Points to Remember - Rational Numbers Class 8 Notes | EduRev

Class 8 Mathematics by VP Classes

Class 8 : Points to Remember - Rational Numbers Class 8 Notes | EduRev

The document Points to Remember - Rational Numbers Class 8 Notes | EduRev is a part of the Class 8 Course Class 8 Mathematics by VP Classes.
All you need of Class 8 at this link: Class 8

NATURAL NUMBERS
The counting numbers 1, 2, 3, 4, 5, … are called ‘natural numbers’. The smallest natural number is 1, but there is no last (or the greatest) natural number. There are infinitely many natural numbers.

WHOLE NUMBERS
The number ‘0’ together with the natural numbers gives us the numbers 0, 1, 2, 3, 4, ... which are called ‘whole numbers’. Every natural number is a whole number but every whole number is not a natural number (because ‘0’ is not a natural number).

Points to Remember - Rational Numbers Class 8 Notes | EduRevINTEGERS
The whole numbers together with the negatives of counting numbers are known as ‘integers’. Thus, …, –4, –3, –2, –1, 0, 1, 2, 3, 4, … are integers.

Points to Remember - Rational Numbers Class 8 Notes | EduRevNote:

 I. The positive integers are 1, 2, 3, 4, …, i.e. the positive integers are the same as natural numbers.
 II. The negative integers are …, –4, –3, –2, –1.
 III. The greatest negative integer is –1, and the smallest positive integer is 1.
 IV. ‘0’ is neither positive nor negative. It is greater than every negative integer and smaller than every positive integer.

RATIONAL NUMBERS
The numbers which can be expressed as ratio of integers are known as rational numbers. The examples of rational numbers will be 1/4, 2/7, - 3/10, 34/7, etc. 

On solving equations like 3x + 5 = 0, we get the solution as x = -5/3. The solution -5/3 is neither a natural number or whole number or integer.
This leads us to the collection of Rational Numbers. These are the numbers which can be expressed in x/y form; where y ≠ 0.


PROPERTIES OF RATIONAL NUMBERS

(i) Closure Property: When any operation is performed between two or more rational numbers and their result is also a rational number then we say that the rational numbers follow the closure property for that operation. 

Operation

Numbers

Remark

Addition

(a) 5/3 + 3/2 = 19/6 (Rational No);
(b) 7/3 +(– 5/2) = -1/6 (Rational No);
……

We can observe that addition of two rational numbers x and y, i.e. x + y is always a rational number.
Hence, rational numbers are closed under addition.

Subtraction

(a) 5/3 - 3/2 = 1/6 (Rational No);
(b) -7/3 – 5/2 = -29/6 (Rational No);
……

We can observe that subtraction of two rational numbers x and y, i.e. x - y is always a rational number.
Hence, rational numbers are closed under subtraction.

Multiplication

(a) 5/ 3 × 3/2 = 5/2 (Rational No);

(b) -2/7 × 14/5 = -4/5 (Rational No);
……

We can observe that multiplication of two rational numbers x and y, i.e. x × y is always a rational number.
Hence, rational numbers are closed under multiplication.

Division

(a) 5/3 ÷ 3/2 = 10/9 (Rational No);
(b) 12/3 ÷ 0 = ∞ (Not a rational no);

…..

We can observe that division of two rational numbers x and y, i.e. x ÷ y is not always a rational number.
Hence, rational numbers are not closed under division.

(ii) Commutative Property: When two rational numbers are swapped between one operator and still their result does not change then we say that the rational numbers follow the commutative property for that operation. 

Operation

Numbers

Remark

Addition

(a) 5/3 + 3/2 = 19/6;
3/2 + 5/3 = 19/6
Here, both answers are same
(b) 7/3 + (-5/2) = -1/6;
(-5/2) + 7/3 = -1/6
Here, both answers are same
……

We can observe that addition of two rational numbers x and y when inter changed yields the same answer, i.e. x + y = y + x.
Hence, rational numbers are commutative under addition.

Subtraction

(a) 5/3 - 3/2 = 1/6;
3/2 – 5/3 = -1/6
Here, both answers are different(b) 7/3 – 5/2 = -1/6;
5/2 – 7/3 = 1/6
Here, both answers are different
……

We can observe that subtraction of two rational numbers x and y when inter changed does not yield the same answer, i.e. x - y ≠ y - x.
Hence, rational numbers are not commutative under subtraction.

Multiplication

(a) 5/3 × 3/2 = 5/2;
3/2 × 5/3 = 5/2
Here, both answers are same
(b) -2/7 × 14/5 = -4/5;
14/5 × (-2/7) = -4/5
Here, both answers are same
……

We can observe that multiplication of two rational numbers x and y when inter changed yields the same answer, i.e. x × y = y × x.
Hence, rational numbers are commutative under multiplication.

Division

(a) 5/3 ÷ 3/2 = 10/9;
3/2 ÷ 5/3 = 9/10
Here, both answers are different(b) 12/3 ÷ 0 = ∞ ;
0 ÷ 12/3 = 0 ;
Here, both answers are different
…..

We can observe that division of two rational numbers x and y when inter changed does not yield the same answer, i.e. x ÷ y ≠ y ÷ x.
Hence, rational numbers are not commutative under division.

