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.

● **Rational Numbers**

Introduction of Rational Numbers

Is Every Rational Number a Natural Number?

Is Every Rational Number an Integer?

Is Every Rational Number a Fraction?

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

Rational Numbers in Descending Order

Representation of Rational Numbers on the Number Line

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 Same Denominator

Subtraction of Rational Number with Different Denominator

Subtraction of Rational Numbers

Properties of Subtraction of Rational Numbers

Rational Expressions Involving Addition and Subtraction

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

Reciprocal of a Rational Number

Rational Expressions Involving Division

Properties of Division of Rational Numbers

Rational Numbers between Two Rational Numbers

**8th Grade Math Practice****From Properties of Addition of Rational Numbers to HOME PAGE**

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● **Rational Numbers - Worksheets**

Worksheet on Equivalent Rational Numbers

Worksheet on Lowest form of a Rational Number

Worksheet on Standard form of a Rational Number

Worksheet on Equality of Rational Numbers

Worksheet on Comparison of Rational Numbers

Worksheet on Representation of Rational Number on a Number Line

Worksheet on Adding Rational Numbers

Worksheet on Properties of Addition of Rational Numbers

Worksheet on Subtracting Rational Numbers

Worksheet on Addition and
Subtraction of Rational Number

Worksheet on Rational Expressions Involving Sum and Difference

Worksheet on Multiplication of Rational Number

Worksheet on Properties of Multiplication of Rational Numbers

Worksheet on Division of Rational Numbers

Worksheet on Properties of Division of Rational Numbers

Worksheet on Finding Rational Numbers between Two Rational Numbers

Worksheet on Word Problems on Rational Numbers

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