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
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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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