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# Class-8 Rational Numbers

Introduction to Rational Number

Facts of Rational Numbers

Subtraction of Rational Number

Subtraction Properties of Rational Number

Multiplication of Rational Numbers

Multiplication Properties of Rational Number

Division of Rational Number

Division Properties of Rational Number

Rational Number Between any Two Given Rational Numbers

Rational Numbers Test

Rational Number Worksheet

## Introduction to Rational Numbers

In class 7, we learnt following things about Rational numbers.

1. Define rational number
2. Number line representation of rational numbers
3. Comparison of rational numbers
4. Operation like addition, subtraction, multiplication, and division
Now, in class 8 we will study the recap of all operation, properties of all operation and find rational number between any two given rational numbers.

## Facts of Rational Numbers

Numbers in form of ab where 'a' and 'b' are integer. Here, 'b' is not equal to zero, known as rational numbers.

some examples are 59, -611, 9, -3, 0, 3-11 etc.

Every integer is a rational number, but every rational number is not an integer. It is because, every integer 'i' can be written as i1 which is in fraction form. For example, -38 is a rational number but not an integer.

Every rational number is not a fraction, but every fraction is a rational number. For example, -59 is a rational number but not a fraction.

The rational number -ab is same as −ab.

Every rational number should be written with positive integer as denominator.

Two rational numbers ab = cd, if a × d = b × c.

Rational number is positive if both numerator and denominator are positive integers, or both are negative integers. If one is positive integer and other is negative integer then rational number is said to be negative.

All finite decimal numbers are rational numbers. Let's see some examples

0.5 = 12, 0.26 = 1350 and 0.625 = 58.

When we add two rational numbers, first convert each rational number with positive denominator.

Rational Numbers with same denominator

If ab and cd are two rational numbers, then (ab) + (cb) = (a+c)b.

Example 1. Add 69 and 8-9.

Solution. 69 + 8-9

Here, first convert 8-9 into positive denominator.

8-9 = {8×(-1)}{-9×(-1)}

= -89

Then, 69 + (-89) = 6989 = (6-8)9 = -29.

Rational Numbers With Different Denominator

Here we first find LCM of denominators of two rational numbers. Then we covert the denominator of each rational number to have same value as LCM. At last, add the rational numbers following the process of same denominator.

Example 1. Add -57 + 73.

Solution.LCM of 7 and 3 = 21

-57 = (-5×3)(7×3) = -1521

73 = (7×7)(3×7) = 4921

Then, -1521 + 4921 = (-15+49)21 = 3421.

## Addition Properties of Rational numbers

1. Closure property
2. Commutative
3. Associative
4. Special properties

### Closure Property

When two rational numbers are added, then sum of two rational numbers always a rational number. Here, if we take ab and cd are two rational numbers, then (ab + cd) is a rational number. Let's see some example

Example 1. Add 12 and 25.

Solution. 12 + 25 = (5+4)10 = 910

So, 910 is a rational number.

Example 2. Add -59 and -23

Solution. -59 + -23 = {-5+(-6)}9 = -119

Example 3. Add -65 and 14.

Solution. -65 + 14 = (-6×4+1×5)20 = (-24+5)20 = -1920

### Commutative

Addition of rational numbers are commutative that is ab + cd = cd + ab, here, b and d are not equal to 0. Let's see some examples.

Example 1. -57 + 35 = 35 + -57

Solution. We have to prove LHS = RHS

LHS = -57 + 35 = (-25+21)35 = -435

RHS = 35 + -57 = {21+(-25)}35 = -435

LHS = RHS. So, it is following commutative properties.

### Associative

Addition of rational number is associative, that is (ab + cd) + ef = ab + (cd + ef). Here b, d, f are not equal to 0. Let's see some examples.

Example 1. (12 + 23) + -35 = 12 + (23 + -35).

Solution. We have to prove LHS = RHS

LHS = (12 + 23) + -35 = (3+4)6 + (-35) = 76 - 35 = (35−18)30 = 1730

RHS = 12 + (23 + -35) = 12 + {10+(-9)}15 = 12 + 115 = (15+2)30 = 1730

LHS = RHS, hence it is proved that it follows associative property.

### Special Properties of Rational Numbers

The number zero is known as additive identity of rational number, that is, ab + 0 = ab = 0 + ab.

Additive inverse of rational number can be calculated by changing its sign, that is additive inverse of ab is (-ab) and additive inverse of -ab is ab. Let's see some examples.

Example 1. Additive inverse of -35 = −(-35) = 35.

Example 2. Additive inverse of -4-9 = −(-4-9) = -49.

## Subtraction of Rational Number

Subtraction of rational numbers can be done by adding additive inverse of subtrahend with minuend. That is, abcd = ab + (-cd).here -cd is additive inverse of cd.

Example 1. Subtract 59 from -37.

Solution. -3759

= -37 + additive inverse of 59

= -37 + (-59)

= (-3×9−5×7)63

= {-27+(-35}63

= -6263

Example 2. subtract -325 from 315.

Solution. 315 − (-325)

= 165 + additive inverse of (-325)

= 165 + 325

= (80+3)25

= 8325

= 3825

## Subtraction Properties of Rational Number

1. Closure Property
2. Not Commutative
3. Not Associative

### Closure Property

Like addition, difference of two rational number is rational number. Let's see some examples.

Example 1. 6934 = (24-27)36 = -336 = -112
-112 is a rational number.

Example 2. -7943 = (-7-12)9 = -199 = −219
−219 is a rational number.

### Not Commutative

If ab and cd are two rational number, then, (abcd ) is not equal to (cdab).
Let's see some example.

Example 1. Prove 13565613.

