It is a very important chapter for the students from the examination point of view and also to build basic knowledge about the numbers. Chapter solutions will give you a strong idea to categorize numbers into rational and Irrational. To help the students, the NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers are solved by the experts.
In this chapter, we learn about the rational numbers, real numbers, whole numbers, integers, and natural numbers and also study their properties like Closure, Commutativity, Associativity. For the students to understand in an easy way, properties of rational numbers explained in tabular form, and also compare with integers and whole numbers in NCERT Grade 8 Maths Chapter 1 Rational Numbers. Rational Number is a very important number category under the topic number system. The following discussion is brief about the chapter
Rational numbers are those numbers which we can represent in the form of a fraction or numerator(p) upon a denominator(q) where the denominator can be any value except 0 or we can say that rational numbers are those numbers which can be represented in p/q form where q ≠0 and p & q are integers. From this definition, we are saying that all the integers come under the category of a rational number.
To compare commutative property over addition, subtraction, multiplication and division of rational numbers with integers, whole numbers and natural numbers lets take a table of NCERT Class 8 Maths Chapter 1 Rational Numbers
Addition  Subtraction  Multiplication 
Division 

Rational numbers 
Yes  No  Yes 
No 
Integers 
Yes  No  Yes 
No 
Whole numbers 
Yes  No  Yes 
No 
Natural numbers 
Yes  No  Yes 
No 
Let’s understand the concept by taking some examples:
6: It is a rational number because it can be written as 6/1 and where the denominator is not zero.
3/2: It is also a rational number because it is already in the form of A/B where the denominator is also not equal to 0.
0.333333333: It is the third type of rational number where the decimal number is recurring and thus if we convert this recurring number into fraction then it will become 1/3 and we already know that any number which is infraction where the denominator is not zero is a rational number. 0.324569576: In this decimal number the digits after the decimal point are not recurring and every next digit is different from the previous digit. Thus we can not represent in a fraction and anything which cannot be represented in the fraction is not a rational number or we can say that it is an irrational number.
Most of the students are in doubt whether fall under the category of rational number or irrational number? This is the perfect value is 3.14159 26535 89793 23846 26433 83279…... So if we want to convert this decimal number into fraction then we cannot convert it because the numbers are not recurring and digits coming in the value are endless. Now you will think that 22/7 is the exact value of but it is not then It is just an approximation of abovewritten value. Thus, it is not a rational number.
In this chapter, there are 2 exercises with 18 questions. Following are the important topics and subtopics of NCERT Class 8 Maths Chapter 1 Rational Numbers.
1.1 Introduction
1.2 Properties of Rational Numbers
1.2.1 Closure
1.2.2 Commutativity
1.2.3 Associativity
1.2.4 The role of zero (0)
1.2.5 The role of 1
1.2.6 Negative of a number
1.2.7 Reciprocal
1.2.8 Distributivity of multiplication over addition for rational numbers
1.3 Representation of Rational Numbers on the Number Line
1.4 Rational Numbers between Two Rational Numbers
NCERT Solutions for Class 8 Maths Chapter 1 Rational numbers, Will give you a strong idea to categorize numbers into the Rational and Irrational. Along with you will find the detailed solutions of all the question of the topic.
Below mentioned are the questions and solutions for Class 8 Maths Rational Numbers:
NCERT Solutions for Class 8 Maths Rational Numbers In Text Question
NCERT Solutions for Class 8 Maths Rational Numbers Exercise 1.1
NCERT Solutions for Class 8 Maths Rational Numbers Exercise 1.2
Chapter 2  
Chapter3  
Chapter4  
Chapter5  
Chapter6  
Chapter7  
Chapter8  
Chapter9  
Chapter10  
Chapter11  
Chapter12  
Chapter13  
Chapter14  
Chapter15  
Chapter16 
Q.1 Using appropriate properties find.
By using commutativity property of numbers, we get,
(Now we will use distributivity of numbers)
=
Q.1 Using appropriate properties find.
By using commutativity, we get
Now by distributivity,
Write the additive inverse of each of the following:
(i) (ii) (iii) (iv) (v)
(i) The additive inverse of is because
(ii) The additive inverse of is because
(iii) The additive inverse of is because
(iv) The additive inverse of is because
(v) The additive inverse of is because
3. Verify that – (– x) = x for
(i) x = (ii) x =
(i) We have x =
The additive inverse of x = is x =
The same equality
which implies shows that (x) = x
(ii) Additive inverse of x = is x = (since )
The same quality shows that the additive inverse of is
i.e., (x) = x
4. Find the multiplicative inverse of the following.
