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A high school math course on complex numbers.
This lesson involves the following concepts:
- definition of the complex number;
- algebraic form;
- geometric form;
- Euler’s formula;
- Moivre’s formula;
- complex equations;
- geometric representation of a complex number;
- real and imaginary part of a complex number;
- operations on complex numbers.
I. Algebraic form of a complex number
There exists a set of numbers noted , whose elements are called the complex numbers, such that :
contains the set
of real numbers;
- the calculation rules in
are the same as in
;
contains an element noted
such that
;
- any complex number z can be written as in a unique way in the form
with
and
The number x is called the real part (noted Re(z)) of the number z and the number y is called the imaginary part (noted Im(z)) of the complex number z.
Example:
The number is a complex number.
is its real part and 2 is its imaginary part.
- z is a real number if and only if Im(z)=0.
- z is a pure imaginary if and only if Re(z)=0.
II. conjugate of a complex number
We consider z a complex number whose algebraic form is z=x+iy with x and y two real numbers.We call conjugate of the number z, the complex number, noted , such as
.
Example:
and
.
We consider two complex numbers and
. We have the following properties:
with
is a pure imaginary
with
with
with
III.graphical representation of complex numbers
1. Affix of a point
Consider the complex plane with a direct orthonormal reference frame
To any complex number z=x+iy , we associate the point M(x;y).
M is called the image point of z and z is called theaffix of the point M in the direct orthonormal reference frame . Let M(z) be the point M with affix z.
Example:
The point M with affix has coordinates
.
The point N with affix has coordinates
.
2. affix of a vector
To any complex number z affix of the point M(x,y), we associate the vector such that
.and we note
, the vector
of affix z.
Examples:
The vector with affix z=1+2i has coordinates
.
The vector with affix 1-3i has coordinates
.
We consider two vectors and
of respective affixes
and
The vector
has the affix
.
The vector has affix
with
.
3. second degree equations in 
Consider a real number .
- If a>0, the solutions are
and
;
- If a<0, the solutions are
and
;
- If a=0, the solution is z=0.
Example:
The equation has solutions in
:
and
.
4. equations of the type az²+bz+c=0
We consider real numbers a, b and c with . We consider in
, the equation (E) :
of discriminant
.
- If
>0, the solutions are
and
;
- If
<0, the solutions are
and
;
- If
=0, the solution is
.
Example:
Solve in , the equation (E) :
.
.
The solutions are:
and .
5.factoring of a second degree trinomial
We consider real numbers a,b and c with . For any number
, we pose
.
We note and
the two solutions of
in (with possibly
=
when
=0).
We have for all ,
.
Example:
Let’s go back to the previous example, .
Cette publication est également disponible en :
Français (French)
العربية (Arabic)