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 \[ax^2+bx+c\], where \[D=b^2-4ac < 0\] which is not possible in the system of real numbers.
Let us denote \(\sqrt{-1}\) by the symbol \(\huge{\iota}\).
Here \(\huge \iota\) is read as \(\huge\text{iota}\).
Then we have \[\huge{\iota}^2=-1\]
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 \(\text{Im } z = 5\).
