1.2    PROPERTIES OF RATIONAL NUMBERS:

  1. Closure Property

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

  1. 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

  1. )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:

 

  1. 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