In this “Rational Number Class 8 Notes”, you will learn the following topics.

- Rational number
- Properties of rational numbers: Closure property, Commutative Property, Associative Property, and Distributive Property.
- Additive identity and multiplicative identity
- Additive inverse and multiplicative inverse
- Rational numbers between two rational number.
- word problem

## What is a rational number?

The numbers which can be written in form of p/q , where p and q are integers and q≠0 are called rational numbers.

Example: 0/1,3/3,-8/-2……etc are Rational Numbers.

Now ,you may think why q ≠0?

It is because if the value of q will zero i.e q=0 then, P/q will become P/0 =∞ which is not defined

That is why the value of q cannot be zero but p can be zero because 0/q =0 which is defined.

Notes:

- Zero is a rational number.
- Every integer is a rational number.
- Natural number, whole number, integers, and decimals numbers are not in form of p/q but can be written in the form p/q and this is why these numbers are also rational numbers.
- Every fraction is a rational number.

### Equivalent rational numbers

If p/q is any rational number and m is a non-integer(cannot be zero)** ,then**

\frac{p}{q}=\frac{p×m}{q×m} is called an Equivalent Rational Number.

For Example: If 5/6 is a rational number and 3 is an integer, then

\frac{5×3}{6×3}=\frac{15}{18} is an Equivalent Rational number.

We can form a lot of equivalent rational numbers from a single rational number. In given example 15/18 is an equivalent rational number which formed due to multiplication by 3. Like this, we can form more numbers by multiplication of 3,4,5 etc.

#### The Standard Form of a Rational number

A rational number p/q is said to be in standard form if p and q have no common divisor other than 1. In simple language, the Simplest Form of any rational number*(equivalent rational number)* is called the *Standard Form of a Rational number. *Let us understand it with an example.

Example: We have 15/18 as an equivalent rational number when we divide by 3 in the numerator and denominator it becomes 5/6 which is the simplest form of 15/18. So 5/6 is the standard form of the rational number 15/18

Another way to convert it ,is through** “HCF of 15 and 18”.** The HCF of 15 and 18 is 3, when we divide Numerator and Denominator by 3, we will get the same result.

#### 1. Closure property

We have 5/6 and 3/5 as a rational number

**Addition**

**Subtraction**

**Multiplication**

**Division**

**Rational number are closed under addition, subtraction, multiplication, and division (excluding division by zero)**

#### 2.C**ommulative property**

**Addition**

1/2+4/5=13/10

4/5 +1/2=13/10

so, 1/2+4/5=4/5 +1/2

**Subtraction**

Is ,5/7-3/7=3/7-5/7

you will observe that 5/7-3/7≠3/7-5/7

**Multiplication**

3/8×5/7=15/56

5/7×3/8=15/56

3/8×5/7=5/7×3/8

**Division**

Is 1/2÷-3/4=-3/4÷1/2?

you will observe that 1/2÷-3/4≠-3/4÷1/2

So, Commutative property holds for Addition and Multiplication of Rational numbers only . This property does not hold for subtraction and division of rational numbers.

#### 3.Associative property

**Addition**

1/2+(-4/5+2/3)=11/30 [using **a+(b+c)=(a+b)+c]**

(1/2+-4/5)+2/3=11/30

1/2+(-4/5+2/3)=(1/2+-4/5)+2/3

**Subtraction**

Is -1/3-(6/5-2/3)=(-1/3-6/5)-2/3) ? [using **a-(b-c)=(a-b)-c]**

you will observe that -1/3-(6/5-2/3)=(-1/3-6/5)-2/3)

**Multiplication**

(5/3×-3/4)×1/2=-15/56 [Using **a×(b×c)=(a×b)×c]**

5/7×(-3/4×1/2) =-15/56

Hence,(5/3×-3/4)×1/2=5/7×(-3/4×1/2)

**Division ** [Using **a÷(b÷c)=(a÷b)÷c]**

Is 4/5÷(6/5÷3/2)=(-4/5÷6/5)÷3/2?

You will observe that 4/5÷(6/5÷3/2)=(-4/5÷6/5)÷3/2?

