# Rational Numbers

## NCERT Exercise 1.1 (Part 2)

Question 3: Verify that -(-x)=x for:

(i) x=(11)/(15)

Solution: Given, x=(11)/(15)

The additive inverse of x=(11)/(15) is -x=(-11)/(15)

Similarly, the additive inverse of (-11)/(15) is (11)/(15)

Or, -((-11)/(15))=(11)/(15)

Or, -(-x)=x proved

(ii) x=-(13)/(17)

Solution: Given, x=-(13)/(17)

The additive inverse of x=-(13)/(17) is -x=(13)/(17)

Similarly, the additive inverse of (13)/(17) is -(13)/(17)

Or, -(13)/(17)+(13)/(17)=0

Or, -(-x)=x proved

Question 4: Find the multiplicative inverse of the following

(i) -13

Solution: We know that multiplicative inverse of a number is reciprocal of the number.

Thus, multiplicative inverse of -13 is equal to (1)/(-13)

(ii) (-13)/(19)

Solution: We know that multiplicative inverse of a number is reciprocal of the number.

Thus, multiplicative inverse of (-13)/(19) is equal to (19)/(-13)

(iii) 1/5

Solution: We know that multiplicative inverse of a number is reciprocal of the number.

Thus, multiplicative inverse of 1/5 is equal to 5

(iv) -5/8xx(-3)/(7)

Solution: Given, -5/8xx(-3)/(7)

=((-5)xx(-3))/(8xx7)=(15)/(56)

We know that multiplicative inverse of a number is reciprocal of the number.

Thus, multiplicative inverse of (15)/(56) is equal to (56)/(15)

(v) -1xx(-2)/(5)

Solution: Given, -1xx(-2)/(5)=2/5

We know that multiplicative inverse of a number is reciprocal of the number.

Thus, multiplicative inverse of 2/5 is equal to 5/2

(vi) -1

Solution: We know that multiplicative inverse of a number is reciprocal of the number.

Thus, multiplicative inverse of -1 is equal to (1)/(-1) or -1

Alternate Method:

The product of a number and its multiplicative inverse is equal to 1

Here, -1xx-1=1

Thus, multiplicative inverse of -1 is -1

Question 5: Name the property under multiplication used in each of the following.

(i) (-4)/(5)xx1=1xx(-4)/(5)=-4/5

Solution: Here, 1 is the multiplicative identity.

Thus, property of multiplicative identity is used.

(ii) -(13)/(17)xx(-2)/(7)=(-2)/(7)xx(-13)/(17)

Solution: Here, multiplicative commutativity is used.

(iii) (-19)/(29)xx(29)/(-19)=1

Solution: Since, the product of given numbers is 1, so (29)/(-19) is the multiplicative inverse of (-19)/(29)

Thus, property of multiplicative inverse is used.

Question 6: Multiply (6)/(13) by the reciprocal of (-7)/(16)

Solution: Reciprocal of (-7)/(16) is (16)/(-7)

So, (6)/(13)xx(16)/(-7)

=(6xx16)/(13xx(-7))=(96)/(-91)

Question 7: Tell what property allows you to compute 1/3xx(6xx4/3) as (1/3xx6)xx4/3

Solution: The property of associativity

Question 8: Is 8/9 the multiplicative inverse of -1\1/8? Why or why not?

Solution: -1\1/8=-7/8

Since, 8/9xx(-7)/(8)=-7/9≠1

So, -1\1/8 is not the multiplicative inverse of 8/9

Question 9: Is 0.3 the multiplicative inverse of 3\1/3 Why or why not?

Solution: 0.3=(3)/(10)

The multiplicative inverse of (3)/(10) is (10)/(3)=3\1/3

Thus, 3\1/3 is the multiplicative inverse of 0.3.

Question 10: Write

(i) The rational number that does not have a reciprocal.

Solution: 0 (zero) is the rational number which does not have a reciprocal.

(ii) The rational numbers that are equal to their reciprocals.

Solution: 1 and – 1 are the rational numbers which are equal to their reciprocals.

(iii) The rational number that is equal to its negative.

Solution: 0 (zero) is the rational number which is equal to its negative.

Question 11: Fill in the blanks:

(i) Zero has __________ reciprocal.

Solution: no

(ii) The numbers ________ and ________ are their own reciprocals.

Solution: 1 and – 1

(iii) The reciprocal of – 5 is _____________.

Solution: (1)/(-5)

(iv) Reciprocal of 1/x where x≠0 is ______________.

Solution: x`

(v) The product of two rational numbers is always a _____________.

Solution: rational number

(vi) The reciprocal of a positive rational number is ____________

Solution: positive