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. 

Rational Numbers

Introduction of 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

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

Addition of Rational Numbers

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

Product of Rational Numbers

Properties of Multiplication of Rational Numbers

Rational Expressions Involving Addition, Subtraction and Multiplication

Reciprocal of a Rational  Number

Division of Rational Numbers

Rational Expressions Involving Division

Properties of Division of Rational Numbers

Rational Numbers between Two Rational Numbers

To Find Rational Numbers





8th Grade Math Practice 

From Properties of Addition of Rational Numbers to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.




Share this page: What’s this?

Recent Articles

  1. Worksheet on Addition of Mass | Word problems on Addition of Mass

    Nov 13, 24 10:17 AM

    Practice the third grade math worksheet on addition of mass/weight. This sheet provides different types of questions where you need to arrange the values of mass under different columns

    Read More

  2. Worksheet on Addition of Length | Word Problems on Addition of Length

    Nov 13, 24 09:23 AM

    Practice the third grade math worksheet on addition of length. This sheet provides different types of questions where you need to arrange the values of length under different columns to find their sum

    Read More

  3. Addition of Mass |Add the Different Units of Mass |Worked-out Examples

    Nov 12, 24 01:36 PM

    Adding Weight
    In addition of mass we will learn how to add the different units of mass or weight together. While adding we need to follow that the units of mass i.e., kilogram and gram are converted into grams

    Read More

  4. Measuring Mass | Addition and Subtraction of Mass | Measure of Mass

    Nov 12, 24 12:07 PM

    Standard Units to Measure Weight
    We will discuss about measuring mass. We know the vegetable seller is weighing potatoes in kilogram. The goldsmith is weighing a ring in grams. The wheat bags are weighing in quintals.

    Read More

  5. Subtraction of Length | Learn How the Values of Length are Arranged

    Nov 11, 24 02:08 PM

    Subtraction of Length
    The process of subtraction of units of length is exactly similar to that of subtraction of ordinary numbers. Learn how the values of length are arranged in different columns for the subtraction of len…

    Read More

Rational Numbers - Worksheets

Worksheet on Rational Numbers

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

Worksheet on Operations on Rational Expressions

Objective Questions on Rational Numbers