Introduction of complex numbers

The equation $x^2+1=0$ has no real Solution $$x^2+1=0$$ gives $$x^2=-1$$ and square of every real number is non negative.

So we need to extend the real number system to a larger system so that we can find the solution of the equation $$x^2=-1$$

In fact the main objective is to solve the equation  $$a{x}^2+bx+c$$, where $$D=b^2-4ac<0$$  which is not possible in the system of real numbers.

Let us denote $\sqrt{-1}$ by the symbol $\iota$. 

Here $\iota$ is read as $\text{iota}$.

Then we have  $${\iota }^2=-1$$

This means that $\iota$ is a solution of the equation $x^2+1=0$ 

A number of the form $a+\iota b$, where $a$ and $b$ are real numbers, is defined to be a complex number.

Complex number $z=a+\iota b$,

$a$ is called the real part, denoted by $\text{Re }z$ and

$b$ is called the imaginary part denoted by $\text{Im }z$ of the complex number $z$.

For example,

if $z=2+\iota 5$, then $\text{Re }z=2$ and $\text{Im }z=5$.