# 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. NCERT solutions for 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. 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. Here you will get NCERT solutions for class 9 also.

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

** Q1 ** Is zero a rational number? Can you write it in the form , where p and q are integers and q ≠ 0?

** Answer: **

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.

** Q2 ** Find six rational numbers between 3 and 4.

** Answer: **

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

** Q3 ** Find five rational numbers between and .

** Answer: **

We can write

And

Therefore, five rational numbers between and . are

** Answer: **

(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.

** Answer: **

(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.

** Answer: **

(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 **

** Answer: **

(i) ** TRUE **

Since the real numbers are the collection of all rational and irrational numbers.

** Answer: **

(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

** Answer: **

(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

** Answer: **

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

** Q3 ** Show how can be represented on the number line.

** Answer: **

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 **

** Q1 (i) ** Write the following in decimal form and say what kind of decimal expansion each has : (i)

** Answer: **

We can write as

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

** Q1 (ii) ** Write the following in decimal form and say what kind of decimal expansion each has : (ii)

** Answer: **

We can rewrite as

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

** Q1 (iii) ** Write the following in decimal form and say what kind of decimal expansion each has : (iii)

** Answer: **

We can rewrite as

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

** Q1 (iv) ** Write the following in decimal form and say what kind of decimal expansion each has : (iv)

** Answer: **

We can rewrite as

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

** Q1 (v) ** Write the following in decimal form and say what kind of decimal expansion each has: (v)

** Answer: **

We can rewrite as

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

** Q1 (vi) ** Write the following in decimal form and say what kind of decimal expansion each has : (vi)

** Answer: **

We can rewrite as

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

** Answer: **

It is given that

Therefore,

Similarly,

** Q3 (i) ** Express the following in the form , where p and q are integers and q ≠ 0. (i)

** Answer: **

Let -(i)

Now, multiply by 10 on both sides

Therefore, form of is

** Q3 (ii) ** Express the following in the form , where p and q are integers and q ≠ 0. (ii)

** Answer: **

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

** Q3 (iii) ** Express the following in the form , where p and q are integers and q ≠ 0. (iii)

** Answer: **

Let -(i)

Now, multiply by 1000 on both sides

Therefore, form of is

** Q4 ** Express 0.99999 .... in the form . Are you surprised by your answer?

** Answer: **

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

** Answer: **

We can rewrite as

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

** Answer: **

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.

** Q7 ** Write three numbers whose decimal expansions are non-terminating non-recurring.

** Answer: **

Three numbers whose decimal expansions are non-terminating non-recurring are

1) 0.02002000200002......

2) 0.15115111511115.......

3) 0.27227222722227.......

** Q8 ** Find three different irrational numbers between the rational numbers and .

** Answer: **

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....

** Q9 (i) ** Classify the following numbers as rational or irrational :

** Answer: **

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.

** Q9 (ii) ** Classify the following numbers as rational or irrational :

** Answer: **

We can rewrite as

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

** Q9 (iii) ** Classify the following numbers as rational or irrational : 0.3796

** Answer: **

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.

** Q9 (iv) ** Classify the following numbers as rational or irrational : 7.478478....

** Answer: **

We can rewrite 7.478478.... as

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

** Q9 (v) ** Classify the following numbers as rational or irrational : 1.101001000100001...

** Answer: **

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 **

** Q1 ** Visualise 3.765 on the number line, using successive magnification.

** Answer: **

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.

** Q2 ** Visualise on the number line, up to 4 decimal places.

** Answer: **

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 **

** Q1 (i) ** Classify the following numbers as rational or irrational:

** Answer: **

Value of is 2.23606798....

Now,

Now,

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

** Q1 (ii) ** Classify the following numbers as rational or irrational:

** Answer: **

Given number is

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

** Q1 (iii) ** Classify the following numbers as rational or irrational:

** Answer: **

Given number is

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

** Q1 (iv) ** Classify the following numbers as rational or irrational:

** Answer: **

Given number is

Now,

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

** Q1 (v) ** Classify the following numbers as rational or irrational:

** Answer: **

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.

** Q2 (i) ** Simplify each of the following expressions:

** Answer: **

Given number is

Now, we will reduce it into

Therefore, answer is

** Q2 (ii) ** Simplify each of the following expressions:

** Answer: **

Given number is

Now, we will reduce it into

Therefore, answer is ** 6 **

** Q2 (iii) ** Simplify each of the following expressions:

** Answer: **

Given number is

Now, we will reduce it into

Therefore, the answer is

** Q2 (iv) ** Simplify each of the following expressions:

** Answer: **

Given number is

Now, we will reduce it into

Therefore, the answer is ** 3 ** .

** Answer: **

There is no contradiction.

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.

** Q4 ** Represent on the number line.

** Answer: **

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.

