# NCERT Solutions for Class 9 Maths Chapter 1 Number Systems

NCERT solutions for class 9 maths chapter 1 Number Systems- Are you looking for in-depth knowledge of numbers and its applications? Then you must read this chapter. You will have a great conceptual clarity related to number system after going through it. Solutions of NCERT class 9 maths chapter 1 Number Systems are also there to provide assistance whenever you feel trouble while facing any problem related to this particular chapter. In this particular chapter, there is a total of 6 exercises which consist of 27 questions. It has a weightage of 8 marks in the CBSE class 9 final examination. CBSE NCERT solutions for class 9 maths chapter 1 Number Systems are covering the solutions to each and every question present in the practice exercises. Apart from school exams, the solutions are also beneficial if you are aiming for exams like- National Talent Search Examination (NTSE), Indian National Olympiad (INO), SSC, CAT, etc. In NCERT class 9 maths chapter 1 Number Systems, you will learn the extended form of the number line and learn how to represent various types of the number on it. The Chapter starts with an introduction which includes rational numbers, whole numbers, and integers followed by important topics like irrational numbers, real numbers, how to represent real numbers on the number line, operations on real numbers and many more. With the understanding of rational numbers, we will also study how to represent square roots of 2, 3, 5 and other non-rational numbers. NCERT solutions for class 9 maths chapter 1 Number Systems are written keeping important basics in mind so that a student can get 100% out of it. NCERT solutions are also available for different subjects and classes which you can get by clicking on the link.

NCERT solutions for class 9 maths chapter 1 Number Systems Excercise: 1.1

Any number that can represent  in the form of    is a rational number

Now, we can write 0 in the form of    for eg.   etc.

Therefore, 0 is a  rational number.

There are an infinite number of rational numbers between 3 and 4. one way to take them is

Therefore, six rational numbers between 3 and 4 are

We can write

And

Therefore, five rational numbers between   and  . are

(i) TRUE
The number that is starting from 1, i.e 1, 2, 3, 4, 5, 6, .................. are natural numbers
The number that is starting from 0. i.e, 0, 1, 2, 3, 4, 5.............are whole numbers
Therefore, we can clearly see that the collection of whole numbers contains all natural numbers.

(ii) FALSE
Because integers may be negative or positive but whole numbers are always positive. for eg. -1 is an integer but not a whole number.

(iii) FALSE
Numbers that can be represented in the form of   are a rational number.
And numbers that are starting from 0 i.e. 0,1,2,3,4,......... are whole numbers
Therefore,  we can clearly see that every rational number  is not a whole number  for eg.   is a rational number but not a whole number

NCERT solutions for class 9 maths chapter 1 Number Systems Excercise: 1.2

(i) TRUE
Since the real numbers are the collection of all rational and irrational numbers.

(ii) FALSE
Because negative numbers are also present on the number line and no negative number can be the square root of any natural number

(iii) FALSE
As real numbers include both rational and irrational numbers. Therefore, every real number cannot be an irrational number.

For eg. 4 is a real number but not an irrational number

NO, Square root of all positive integers is not irrational. For the eg  square of 4 is 2 which is a rational number.

We know that

Now, Let OA be a line of length 2 unit on the number line. Now, construct AB of unit length perpendicular to OA. and join OB.
Now, in right angle triangle OAB, by Pythagoras theorem

Now,  take O as centre and OB as radius, draw an arc intersecting number line at C.  Point C represent  on a number line.

NCERT solutions for class 9 maths chapter 1 Number Systems Excercise: 1.3

We can write   as

Since the decimal expansion ends after a finite number of steps. Hence, it is terminating

We can rewrite    as

Since decimal expansion repeats itself so it is a non-terminating recurring decimal expansion.

We can rewrite   as

Since the decimal expansion ends after a finite number. Therefore, it is terminating

We can rewrite   as

Since decimal expansion repeats itself so it is a non-terminating recurring decimal expansion.

We can rewrite  as

Since decimal expansion repeats itself so it is a non-terminating recurring decimal expansion.

We can rewrite  as

Since decimal expansion ends after finite no. of figures. Hence, it is terminating.

It is given  that

Therefore,

Similarly,

Let              -(i)

Now, multiply by 10 on both sides

Therefore,    form of    is

We can write  as

-(i)

Now,

Let              -(ii)

Now, multiply by 10 on both sides

Now, put the value of x in equation (i). we will get

Therefore,    form of    is

Let              -(i)

Now, multiply by 1000 on both sides

Therefore,    form of    is

Let              -(i)

Now, multiply by 10 on both sides

Therefore,    form of    is  1

The difference between 1 and 0.999999 is o.000001 which is almost negligible.

Therefore, 0.999 is too much closer to 1. Hence, we can write 0.999999.... as 1

We can rewrite    as

Therefore, there are total  16  number of digits be in the repeating block of digits in the decimal expansion of

We can observe that when q is 2, 4, 5, 8, 10… then the decimal expansion is terminating. For example:

, denominator

, denominator

, denominator

Therefore,

It can be observed that the terminating decimal can be obtained in a condition where prime factorization of the denominator of the given fractions has the power of 2 only or 5 only or both.

Three numbers whose decimal expansions are non-terminating non-recurring are
1) 0.02002000200002......
2) 0.15115111511115.......
3) 0.27227222722227.......

We can write    as

And   as

Therefore,  three different irrational numbers between the rational numbers  and   are

1) 0.72737475....
2) 0.750760770780...
3) 0.790780770760....

