Rational Number

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Introduction To Rational Numbers

In the chapter Rational Numbers for Class 8, we will learn about rational numbers, their properties, how to represent rational numbers on a number line, and also to find rational numbers between any two rational numbers with the help of examples.

Whole Numbers and Natural Numbers

Natural Numbers are set of numbers starting from 1 counting up to infinity. The set of natural numbers is denoted as 'N'. Set of Natural Numbers (N) = {1,2,3,...}

Whole Numbers are set of numbers starting from 0 and going up to infinity. So basically they are natural numbers with the zero added to the set. The set of whole numbers is denoted as 'W'. Whole Numbers (W) = {0,1,2,3,...}

Properties of Natural Numbers and Whole Numbers

Closure Property Closure property is applicable for whole numbers in the case of addition and multiplication while it isn't in the case of subtraction and division. This applies to natural numbers as well.

Commutative Property Commutative property applies for whole numbers and natural numbers in the case of addition and multiplication but not in the case of subtraction and division.

Associative Property Associative property applies to whole numbers and natural numbers in the case of addition and multiplication but not in the case of subtraction and division.

Integers

In simple terms Integers are natural numbers and their negatives .

The set of Integers is denoted as 'Z' or 'I'. Set of Integer Numbers (Z) = {...,-3, -2, -1, 0, 1, 2, 3,...}

Properties of Integers

Closure Property: Closure Property applies to integers in the case of addition, subtraction and multiplication but not division.

Commutative Property: Commutative property applies to integers in the case of addition and multiplication but not subtraction and division.

Associative Property: Associative property applies to integers in the case of addition and multiplication but not subtraction and division.

Rational Numbers

A rational number is a number that can be represented as a fraction of two integers in the form of p q , where q must be non-zero. The set of rational numbers is denoted as Q.

For example, - 5 7 is a rational number where -5 and 7 are integers. Even 2 is a rational number since it can be written as 2 1 where 2 and 1 are integers.

Properties of Rational Numbers

Closure Property of Rational Numbers

For any two rational numbers a and b, a∗b=c∈Q i.e. For two rational numbers say a and b the results of addition, subtraction and multiplication operations give a rational number. Since the sum of two numbers ends up being a rational number, we can say that the closure property applies to rational numbers in the case of addition .

For example: The sum of 2 3 + 3 4 = (8+9) 12 = 17 12 is also a rational number where 17 and 12 are integers. The difference between two rational numbers results in a rational number. Therefore, the closure property applies for rational numbers in the case of subtraction .

For example: The difference between 4 5 - 3 4 = (16-15) 20 = 1 20 is also a rational number where 1 and 20 are integers. The multiplication of two rational numbers results in a rational number. Therefore we can say that the closure property applies to rational numbers in the case of multiplication as well.

For example: The product of 1 2 × (- 4 5 ) = (- 4 10 ) = (- 2 5 ) which is also a rational number where -2 and 5 are integers. In the case with the division of two rational numbers, we see that for a rational number a, a0 is not defined. Hence we can say that the closure property does not apply for rational numbers in the case of division .

Commutative Property of Rational Numbers

For any two rational numbers a and b, a∗b = b∗a. i.e., Commutative property is one where in the result of an equation must remain the same despite the change in the order of operands . Given two rational numbers a and b, (a+b) is always going to be equal to (b+a). Therefore addition is commutative for rational numbers.

For example:
2 3 + 4 3 = 4 3 + 2 3
⇒ 6 3 = 6 3
⇒ 2 = 2

Considering the difference between two rational numbers a and b, (a−b) is never the same as (b−a). Therefore, subtraction is not commutative for rational numbers.

For example:
2 3 - 4 3 = - 2 3 but
4 3 - 2 3 = 2 3

When we consider the product of two rational numbers a and b, (a×b) is the same as (b×a). Therefore, multiplication is commutative for rational numbers.

For example:
2 3 × 4 3 = 8 9 and
4 3 × 2 3 = 8 9

Considering the division of two numbers a and b, (ab) is different from (ba). Therefore division is not commutative for rational numbers.

For example:
2 3 is definitely different from 3 2 .

Associative Property of Rational Numbers

For any three rational numbers a,b and c, (a∗b)∗c=a∗(b∗c). i.e., Associative property is one where the result of an equation must remain the same despite a change in the order of operators.

Given three rational numbers a,b and c, it can be said that:

(a+b)+c = a+(b+c).
Therefore addition is associative .

(a−b)−c≠a−(b−c).
Because (a-b)-c = a-b-c whereas a-(b-c) = a-b+c. Therefore we can say that subtraction is not associative .

(a×b)×c=a×(b×c).
Therefore multiplication is associative .

ab ÷ c ≠ a ÷ bc
Therefore division is not associative.

