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.
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 .
We consider two complex numbers and . We have the following properties:
- is a pure imaginary
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.
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.
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 andThe 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.
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 .
Solve in , the equation (E) : .
The solutions are:
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 , .
Let’s go back to the previous example, .
Cette publication est également disponible en : Français (French) العربية (Arabic)