**1.2 PROPERTIES OF RATIONAL NUMBERS:**

**Closure Property**

**This shows that the operation of any two same types of numbers is also the same type or not.**

**Associative Property**

This shows that the grouping of numbers does not matter i.e. we can use operations on any two numbers first and the result will be the same.

**c). Rational Numbers**

**If p, q and r are three rational numbers then**

**The Role of Zero in Numbers (Additive Identity)**

Zero is the additive identity for whole numbers, integers and rational numbers.

**Negative of a Number (Additive Inverse)**

**Distributivity of Multiplication over Addition and Subtraction for Rational Numbers**

This shows that for all rational numbers p , q and r

- )p ( q + r ) = p q + pr 2). p ( q – r ) = p q – pr

**Example: Check the distributive property of the three rational numbers **

** 4 / 7 , – ( 2 ) / 3 and 1 / 2.**

Solution: Let’s find the value of

This shows that

Q1. Using appropriate properties find:

**Q2. Write the additive inverse of each of the following:**

**Name the property under multiplication used in each of the following.**

Solution:

(i) – 4 / 5 × 1 = 1 × (- 4 / 5) = – 4 / 5

Here 1 is the multiplicative identity.

(ii) -13 / 17 × (- 2 / 7) = – 2 / 7 × (- 13 / 17)

The property of commutativity is used in the equation

(iii) – 19 / 29 × 29 / – 19 = 1

Multiplicative inverse is the property used in this equation.

6. Multiply 6 / 13 by the reciprocal of – 7 / 16

Solution:

Reciprocal of – 7 / 16 = 16 / – 7 = – 16 / 7

According to the question,

6 / 13 × ( Reciprocal of – 7 / 16)

6 / 13 × ( – 16 / 7) = – 96 / 91 )

7. Tell what property allows you to compute

1 / 3 × ( 6 × 4 / 3) as ( 1 / 3 × 6 ) × 4 / 3

Solution:

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.

**10. Write**

Solution: (i) The rational number that does not have a reciprocal is 0.

Reason: 0 = 0 / 1

Reciprocal of 0 = 1 / 0 , which is not defined.

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

1 and – 1.

Reason:

1 = 1 / 1

Reciprocal of 1 = 1 / 1 = 1 Similarly, Reciprocal of – 1 = – 1

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

Reason:

Negative of 0 = – 0 = 0

**11. Fill in the blanks.**

Solution:

(i) Zero has no reciprocal.

(ii) The numbers – 1 and 1 are their own reciprocals

(iii) The reciprocal of – 5 is – 1 / 5.

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

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

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

Additional Questions:

1. Identify the Properties Associated with the following :–

2. Find the multiplicative inverse of the following:

(i) 2 / 3 (ii) 5 / 12 (iii) 1 / 12 (iv) 1 / 18

3. Write the additive inverse of each of the following rational numbers:

(i) 9 / 4 (ii) 7 / −13 (iii) − 1 / 15 (iv) −14 / − 11