What is a Real Number?

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

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
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,

$\displaystyle 12,\,\,\frac{3}{2},\,-5,\,\,2\frac{1}{4},\,\,0$

There are two kinds of decimal representations of rational numbers.

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

$\displaystyle \frac{3}{2}=1.5$

$\displaystyle \frac{6}{5}=1.2$

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,

$\displaystyle \frac{6}{7}=0.857142....$

$\displaystyle \frac{4}{9}=0.44444....$

1.1 – Fractions

A fraction can be written as,

$\displaystyle \frac{{Numerator}}{{Deno\min ator}}$

For examples,

$\displaystyle \frac{3}{2},\,\,\,2\frac{1}{4},\,\,\frac{{12}}{{25}}$

Types of fractions are;

Proper Fractions – In proper fractions, numerator is always less than denominator, e.g. $\displaystyle \frac{1}{4},\,\,\frac{{12}}{{25}}$
Improper Fractions – In improper fractions, numerator is always equal or greater than denominator, e.g. $\displaystyle \frac{5}{5},\,\,\frac{{21}}{4},\,\,\frac{{51}}{{25}}$
Mixed Fractions – In mixed fractions, there is a proper fraction and a whole number as well, e.g. $\displaystyle 2\frac{5}{{12}},\,\,6\frac{{21}}{{25}}$

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. $\displaystyle Z=\{........,-3,-2,-1,0,1,\,2,\,3,.........\}$

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. $\displaystyle W=\{0,1,\,2,\,3,.........\}$

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.

$\displaystyle {{Z}^{+}}=\{1,\,2,\,3,4,5,.........\}$

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.

$\displaystyle N=\{1,\,2,\,3,4,5,.........\}$

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.

$\displaystyle {{Z}^{-}}=\{..........,-5,-4,-3,-2,-1\}$

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,

$\displaystyle \,\pi \,,\,\sqrt{2},\,\sqrt{5}$

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,

$\displaystyle a-b\ne b-a$

For example,

a = 5, b = 3

$\displaystyle 5-3=2\ne 3-5=-2$

2.3 – Associative Property

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

$\displaystyle a-(b-c)\ne (a-b)-c$

For example,

a = 3, b = 5, c = 4

$\displaystyle 3-(5-4)=2\ne (3-5)-4=-6$

3 – Properties of Multiplication

3.1 – Closure Property

If a, b are two whole numbers and $\displaystyle a\times b=c$ , then c is also a whole number. For example,

Let, a = 5 and b = 3, then

$\displaystyle 5\times 3=15$ (a whole number) and

$\displaystyle 3\times 5=15$ , which is also a whole number.

3.2 – Commutative Property

If a, b are two real numbers, then,

$\displaystyle a\times b=b\times a$

For example,

a = 5, b = 3

$\displaystyle 5\times 3=15=3\times 5$

3.3 – Associative Property

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

$\displaystyle a\times (b\times c)=(a\times b)\times c=(a\times c)\times b$

For example,

a = 3, b = 5, c = 4

$\displaystyle 3\times (5\times 4)=(3\times 5)\times 4=(3\times 4)\times 5=60$

3.4 – Multiplicative property of zero

Multiplicative property of zero is,

$\displaystyle a\times 0=0\times a$

For example,

$\displaystyle 3\times 0=0=0\times 3$

3.5 – Multiplicative Identity

Multiplicative identity is given by,

$\displaystyle a\times 1=a=1\times a$

For example,

$\displaystyle 5\times 1=5=1\times 5$

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,

$\displaystyle \frac{{25}}{5}=5$ (a whole number), whereas,

$\displaystyle \frac{{14}}{{28}}=\frac{1}{2}$

4.2 – Commutative Property

If a, b are two real numbers, then,

$\displaystyle a\div b\ne b\div a$

For example,

a = 20, b = 10

$\displaystyle \frac{{20}}{{10}}=2\ne \frac{{10}}{{20}}=\frac{1}{2}$

4.3 – Associative Property

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

$\displaystyle a\div (b\div c)\ne (a\div b)\div c$

For example,

a = 3, b = 5, c = 4

$\displaystyle 3\div (5\div 4)=2.4\ne (3\div 5)\div 4=0.15$