We can rewrite  in decimal form as

Now, as the decimal expansion of this number is non-terminating non-recurring.

Therefore, it is an irrational number.

We can rewrite    as

We can clearly see that it is a rational number because we can represent it in    form

We can rewrite 0.3796 as

Now, we can clearly see that it is a rational number as the decimal expansion of this number is terminating and we can also write it in   form.

We can rewrite  7.478478.... as

Now,  as the decimal expansion of this number is non-terminating recurring. Therefore, it is a rational number.

In the case of number  1.101001000100001...
As the decimal expansion of this number is non-terminating non-repeating. Therefore, it is an irrational number.

NCERT solutions for class 10 maths chapter 1 Number Systems Excercise: 1.4

3.765 can be visualised as in the following steps.
First, we draw a number line and mark points on it after that we will divide the number line between points 3 and 4. And then we will divide the points between 3.7 and 3.8 as the number is between both of them. We can rewrite   as

Now,  4.2626 can be visualised as in the following steps. NCERT solutions for class 9 maths chapter 1 Number Systems Excercise: 1.5

Value of    is  2.23606798....
Now,

Now,
Since the number is in non-terminating non-recurring. Therefore, it is an irrational number.

Given number is

Now, it is clearly a rational number because we can represent it in the form of

Given number is

As we can clearly see that it can be represented in    form. Therefore, it is a rational number.

Given number is

Now,
Clearly, as the decimal expansion of this expression is non-terminating and non-recurring. Therefore, it is an irrational number.

Given number is
We know that the value of
Now,

Now,
Clearly, as the decimal expansion of this expression is non-terminating and non-recurring. Therefore, it is an irrational number.

Given number is
Now, we will reduces it into

Given number is

Now, we will reduces it into

Given number is

Now, we will reduce it into

Given number is

Now, we will reduce it into

When we measure a length with scale or any other instrument, we only obtain an approximate rational value. We never obtain an exact value.
For this reason, we cannot say that either c or d is irrational.
Therefore, the fraction    is irrational. Hence, the value of  is approximately equal to

Therefore, is irrational. Draw a line segment OB of 9.3 unit. Then, extend it to C so that BC = 1 unit. Find the mid-point D of OC and draw a semi-circle on OC while taking D as its centre and OD as the radius. Now, Draw a perpendicular to line OC passing through point B and intersecting the semi-circle at E. Now, Take B as the centre and BE as radius, draw an arc intersecting the number line at F.  the length BF is units.

Given number is

Now, on rationalisation, we will get

Given number is

Now, on rationalisation, we will get

Given number is

Now, on rationalisation, we will get

Given number is

Now, on rationalisation we will get

NCERT solutions for class 9 maths chapter 1 Number Systems Excercise: 1.6

Q1 (i) Find :

Given number is

Now, on simplifying it we will get

Q1 (ii) Find :

Given number is

Now, on simplifying it we will get

Q1 (iii) Find :

Given number is

Now, on simplifying it we will get

Q2 (i) Find :

Given number is

Now, on simplifying it we will get

Q2 (ii) Find :

Given number is

Now, on simplifying it we will get

Q2 (iii) Find :

Given number is

Now, on simplifying it we will get

Q2 (iv) Find :

Given number is

Now, on simplifying it we will get

Q3 (i) Simplify :

Given number is

Now, on simplifying it we will get

Q3 (ii) Simplify :

Given number is

Now, on simplifying it we will get

Q3 (iii) Simplify :

Given number is

Now, on simplifying it we will get

Q3 (iv) Simplify :

Given number is

Now, on simplifying it we will get

## NCERT solutions for class 9 maths chapter wise

 Chapter No. Chapter Name Chapter 1 NCERT solutions for class 9 maths chapter 1 Number Systems Chapter 2 CBSE NCERT solutions for class 9 maths chapter 2 Polynomials Chapter 3 Solutions of NCERT class 9 maths chapter 3 Coordinate Geometry Chapter 4 NCERT solutions for class 9 maths chapter 4 Linear Equations In Two Variables Chapter 5 CBSE NCERT solutions for class 9 maths chapter 5 Introduction to Euclid's Geometry Chapter 6 Solutions of NCERT class 9 maths chapter 6 Lines And Angles Chapter 7 NCERT solutions for class 9 maths chapter 7 Triangles Chapter 8 CBSE NCERT solutions for class 9 maths chapter 8 Quadrilaterals Chapter 9 Solutions of NCERT class 9 maths chapter 9 Areas of Parallelograms and Triangles Chapter 10 NCERT solutions for class 9 maths chapter 10 Circles Chapter 11 CBSE NCERT solutions for class 9 maths chapter 11 Constructions Chapter 12 Solutions of NCERT class 9 maths chapter 12 Heron’s Formula Chapter 13 NCERT solutions for class 9 maths chapter 13 Surface Area and Volumes Chapter 14 CBSE NCERT solutions for class 9 maths chapter 14 Statistics Chapter 15 Solutions of NCERT class 9 maths chapter 15 Probability

## How to use NCERT Solutions for class 9 maths chapter 1 Number Systems?

• First of all, go through the conceptual text given in the book before the exercises.

• After covering the conceptual theory, you must go through some examples to understand the application of that particular concept.

• After observing the application, come to the practice exercises available in the textbook

• While solving the practice exercises, you can take the help of NCERT solutions for class 9 maths chapter 1 Number Systems to boost your preparation.

Keep working hard & happy learning!