NCERT solutions for class 9 maths chapter 1 Number Systems Are you looking for indepth 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 nonrational 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.
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:
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.
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 nonterminating 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 nonterminating 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 nonterminating 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 nonterminating nonrecurring.
Answer:
Three numbers whose decimal expansions are nonterminating nonrecurring 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 nonterminating nonrecurring.
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 nonterminating 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 nonterminating nonrepeating. 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 nonterminating nonrecurring. 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 nonterminating and nonrecurring. 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 nonterminating and nonrecurring. 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 midpoint D of OC and draw a semicircle 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 semicircle 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
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 

Chapter 7 

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 

Chapter 11 
CBSE NCERT solutions for class 9 maths chapter 11 Constructions 
Chapter 12 

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 
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!