# 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)}

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

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.

Rational Numbers

What is Rational Numbers?

Is Every Rational Number a Natural Number?

Is Zero a Rational Number?

Is Every Rational Number an Integer?

Is Every Rational Number a Fraction?

Positive Rational Number

Negative Rational Number

Equivalent Rational Numbers

Equivalent form of Rational Numbers

Rational Number in Different Forms

Properties of Rational Numbers

Lowest form of a Rational Number

Standard form of a Rational Number

Equality of Rational Numbers using Standard Form

Equality of Rational Numbers with Common Denominator

Equality of Rational Numbers using Cross Multiplication

Comparison of Rational Numbers

Rational Numbers in Ascending Order

Representation of Rational Numbers on the Number Line

Addition of Rational Number with Same Denominator

Addition of Rational Number with Different Denominator

Properties of Addition of Rational Numbers

Subtraction of Rational Number with Different Denominator

Subtraction of Rational Numbers

Properties of Subtraction of Rational Numbers

Simplify Rational Expressions Involving the Sum or Difference

Multiplication of Rational Numbers

Properties of Multiplication of Rational Numbers

Rational Expressions Involving Addition, Subtraction and Multiplication

Division of Rational Numbers

Rational Expressions Involving Division

Properties of Division of Rational Numbers

To Find Rational Numbers

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