Effectively, the equation
$$ x^{2} + 1 = 0 $$
Has solution, in fact it has two solutions wich are \( x = \pm \sqrt{-1} = \pm i \). Positive solution from them is called as imaginary unit.

In general, numbers on the form
$$ z=a+bi $$
Are called complex numbers, the a is called real part of b and b is the imaginary part of z. They are denoted respectively for \( Re z=a, Im z=b \).

The complex plane consists of all complex numbers, that is, pairs of real numbers (a, b), formally
$$ \mathbb{C} = \{ z = a+bi : a, b \in \mathbb{R} \}$$
This notation, used for the first time by Euler in 1773, gives way to a precious and prosperous branch of analysis called Complex Variable analysis.

If we change the sign to the imaginary part we have the

conjugate complex . Ie, the number
$$ \bar{z}=a-bi $$
It is called

conjugate complex .

Note that an attentive student could claim that \( \mathbb{C} = \mathbb{R} ^2 \), which is true from a topological point of view, but not from an
algebraic point of view because in the plane\( \mathbb{C} \) there are addition and multiplication operations which do not exist a priori in the usual plane of \( \mathbb{R} ^2 \).
Formally:

### Complex plane definition

Complex plane is defined as
$$ \mathbb{C} = \{ z = a+bi : a, b \in \mathbb{R} \}$$
and if \( z, w \in \mathbb{C} : z = a+bi, w = c + di \) then the addition and multiplication operations respectively are defined as:
$$ z+w = a+c +(b+d)i $$
$$ zw = (ac - bc) +(bc+ad)i $$
The complex plane endowed with these two operations has a Field structure.
Every complex number can be written in the

polar form , so

\(z = a+bi = re^{i\theta}\)

by

polar form of a complex number we have

Figura 1: Polar form of complex number.

\(a=r \,cos\,\theta,\: b =r \,sin\,\theta\)

being r

modulus of z. It is calculated as

\(|z| = \sqrt{x^{2}+y^{2}} \)

The argument of a complex number is well defined except multiples of \(2\pi\), ie
$$ arg\, z = \theta + 2k\pi,\,\, k=1, 2, 3 ...$$
or
$$ arg\,z = \theta ,\,\,\,\ mod\,2\pi$$
We will refer, from now on, to \(Arg z = \theta \), then
$$ arg\,z = Arg\,z + 2k\pi,\,\, k=1, 2, 3 ...$$
Suppose we want to calculate the square of z, that is, \(z^{2}\), then:
$$z = re^{i \theta} \Rightarrow z^{2} = r^2 e^{2i \theta} \Rightarrow arg z^{2} = 2 Arg\,z + 4k\pi,\,\, k=1, 2, 3 ...$$
When we do the inverse operation, that is, the square root, it happens that
$$ arg \, z^{\frac{1}{2}} = \frac{arg \, z }{2} + \frac{2k\pi}{2} $$
So, now we can calculate square roots of negative numbers, for example, to calculate the square root of -1
$$ arg \, (-1)^{\frac{1}{2}} = \frac{\pi}{2} + k\pi $$.
because this i and -i are both square roots of -1.

### Complex numbers operations

Given two complex numbers

\( z_{1} = r_{1} e^{i \theta_{1} }\)

\( z_{2} = r_{2} e^{i \theta_{2} }\)

It is easy

\(arg\, z_{1}z_{2} = arg\, z_{1} + arg\,z_{2}\)

\(arg\, \frac{z_{1}}{z_{2}} = arg\, z_{1} - arg\,z_{2}\)

this is because

\( z_{1}z_{2} = r_{1} e^{i \theta_{1} } . r_{2} e^{i \theta_{2} } = r_{1} r_{2} e^{i (\theta_{1}+ \theta_{2}) } \)

Lets use this link to see how to calculate power of complex numbers.