Let us recall the necessary information about complex numbers.
Complex number is an expression of the form a + bi, Where a, b are real numbers, and i- the so-called imaginary unit, a symbol whose square is equal to –1, that is i 2 = –1. Number a called real part, and the number b - imaginary part complex number z = a + bi. If b= 0, then instead a + 0i they write simply a. It can be seen that real numbers are a special case of complex numbers.
Arithmetic operations on complex numbers are the same as on real numbers: they can be added, subtracted, multiplied and divided by each other. Addition and subtraction occur according to the rule ( a + bi) ± ( c + di) = (a ± c) + (b ± d)i, and multiplication follows the rule ( a + bi) · ( c + di) = (ac – bd) + (ad + bc)i(here it is used that i 2 = –1). Number = a – bi called complex conjugate To z = a + bi. Equality z · = a 2 + b 2 allows you to understand how to divide one complex number by another (non-zero) complex number:
(For example, 
.)
Complex numbers have a convenient and visual geometric representation: number z = a + bi can be represented by a vector with coordinates ( a; b) on the Cartesian plane (or, which is almost the same thing, a point - the end of a vector with these coordinates). In this case, the sum of two complex numbers is depicted as the sum of the corresponding vectors (which can be found using the parallelogram rule). According to the Pythagorean theorem, the length of the vector with coordinates ( a; b) is equal to . This quantity is called module complex number z = a + bi and is denoted by | z|. The angle that this vector makes with the positive direction of the x-axis (counted counterclockwise) is called argument complex number z and is denoted by Arg z. The argument is not uniquely defined, but only up to the addition of a multiple of 2 π
radians (or 360°, if counted in degrees) - after all, it is clear that a rotation by such an angle around the origin will not change the vector. But if the vector of length r forms an angle φ
with the positive direction of the x-axis, then its coordinates are equal to ( r cos φ
; r sin φ
). From here it turns out trigonometric notation complex number: z = |z| · (cos(Arg z) + i sin(Arg z)). It is often convenient to write complex numbers in this form, because it greatly simplifies the calculations. Multiplying complex numbers in trigonometric form is very simple: z 1 · z 2 = |z 1 | · | z 2 | · (cos(Arg z 1 + Arg z 2) + i sin(Arg z 1 + Arg z 2)) (when multiplying two complex numbers, their modules are multiplied and their arguments are added). From here follow Moivre's formulas: z n = |z|n· (cos( n· (Arg z)) + i sin( n· (Arg z))). Using these formulas, it is easy to learn how to extract roots of any degree from complex numbers. nth root of z- this is a complex number w, What w n = z. It is clear that 
, and , where k can take any value from the set (0, 1, ..., n– 1). This means that there is always exactly n roots n th degree of a complex number (on the plane they are located at the vertices of the regular n-gon).
Theorem on the existence of a field of complex numbers. Operations on complex numbers in algebraic form. Geometric interpretation of complex numbers, trigonometric notation, operations on complex numbers in trigonometric form.
Basic skills and abilities that students should master in the process of studying this topic:
Be able to represent complex numbers on the coordinate plane in the form of points and radius vectors and vice versa;
Be able to find the modulus and argument zC, move from the algebraic form of writing a complex number to the trigonometric one and vice versa;
Be able to perform arithmetic operations on complex numbers in algebraic and trigonometric form, understand their geometric meaning;
Use the acquired knowledge to solve algebra and geometry problems.
All theoretical positions and practical conclusions of this topic follow from the theorem on the existence of a field of complex numbers.
Theorem 1: There is a unique, up to isomorphism, field C in which the following conditions are satisfied:
1. Field R is a subfield of field C.
3. zC х, yR: z = x + iy
Writing a complex number z in the form x + iy is called its algebraic form, while (x) is called the real part of the complex number, iy is the imaginary part, and (y) is the coefficient of the imaginary part. Designation: Re z - real part, Im z - imaginary part of a complex number.
Since (z = x + iy) (C = R x R), then from a geometric point of view, any complex number has two equal geometric interpretations (models).
a) point of the coordinate plane A (x, y);
b) radius vector with end at a point with coordinates (x, y)
The geometric approach to the concept of a complex number allows it to be written in the so-called trigonometric form.
To do this, the concepts of modulus and argument of a complex number are introduced.
Definition 1: The modulus of a complex number z is the arithmetic value of the square root of x 2 + y 2, that is
This concept is a generalization of the concept “absolute value of a real number”, since if z = x + 0i, then 
.
From a geometric point of view, the modulus of a complex number is the length of the radius vector OA or the distance from the origin to the point with coordinates (x, y).
Definition 2: The argument of the complex number z is the angle between the positive direction of the axis 
and radius - vector 
, counted counterclockwise.
From this definition it follows that the argument of a complex number is determined ambiguously, but up to a multiple of 2 . Therefore, in practice, as an argument, they usually take the smallest positive or smallest absolute angle, which is denoted by =arg z and is found from the relations:
cos = 
, sin = 
, 0 2 
then x + iy = 
and we get the trigonometric formula for writing a complex number.
