Class 8 Maths Chapter 1 Notes - Rational Numbers


Introduction to Rational Numbers


A rational number is a number that can be written as a fraction, where, the numerator and denominator are integers and the denominator is non-zero. If the denominator of a rational number is greater than 0 and the common divisor of the numerator and the denominator is 1 only, then the rational number is said to be in its standard form.

A rational number is positive if the numerator and the denominator are of the same sign. A rational number is negative if the numerator and the denominator are of opposite signs.

We can perform various operations on rational numbers. Let us see an example.

Solve:
Introduction to rational number

Let us learn some properties of rational numbers.

Properties of Rational Numbers


1. Closure Property


For two rational numbers, if an operation performed (like addition, subtraction, multiplication), gives a rational number as the result, then the set of rational numbers is closed under that operation. The result of adding, subtracting and multiplying two rational numbers is always a rational number.

Example: Closure-Property-example is a rational number.

2. Commutative Property


If we swap the order of operation for any two rational numbers and the result does not change, then the rational numbers follow commutative property for that operation. Rational numbers follow this property for addition and multiplication, but not for subtraction and division.

Example:
commutative-Property-example

So, rational numbers follow commutative property for addition.

3. Associative Property


If we rearrange a set of rational numbers among two or more same operations and their result does not change, rational numbers follow the associative property for that operation. Rational numbers follow this property for addition and multiplication only.

Example:
associative-Property-example

So, rational numbers follow associative property for multiplication.


You may also need this
NCERT Solutions for class 8 Maths Chapter 1
NCERT Solutions for class 8 Maths Chapter 1 Exercise 1.1
NCERT Solutions for class 8 Maths Chapter 1 Exercise 1.2

4. Distributive Property


When we multiply a sum of variables by a number, we get the same result as when we multiply each variable by the number and then add their products together, that is, if a, b and c are three rational numbers, then a × (b + c) = (a × b) + (a × c).
Similarly, if we multiply a difference of variables by a number, we get the same result as when we multiply each variable by the number and then find the difference between the products, that is, if a, b, and c are three rational numbers, then a × (b − c) = (a × b) − (a × c).

Example:
Distributive-Property-example

5. Identity Property


0 is the additive identity for the rational numbers because adding 0 to a rational number does not change it.
1 is the multiplicative identity for the rational numbers because multiplying a rational number with 1 does not change the result.
Additive-Property (Additive Identity)

Multiplicative-Property (Multiplicative Identity)

6. Inverse Property


The additive inverse is what we add to a number to get 0. For any rational number x/y, the additive inverse is –x/y.
The additive inverse of 3/4 is -3/4.
The multiplicative inverse is what we multiply to a number to get 1. It is the reciprocal of the number.
For any rational number x/y, y/x is the multiplicative inverse.
The multiplicative inverse of 3/4 is 4/3.

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Representation on Number Line


We can represent rational numbers on a number line. A number line is a straight line with three parts: negative side, zero (origin), and positive side. Negative rational numbers lie on the negative side, that is to the left of 0, whereas, positive rational numbers lie on the positive side, that is to the right of 0. Let us see how to represent 3/4 on a number line.
3/4 is greater than 0 and less than 1. So, it will lie in between 0 and 1. Draw a number line and mark 0 and 1. Divide the gap between 0 and 1 into four equal parts and mark the third point from 0 towards the right as 3/4.

Representation of number line

Finding Rational Numbers


There are infinite rational numbers between two rational numbers. We can use the methods given below to find rational numbers between two rational numbers.
Mid-point method:

If a and b are two rational numbers, a rational number between a and b is their mid-point.
Mid-point = mid point method
When denominators are the same:
If the denominators are same, we can easily find the rational numbers between the given rational numbers.
3/7, 4/7, 5/7are rational numbers between 2/7 and 6/7.

When denominators are not the same:

If the denominators are not the same, we have to use the LCM method to find the rational numbers between the given rational numbers. Let us see an example.

Example: Find rational numbers between 3/11 and 5/9.
LCM of 11 and 9 = 99
lcm-of-11-and-9

So, lcm-of-11-and-9are rational numbers in between 3/11 and 5/9.



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