Properties of Addition of Rational Numbers

We will learn the properties of addition of rational numbers i.e. closure property, commutative property, associative property, existence of additive identity property and existence of additive inverse property of addition of rational numbers.

Closure property of addition of rational numbers:

The sum of two rational numbers is always a rational number.

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

For example:

(i) Consider the rational numbers 1/3 and 3/4 Then,

(1/3 + 3/4)

= (4 + 9)/12

= 13/12, is a rational number

(ii) Consider the rational numbers -5/12 and -1/4 Then,

(-5/12 + -1/4)

= {-5 + (-3)}/12

= -8/12

= -2/3, is a rational number

(iii) Consider the rational numbers -2/3 and 4/5 Then,

(-2/3 + 4/5)

= (-10 + 12)/15

= 2/15, is a rational number

Commutative property of addition of rational numbers:

Two rational numbers can be added in any order.

Thus for any two rational numbers a/b and c/d, we have

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

For example:

(i) (1/2 + 3/4)

= (2 + 3)/4

=5/4

and (3/4 + 1/2)

= (3 + 2)/4

= 5/4

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

(ii) (3/8 + -5/6)

= {9 + (-20)}/24

= -11/24

and (-5/6 + 3/8)

= {-20 + 9}/24

= -11/24

Therefore, (3/8 + -5/6) = (-5/6 + 3/8)

(iii) (-1/2 + -2/3)

= {(-3) + (-4)}/6

= -7/6

and (-2/3 + -1/2)

= {(-4) + (-3)}/6

= -7/6

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

Associative property of addition of rational numbers:

While adding three rational numbers, they can be grouped in any order.

Thus, for any three rational numbers a/b, c/d and e/f, we have

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

For example:

Consider three rationals -2/3, 5/7 and 1/6 Then,

{(-2/3 + 5/7) + 1/6} = {(-14 + 15)/21 + 1/6} = (1/21 + 1/6) = (2 + 7)/42

= 9/42 = 3/14

and {(-2/3 + (5/7 + 1/6)} = {-2/3 + (30 + 7)/42} = (-2/3 + 37/42)

= (-28 + 37)/42 = 9/42 = 3/14

Therefore, {(-2/3 + 5/7) + 1/6} = {-2/3 + (5/7 + 1/6)}

Existence of additive identity property of addition of rational numbers:

0 is a rational number such that the sum of any rational number and 0 is the rational number itself.

Thus, (a/b + 0) = (0 + a/b) = a/b, for every rational number a/b

0 is called the additive identity for rationals.

For example:

(i) (3/5 + 0) = (3/5 + 0/5) = (3 + 0)/5 = 3/5 and similarly, (0 + 3/5) = 3/5

Therefore, (3/5 + 0) = (0 + 3/5) = 3/5

(ii) (-2/3 + 0) = (-2/3 + 0/3) = (-2 + 0)/3 = -2/3 and similarly, (0 + -2/3)

= -2/3

Therefore, (-2/3 + 0) = (0 + -2/3) = -2/3

Existence of 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) = {a + (-a)}/b = 0/b = 0 and similarly, (-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

For example:

(4/7 + -4/7) = {4 + (-4)}/7 = 0/7 = 0 and similarly, (-4/7 + 4/7) = 0

Thus, 4/7 and -4/7 are additive inverses of each other.