We start this section by recalling the concept of imaginary unit.
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:
A complex number can be represented in the form:
This is the algebraic form.
Given , we call, respectively, real part and imaginary part of the two real numbers and and we refer to them as:
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 .
We define the module of as the number
So, it derives naturally the following:
is a metric space.