To avoid errors when finding the argument of a complex number z, it must first be depicted on the coordinate plane.
The modulus and argument of a complex number z are the polar coordinates of the point corresponding to the number z, known from the geometry course. In this case, the polar axis is the axis 
The concept of the modulus and argument of a complex number z allows us to write this number in trigonometric form.
Example 1. Complex number z = 
written in trigonometric form.
1. Let us depict this complex number on the coordinate plane. This will be a radius vector ending at ( 
, -1) (see figure)
2. Let's find its module | z | = | 
| =
3. Find the argument from the relations
or j = 11p/6
Thus,
The existence of two forms of writing the same complex number z = x + iy = |z| (cosj + sinj) allows you to perform algebraic operations on the set C in the form that is most convenient in each specific case.
Theorem 2. If z 1 =x 1 + iy 1, z 2 =x 2 + iy 2, then
1. z 1 +z 2 = (x 1 +x 2) + i(y 1 +y 2)
2. z 1 -z 2 = (x 1 -x 2) + i(y 1 -y 2)
3. z 1 z 2 = (x 1 x 2 -y 1 y 2) + i(x 1 y 2 + x 2 y 1)
4.
5. In practice, formulas 3), 4) are usually not memorized, but are guided by the following mnemonic rules:
a) to multiply two complex numbers, you need to multiply them as two binomials;
b) to divide z 1 by z 2 0, you need to multiply the numerator and denominator by the complex number conjugate to the denominator and perform the indicated actions (z = x – iy is called conjugate with respect to z = x + iy).
Extraction 
in algebraic form it is practically not used.
Example 2. z 1 = -2 + 5i, z 2 = 1 + 3i
z 1 + z 2 = -1 + 8i
z 1 - z 2 = -3 + 2i
z 1 z 2 = (-2 + 5i)(1 + 3i) = (-2 - 6i + 5i + 15i 2) = -17 – i
Theorem 3. If z 1 =|z 1 |(cosj+i sinj), z 2 =|z 2 |(cos+i sin), then
1) z 1 z 2 = |z 1 | |z 2 |
2) z 1 n = |z 1 | n
3)
4)k = 0,1, 2,.., (n-l)
The operations of addition and subtraction in trigonometric form are not performed in practice.
Example 3.
k = 0, 1, 2
Operations on complex numbers can also be thought of from a geometric point of view. Thus, the operations of adding and subtracting two complex numbers correspond to the operations of adding and subtracting two vectors according to the parallelogram rule.
Example 4. Let z 1 = 3 + i, z 2 = 1 – 3i. Find z 1 + z 2 and z 1 - z 2 geometrically.
We depict each complex number as a radius vector, then we build a parallelogram and find the vector corresponding to z 1 + z 2 and z 1 - z 2 (see Fig. 1)
The geometric meaning of the product of two complex numbers can be found out by multiplying these numbers in trigonometric form. Let Z 1 = |Z 1 |(cos + i sin), Z 1 0,
Z 2 = |Z 2 | (cos + i sin), Z 2 0, then
z 1 z 2 = |z 1 ||z 2 |
When multiplying two complex numbers in trigonometric form, their modules are multiplied and their arguments are added. Therefore, to multiply a complex number z 1 by z 2, you need the length of the vector z 1 change to |z 2 | times (stretch or compress), and then rotate the resulting vector around the origin by an angle arg z 2 (see Fig. 2).
Geometric meaning of the operation 
consists of dividing a circle of radius 
into n equal parts.
Example 5. Calculate 
and depict all its meanings geometrically.
Let's represent the complex number z = - 4 in trigonometric form. To do this, we find its module and argument. |-4|=4arg(-4)=, -4 = 4 (cos + i sin)
Then k = 0,1,2,3
Giving parameter (k) the values 0, 1.2, 3, we get four values of the fourth root of -4.
AND 
Let us represent the found roots on the complex plane; they divide
a circle of radius 2 into four equal parts. In addition, we inscribed a regular quadrangle (square) into this circle.
Often, when solving problems, the geometric meaning of the modulus of the difference of two complex numbers is used, as the distance between two points on a plane. |z 1 - z 2 | = (z 1 , z 2)
Task 1. Find the locus of points for which |z - (2 + i)|< 3
The geometric meaning of this inequality is that the distance from point (2, 1) to points (x, y) should not be more than three units.
This means that the required locus of the points is an open circle (the points of the circle do not belong to it) with the center at the point (2, 1) and radius r = 3
This answer to the problem question could also be obtained algebraically, using the definition of the module
|(x + iy) - (2 + i)|< 3 |(x- 2) + i(y -1)| < 3
(x-2) 2 + (y-l) 2<9
From the geometry course we know that the equality (x-2) 2 + (y - 1) 2 =3 2 is the equation of a circle with center at point (2, 1) and r = 3, then the required g.m.t. will be the inner part of this circle.
