Properties of Addition of Rational Numbers

Learn the Properties of Rational Numbers such as Closure Property, Commutative Property, Identity Property, Associative Property, Additive Inverse Property, etc. We tried explaining each and every Property of Rational Numbers Addition in the following sections. Check them and learn the concepts easily. To help you understand the concept, even more, better we have provided a few examples.

Closure Property of Addition of Rational Numbers

Closure Property is applicable for the Addition Operation of Rational Numbers. The Sum of Two Rational Numbers always yields in a Rational Number. Let a/b, c/d be two rational numbers then (a/b+c/d) is also a Rational Number.

Examples

(i) Consider the Rational Numbers 5/4 and 1/3

= (5/4+1/3)

= (5*3 +1*4)/12

= (15+4)/12

= 19/12

Therefore, the Sum of Rational Numbers 5/4 and 1/3 i.e. 19/12 is also a Rational Number.

(ii) Consider the Rational Numbers -4/3 and 2/5

= -4/3+2/5

= (-4*5+2*3)/15

= (-20+6)/15

= -14/15 is also a Rational Number.

Commutative Property of Addition of Rational Numbers

Commutative Property is applicable for the Addition Operation of Rational Numbers. Two Rational Numbers can be added in any order. Let us consider two rational numbers a/b, c/d then we have

(a/b+c/d) = (c/d+a/b)

Examples

(i) 1/3+4/5

= (5+12)/15

= 17/15

and 4/5+1/3

= (12+5)/15

= 17/15

Therefore, (1/3+4/5) = (4/5+1/3).

(ii) -1/2+3/2

= (-1+3)/2

= 2/2

and 3/2+(-1/2)

= (3-1)/2

= 2/2

Therefore, (-1/2+3/2) = (3/2+-1/2)

Associative Property of Addition of Rational Numbers

While adding Three Rational Numbers you can group them in any order. Let us consider three Rational Numbers a/b, c/d, e/f we have

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

Example

Consider Three Rational Numbers 1/2, 3/4 and 5/6 then

(1/2+3/4)+5/6 = (2+3)/4+5/6

= 5/4+5/6

= (15+10)/12

= 25/12 and

1/2+(3/4+5/6) = 1/2+(9+10)/12

= 1/2+19/12

= (6+19)/12

=25 /12

Therefore, (1/2+3/4)+5/6 = 1/2+(3/4+5/6)

Additive Identity Property of Addition of Rational Numbers

0 is a Rational Number and any Rational Number added to 0 results in a Rational Number.

For every Rational Number a/b (a/b+0)=(0+1/b)= a/b and 0 is called the Additive Identity for Rationals.

Example

(i) (4/5+0) = (4/5+0/5) =(4+0)/5 =4/5 and similarly (0+4/5) = (0/5+4/5) = (0+4)/5 = 4/5

Therefore, (4/5+0) = (0+4/5) = 4/5

(ii)(-1/3+0) =(-1/3+0/3) =(-1+0)/3 = -1/3 and similarly (0-1/3) = (0/3-1/3) =(0-1)/3 = -1/3

Therefore, (-1/3+0) =(0+-1/3) = -1/3

Additive Inverse Property of Addition of Rational Numbers

For every Rational Number a/b there exists a Rational Number -a/b such that (a/b+-a/b)=0 and (-a/b+a/b)=0

Thus, (a/b+-a/b) = (-a/b+a/b) = 0

-a/b is called the Additive Inverse of a/b

Example

(4/3+-4/3) = (4+(-4))/3 = 0/3 = 0

Similarly, (-4/3+4/3) = (-4+4)/3 = 0/3 = 0

Thus, 4/3 and -4/3 are additive inverse of each other.