero can be represented as $\dfrac{0}{1},\dfrac{0}{2},\dfrac{0}{3},\dfrac{0}{4},\dfrac{0}{5}.....$

Also, it can be expressed as $\dfrac{0}{-1},\dfrac{0}{-2},\dfrac{0}{-3},\dfrac{0}{-4}.....$

Therefore, it is concluded from here that $0$ can be expressed in the form of $\dfrac{p}{q}$, where $p$ and $q$ are integers.

Hence, zero must be a rational number.

2. Find any six rational numbers between $ {3}$ and $ {4}$. 

Ans: It is known that there are infinitely many rational numbers between any two numbers. Since we need to find $6$ rational numbers between $3$ and $4$, so multiply and divide the numbers by $7$ (or by any number greater than $6$)

Then it gives, 

$\begin{align} & 3=3\times \dfrac{7}{7}=\dfrac{21}{7} \\ & 4=4\times \dfrac{7}{7}=\dfrac{28}{7} \\ \end{align}$

Hence, $6$ rational numbers found between $3$ and $4$ are $\dfrac{22}{7},\dfrac{23}{7},\dfrac{24}{7},\dfrac{25}{7},\dfrac{26}{7},\dfrac{27}{7}$.

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

Ans: It is known that there are infinitely many rational numbers between any two numbers.

Since here we need to find five rational numbers between $\dfrac{3}{5}$ and $\dfrac{4}{5}$,  so multiply and divide by $6$ (or by any number greater than $5$).

Then it gives,

$\dfrac{3}{5}=\dfrac{3}{5}\times \dfrac{6}{6}=\dfrac{18}{30}$,

$\dfrac{4}{5}=\dfrac{4}{5}\times \dfrac{6}{6}=\dfrac{24}{30}$.

Hence, $5$ rational numbers found between $\dfrac{3}{5}$ and $\dfrac{4}{5}$ are  $\dfrac{19}{30},\dfrac{20}{30},\dfrac{21}{30},\dfrac{22}{30},\dfrac{23}{30}$.

4. State whether the following statements are true or false. Give reasons for your answers.

(i) Every natural number is a whole number. 

Ans: Write the whole numbers and natural numbers in a separate manner.

It is known that the whole number series is $0,1,2,3,4,5.....$. and

the natural number series is $1,2,3,4,5.....$.

Therefore, it is concluded that all the natural numbers lie in the whole number series as represented in the diagram given below.

Thus, it is concluded that every natural number is a whole number.

Hence, the given statement is true.

(ii) Every integer is a whole number.

Ans: Write the integers and whole numbers in a separate manner.

 It is known that integers are those rational numbers that can be expressed in the form of $\dfrac{p}{q}$, where $q=1$.

Now, the series of integers is like $0,\,\pm 1,\,\pm 2,\,\pm 3,\,\pm 4,\,...$.

But the whole numbers are $0,1,2,3,4,...$. 

Therefore, it is seen that all the whole numbers lie within the integer numbers, but the negative integers are not included in the whole number series. 

Thus, it can be concluded from here that every integer is not a whole number.

Hence, the given statement is false.

(iii) Every rational number is a whole number.

Ans: Write the rational numbers and whole numbers in a separate manner. 

It is known that rational numbers are the numbers that can be expressed in the form  $\dfrac{p}{q}$, where $q\ne 0$ and the whole numbers are represented as $0,\,1,\,2,\,3,\,4,\,5,...$

Now, notice that every whole number can be expressed in the form of $\dfrac{p}{q}$

as  \[\dfrac{0}{1},\text{ }\dfrac{1}{1},\text{ }\dfrac{2}{1},\text{ }\dfrac{3}{1},\text{ }\dfrac{4}{1},\text{ }\dfrac{5}{1}\],…

Thus, every whole number is a rational number, but all the rational numbers are not whole numbers. For example,

$\dfrac{1}{2},\dfrac{1}{3},\dfrac{1}{4},\dfrac{1}{5},...$ are not whole numbers.

Therefore, it is concluded from here that every rational number is not a whole number.

Hence, the given statement is false.

What Do You Learn About Number System

Number System is amongst the most crucial parts of NCERT Solutions for Class 9 Maths. Number System is the mathematical notations created using digits and symbols. From understanding the rational and irrational numbers to solving problems of them, Vedantu will be there for you in every step.

There are 8 different types of number as follows:

1. Complex Number

2. Imaginary Number

3. Real Number

4. Rational number

5. Irrational number

6. Integers

7. Whole numbers

8. Natural numbers

The NCERT Solution for Class 9 Maths Chapter 1 covers all these number types with vivid explanation. 

