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

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

${\displaystyle (0,1)(0,1)=(-1,0)\equiv -1\ ,}$

where we have used the isomorphism between ${\displaystyle \mathbb {C} }$ and ${\displaystyle \mathbb {R} ^{2}\ .}$

Therefore, the imaginary unit is such that ${\displaystyle i^{2}=-1}$: so, it solves the equation ${\displaystyle x^{2}+1=0}$.

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

So, we can write: ${\displaystyle i={\sqrt {-1}}\ .}$

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

${\displaystyle \forall (a,b)\in \mathbb {C} {\mbox{ we have that }}(a,b)=(a,0)+(0,b)=(a,0)+(b,0)(0,1)\ .}$

In this way, a complex number ${\displaystyle (a,b)}$ can be written, by the previous isomorphism and the definition of imaginary unit, in the following way:

Definition

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

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

Definition

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

${\displaystyle Re(z)\equiv a\ \ ,\qquad Im(z)\equiv b\ .}$

Definition

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

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

• ${\displaystyle z+{\bar {z}}=2Re(z)}$,
• ${\displaystyle z-{\bar {z}}=2iIm(z)}$,
• ${\displaystyle z{\bar {z}}=a^{2}+b^{2}\equiv \mid z\mid ^{2}.}$

The following definition of module is an extension of its usual concept in ${\displaystyle \mathbb {R} }$.

Definition

We define the module of ${\displaystyle z}$ as the number ${\displaystyle r=\mid z\mid \in \mathbb {R} \colon r={\sqrt {(Re(z))^{2}+(Im(z))^{2}}}\ .}$

So, it derives naturally the following:

Theorem

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