1 . Rational Numbers

Chapter 1 : Rational Numbers

1. Rational Numbers

Exercise 1.1

1. Using appropriate properties find.
(i)

(ii)

Solution: (i) We have,

(ii) We have,

2. Write the additive inverse of each of the following.
(i)

(ii)

(iii)

(iv)

(v)

Solution: (i)

The additive inverse of is .

(ii)

The additive inverse of is .
(iii)

The additive inverse of is .

(iv)

The additive inverse of is .

(v)

The additive inverse of is

3. Verify that – (– x) = x for.
(i)

(ii)

Solution: (i) Given

We have, verified .

(ii) Given,

We have , verified .

4. Find the multiplicative inverse of the following.
(i) – 13 (ii)

(iii)

(iv)

(v)

(vi) – 1

Solution: (i) We have, the multiplicative inverse of – 13 is .

(ii) We have, the multiplicative inverse of is .

(iii) We have ,the multiplicative inverse of is 5.

(iv) We have, the multiplicative inverse of is .
(v) We have , the multiplicative inverse of is .
(vi) We have , the multiplicative inverse of – 1 is .

5. Name the property under multiplication used in each of the following.
(i)

(ii)

(iii)

Solution: (i) We have,

This is the multiplicative identity Property . Because , 1 is the multiplicative identity .
(ii) We have,

The multiplication is commutative .
(iii) We have,

This is the multiplicative inverse .

6. Multiply

by the reciprocal of

Solution: We have , the reciprocal of is

7. Tell what property allows you to compute

Solution: The multiplication is associative . [Because, ]

8. Is

the multiplicative inverse of

? Why or why not?

Solution: We have ,

is not the multiplicative inverse of , Because the product is not 1 .

9. Is 0.3 the multiplicative inverse of

? Why or why not ?

Solution: We have ,

The multiplicative inverse of is

Yes , 0.3 is the multiplicative inverse of .

10. 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) 0 .

(ii) 1 and – 1 .

(iii) 0 .

11. Fill in the blanks.
(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 reciprocal.
(ii) The numbers and are their own reciprocals
(iii) The reciprocal of – 5 is
(iv) Reciprocal of , where is .
(v) The product of two rational numbers is always a .
(vi) The reciprocal of a positive rational number is .

Exercise 1.2

1. Represent these numbers on the number line. (i)

(ii)

Solution: (i) We draw the number line ,

(ii) We draw the number line,

2. Represent

on the number line.

Solution: We draw the number line ,

3. Write five rational numbers which are smaller than 2.

Solution: The five rational numbers which are smaller than 2 are : – 4 , – 3 , – 2 , – 1 and 1 .

4. Find ten rational numbers between

and

Solution: We have ,

and

Therefore , the ten rational numbers between and are : and .

5. Find five rational numbers between.
(i)

and

(ii)

and

(iii)

and

Solution: (i) and

We have , and

Therefore ,the five rational numbers between and are : and .

(ii) and

We have , and

Therefore ,the five rational numbers between and are : and .

(iii) and

We have , and

Therefore ,the five rational numbers between and are : and .

6. Write five rational numbers greater than –2.

Solution: The five rational numbers greater than –2 are : – 1 , 1 , 2 , 3 and 4 [There are infinitely many rational between two rational numbers .]

7. Find ten rational numbers between

and

Solution: We have ,

And

The rational numbers are :

The ten rational numbers between and are

and