Eleven Standard >> Complex number | Introduction

What is a complex number?

The Square of any real number is positive. 

Ex: \(2^{2}=4\)
     \((-2)^{2}=4\)
     \(( -\sqrt{3})^{2}=3\)

What are the solutions or roots of the following equations?
i) \(x^{2}-1=0\)
ii) \(x^{2}+1=0\)

Solution:
i) \(x^{2}-1=0\)
  \(\Rightarrow\) (x-1)(x+1)=0
 \(\Rightarrow\) (x-1)=0 or (x+1)=0
 \(\therefore\) x=1, x=-1
The real solution of \(x^{2}-1\)=0 are 1, -1

ii) \(x^{2}+1=0\)
  \(\Rightarrow\) \(x^{2}=-1\)
 But we know squire of any real number is positive. That means it has no solution. So in real number system we can't solve \(x^{2}+1=0\)

So we extend Real number system to a large number system. This lage number system is called Complex number system.

In a complex number system, we are able to solve any equations like \(ax^{2}+bx+c=0\)

Where discriminant is positive negative or zero.

Mathematician Euler introduce the symbol iota (\(i\) for \(\sqrt{-1}\))

 i.e, \((\sqrt{-1})^{2}\)=(\(i^{2}\))=-1

\(\sqrt{-4}\)=\(\sqrt{4}\)\(\sqrt{i}\)=(\2i\)

Now we can solve \(x^{2}+1=0\)

     \(x^{2}\)+1=0\)
\(x^{2}=-1\)  
\(x=\pm \sqrt{-1}\)
     =\(\pm i\) 

Defination of a complex number:

Anumber of the form x+iy where x and y are real numbers is called complex number. The set of cmplex number is denoted by 'C'.

a+ib \(\in\) C or C={(a, b): z=a+ib where z\(\in\)C and a\(\in\)R, b\(\in\)R}

Example:
2+3i is in the form a+ib

2+3i\(\equiv\)a+ib, a=2, b=3 \(\in\)R

z=a+ib=Re(z)+Im(z)
z has two parts. First one is real z and second one is imazinary z.