**Note: **Associative property is also called Re-arrangement property

**4. Distributive property of multiplication over addition for rationals number**

4/5×(2/3+7/15)=4/5×(10+7/15) =4/5×17/15=68/75 [Using :a×(b+c)=a×b+b×c]

4/5×2/3+4/5×7/15=68/75

Thus,4/5×(2/3+7/15)=4/5×2/3+4/5×7/15

**Additive identity and multiplicative identity**

Additive identity: **The adding result which is true for all numbers is called additive identity.** S**o 0 is an only additive identity** because when we add 0 to any number the result will always the same as that number.

e.g 5+0=5 , -3+0=-3

Multiplicative identity: The multiplication result which is true for all numbers is called multiplicative identity. Here *1 is a multiplicative identity* because when we multiply any number by 1 the result will be the same number.

e.g 5×1=5 ,-3×1=-3

**Additive inverse and multiplicative inverse**

Additive inverse: Additive inverse of any number is the opposite number which when add to the number gives 0 as a result.

e.g The additive inverse of 5 is -5.so 5+(-5)=0 result is zero.

The additive inverse of -5 is 5.so -5+(5)=0 result is zero.

Multiplicative inverse: Multiplicative inverse of any number is the number which when multiplied to the original number gives 1 as a result.

e.g The multiplicative inverse of 2/3 is 3/2.2/3×3/2=1 as a result

**Notes**

- The additive inverse of zero is zero.
- 1 and-1 are only rational numbers which are their own reciprocals.

** Rational number between two number**

There are countless rational numbers between two rational number. It is not possible to find all the rational numbers but we can find how many we want. There are two methods to find a rational number between two numbers. Let us understand them one-by-one.

Q.1 Find three rational numbers lying between 1/3 and 1/2.

Solution:

First method

Step.1 Take the Lcm of both denominators i.e Lcm of 3 and 2 is 6.

Step.2 Make the denominator of fractions 1/3 and 1/2 equal by multiplying 2 and 3 respectively.

Step.3 After multiplication, the fraction will 2/6 and 3/6.

Step.4 Multiply by 10 in both fraction’s numerator and denominator, it become 20/60 and 30/60.

Step.5 Now we can easily find ten rational numbers between these i.e 21/60,22/60,23/60,24/60………….29/60.

Step.6 We can choose any three rational numbers among these. If we want find more than 10 rational number then, again multiply by 10.

**Second method **

This method also known as the “Arithmetic Mean Method”. We find the arithmetic mean of two given rational numbers to find the mid-value of the rational numbers.

Let understand it with the help of an Example :

We have \frac{1}{3} and \frac{1}{2} are two rational number .The mid-value of these numbers will be \frac{\frac{1}{3} +\frac{1}{2}}{2}=\frac{5}{12}.Similarly ,You can find more arithmetic mean of numbers such as \frac{5}{12} and \frac{1}{2}. But I will recommend you to solve this type of problem with the help of first method . First Method is easy to use and less time consuming .

### Rational Number Word problem with Solution

Problem:01 The product of two rational numbers is \frac{-56}{25}.If one number is \frac{-8}{15},Find the other

Solution:

Product of two rational numbers=\frac{-56}{25}

One number is given=\frac{-8}{15}

Let the other number will=x

A/Q

\frac{-8}{15}×x=\frac{-56}{25}

x=\frac{21}{5}

Problem:02 Using distributive property, evaluate \frac{-5}{3}×\frac{5}{7}-\frac{4}{7}×\frac{5}{3}

Solution:

\frac{-5}{3}×\frac{5}{7}-\frac{4}{7}×\frac{5}{3}

\frac{-5}{3}×(\frac{5}{7}+\frac{4}{7})

\frac{-5}{3}×\frac{9}{7}

=\frac{-15}{7} is the answer

### Check these stuff as well :

- Square and square root class 8 Notes ,Worksheet PDF
- Linear Equation in One Variable class 8 Notes
- Linear Equation in One Variable worksheet [Answer Key Included]
- Practice test : Linear Equation in One Variable