Do you know zero is a rational number too? If not, don’t worry! The 9th class Maths exercise 1.1 Solution will explain to you the answer with a perfect illustration. 

Natural Numbers: Numbers which are used for counting is called natural numbers like 1, 2, 3 etc. In other words, we can say that all positive integers are a natural number. There are infinite natural numbers and 1 is of the least value among them.

The Class 9 Maths Chapter 1 exercise 1.1 also makes you understand that all whole numbers are rational numbers but all rational numbers are not always whole numbers.

The examples of numbers that are not natural are 8, 3.49, 0, etc.

Rational Number: Ratio of two integers where the denominator is not zero is called a rational number. One surprising fact is that a repeating decimal is also called a rational number. 

Ex:5/3, 16/7 these are rational numbers.

Irrational Number: An irrational number is a real number that cannot be written as a simple fraction. ‘’ is a very famous irrational number.

Decimals: Decimal is very interesting and fun part of the Number System. Decimal fractions were first introduced and used by Chinese at the end of the 4th century and then spread to the Middle East and from there to Europe. We use decimal in our daily life without our consciousness. For example – counting money, putting fuel into our vehicle or while measuring our weight. Decimals can never be whole numbers. Few example: 1.8, 000.23 etc.

Decimals classifications

Terminating decimal fractions are 17/4= 4.25, 21/5 = 4.2 and so on.

Non- terminating decimal fractions are 16/3 = 5.33333 , 15.35353535 etc.

Integer: All numbers which do not have a decimal in them are called integers. Thus, -9, 4, 1476 etc. Do you know? All integers are whole numbers including the negative numbers.

These are a few fundamental topics of the Number System. Class 9 Maths Chapter 1 will give you detailed explations of all of them with important exercises to practice in the NCERT Solutions for Class 9 Maths Chapter 1, exercise 1.1. Apart from this, you will get to know how to use these numbers in a logical manner to solve problems. There will be ‘n’ number of solved questions in the exercise 1.1 Class 9 Maths NCERT Solutions. These will help you have complete clarity on the subject.

Class 9 Maths Chapter 1 Solutions - Vedantu: 

Vedantu has created a harmonious balance between theory and practical knowledge in the Number System Class 9 Solution. The course is created by expert teachers who have vast knowledge and experience in this field. Hence, study materials are easily understandable by students. They can prepare on their own without any extra guidance. All the important topics are jotted down in a structured manner so it can satisfy your inquisitiveness. In NCERT Solution Class 9 Maths Chapter 1, you will get everything related to Number System that you need to outshine your scores in previous exams.

You can also download NCERT Solutions Class 9 Maths to help you to revise complete syllabus and score more marks in your examinations.

NCERT Solutions for Class 9 Maths

• Chapter 1 - Number System

• Chapter 2 - Polynomials 

• Chapter 3 - Coordinate Geometry

• Chapter 4 - Linear Equations in Two Variables

• Chapter 5 - Introductions to Euclid's Geometry

• Chapter 6 - Lines and Angles

• Chapter 7 - Triangles

• Chapter 8 - Quadrilaterals

• Chapter 9 - Areas of Parallelogram and Triangles

• Chapter 10 - Circles

• Chapter 11 - Constructions

• Chapter 12 - Heron's formula

• Chapter 13 - Surface area and Volumes

• Chapter 14 - Statistics

• Chapter 15 - Probability

NCERT Solution Class 9 Maths of Chapter 1 All Exercises

Important Points Covered under NCERT Solutions Class 9 Maths Chapter 1 – Number Systems Exercise 1.1

Chapter 1 – Number Systems Exercise 1.1 is the first exercise of NCERT Solutions Class 9 Maths. This chapter is more about the calculation of rational numbers between any two given numbers. So, students are advised to focus on the step-by-step solutions of the problems provided in this chapter which will help them get ready for their board exam. So, let's jump to the key points that the students will learn from this chapter.

Below are some of the important points you will learn from this chapter.

1. The difference between natural, integer, and whole numbers.

2. Students will learn to calculate the rational numbers between any two given numbers. 

3. Every natural number is a whole number.

4. Every integer is not a whole number.

5. Every rational number is not a whole number.

Benefits of NCERT Solutions Class 9 Maths Chapter 1 – Number Systems (Exercise 1.1) 

• After going through the step-by-step explanations of the problems, you will have a better understanding of the concepts.

• NCERT Solutions for Class 9 Maths Chapter 1 (Exercise 1.1) will help students revise all the questions of Exercise 1.1.

• To score good marks, you need to follow the stepwise solutions given by our subject expert teachers.

• These NCERT solutions consist of all the important questions from the examination point of view.

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