Example:
1 2 + ( 1 4 + 2 3 ) = ( 1 2 + 1 4 ) + 2 3
⇒ 17 12 = 17 12

Also,
1 2 x ( 1 4 x 2 3 ) = ( 1 2 x 1 4 ) x 2 3
⇒ 2 24 = 2 24
⇒ 1 12 = 1 12

Distributive Property of Rational Numbers

Given three rational numbers a,b and c, the distributivity of multiplication over addition and subtraction is respectively given as:
a(b + c) = ab + ac
a(b − c) = ab − ac

Negatives and Reciprocals

Negation of a Number

For a rational number, a b , a b +0 = a b , i.e., when zero is added to any rational number the result is the same rational number. Here '0' is known as additive identity for rational numbers.

If ( a b ) + (- a b ) = (- a b ) + ( a b ) = 0, then it can be said that the additive inverse or negative of a rational number a b is - a b . Also, (- a b ) is the additive inverse or negative of a b .
For example: The additive inverse of - 21 8 is -(- 21 8 ) = 21 8

Reciprocal of a Number

For any rational number a b , a b × 1 = a b , i.e., When any rational number is multiplied by '1' , the result is the same rational number. Therefore '1' is called multiplicative identity for rational numbers.

If a b × c d = 1, then it can be said that the c d is reciprocal or the multiplicative inverse of a rational number a b . Also, a b is reciprocal or the multiplicative inverse of a rational number c d .

For example: The reciprocal of 2 3 is 3 2 as 2 3 × 3 2 = 1.

Representing on a Number Line

Representation of Rational Numbers on the Number Line

In order to represent a given rational number a n , where a and n are integers, on the number line:

Step 1 : Divide the distance between two consecutive integers into 'n' parts.

For example: If we are given a rational number 5 3 , we divide the space between 0 and 1, 1 and 2 into three parts.

Step 2: Label the rational numbers till the range includes the number you need to mark

-2

-6

-5

-4

-1

-3

-2

-1

Rational Numbers between Two Rational Numbers

The number of rational numbers between any two given rational numbers aren't definite , unlike that of whole numbers and natural numbers.

For example: Between natural numbers 2 and 10 there are exactly 7 numbers but between 2 10 and 8 10 there are infinite numbers that could exist. Method 1 Given two rational numbers, ensure both of them have the same denominators . Once there is a common denominator, we can pick out any rational number that lies in between.

Example: Find the rational numbers between 1 2 and 2 3 .
LCM of denominators (2 and 3) = 6
1 2 × 3 3 = 3 6
2 3 × 2 2 = 4 6

We cannot find any number between 3 and 4. So will multiply each rational number 3 6 and 4 6 by 3 3 .
3 6 × 3 3 = 9 18
4 6 × 3 3 = 12 18

Hence, few rational numbers between 1 2 and 2 3 are 10 18 and 11 18 .

Method 2 Given two rational numbers, we can always find a rational number between them by calculating their mean or midpoint .

Example: Find the rational numbers between 3 and 4.
Mean of 3 and 4 = (3+4) 2 = 7 2
Mean of 3(same as 6 2 ) and 7 2 = 13 4
Mean of 7 2 and 4(same as 8 2 ) = 15 4

Hence, the three rational numbers between 3 and 4 are 7 2 , 13 4 and 15 4 .

MCQs

1. Which of the following is a rational number?

(a) √2

(b) π

(d) 0.333…

► (c) 34

2. The rational number −58 is:

(a) Positive

(b) Negative

(d) Natural number

► (b) Negative

3. Which of the following is equal to 46?

(a) 23

(b) 34

(d) 12

► (a) 23

4. The reciprocal of −37 is:

(a) 37

(b) −73

(d) −37

► (b) −73

5. Rational numbers are:

(a) Only fractions

(b) Only integers

(d) Only decimals

► (c) Numbers expressed as p/q where q ≠ 0

6. Add: 13 + 25

(a) 78

(b) 1115

(d) 28

► (c) 1315

7. Which rational number lies between 14 and 12?

(a) 15

(b) 38

(d) 45

► (b) 38

8. Multiply: (−23) × (34)

(a) −612

(b) −12

(d) −37

► (b) −12

9. Which is the smallest rational number?

(a) −25

(b) −14

(d) −56

► (d) −56

10. Subtract: 78 − 34

(a) 18

(b) 28

(d) 14

► (a) 18

11. Rational numbers are closed under:

(a) Addition

(b) Subtraction

(d) All of the above

► (d) All of the above

12. Which of these is NOT a rational number?

(a) 0

(b) 2.5

(d) −35

► (c) 40

13. Which is equivalent to 0.75?

(a) 34

(b) 13

(d) 58

► (a) 34

14. Which is greater: −37 or −14?

(a) −37

(b) −14

(d) Cannot be determined

► (b) −14

15. Divide: (23) ÷ (45)

(a) 1012

(b) 815

(d) 158

► (c) 56

16. 0 can be expressed as a rational number?

(a) Yes, 01

(b) No

(d) None of these

► (a) Yes, 01

17. The additive inverse of 58 is:

(a) −58

(b) 58

(d) −85

► (a) −58

18. Which of the following is in standard form?

(a) 6−12

(b) −37

(d) −1836

► (b) −37