(iii) Associative Property: When rational numbers are rearranged among two or more same operations and still their result does not change then we say that the rational numbers follow the associative property for that operation. 

Operation

Numbers

Remark

Addition

(a) 5/3 + (3/2 + 1/3) = 7/2;
(5/3 + 3/2) + 1/3 = 7/2
Here, both answers are same
(b) 7/3 + (-5/2 + 1/4) = 1/12; (7/3 + -5/2) + 1/4 = 1/12;
Here, both answers are same
……

We can observe that addition of rational numbers x, y, and z in any order yields the same answer, i.e. x + (y + z) = (x + y) + z.
Hence, rational numbers are associative under addition.

Subtraction

(a) 5/3 – (3/2 – 1/3) = 3/2;
(5/3 – 3/2) – 1/3 = -1/2
Here, both answers are different
……

We can observe that subtraction of rational numbers x, y, and z in any order does not yields the same answer, i.e. x - (y - z) ≠ (x - y) - z.
Hence, rational numbers are not associative under subtraction.

Multiplication

(a) 5/ 3 × (3/2 × 2/3) = 5/3;
(5/ 3 × 3/2) × 2/3 = 5/3
Here, both answers are same
(b) -2/7 × (14/5 × 10/2) = -4; (-2/7 × 14/5) × 10/2 = -4
Here, both answers are same
……

We can observe that multiplication of rational numbers x, y, and z in any order yields the same answer, i.e. x × (y × z) = (x × y) × z.
Hence, rational numbers are associative under multiplication.

Division

(a) 5/3 ÷ (3/2 ÷ 1/4) = 5/18;
(5/3 ÷ 3/2) ÷ 1/4 = 40/9
Here, both answers are different
…..

We can observe that division of rational numbers x, y, and z in any order does not yields the same answer, i.e. x ÷ (y ÷ z) ≠ (x ÷ y) ÷ z.
Hence, rational numbers are not associative under division.


THE ROLE OF ZERO (0) AND ONE (1)
(i) The zero is called the identity element for the addition of rational numbers, i.e. the sum of 0 and a rational number is the rational number itself.
Note: Zero is also the additive identity for integers and whole numbers as well. 

(ii) 1 is the multiplicative identity for rational numbers, i.e. the product of 1 and a rational number is the rational number itself.
Note: 1 is the multiplicative identity for integers and whole numbers also.


NEGATIVE OF A NUMBER
The negative of a rational number a/b, is (-a/b) and the negative of (-a/b), is a/b such that
Points to Remember - Rational Numbers Class 8 Notes | EduRev

Example: The negative of (5/8) is  (-5/8) since, (5/8) + (-5/8) = 0

Note: The negative of a number is also called its ‘additive inverse’.

MULTIPLICATIVE INVERSE
If the product of two rational numbers is 1, then they are called multiplicative inverse of each other.

For example: 

Points to Remember - Rational Numbers Class 8 Notes | EduRev

Therefore, 5/3 in the multiplicative inverse of  3/5 and 3/5 is the multiplicative inverse of 5/3
Note:
I. The multiplicative inverse of a number is also called the reciprocal of that number.
II. There is no rational number which when multiplied by 0 gives 1.
III. Zero has no reciprocal


DISTRIBUTIVITY OF MULTIPLICATION OVER ADDITION AND SUBTRACTION
If the rational numbers a, b, and c obey property of a × (b + c) = ab + ac, then it is said to follow Distributive property of multiplication over addition.
Example:
1/3 × (2/3 + 1/4) = 1/3 × 11/12 = 11/36…………………..(i)
(1/3 × 2/3) + (1/3 × 1/4) = 2/9 + 1/12 = 11/36…………(ii)
Here, answer for both the equations (i) and (ii) are same.
Hence, rational numbers follow distributive property of multiplication over addition.

Points to Remember:

A number of the form p/q , where ‘p’ and ‘q’ are any integers and q ≠ 0 is called a rational number.

  • Rational numbers are closed under the operations of addition, subtraction and multiplication.
  • Rational numbers are commutative under addition and multiplication.
  • Rational numbers are associative under addition and multiplication.
  • The rational number ‘0’ is the additive identity for rational numbers i.e. x/y + 0 = x/y. 
  • The rational number 1 is the multiplicative identity for rational numbers  i.e. x/y × 1 = x/y. 
  • For all rational numbers, a, b and c, we have:
     a(b + c) = ab + ac
     a(b – c) = ab – ac
Offer running on EduRev: Apply code STAYHOME200 to get INR 200 off on our premium plan EduRev Infinity!

Related Searches

Viva Questions

,

Previous Year Questions with Solutions

,

Points to Remember - Rational Numbers Class 8 Notes | EduRev

,

MCQs

,

past year papers

,

study material

,

video lectures

,

Important questions

,

Sample Paper

,

mock tests for examination

,

Points to Remember - Rational Numbers Class 8 Notes | EduRev

,

Objective type Questions

,

Extra Questions

,

shortcuts and tricks

,

Exam

,

practice quizzes

,

Summary

,

ppt

,

Points to Remember - Rational Numbers Class 8 Notes | EduRev

,

Semester Notes

,

pdf

,

Free

;