## 1. Properties of rational number

- Books Name
- Sagar Daksh Mathematics Book

- Publication
- Carrier Point

- Course
- CBSE Class 8

- Subject
- Mathmatics

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Mathematics > Chapter 1 Rational Numbers > Introduction

Notes by Teacher Account : sudhanshu@kaysosneducation.co.in

## 1. Properties of rational number

- Books Name
- Mr. Sudhanshu Sharma Mathematics Book

- Publication
- Success Academy

- Course
- CBSE Class 8

- Subject
- Mathmatics

Alpha beta classes

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## 1. Properties of rational number

- Books Name
- class 8 th Mathematics Book

- Publication
- ReginaTagebücher

- Course
- CBSE Class 8

- Subject
- Mathmatics

**Rational Numbers**

**Properties of rational number**

**Introduction to rational numbers**,

**Rational Numbers **

A number is called Rational if it can be expressed in the form p/ q where p and q are integers (q> 0). It includes all natural, whole number, and integers.

Case1/2, 4/3, 5/7, 1 etc.

**Natural Numbers - **All the positive integers from 1, 2, 3,, ∞.

**Whole Numbers - **All the natural numbers including zero are called Whole Numbers.

**Integers - **All negative and positive numbers including zero are called Integers.

**Closure property, **

**Closure-** Rational numbers are closed under addition, subtraction and multiplication. For eg.- If p and q are any two rational numbers, then and the sum, difference and product of these rational numbers is also a rational number. This is known as the closure law

**Commutative property, **

**Commutativity-** Rational numbers are commutative under addition and multiplication. If p and q are two rational numbers, then:

Commutative law under addition says- p + q = q + p.

Commutative law under multiplication says p x q = q x p.

Note- Rational numbers, integers and whole numbers are commutative under addition and multiplication. Rational numbers, integers and whole numbers are non commutative under subtraction and division.

**Associativity property, **

**▪ **Associativity- Rational numbers are associative under addition and multiplication. If a, b, c are rational numbers, then:

Associative property under addition: p + (q + r) = (p + q) + r

Associative property under multiplication: p(qr) = (pq)r

**The role of 0 and 1**

**Role of zero and one-** 0 is the additive identity for rational numbers. 1 is the multiplicative identity for rational numbers.

Zero is the additive identity for whole numbers, integers and rational numbers.

**▪ Multiplicative inverse**- If the product of two rational numbers is 1, then they are called multiplicative inverse of each other.

Eg. 4/9 * 9/4 = 1

## 2. Negative numbers

- Books Name
- Sagar Daksh Mathematics Book

- Publication
- Carrier Point

- Course
- CBSE Class 8

- Subject
- Mathmatics

sudhanshusharma259438@gmail.com

Accountancy point.

## 2. Negative numbers

- Books Name
- class 8 th Mathematics Book

- Publication
- ReginaTagebücher

- Course
- CBSE Class 8

- Subject
- Mathmatics

**Negative numbers**

**Negative of a numbers, **

**Reciprocal of rational numbers,**

The multiplicative inverse of any rational number

**Example**

The reciprocal of 4/5 is 5/4.

**Distributivity of multiplication over addition for rational numbers,**

This shows that for all rational numbers p, q and r

1. p(q + r) = pq + pr

2. p(q – r) = pq – pr

**Example**

Check the distributive property of the three rational numbers 4/7,-( 2)/3 and 1/2.

**Solution**

Let’s find the value of

This shows that

**Distributivity of multiplication over subtraction for rational numbers**

The distributive property of rational numbers states that if any expression with three rational numbers A, B, and C is given in form A (B + C), then it can be solved as A × (B + C) = AB + AC. This applies to subtraction also which means A (B - C) = AB - AC. This means operand A is distributed between the other two operands, i.e., B and C. This property is also known as the distributive property of multiplication over addition or subtraction.

## 3. Rational numbers in number line and between two numbers

- Books Name
- class 8 th Mathematics Book

- Publication
- ReginaTagebücher

- Course
- CBSE Class 8

- Subject
- Mathmatics

**Rational numbers in number line and between two numbers**

**Representation of rational numbers on the number line,**

On the number line, we can represent the Natural numbers, whole numbers and integers as follows

**Rational Numbers can be represented as follows**

**Rational numbers between two rational numbers with same denominator,**

There could be n number of rational numbers between two rational numbers. There are two methods to find rational numbers between two rational numbers.

**Method 1 **

We have to find the equivalent fraction of the given rational numbers and write the rational numbers which come in between these numbers. These numbers are the required rational numbers.

**Example**

Find the rational number between 1/10 and 2/10.

**Solution**

As we can see that there are no visible rational numbers between these two numbers. So we need to write the equivalent fraction.

2/10 = 20/100((multiply the numerator and denominator by 10)

Hence, 2/100, 3/100, 4/100……19/100 are all the rational numbers between 1/10 and 2/10.

**Method 2**

We have to find the mean (average) of the two given rational numbers and the mean is the required rational number.

**Example**

Find the rational number between 1/10 and 2/10.

**Solution**

To find mean we have to divide the sum of two rational numbers by 2.

3/20 is the required rational numbers and we can find more by continuing the same process with the old and the new rational number.

**Remark**: 1. This shows that if p and q are two rational numbers then (p + q)/2 is a rational number between p and q so that p < (p + q)/2 < q.

2. There are infinite rational numbers between any two rational numbers

**Rational numbers between two rational numbers with different denominator**

Let us suppose the rational number f1 = p1/q1 and rational number f2 = p2/q2.

The following steps are performed to find one or more rational numbers between a pair of given rational numbers f1 and f2:

**Step 1: **Check the denominator values of both the fractions, that is compare the values of q1 and q2.

**Step 2:** If both the denominators are equal, that is q1 = q2, the numerators are then compared, that is the values of p1 and p2 are checked.

**Step 3:** If numerators differ by a large number, then we add any small constant integer value to the smaller numerator, keeping the denominator same. The rational numbers thus become p1/q1 and p2/q1 (since q1 = q2). Two cases may arise :

- p1 > p2 by a large amount, then we can add any numerical value to p1 such that it is less than p2, keeping the denominator same.
- p1 < p2 by a large amount, then we can add any numerical value to p2 such that it is less than p1, keeping the denominator same.

**Step 4:** If numerators differ by small value, then we can multiply both the rational numbers by a large constant value and follow first sub point, of adding a small constant integer to the smaller numerator. Multiplying the rational numbers by a large constant values increases the gap in the values of p1 and p2.

*Since the denominators are the same, we find fractions by adding 1 to a smaller numerator, that is, f1 numerator = 2. *

*3/9, 4/9, 5/9, 6/9*