Complex numbers: senior math course download pdf.

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 i such that i^2=-1;
  • any complex number z can be written as in a unique way in the form z=x+iy with x and y 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.

sets of numbers

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 z and z'. 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
  • 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.

Graphical representation of complex numbers

Example:

The point M with affix z=3+i has coordinates M(3,1).

The point N with affix z=-1-i has coordinates M(-1,-1).

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).

vector affix

Properties:

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

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

3. second degree equations in \mathbb{C}

Ownership:

Consider a real number a.

  • 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 z^2=-4 has solutions in \mathbb{C}: z=2i and z=-2i.

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) : az^2+bz+c=0 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}.

Example:

Solve in \mathbb{C}, the equation (E) : z^2+4z+5=0.

\Delta\,=b^2-4ac=4^2-4\times  \,1\times  \,5=16-20=-4<0.

The solutions are:

z_1=\frac{-b+i\sqrt{-\,\Delta\,}}{2a}=\frac{-4+i\sqrt{4\,}}{2}=\frac{-4+2i}{2}=-2+i

and .

5.factoring of a second degree trinomial

Ownership:

We consider real numbers a,b and c with a\neq\,0. For any number z\in\,\mathbb{C}, we pose P(z)=az^2+bz+c.

We note z_1 and z_2 the two solutions of P(z)=0 in (with possibly z_1= z_2 when \Delta=0).

We have for all z\in\,\mathbb{C}, P(z)=a(z-z_1)(z-z_2).

Example:

Let’s go back to the previous example, P(z)=z^2+4z+5=\,(z+2-i)(z+2+i).

Cette publication est également disponible en : Français (French) العربية (Arabic)

Télécharger puis imprimer cette fiche en PDF

Télécharger ou imprimer cette fiche «complex numbers: senior math course download pdf.» au format PDF afin de pouvoir travailler en totale autonomie.


D'autres fiches dans la section High school math course




Télécharger nos applications gratuites Mathématiques Web avec tous les cours,exercices corrigés.

Application Mathématiques Web sur Google Play Store.    Application Mathématiques Web sur Apple Store.    Suivez-nous sur YouTube.


D'autres articles analogues à complex numbers: senior math course download pdf.


  • 93
    Powers: 4th grade math course download PDF.A math course on powers in the fourth grade. The student must have assimilated the definition of a power but also know the different formulas such as the product of two powers, the quotient of powers or the power of a power. Develop computational skills with scientific writing (or scientific…
  • 93
    Translation and rotation: 4th grade math lessons for download PDF.This course on translation and rotation in the fourth grade (4e) with the definition and properties of these two transformations of the plane studied in college must be well learned. Thus, this lesson is written by a team of national education teachers and it is in accordance with the current…
  • 93
    Probability and binomial distribution: senior high school math course in PDFProbabilities are applicable in everyday life. This chapter helps you to progress and solve problems with a new method. Probabilities are very interesting and you will encounter them everywhere. Follow the course on probability and the binomial distribution to fully understand this chapter. I. Proof, binomial distribution and Bernoulli's scheme.…
Les dernières fiches mises à jour

Voici la liste des derniers cours et exercices ajoutés au site ou mis à jour et similaire à complex numbers: senior math course download pdf. .

  1. Integrals : corrected high school math exercises in PDF.
  2. Intégrales : exercices de maths en terminale corrigés en PDF.
  3. الوظائف والحدود: تمارين الرياضيات في السنة النهائية مصححة بتنسيق PDF.
  4. Functions and limits: senior math exercises corrected in PDF.
  5. Fonctions et limites : exercices de maths en terminale corrigés en PDF.


Inscription gratuite à Mathématiques Web. Mathématiques Web c'est 2 146 115 fiches de cours et d'exercices téléchargées.

Copyright © 2008 - 2023 Mathématiques Web Tous droits réservés | Mentions légales | Signaler une Erreur | Contact

.
Scroll to Top
Mathématiques Web

FREE
VIEW