# Introduction

In this first section we recall some basic concepts about complex numbers. A deep knowledge of these notions is essential for the comprehension of the following topics of the course.

The need of extending the real field $\mathbb {R}$ has its origins in the impossibility of finding a solution of particular equations, such as:

$x^{2}+1=0\ ,$ for which it is clear that $\nexists \ x\in \mathbb {R\colon } \ x^{2}+1=0$ .

We define the complex field $\mathbb {C}$ , so that we can solve equations such as the previous one.

$\mathbb {C}$ is defined through the extension of the well-known properties of $\mathbb {R}$ .

We remember that $\mathbb {R}$ is a vector field if it is equipped with the following operations:

1. Addition

$\forall x,y\in \mathbb {R} {\mbox{ we have that }}x+y\in \mathbb {R} \ ,$ 1. Multiplication

$\forall x,y\in \mathbb {R} {\mbox{ we have that }}xy\in \mathbb {R} \ ,$ defined in an axyomatic way, in order to satisfy the following properties:

• Commutative property: $\forall x,y\in \mathbb {R}$ we have that:

$x+y=y+x\ ,$ $xy=yx\ .$ • Associative property: $\forall x,y,z\in \mathbb {R}$ we have that:

$x+(y+z)=(x+y)+z\ ,$ $x(yz)=(xy)z\ .$ • Distributive property: $\forall x,y,z\in \mathbb {R}$ we have that:

$x(y+z)=xy+xz\ .$ • Identity element: $\forall x\in \mathbb {R}$ we have that:

$x+0=0+x=x\ ,$ $1x=x1=x\ .$ • Inverse element (addition):

$\forall x\in \mathbb {R} \,\exists x\in \mathbb {R} \,\colon \,x+(-x)=(-x)+x=0\ .$ • Inverse element (multiplication):

$\forall x\in \mathbb {R} \setminus \{0\}\,\exists {\frac {1}{x}}\in \mathbb {R} \,\colon \,x\left({\frac {1}{x}}\right)=\left({\frac {1}{x}}\right)x=1\ .$ We define the complex field $\mathbb {C}$ as the set of the pairs $(a,b)$ with $a,b\in \mathbb {R}$ equipped with the properties:

1. Equality:

$\forall (a,b)\,\&\,(c,d)\,\in \mathbb {C} {\mbox{ we have that }}(a,b)=(c,d)\,\Longleftrightarrow {\begin{cases}a=c\\b=d\end{cases}}\ ,$ 1. Addition:

$\forall (a,b)\,\&\,(c,d)\,\in \mathbb {C} {\mbox{ we have that }}(a,b)+(c,d)=(a+c,b+d)\ ,$ 1. Multiplication:

$\forall (a,b)\,\&\,(c,d)\,\in \mathbb {C} {\mbox{ we have that }}(a,b)(c,d)=(ac-bd,ad+bc)\ .$ we see that if we define $\mathbb {C} _{0}\equiv \{(a,b)\in \mathbb {C} \colon b=0\}$ we have that:

• $\mathbb {C} _{0}\subset \mathbb {C} \ ;$ • $\mathbb {C} _{0}$ is a field, with theoperations and properties inherited by $\mathbb {C} \ ;$ • $\mathbb {C} _{0}$ is isomorphic to $\mathbb {R} \ ,$ that is:

$\exists f:\mathbb {R} \rightarrow \mathbb {C} \;x\mapsto f(x)=(x,0){\mbox{ is an isomorphism.}}$ Moreover, it can be proved that the function:

$g:\mathbb {R} ^{2}\rightarrow \mathbb {C} {\mbox{ definita come: }}\ g(x,y)=(x,y)\,\forall x,y\in \mathbb {R}$ is an isomorphism, that is $\mathbb {C} {\mbox{ is isomorphic to }}\mathbb {R} ^{2}$ .