1 - Rational Numbers

Exercise 1.1

Question 1- Using appropriate properties find:

(i) 

(ii) 

Answer - (i)

(ii)

 (By commutativity)

Question 2- Write the additive inverse of each of the following:

(i) (ii) (iii) (iv) (v)

Answer - (i)

Additive inverse = 

(ii)

Additive inverse = 

(iii)

Additive inverse = 

(iv)

Additive inverse 

(v)

Additive inverse 

Question 3- Verify that −(−x) = x for.

(i) (ii)

Answer - (i)

The additive inverse of  is  as 

This equality  represents that the additive inverse of  is  or it can be said that  i.e., −(−x) = x

(ii)

The additive inverse of  is  as 

This equality  represents that the additive inverse of  is − i.e., −(−x) = x

Question 4- Find the multiplicative inverse of the following.

(i) (ii) (iii)

(iv) (v) (vi) −1

Answer - (i) −13

Multiplicative inverse = −

(ii)

Multiplicative inverse = 

(iii)

Multiplicative inverse = 5

(iv)

Multiplicative inverse 

(v)

Multiplicative inverse 

(vi) −1

Multiplicative inverse = −1

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

(i)

(ii)

(iii)

Answer - (i)

1 is the multiplicative identity.

(ii) Commutativity

(iii) Multiplicative inverse

Question 6- Multiply by the reciprocal of.

Answer - 

Question 7- Tell what property allows you to compute

Answer - Associativity

Question 8- Is the multiplicative inverse of? Why or why not?

Answer - If it is the multiplicative inverse, then the product should be 1.

However, here, the product is not 1 as

Question 9- Is 0.3 the multiplicative inverse of? Why or why not?

Answer - 

0.3 ×  = 0.3 × 

Here, the product is 1. Hence, 0.3 is the multiplicative inverse of.

Question 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 is a rational number but its reciprocal is not defined.

(ii) 1 and −1 are the rational numbers that are equal to their reciprocals.

(iii) 0 is the rational number that is equal to its negative.

Question 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 is __________.

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

(vi) The reciprocal of a positive rational number is __________.

Answer - (i) No

(ii) 1, −1

(iii)

(iv) x

(v) Rational number

(vi) Positive rational number

Exercise 1.2

Question 1- Represent these numbers on the number line.

(i) (ii)

Answer - (i)  can be represented on the number line as follows.

(ii)  can be represented on the number line as follows.

Question 2- Represent on the number line.

Answer -  can be represented on the number line as follows.

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

Answer - 2 can be represented as.

Therefore, five rational numbers smaller than 2 are

Question 4- Find ten rational numbers between and.

Answer - and can be represented as  respectively.

Therefore, ten rational numbers between andare

Question 5- Find five rational numbers between

(i)

(ii)

(iii)

Answer - (i)  can be represented as respectively.

Therefore, five rational numbers between  are

(ii)  can be represented as  respectively.

Therefore, five rational numbers between  are

(iii)  can be represented as  respectively.

Therefore, five rational numbers between are

Question 6- Write five rational numbers greater than − 2.

Answer - −2 can be represented as −.

Therefore, five rational numbers greater than −2 are

Question 7- Find ten rational numbers between and.

Answer - and can be represented as  respectively.

Therefore, ten rational numbers between and are