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

Theorem and definition:

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.

Properties:
• 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

Definition:

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 .

Properties:

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

Definition:

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

Definition:

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 .

Properties:

We consider two vectors and of respective affixes andThe vector has the affix .

The vector has affix with .

### 3. second degree equations in $\mathbb{C}$

Ownership:

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

Ownership:

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

Ownership:

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, .

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