(i)  13 (ii) (iii) (iv) (v) (vi)  1
(i) The multiplicative inverse of 13 is because
(ii) The multiplicative inverse of is because of i
(iii) The multiplicative inverse of is 5 because
(iv) The multiplicative inverse of is because
(v) The multiplicative inverse of is because
(vi) The multiplicative inverse of 1 is 1 because
Q5. Name the property under multiplication used in each of the following.
(i) (ii) (iii)
(i) Multiplying any number with 1 we get the same number back.
i.e., a1 = 1a = a
Hence 1 is the multiplicative identity for rational numbers.
(ii) Commutavity property states that ab = ba
(iii) It is the multipicative inverse identity, i.e.,
Q6. Multiply by the reciprocal of .
We know that the reciprocal of is .
Now,
Q7. Tell what property allows you to compute as .
By the Associativity property for multiplication, we know that
a × (b × c) = (a × b) × c
Thus property used here is the associativity.
8. Is the multiplicative inverse of ? Why or why not?
We know that A is the multiplicative inverse of B if BA = 1
Applying this in given question, we get,
Thus is not the multiplicative inverse of .
9. Is 0.3 the multiplicative inverse of ? Why or why not?
We know that if A is multiplicative inverse of B then BA = 1
In this question, (Since 0.3 = )
This 0.3 is the multiplicative inverse of .
Q10.Write.
(i) The rational number that does not have a reciprocal.
(ii) The rational numbers that are equal to their reciprocals.
(iii) The rational number that is equal to its negative.
(i) Zero(0). We know that reciprocal of A is . So for 0, its reciprocal is not defined.
(ii) 1 and 1 . (Since and )
(iii) Zero,0. (as 0=0)
(i) Zero has ________ reciprocal.
(ii) The numbers ________ and ________ are their own reciprocals.
(iii) The reciprocal of – 5 is ________.
(iv) Reciprocal of , where x ≠ 0 is ________.
(v) The product of two rational numbers is always a _______.
(vi) The reciprocal of a positive rational number is ________.
(i) Zero has no reciprocal as it's reciprocal is not defined.
(ii) The numbers 1 and 1 are their own reciprocals as and
(iii) . We know that reciprocal of A is .
(iv) Since .
(v) rational number. We know that if p and q are 2 rational numbers then pq is also a rational number.
(vi) Positive. Since reciprocal of A is , now if A is positive then reciprocal is also positive.
1. Represent these numbers on the number line. (i) (ii)
(i) To represent on number line, firstly we will divide 1 in 4 parts and draw it on a line such as 1/4, 2/4, 3/4, ........, 9/4. Then will mark the required number.
(ii) To represent on number line, firstly we will divide 1 in 6 parts and draw it on the left side of zero on number line such as 1/6, 2/6, .......,9/4. Then mark the required number on the number line.
Q2. Represent ,, on the number line.
We will divide 1 into 11 parts, then start marking numbers on left side of zero such as 1/11, 2/11, 3/11,.........,12/11. Mark the required numbers on the drwan number line.
Q3. Write five rational numbers which are smaller than 2.
The 5 rational numbers smaller than 2 can be any number any number in form of p/q where
q 0. Hence infinite numbers are possible.
Examples of 5 such numbers are 1, 1/3, 0, 1, 2
4. Find ten rational numbers between and .
Rational numbers between any 2 numbers can easily be find out by taking their means.
i.e., For and
Their mean is . Hence 1 rational number between and is .
Now we will find mean between and .
This implies new required rational number is .
Similarly, we will find mean between and
New required rational number is
Similarly, we will take means of new numbers generated between and .
5. Find five rational numbers between.
(i) and (ii) and (iii) and
(i) For finding rational numbers between 2 numbers one method is to find means between the numbers repeatedly.
Another method is: For and
can be written as
and can be written as
Thus numbers between and are the required numbers.
Now since we require 5 numbers in between, thus we multiply numerator and denomenator both by 4.
It becomes numbers between and .
Thus numbers are .
(ii) Similarly for and
Required numbers fall between and
Thus numbers are
(iii) For and
Required numbers lie between and or we can say between and
Thus numbers are
6. Write five rational numbers greater than –2.
There exist infinitely many rational numbers (can be expressed in the form of p/q where q 0) greater than 2 .
Few such examples are 1, 1/2, 0, 1, 1/3 etc.
7. Find ten rational numbers between and .
Finding rational numbers between and is equivalent to find rational numbers between rational numbers between and ,since these numbers are obtained by just making their denomenators equal.
Further it is equivalent to find rational number between and
(We obtained above numbers by multiplying and dividing numbers by 8 to create differnce of atleast 10 numbers).
Thus required numbers are
Alternate: Rational numbers can also be found by taking mean of the given numbers and the newly obtained number.
Fill in the blanks in the following table.
Addition  Substraction  Multiplication  Division  

