# Algebraic form of complex numbers

We start this section by recalling the concept of *imaginary unit.*

**Definition**

We define the imaginary unit as the element , and we call it .

Now we compute the multiplication, according to the previous definition given in , of by itself:

where we have used the isomorphism between and

Therefore, the imaginary unit is such that : so, it solves the equation .

Analogously, we refer to the complex number as ; we remark that this is another solution of the previous equation.

So, we can write:

Through we can give an algebraic form of complex number, that alloes us to simplify operations between them. In particular:

In this way, a complex number can be written, by the previous isomorphism and the definition of imaginary unit, in the following way:

**Definition**

A complex number can be represented in the form:

**Definition**

Given , we call, respectively, real part and imaginary part of the two real numbers and and we refer to them as:

**Definition**

Given we define the *complex conjugate* of the complex number , that has the same real part and the opposite imaginary part.

the following properties are valid:

- ,
- ,

The following definition of module is an extension of its usual concept in .

**Definition**

We define the module of as the number

So, it derives naturally the following:

**Theorem**

is a metric space.