PROPERTIES OF ADDITION OF RATIONAL NUMBERS

There are some properties of adding rational numbers.

They are closure, commutative, associative, identity, inverse and distributive. 

Closure Property

The sum of any two rational numbers is always a rational number. This is called ‘Closure property of addition’ of rational numbers. Thus, Q is closed under addition

If a/b and c/d are any two rational numbers, then (a/b) + (c/d) is also a rational number. 

Example :

2/9 + 4/9  =  6/9  =  2/3 is a rational number. 

Commutative Property

Addition of two rational numbers is commutative.

If a/b and c/d are any two rational numbers,

then (a/b) + (c/d)  =  (c/d) + (a/b)

Example :

2/9 + 4/9  =  6/9  =  2/3 

4/9 + 2/9  =  6/9  =  2/3 

Hence, 2/9 + 4/9  =  4/9 + 2/9

Associative Property

Addition of rational numbers is associative.

If a/b, c/d and e/f  are any three rational numbers,

then a/b + (c/d + e/f)  =  (a/b + c/d) + e/f

Example : 

2/9 + (4/9 + 1/9)  =  2/9 + 5/9  =  7/9 

(2/9 + 4/9) + 1/9  =  6/9 + 1/9  =  7/9 

Hence, 2/9 + (4/9 + 1/9)  =  (2/9 + 4/9) + 1/9

Additive Identity

The sum of any rational number and zero is the rational number itself.

If a/b is any rational number,

then a/b + 0 = 0 + a/b  =  a/b

Zero is the additive identity for rational numbers.

Example :

2/7 + 0 = 0 + 2/7 = 27

Additive Inverse

(- a/b) is the negative or additive inverse of (a/b)

If a/b is a rational number,then there exists a rational number (-a/b) such that

a/b + (-a/b) = (-a/b) + a/b  =  0

Example :

Additive inverse of 3/5 is (-3/5)

Additive inverse of (-3/5) is 3/5

Additive inverse of 0 is 0 itself. 

Distributive Property

Distributive Property of Multiplication over Addition :

Multiplication of rational numbers is distributive over addition.

If a/b, c/d and e/f  are any three rational numbers,

then a/b x (c/d + e/f)  =  a/b x c/d  +  a/b x e/f

Example :

1/3 x (2/5 + 1/5)  =  1/3 x 3/5  =  1/5

1/3 x (2/5 + 1/5)  =  1/3 x 2/5  +  1/3 x 1/5  =  (2 + 1) / 15 = 1/5

Hence, 1/3 x (2/5 + 1/5)  =  1/3 x 2/5  +  1/3 x 1/5

Therefore, Multiplication is distributive over addition.

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