# 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 $(0,1)\,\in \mathbb {C}$ , and we call it $i$ .

Now we compute the multiplication, according to the previous definition given in $\mathbb {C}$ , of $(0,1)$ by itself:

$(0,1)(0,1)=(-1,0)\equiv -1\ ,$ where we have used the isomorphism between $\mathbb {C}$ and $\mathbb {R} ^{2}\ .$ Therefore, the imaginary unit is such that $i^{2}=-1$ : so, it solves the equation $x^{2}+1=0$ .

Analogously, we refer to the complex number $(0,-1)$ as $-i$ ; we remark that this is another solution of the previous equation.

So, we can write: $i={\sqrt {-1}}\ .$ Through $i$ we can give an algebraic form of complex number, that alloes us to simplify operations between them. In particular:

$\forall (a,b)\in \mathbb {C} {\mbox{ we have that }}(a,b)=(a,0)+(0,b)=(a,0)+(b,0)(0,1)\ .$ In this way, a complex number $(a,b)$ can be written, by the previous isomorphism and the definition of imaginary unit, in the following way:

Definition

A complex number $(a,b)$ can be represented in the form:

$(a,b)=a+ib\ .$ This is the algebraic form.

Definition

Given $z=a+ib$ , we call, respectively, real part and imaginary part of $z$ the two real numbers $a$ and $b$ and we refer to them as:

$Re(z)\equiv a\ \ ,\qquad Im(z)\equiv b\ .$ Definition

Given $z=a+ib\,\in \mathbb {C} \ ,$ we define the complex conjugate of $z$ the complex number ${\bar {z}}=a-ib$ , that has the same real part and the opposite imaginary part.

$\forall \,z\,\in \mathbb {C}$ the following properties are valid:

• $z+{\bar {z}}=2Re(z)$ ,
• $z-{\bar {z}}=2iIm(z)$ ,
• $z{\bar {z}}=a^{2}+b^{2}\equiv \mid z\mid ^{2}.$ The following definition of module is an extension of its usual concept in $\mathbb {R}$ .

Definition

We define the module of $z$ as the number $r=\mid z\mid \in \mathbb {R} \colon r={\sqrt {(Re(z))^{2}+(Im(z))^{2}}}\ .$ So, it derives naturally the following:

Theorem

$(\mathbb {C} ,\mid \cdot \mid )$ is a metric space.