Rational Numbers  Yes  Yes  ...  No 
Integers  ...  Yes  ...  No 
Whole Numbers  ...  ...  Yes  ... 
Natural Numbers  ...  No  ...  ... 
It can be seen that rational numbers, integers, whole numbers, natural numbers are not closed under division because of Zero is included in these number. Any number divided byzero is not defined.
Addition  Substraction  Multiplication  Division  

Rational Numbers  Yes  Yes  Yes  No 
Integers  Yes  Yes  Yes  No 
Whole Numbers  Yes  No  Yes  No 
Natural Numbers  Yes  No  Yes  Yes 
Complete the following table:
Addition  Subtraction  multiplication  division  

Rational Numbers  Yes  ...  ...  ... 
Integers  ...  No  ...  ... 
Whole numbers  ...  ...  Yes  ... 
Natural numbers  ...  ...  ...  No 
In rational numbers , ab ba
also ab ba
Addition  Subtraction  multiplication  division  

Rational Numbers  Yes  No  Yes  No 
Integers  Yes  No  Yes  No 
Whole numbers  Yes  No  Yes  No 
Natural numbers  Yes  No  Yes  No 
Complete the following table:
Addition  Subtraction  Multiplication  Division  

Rational Numbers  ...  ...  ...  No 
Integers  ...  ...  Yes  ... 
Whole Numbers  Yes  ...  ...  ... 
Natural Numbers  ...  Yes  ...  ... 
For associative in multiplication : a(bc) = (ab)c
Addition  Subtraction  Multiplication  Division  

Rational Numbers  Yes  No  Yes  No 
Integers  Yes  No  Yes  No 
Whole Numbers  Yes  No  Yes  No 
Natural Numbers  Yes  No  Yes  No 
Find using distributivity:
(i) (ii)
(i) Using distributivity, a(b+c) = ab + ac
(ii) Using distributivity of multiplication over addition and subtraction,
Write the rational number for each point labelled with a letter:
(i) In this, we can see that 1 is divided into 5 parts each, so when we are moving from zero to the righthand side, it is easy to observe that
All the numbers should contain 5 in their denominator. Thus, A is equal to , B is equal to , C is equal to , D is equal to , E is equal to
(ii) Here we see that 1 is divided in 6 parts each. So when we move from zero towards left we observe that
All the numbers should contain 6 in their denominator. Thus, F is equal to , G is equal to , H is equal to , I is equal to , J is equal to