The numbers which are involved in many mathematical applications such as addition, subtraction and multiplication which are inherently closed with many mathematical processes are called Rational numbers.

## Introduction to Rational Numbers

### 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′****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′****Closure Property ** Closure property is applicable for whole numbers in the case of **addition **and **multiplication** while it isn’t in the case for 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 for 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′**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 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**** \(\frac{p}{q}\)**, where q must be non-zero. The set of rational numebrs is denoted as Q.

For example: \(\frac{-5}{7}\) is a rational number where -5 and 7 are integers. Even 2 is a rational number since it can be written as \(\frac{2}{1}\) where 2 and 1 are integers.

## Properties of Rational Numbers

### Closure Property of Rational Numbers

For any two rational numbers a and ba∗b=c∈Q i.e. For two rational numbers say a and b the results of addition, subtraction and multiplication operations gives 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 \(\frac{2}{3}\) +\(\frac{3}{4}\) =\(\frac{(8+9)}{12}\) =\(\frac{17}{12}\) is also a rational number where 17 and 12 are integers. The difference between two rational numbers result in a rational number. Therefore, the **closure property applies** for rational numbers in the case of **subtraction**.

For example : The difference between \(\frac{4}{5}\) −\(\frac{3}{4}\) =\(\frac{(16-15)}{20}\) =\(\frac{1}{20}\) 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 \(\frac{1}{2}\) ×\(\frac{-4}{5}\) =\(\frac{-4}{10}\) =\(\frac{-2}{5}\) which is also a rational number where -2 and 5 are integers. In the case with division of two rational numbers, we see that for a rational number a, a÷0 is not defined. Hence we can say that the **closure property does no**t **apply** for rational numbers in the case of **division**.

### Commutative Property of Rational Numbers

For any two rational numbers a and ba∗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: \(\frac{2}{3}\) +\(\frac{4}{3}\) = \(\frac{4}{3}\) + \(\frac{2}{3}\) ⇒\(\frac{6}{7}\) = \(\frac{6}{7}\) 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: \(\frac{2}{3}\) − \(\frac{4}{3}\) = \(\frac{-2}{3}\) Whereas \(\frac{4}{3}\) − \(\frac{2}{3}\) = \(\frac{2}{3}\) 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: \(\frac{2}{3}\) × \(\frac{4}{3}\) = \(\frac{8}{9}\) \(\frac{4}{3}\) × \(\frac{2}{3}\) = \(\frac{8}{9}\) Considering the division of two numbers a and b, (a÷b) is different from (b÷a). Therefore **division** is **not commutative** for rational numbers.

For example: 2÷3=\(\frac{2}{3}\) is definitely different from 3÷2=\(\frac{3}{2}\)

### 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.**(a÷b)÷c≠(a÷b)÷c. Therefore **division** is **not ****associative**.

### 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+aca(b−c)=ab−ac

## Negatives and Reciprocals

### Negation of a Number

For a rational number \(\frac{a}{b}\), \(\frac{a}{b}\) + 0 = \(\frac{a}{b}\). 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 (\(\frac{a}{b}\))+(−\(\frac{a}{b}\))=(−\(\frac{a}{b}\))+(\(\frac{a}{b}\))=0, then it can be said that the **additive inverse** or **negative** of a rational number ab is −ab. Also −\(\frac{a}{b}\) is the **additive inverse** or **negative** of \(\frac{a}{b}\).

For example : The additive inverse of \(\frac{-21}{8}\) is −(\(\frac{-21}{8}\))=\(\frac{21}{8}\)

### Reciprocal of a Number

For any rational number \(\frac{a}{b}\), \(\frac{a}{b}\)×1=\(\frac{a}{b}\). i.e., When any rational numbers is multiplied by **‘1’ **,the result is same rational number. Therefore **‘1’ ** is called **multiplicative identity **for rational numbers. If \(\frac{a}{b}\)×\(\frac{c}{d}\)=1, then it can be said that the cd is **reciprocal** or the **multiplicative inverse** of a rational number \(\frac{a}{b}\). Also \(\frac{a}{b}\) is reciprocal or the multiplicative inverse of a rational number \(\frac{c}{d}\)
For example : The reciprocal of \(\frac{2}{3}\) is \(\frac{3}{2}\) as \(\frac{2}{3}\)×\(\frac{3}{2}\)=1

## Representing on a Number Line

### Representation of Rational Numbers on the Number Line

In order to represent a given rational number an, 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 23, we divide the space between 0 and 1, 1 and 2 etc. into three parts.

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

Similar steps can be followed for negative rational numbers by repeating the steps towards negative direction.

## Rational Numbers between Two Rational Numbers

### 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 \(\frac{2}{10}\) and \(\frac{8}{10}\) 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. **Method 2** Given two rational numbers, we can always find a rational number between them by calculating their **mean **or **midpoint**.

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