NCERT solutions for class 8 maths chapter 1 Rational Numbers- It is a very important chapter for the students from the examination point of view and also to build basic knowledge about the numbers. The number system is the foundation of mathematics as mathematics starts with numbers. Solutions of NCERT for class 8 maths chapter 1 Rational Numbers is an important tool to enhance your preparation of the number system. Rational Numbers is the subpart of the unit numbers system. Rational numbers are those numbers which you 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 can conclude that all the integers come under the category of a rational number. CBSE NCERT solutions for class 8 maths chapter 1 Rational Numbers are designed to provide you proper step by step solutions to a particular question. In this chapter, you will 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 are explained in tabular form, and also the comparison is given with integers and whole numbers. In this particular chapter, there are a total of 24 questions in 2 exercises. NCERT solutions for class 8 maths chapter 1 Rational Numbers are covering detailed answers to all the 24 questions. NCERT solutions are downloadable for free through the link.
Q1 Fill in the blanks in the following table.
Addition | Subtraction | Multiplication | Division | |
Rational Numbers | Yes | Yes | ... | No |
Integers | ... | Yes | ... | No |
Whole Numbers | ... | ... | Yes | ... |
Natural Numbers | ... | No | ... | ... |
Answer:
It can be seen that rational numbers, integers, whole numbers, natural numbers are not closed under division because of Zero is included in these numbers. Any number divided by zero is not defined.
Addition | Subtraction | 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 |
Q2 Complete the following table:
Commutative for
Addition | Subtraction | Multiplication | Division | |
Rational Numbers | Yes | .. | ... | ... |
Integers | ... | No | ... | ... |
Whole Numbers | ... | ... | Yes | ... |
Natural Numbers | ... | ... | ... | No |
Answer:
In rational numbers, ab ba
also a-b b-a
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 |
Q3 Complete the following table:
Associative for
Addition | Subtraction | Multiplication | Division | |
Rational Numbers | ... | ... | ... | No |
Integers | ... | ... | Yes | ... |
Whole Numbers | Yes | ... | ... | ... |
Natural Numbers | ... | No | ... | ... |
Answer:
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 |
Answer:
(i) Using distributivity, a(b+c) = ab + ac
(ii) Using distributivity of multiplication over addition and subtraction,
Q5 Write the rational number for each point labeled with a letter:
Answer:
(i) In this, we can see that 1 is divided into 5 parts each, so when we are moving from zero to the right-hand 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
Answer:
By using commutativity property of numbers, we get,
(Now we will use distributivity of numbers)
=
Q1 (ii) Using appropriate properties find.
Answer:
By using commutativity, we get
Now by distributivity,
Q2 Write the additive inverse of each of the following:
Answer:
(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
Q3 Verify that – (– x) = x for (i) x = (ii) x =
Answer:
(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
Q4 Find the multiplicative inverse of the following. (i) - 13 (ii) (iii) (iv) (v) (vi) - 1
Answer:
(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.
Answer:
(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 .
Answer:
We know that the reciprocal of is .
Now,
Q7 Tell what property allows you to compute as .
Answer:
By the Associativity property for multiplication, we know that
a × (b × c) = (a × b) × c
Thus property used here is the associativity.
Q8 Is the multiplicative inverse of ? Why or why not?
Answer:
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 .
Q9 Is 0.3 the multiplicative inverse of ? Why or why not?
Answer:
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.
Answer:
(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 ________.
Answer:
(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.
Q1 Represent these numbers on the number line. (i) (ii)
Answer:
(i) To represent on a 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 the 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.
Answer:
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 drawn number line.
Q3 Write five rational numbers which are smaller than 2.
Answer:
The 5 rational numbers smaller than 2 can be any number in the form of p/q where q 0. Hence infinite numbers are possible.
Examples of 5 such numbers are 1, 1/3, 0, -1, -2
Q4 Find ten rational numbers between and .
Answer:
Rational numbers between any 2 numbers can easily find out by taking their means.
i.e., For and
Their mean is . Hence 1 rational number between and is .
Now we will find the mean between and .
This implies a new required rational number is .
Similarly, we will find a mean between and
New required rational number is
Similarly, we will take means of new numbers generated between and .
Q5 Find five rational numbers between.
Answer:
(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 denominator 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
Q6 Write five rational numbers greater than –2.
Answer:
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.
Q7 Find ten rational numbers between and .
Answer:
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 denominators equal.
Further, it is equivalent to find rational number between and
(We obtained above numbers by multiplying and dividing numbers by 8 to create a difference of at least 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.
Keep working hard and happy learning!
Complete the following table:
Addition | Subtraction | Multiplication | Division | |
---|---|---|---|---|
Rational Numbers | ... | ... | ... | No |
Integers | ... | ... | Yes | ... |
Whole Numbers | Yes | ... | ... | ... |
Natural Numbers | ... | Yes | ... | ... |
4. Find the multiplicative inverse of the following.
(i) - 13 (ii) (iii) (iv) (v) (vi) - 1
Q.1 Using appropriate properties find.