In this post, we are going to discuss; what is a real number, its definition, classification, and properties associated with real numbers.

## What is a Real Number?

A real number is a value that can be represented along the number line as

## Classification of Real Numbers

Real Numbers are classified into the following,

- Rational Numbers
- Fractions
- Integers
- Whole Numbers
- Natural Numbers
- Zero

- Negative Integers

- Whole Numbers

- Irrational Numbers

Now we discuss all of the above types of real numbers one by one….

### 1 – Rational Numbers

Rational numbers can be written as a ratio of two integers in the form p/q, where p and q are both integers and q is not equal to zero. It is denoted by Q. few examples of rational numbers are,

There are two kinds of decimal representations of rational numbers.

**Terminating decimal fractions**– fixed number of digits in its decimal part. For example,

**Non-terminating or recurring decimal fractions**– no fixed number of digits in its decimal part or some digits are recurring again and again. For example,

#### 1.1 – Fractions

A fraction can be written as,

For examples,

Types of fractions are;

**Proper Fractions –**In proper fractions, numerator is always less than denominator, e.g.**Improper Fractions**– In improper fractions, numerator is always equal or greater than denominator, e.g.**Mixed Fractions**– In mixed fractions, there is a proper fraction and a whole number as well, e.g.

#### 1.2 – Integers

It can be a whole number or negative integer having no fractional value after decimal point. It is denoted by Z, i.e.

**1.2.1 – Whole Numbers**

Whole numbers are integers with no negative integer which start from zero to onward on a number line.

Whole numbers are denoted by W. i.e.

It is pertinent to mention here that, positive integers are just like whole numbers but these start from 1 to onward without 0. i.e.

**1.2.1 (a) Natural Numbers**

Natural numbers are all whole numbers start from 1 to onward i.e. 1, 2, 3, 4, 5,……………. Natural Numbers are denoted by N. i.e.

**1.2.1 (b) Zero**

Zero is considered as a whole number but without having a positive or negative value.

**1.2.2 – Negative Integers**

Negative Integers start from minus one to onward, i.e.

### 2 – Irrational Numbers

Irrational numbers are real numbers that cannot be written as a ratio of two integers. It is denoted by Q’. Few examples of irrational numbers are,

## Properties of Real Numbers

Properties of real numbers with respect to addition, subtraction, multiplication and are:

### 1 – Properties of Addition

#### 1.1 – Closure Property

If *a, b* are two real numbers and a + b = c, then c is
also a whole number.

5 + 3 = 8. Where, 8 is also a whole number.

#### 2.2 – Commutative Property

If a, b are two real numbers, then,

a + b = b + a.

For example,

a = 5, b = 3

⇒ 5 + 3 = 8 = 3 + 5

#### 2.3 – Associative Property

Let, a, b, c are 3 whole numbers, then,

a + (b + c) = (a + b) + c = (a + c) + b.

For example,

a = 3, b = 5, c = 4

3 + (5 + 4) = (3 + 5) + 4 = (3 + 4) + 5 = 12

#### 2.4 – Additive Identity

In additive identity, there exists a distinct real number i.e. 0, therefore,

a + 0 = a = 0 + a

For example,

3 + 0 = 3 = 0 + 3

#### 2.5 – Additive Inverse

Additive inverse of a is denoted by – a, then

a + (–a) = 0 = (–a) + a

It means, additive inverse of 1 is – 1. For example,

5 + (–5) = 0 = (–5) + 5

### 2 – Properties of Subtraction

#### 2.1 – Closure Property

If *a, b* are two real numbers and a – b = c, then c is not
always a whole number. For example,

Let, a = 5 and b = 3, then

5 – 3 = 2 (a whole number) whereas,

3 – 5 = -2 which is not a whole number.

#### 2.2 – Commutative Property

If a, b are two real numbers, then,

For example,

a = 5, b = 3

#### 2.3 – Associative Property

Let, a, b, c are 3 whole numbers, then,

For example,

a = 3, b = 5, c = 4

### 3 – Properties of Multiplication

#### 3.1 – Closure Property

If *a, b* are two whole numbers and , then c is also a whole number. For example,

Let, a = 5 and b = 3, then

(a whole number) and

, which is also a whole number.

#### 3.2 – Commutative Property

If a, b are two real numbers, then,

For example,

a = 5, b = 3

#### 3.3 – Associative Property

Let, a, b, c are three whole numbers, then,

For example,

a = 3, b = 5, c = 4

#### 3.4 – Multiplicative property of zero

Multiplicative property of zero is,

For example,

#### 3.5 – Multiplicative Identity

Multiplicative identity is given by,

For example,

### 4 – Properties of Division

#### 4.1 – Closure Property

This property tells that the result of division of two numbers is not always a whole number. For example,

(a whole number), whereas,

#### 4.2 – Commutative Property

If a, b are two real numbers, then,

For example,

a = 20, b = 10

#### 4.3 – Associative Property

Let, a, b, c are three whole numbers, then,

For example,

a = 3, b = 5, c = 4