** Q5 (i) ** Rationalise the denominators of the following:

** Answer: **

Given number is

Now, on rationalisation, we will get

Therefore, the answer is

** Q5 (ii) ** Rationalise the denominators of the following:

** Answer: **

Given number is

Now, on rationalisation, we will get

Therefore, the answer is

** Q5 (iii) ** Rationalise the denominators of the following:

** Answer: **

Given number is

Now, on rationalisation, we will get

Therefore, the answer is

** Q5 (iv) ** Rationalise the denominators of the following:

** Answer: **

Given number is

Now, on rationalisation, we will get

Therefore, answer is

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

** Q1 (i) ** Find :

** Answer: **

Given number is

Now, on simplifying it we will get

Therefore, answer is ** 8 **

** Q1 (ii) ** Find :

** Answer: **

Given number is

Now, on simplifying it we will get

Therefore, the answer is ** 2 **

** Q1 (iii) ** Find :

** Answer: **

Given number is

Now, on simplifying it we will get

Therefore, the answer is ** 5 **

** Q2 (i) ** Find :

** Answer: **

Given number is

Now, on simplifying it we will get

Therefore, the answer is ** 27 **

** Q2 (ii) ** Find :

** Answer: **

Given number is

Now, on simplifying it we will get

Therefore, the answer is ** 4 **

** Q2 (iii) ** Find :

** Answer: **

Given number is

Now, on simplifying it we will get

Therefore, the answer is ** 8 **

** Q3 (i) ** Simplify :

** Answer: **

Given number is

Now, on simplifying it we will get

Therefore, the answer is

** Q3 (ii) ** Simplify :

** Answer: **

Given number is

Now, on simplifying it we will get

Therefore, the answer is

** Q3 (iii) ** Simplify :

** Answer: **

Given number is

Now, on simplifying it we will get

Therefore, the answer is

** Q3 (iv) ** Simplify :

** Answer: **

Given number is

Now, on simplifying it we will get

Therefore, the answer is

** NCERT solutions for class 9 maths chapter wise **

** NCERT solutions for class 9 subject wise **

** 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! *

## Frequently Asked Question (FAQs) - NCERT Solutions for Class 9 Maths Chapter 1 Number Systems

**Question: **What are the important topics in chapter Number Systems ?

**Answer: **

The basic concept of the number system, rational numbers, whole numbers, and integers, irrational numbers, real numbers, representation of real numbers on the number line, operations on real numbers are the important topics of this chapter.

**Question: **Which are the most difficult chapters of NCERT Class 9 Maths syllabus?

**Answer: **

Most of the students consider geometry especially triangles as the most difficult chapter in the CBSE class 9 maths.

**Question: **Does CBSE provides the solutions of NCERT for class 9 maths ?

**Answer: **

No, CBSE doesn’t provide NCERT solutions for any class or subject.

**Question: **Where can I find the complete solutions of NCERT for class 9 maths ?

**Answer: **

Here you will get the detailed NCERT solutions for class 9 maths by clicking on the link.

**Question: **Which is the official website of NCERT ?

**Answer: **

NCERT official is the official website of the NCERT where you can get NCERT textbooks and syllabus from class 1 to 12.

**Question: **How does the NCERT solutions are helpful ?

**Answer: **

NCERT solutions are helpful for the students if they stuck while solving NCERT problems. Also, these solutions are provided in a very detailed manner which will give them conceptual clarity.

## Latest Articles

##### OFSS Bihar Admission 2020 Started on ofssbihar.in - Apply Now

OFSS Bihar admission 2020 Online - Bihar School Examination Bo...

##### Bihar Board Compartmental Result 2020 Declared!- Check BSEB 10...

Bihar Board Compartmental Result 2020 has released in on Aug 7...

##### NCERT Books for Class 11 English 2020-21 - Download Pdf here

NCERT Books for Class 11 English 2020-21 - NCERT books of Engl...

##### Mahadbt Scholarship 2020 -Dates, Eligibility, Application Form

Mahadbt Scholarship 2020 - Students can check eligibility, ben...

##### PRMO Registration 2020- Check Application Dates, Fee, Details

PRMO Registration 2020 - HBCSE will release the registration f...

##### Looking Ahead - Planning for Post COVID-19 Learning and Teachi...

Looking Ahead - Planning for Post COVID-19 Learning and Teachi...

##### Kerala Plus One Result 2020 Date and Time- Check DHSE Kerala ...

Kerala Plus One Result 2020 - DHSE, Kerala has declared the pl...

##### JAC Board 2021 - Jharkhand Academic Council Exam Date, Admit ...

Jharkhand Academic Council 2021 - JAC board has started 2020-2...

##### JAC 9th Syllabus 2020-21 - Download Jharkhand Board Class 9th ...

JAC Cass 9th Syllabus 2020-21 - Students can download Jharkhan...

##### Meghalaya Board of School Education 2021 - MBOSE Board Date Sh...

Meghalaya Board of School Education (MBOSE) 2021 - Know more d...