Solution. LHS = 1356 = (2-5)6 = -12

RHS = 5613 = (5-2)6 = 12

LHS ≠ RHS i.e. -1212

Hence it is proved that 13565613.

### Not Associative

If ab, cd and ef are three rational numbers, then (abcd) − efab − (cdef).
Let's see some example.

Example 1. prove (-3527) − (-15) ≠ -35 − (27-15).

Solution. LHS = (-3527) − (-15)

= (-21-10)35 + 15

= (-31)35 + 15

= (-31+7)35 = -2435

RHS = -35 − (27-15)

= -35 − (27 + 15)

= -35(10+7)35

= -351735

= (-21-17)35 = -3835

-2435-3835 i.e. LHS ≠ RHS

Hence, it is proved that subtraction does not follow associative property.

## Multiplication of Rational Numbers

To get the product of two rational number, numerators are multiplied, and denominators are multiplied. Let's see some examples.

Example 1. Multiply -37 and 89

Solution. -37 × 89

= (-3×8)7×9

= -821

Example 2. Multiply 317 and -434

Solution. 317 × (-434)

= 227 × (-194)

= {22×(-19)}7×4

= -41828

= -142628

## Multiplication Properties of Rational Number

1. Closure Property
2. Commutative Property
3. Associative Property
4. Distributive Properties
5. Multiplicative Identity
6. Multiplicative Inverse

### Closure Property

Product of two rational number always a rational number. If ab and cd are two rational number ,then ab × cd is also rational numbers. Let's see some examples.

Example 1. 25 × 79 = 2×75×9 = 1445
1445 is a rational number.

Example 2. 34 × 16 = 3×14×6 = 324 = 18
18 is a rational number.

Example 3. 47 × 58 = 4×57×8 = 514
514 is a rational number.

### Commutative Property

If ab and cd are two rational numbers, then ab × cd = cd × ab. Let's see some examples.

Example 1. Prove 23 × (-45) = (-45) × 23.

Solution. LHS = 23 × (-45) = {2×(-4)}(3×5) = -815

RHS = -45 × 23 = (-4×2)5×3 = -815

LHS = RHS i.e. 23 × (-45) = (-45) × 23

Hence, it is proved that multiplication of rational number is commutative.

### Associative Property

Like addition, multiplication have associative property. If ab, cd and ef are three rational numbers, then (ab × cd) × ef = ab × (cd × ef). Let's see some example.

Example 1. Prove (57 × -32) × -17 = 57 × (-32 × -17)

Solution. LHS = (57 × -32) × -17 = -1514 × -17 = 1598

RHS = 57 × (-32 × -17) = 57 × 314 = 1598

LHS = RHS i.e. (57 × -32) × -17 = 57 × (-32 × -17)

Hence, multiplication of fraction is associative.

### Distributive Property

When two rational numbers are added or subtracted and multiply with another rational number, then it should follow distributive laws.

If ab, cd and ef are three rational numbers, then ab × (cd + ef) = ab × cd + ab × ef and

ab × (cdef ) = ab × cdab × ef.

### Multiplicative Identity

The number 1 is multiplicative identity means ab × 1 = ab = 1 × ab.

### Multiplicative Inverse

When the numerator and denominator of non-zero rational number are interchange called multiplicative inverse or reciprocal. Multiplicative inverse of ab is ba.

product of rational number with its reciprocal always 1.

Zero has no multiplicative inverse.

Example 1. Find multiplicative invers of 79 × 107.

Solution. First, we multiply 79 and 107, then convert into its reciprocal

79 × 107

= (7×10)(9×7)

= 7063 = 109

Multiplicative invers of 109 = 910

## Division of Rational Number

When two rational numbers are divided, first we find out reciprocal of the divisor, then it is multiplied with the dividend. In other words, if ab and cd are rational numbers ,then ab ÷ cd = ab × dc (reciprocal of cd) = a×db×c.
Let's see some examples.

Example 1. 215 ÷ (-13).

Solution. 215 ÷ (-13)

= 115 ÷ (-13)

Reciprocal of -13 is equal to -3.

= 115 × (-3)

= -335

Example 2. Divide the sum of 17 and 12 by the product of 92 and 37

Solution. (17 + 12) ÷ (92 × 37)

= (2+7)14 ÷ (9×3)(2×7)

= 914 ÷ 2714

Reciprocal of 2714 is 1427

= 914 × 1427

= (9×14)(14×27)

= 927

= 13

## Division Properties of Rational Number

1. Closure Property
2. Division is Non-commutative
3. Division is Non-associative

### Closure Property

When two non-zero rational numbers are divided their result remain rational number. If ab and cd are non-zero rational number, then ab ÷ cd is rational number. If we do not consider 0 as a rational number, then rational number closed under division.

### Division is Non-commutative

If ab and cd are on-zero rational number, ab ÷ cdcd ÷ ab

### Division is Non-associative

If ab, cd and ef are non-zero rational numbers, ab ÷ (cd ÷ ef) ≠ (ab ÷ cd) ÷ ef.

## Rational Number Between any Two Given Rational Numbers

In order to know rational number between two rational numbers, we have to calculate the mean of two given rational numbers. If ab and cd are two rational numbers. Then mean of ab and cd is written as (ab+cd)2.

Example 1. Find a rational number between -45 and 56.

Solution. First, we calculate mean of (-45 + 56) ÷ 2

= (-24+25)30 ÷ 2

= 130 × 12 = 160

So, 160 is a rational number between -45 and 56.

## Class-8 Rational Numbers Test

Rational Numbers - 1

Rational Numbers - 2

Rational Numbers - 3

## Class-8 Rational Numbers Worksheet

Rational Numbers Worksheet - 1

Rational Numbers Worksheet - 2

Rational Numbers Worksheet - 3

Rational Numbers Worksheet - 4