The root of a complex number in algebraic form. Complex numbers and algebraic operations on them

Consider a quadratic equation.

Let's define its roots.

There is no real number whose square is -1. But if the formula defines the operator i as an imaginary unit, then the solution of this equation can be written in the form . 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 (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 are real numbers, is an 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) called real and imaginary parts numbers z. The expression is called algebraic form notation of 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 the 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 AC circuits, analog electronics, signal theory and processing, automatic control theory, and other applied sciences.

1. Arithmetic of complex numbers

The addition of two complex numbers consists in adding their real and imaginary parts, i.e.

Accordingly, the difference of two complex numbers

Complex number called complex conjugate number z=x +i.y.

The 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 because .

The product (multiplication) of two complex numbers can be calculated as follows:

Division of two complex numbers:

Example:

1. Complex plane

A complex number can be graphically represented in a rectangular coordinate system. Let us set a rectangular coordinate system in the plane (x, y).

on axle Ox we will arrange the real parts x, it is called real (real) axis, on the axis Oy– imaginary parts y complex numbers. She bears the name imaginary axis. Moreover, each complex number corresponds to a certain point of the plane, and such a plane is called complex plane. Point BUT 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 displayed as dots located symmetrically about the real axis.

If on the plane 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.

Complex number modulus always non-negative. The argument of a complex number is not uniquely defined. The main value of the argument must satisfy the condition . Each point of the complex plane also corresponds to the total value of the argument . Arguments that differ by a multiple of 2π are considered equal. The number argument zero is not defined.

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.

1. The exponential form of complex numbers

Decomposition in Maclaurin series for real argument functions looks like:

For the exponential function of a complex argument z 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 formula.

For a negative argument, it looks like

By combining these expressions, we can define the following expressions for sine and cosine

.

Using the Euler formula, from the trigonometric form of the representation of complex numbers

you can get it demonstrative(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 the exponential form, it is easy to define the following formulas for multiplication and division of complex numbers

That is, in exponential form, the product and division of complex numbers is easier 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 the i vector z rotates counterclockwise by 90

In division, the numerator modulus is divided by the denominator modulus, and the denominator argument is subtracted from the numerator argument.

Using the exponential form of complex numbers, one can obtain expressions for well-known trigonometric identities. For example, from the identity

using the Euler 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 the angles

1. Powers, roots and logarithms of complex numbers

Raising a complex number to a natural power n produced according to the formula

Example. Compute .

Imagine a number in trigonometric form

’

Applying the exponentiation formula, we get

Putting the value in the expression r= 1, we get the so-called De Moivre's formula, with which you can determine the 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 .

To do this, we express the complex number () to the trigonometric form

.

According to the formula for calculating the root of a complex number, we get

Logarithm of a complex number z is a 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 real (cosine) and imaginary (sine) parts. Such stress can be represented as a vector of length U m, 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 a real function, then its real and imaginary parts are multiplied by the same factor. Differentiation/integration of such a complex function is reduced to differentiation/integration of the real and imaginary parts.

For example, the differentiation of the complex stress expression

is to multiply it by iω is the real part of the function f(z), and is the 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 pass into the lines of the plane w, figures of one plane into figures of another, but the shapes of lines or figures may change significantly.