Consider a quadratic equation.
Let's determine its roots.
There is no real number whose square is -1. But if we define the operator with a formula i as an imaginary unit, then the solution to this equation can be written as 
. Wherein 
And 
- complex numbers in which -1 is the real part, 2 or in the second case -2 is the imaginary part. The imaginary part is also a real number. The imaginary part multiplied by the imaginary unit means already imaginary number.
In general, a complex number has the form
z = x + iy ,
Where x, y– real numbers, – imaginary unit. In a number of applied sciences, for example, in electrical engineering, electronics, signal theory, the imaginary unit is denoted by j. Real numbers x = Re(z) And y=Im(z) are called real and imaginary parts numbers z. The expression is called algebraic form writing a complex number.
Any real number is a special case of a complex number in the form 
. An imaginary number is also a special case of a complex number 
.
Definition of the set of complex numbers C
This expression reads as follows: set WITH, consisting of elements such that x And y belong to the set of real numbers R and is an imaginary unit. Note that, etc.
Two complex numbers 
And 
are equal if and only if their real and imaginary parts are equal, i.e. And .
Complex numbers and functions are widely used in science and technology, in particular, in mechanics, analysis and calculation of alternating current circuits, analog electronics, in the theory and processing of signals, in the theory of automatic control and other applied sciences.
- Complex number arithmetic
The addition of two complex numbers consists of adding their real and imaginary parts, i.e.
Accordingly, the difference of two complex numbers
Complex number 
called comprehensively conjugate number z =x+iy.
Complex conjugate numbers z and z * differ in the signs of the imaginary part. It's obvious that
.
Any equality between complex expressions remains valid if in this equality everywhere i replaced by -
i, i.e. go to the equality of conjugate numbers. Numbers i And –
i are algebraically indistinguishable, since 
.
The product (multiplication) of two complex numbers can be calculated as follows:
Division of two complex numbers:
Example:
- Complex plane
A complex number can be represented graphically in a rectangular coordinate system. Let us define a rectangular coordinate system in the plane (x, y).
On axis Ox we will place the real parts x, it is called real (real) axis, on the axis Oy–imaginary parts y complex numbers. It's called imaginary axis. In this case, each complex number corresponds to a certain point on the plane, and such a plane is called complex plane. Point A the complex plane will correspond to the vector OA.
Number x called abscissa complex number, number y – ordinate.
A pair of complex conjugate numbers is represented by points located symmetrically about the real axis.
 |
If on the plane we set polar coordinate system, then every complex number z determined by polar coordinates. Wherein module numbers 
is the polar radius of the point, and the angle 
- its polar angle or complex number argument z.
Modulus of a complex number 
always non-negative. The argument of a complex number is not uniquely determined. The main value of the argument must satisfy the condition 
. Each point of the complex plane also corresponds general meaning argument. Arguments that differ by a multiple of 2π are considered equal. The number zero argument is undefined.
The main value of the argument is determined by the expressions:
It's obvious that
Wherein
, 
.
Complex number representation z as
called trigonometric form complex number.
Example.
- Exponential form of complex numbers
Decomposition in Maclaurin series for real argument functions 
has the form:
For an exponential function with a complex argument z the decomposition is similar
.
The Maclaurin series expansion for the exponential function of the imaginary argument can be represented as
The resulting identity is called Euler's formula.
For a negative argument it has the form
By combining these expressions, you can define the following expressions for sine and cosine
.
Using Euler's formula, from the trigonometric form of representing complex numbers
available indicative(exponential, polar) form of a complex number, i.e. its representation in the form
,
Where 
- polar coordinates of a point with rectangular coordinates ( x,y).
The conjugate of a complex number is written in exponential form as follows.
For exponential form, it is easy to determine the following formulas for multiplying and dividing complex numbers
That is, in exponential form, the product and division of complex numbers is simpler than in algebraic form. When multiplying, the modules of the factors are multiplied, and the arguments are added. This rule applies to any number of factors. In particular, when multiplying a complex number z on i vector z rotates counterclockwise 90
In division, the modulus of the numerator is divided by the modulus of the denominator, and the argument of the denominator is subtracted from the argument of the numerator.
Using the exponential form of complex numbers, we can obtain expressions for the well-known trigonometric identities. For example, from the identity
using Euler's formula we can write
Equating the real and imaginary parts in this expression, we obtain expressions for the cosine and sine of the sum of angles
- Powers, roots and logarithms of complex numbers
Raising a complex number to a natural power n produced according to the formula
Example. Let's calculate 
.
Let's imagine a number in trigonometric form
’
Applying the exponentiation formula, we get
By putting the value in the expression r= 1, we get the so-called Moivre's formula, with which you can determine expressions for the sines and cosines of multiple angles.
Root n-th power of a complex number z It has n different values determined by the expression
Example. Let's find it.
To do this, we express the complex number () in trigonometric form
.
Using the formula for calculating the root of a complex number, we get
Logarithm of a complex number z- this is the number w, for which . The natural logarithm of a complex number has an infinite number of values and is calculated by the formula
Consists of a real (cosine) and imaginary (sine) part. This voltage can be represented as a vector of length Um, initial phase (angle), rotating with angular velocity ω .
Moreover, if complex functions are added, then their real and imaginary parts are added. If a complex function is multiplied by a constant or real function, then its real and imaginary parts are multiplied by the same factor. Differentiation/integration of such a complex function comes down to differentiation/integration of the real and imaginary parts.
For example, differentiating the complex stress expression
is to multiply it by iω is the real part of the function f(z), and 
– imaginary part of the function. Examples: 
.
Meaning z is represented by a point in the complex z plane, and the corresponding value w- a point in the complex plane w. When displayed w = f(z) plane lines z transform into plane lines w, figures of one plane into figures of another, but the shapes of the lines or figures can change significantly.