Rational numbers are numbers that can be expressed as the ratio or fraction of two integers, where the denominator is not zero. Examples of rational numbers include:
1/2
5/7
-3/4
0 (since it can be expressed as 0/1)
Let’s have a look at the NCERT Solutions of Class 8 Mathematics Chapter “Rational Numbers”
Exercise 1.1
Q1. Name the property under multiplication used in each of the following: (i) -4/5 × 1 = 1 × (-4/5) = -4/5 Ans: Since, 1 is the multiplicative identity. ∴ 1 is the multiplicative identity and here, property of multiplicative identity is used.
(ii) -13/17 × (-2/7) = -2/7 × (-13/17) Ans: The property of commutativity is used in the equation.
(iii) -19/29 × 29/-19 = 1 Ans: The multiplicative inverse is the property used in this equation.
Q2. Tell what property allows you to compute 1/3 × (6 × 4/3) as (1/3 × 6) × 4/3. Ans: 1/3 × (6 × 4/3) = (1/3 × 6) × 4/3 Here, the way in which factors are grouped in a multiplication problem supposedly does not change the product. Hence, the Associativity Property is used here.
Q3. The product of two rational numbers is always a _______. Ans:The product of two rational numbers is always a rational number. Let’s consider two rational numbers: 1/2 and 3/4. When we multiply these two rational numbers, (1/2) * (3/4) = (1 * 3) / (2 * 4) = 3/8 Here, the product of 1/2 and 3/4 is 3/8, which is also a rational number. Therefore, this example demonstrates that the product of two rational numbers is always a rational number.
Deleted Questions from NCERT
Q.1. Using appropriate properties find: (i) (ii)
Difficulty Level: Easy
What is the known/given?
Rational numbers with addition subtraction and multiplication.
What is the unknown?
Result of addition, subtraction and multiplication of rational numbers.
Reasoning:By using commutativity of multiplication and addition getting the answer.
Solution (i):
[By commutativity of multiplication] [Commutativity of addition] [Rearranging to take a common]
Answer (i): 2
Solution (ii):
Rearranging
Taking 2/5 common [by distributivity]
Answer (ii):
Q.2. Write the additive inverse of each of the following (i) 2/8 (ii) -5/9 (iii) -6/-5 (iv) 2/-9 (v) 19/-6
Difficulty Level: Easy
What is the known/given?
Rational numbers
What is the unknown?
Additive inverse
Reasoning: The negative of a rational number is called additive inverse.
Solution: (i) 2/8
Additive inverse of 2/8 is
(ii) -5/9
Additive inverse of -5/9 is
(iii) -6/-5
The rational number is Additive inverse of -6/-5 is
(iv) 2/-9
Additive inverse of 2/-9 is
(v) 19/-6
Additive inverse of 19/-6 is
Q.3. Verify that − ( − x ) = x for (i) x = 11/15 (ii) x =
Difficulty Level: Easy What is the known/given?
Rational number
What is the unknown?
The negative of the negative of a rational number.
Reasoning:
The negative of the negative of a rational number is that rational number of Self.
Solution:
(i) x = 11/15
= 11/15 = x Proved.
(ii) x =
Proved.
Q.4. Find the multiplicative inverse of the following. (i) −13 (ii) −13/19 (iii) 1/5 (iv) (v) (vi) −1
Difficulty Level: Medium
What is the known/given?
Rational number
What is the unknown?
The multiplicative inverse.
Reasoning:The reciprocal of the given rational number is the multiplicative inverse. [the product of the rational number and its multiplicative inverse is 1]
Solution:
(i) −13
The Multiplicative inverse of −13 is -1/13
(ii) −13/19
The Multiplicative inverse of −13/19 is 19/-13
(iii) 1/5 The Multiplicative inverse of 1/5 is 5/1
(iv)
The Multiplicative inverse of 15/56 is 56/15
(v)
This can be simplified as:
= 2/5 The multiplicative inverse of 2/5 is 5/2
(vi) −1
The multiplicative inverse of −1 is −1. (−1) x (-1) = 1
Q.5. Multiply 6/13 by the reciprocal of -7/16
Difficulty Level: Easy
What is the known/given?
Rational numbers
What is the unknown?
Product of the rational numbers.
Reasoning:
Reciprocal of a rational number is its multiplicative inverse.
Solution:
Answer:
Q6. Is 8/9 the multiplicative inverse of Why or why not?
Difficulty Level: Medium
What is the known/given?
Rational numbers.
What is the unknown?
Multiplicative or not
Reasoning:The product of a rational number with its multiplicative inverse is 1.
Solution:
Now:
Answer: So, 8/9 is not the multiplicative inverse of
8/9 is not the multiplicative inverse of because the product of 8/9 and is -1, and it should be 1 to be a multiplicative inverse.
Q.7. Is 0.3 the multiplicative inverse of Why or why not?
Difficulty Level: Medium
What is the known/given?
Rational number.
What is the unknown?
Multiplicative inverse or not?
Reasoning:The product of the rational number and its multiplicative inverse is 1.
Solution:0.3 can be written as 3/10 Given rational number can be written as 10/3 So,
Answer: Yes, 0.3 is the multiplicative inverse of because their product is 1.
Q.8. Write:
(i) Rational number does not have a reciprocal 0 (zero) is the rational number which does not have a reciprocal.
(ii) The rational number that is equal to its reciprocals.
The rational numbers 1 and (–1) are equal to their own reciprocals.
(iii) The rational number that is equal to its negative.
Rational number 0 is equal to its negative.
Q.9. Fill in the blanks. (i) Zero has ________ reciprocal.
Zero has no reciprocal
(ii) The numbers ________ and ________ are their own reciprocals
The numbers 1 and (–1) are their own reciprocals.
(iii) The reciprocal of –5 is ________.
The reciprocal of (–5) is 1/-5
(iv) Reciprocal of 1/x , where x ≠ 0 is ________.
Reciprocal of 1/x where x ≠ 0 is x .
(v) The reciprocal of a positive rational number is ________.
The reciprocal of a positive rational number is positive.
