Complex numbers: senior math course in PDF.

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1 I. Algebraic form of a complex number
2 II. conjugate of a complex number
3 III.graphical representation of complex numbers

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 $\mathbb{C}$ , whose elements are called the complex numbers, such that :

$\mathbb{C}$ contains the set $\mathbb{R}$ of real numbers;
the calculation rules in $\mathbb{C}$ are the same as in $\mathbb{R}$ ;
$\mathbb{C}$ 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 $z=\sqrt{3}+2i$ is a complex number.

$\sqrt{3}$ 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 $\overline{z}$ , such as $\overline{z}=x-iy$ .

Example:

$\overline{1+3i}=1-3i$ and $\overline{2-5i}=2+5i$ .

Properties:

We consider two complex numbers and . We have the following properties:

$\overline{\overline{z}}=z$
$\overline{z+z'}=\overline{z}+\overline{z'}$
$\overline{(\frac{1}{z})}=\frac{1}{\overline{z}}$ with $z\neq\,0$
$z\in\,\mathbb{R}\Leftrightarrow\,\overline{z}=z$
is a pure imaginary $\Leftrightarrow\,\overline{z}=-z$
$\overline{zz'}=\overline{z}\overline{z'}$
$\overline{(\frac{z}{z'})}=\frac{\overline{z}}{\overline{z'}}$ with $z'\neq\,0$
$\overline{(z^n)}=(\overline{z})\,^n$ with $n\in\,\mathbb{N}$
$\overline{(kz)}=k\,\overline{z}$ with $k\in\,\mathbb{R}$

III.graphical representation of complex numbers

1. Affix of a point

Definition:

Consider the complex plane with a direct orthonormal reference frame $(O,\vec{u},\vec{v})$

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 $(O,\vec{u},\vec{v})$ . 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 $\vec{w}=\vec{OM}$ such that $\vec{w}(x;y)$ .and we note $\vec{\,w}(z)$ , the vector $\vec{\,w}$ of affix z.

Examples:

The vector $\vec{OM}$ with affix z=1+2i has coordinates $\vec{\,OM}(1;2)$ .

The vector $\vec{t}$ with affix 1-3i has coordinates $\vec{\,t}(1;-3)$ .

Properties:

We consider two vectors $\vec{w}$ and $\vec{w'}$ of respective affixes andThe vector $\vec{w}+\vec{w'}$ has the affix z+z' .

The vector $k\vec{w}$ has affix with $k\in\,\mathbb{R}$ .

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

Ownership:

Consider a real number .

If a>0, the solutions are $z=\sqrt{a}$ and $z=-\sqrt{a}$ ;
If a<0, the solutions are $z=i\sqrt{-a}$ and $z=-\sqrt{-a}$ ;
If a=0, the solution is z=0.

Example:

The equation has solutions in $\mathbb{C}$ : and .

4. equations of the type az²+bz+c=0

Ownership:

We consider real numbers a, b and c with $a\neq\,0$ . We consider in $\mathbb{C}$ , the equation (E) : of discriminant $\Delta\,=b^2-4ac$ .

If $\Delta$ >0, the solutions are $z_1=\frac{-b+\sqrt{\,\Delta\,}}{2a}$ and $z_2=\frac{-b-\sqrt{\,\Delta\,}}{2a}$ ;
If $\Delta$ <0, the solutions are $z_1=\frac{-b+i\sqrt{-\,\Delta\,}}{2a}$ and $z_2=\frac{-b-i\sqrt{\,-\Delta\,}}{2a}$ ;
If $\Delta$ =0, the solution is $z\,=\frac{\,\sqrt{\,\Delta\,}}{2a}$ .