Complex Number: Introduction

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In starting, numbers were used only for counting which created the system of Natural Number. Natural Numbers came into existence when man first learnt counting which can also be viewed as adding successively the number 1 to unity. If we add two natural numbers we get a natural number.

 N = {1,2,3,4,5,6................................... \infty  }

 But Natural number had no solution of the equation

 x + 5 = 5

 So, whole number system was evolved

 W = {0,1,2,3,4...................................\infty  }

  If we add two natural numbers we get a natural number.But it has no solution of the equation

 x + 8 = 3

 So, now it became necessary to enlarge the system to get the solution. Thus to every natural number corresponds a unique negative integer designated -n and called the additive inverse of n, and the number '0' written such that n + (-n) = 0, and n + 0 = n for every natural number n. 

Also n is the inverse of (-n). The negative integers, the number 0 and the natural numbers(i.e. positive integers together constitute a system of Integers

 Z or I = {  - \infty ........................._3,-2,-1,0,1,2,3.............................\infty  }

 Similarly for solution of  5x = 2

 Rational Numbers system was evolved

 Q = { Number of the form p/q , q \ne 0 and p,q \in Z }

 But you cant measure the hypotenuse of the right angled triangle having sides 1 and 2 in rational number system.Equation like, {x^2} = 2 cannot be solved under the system of Rational number.

 Irrational Numbers = 2\sqrt 2 ,\pi ,\sqrt[4]{3} ......       are irrational numbers

 Real Numbers = {Rational Numbers}U{Irrational Numbers}

 and, N \subset W \subset I \subset Q \subset R

 Every set is the extension of previous set. But {x^2} + 1 = 0 has no solution in R. Here right hand side is always positive and greater than 1. Therefore, it will never become '0'. So it has no Real solution.

However, it has no solution in Real number, mathematician defined a number i 'iota'  whose square is -1. Though such number does not exist so we can called it imaginary number.

 Therefore,
{i^2} + 1 = 0

 or, {i^2} = - 1

 or, i = \sqrt { - 1}

Consider a Real line.

 _________________________________________________________________
             -3         -2       -1         0          1         2        3         (Real Line)

 We cannot find any imaginary number in this line. For imaginary number system we need a different line. 
We represent the imaginary line or axis, perpendicular to the Real axis because it is independent with Real line.
The above xy-plane is known as 'Argand Plane'. You can represent any number whether real, imaginary or complex by this plane. Any cordinate of this plane let (2,3) gives the complex number 2 + 3i. 

 z = 2 + 3i has both real as well as imaginary part.

 Real part = 2 , Imaginary part = 3

 Any Real number can be written in the complex form by association of 0 in the imaginary part.

 2 = 2 + 0i
-3 = -3 + 0i
0 = 0 + 0i

 '0' is the only number which is purely real and also purely imaginary number.

 so Complex number is the extended set of Real Number

 Therefore,  N \subset W \subset I \subset Q \subset R \subset C






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