Complex numbers

Imaginary And complex numbers. Abscissa and ordinate

complex number. Conjugate complex numbers.

Operations with complex numbers. Geometric

representation of complex numbers. Complex plane.

Modulus and argument of a complex number. Trigonometric

complex number form. Operations with complex

numbers in trigonometric form. Moivre's formula.

Basic information about imaginary And complex numbers are given in the section “Imaginary and complex numbers”. The need for these numbers of a new type arose when solving quadratic equations for the caseD< 0 (здесь D– discriminant quadratic equation). For a long time these numbers were not found physical application, which is why they were called “imaginary” numbers. However, now they are very widely used in various fields of physics.

and technology: electrical engineering, hydro- and aerodynamics, elasticity theory, etc.

Complex numbers are written in the form:a+bi. Here a And b – real numbers , A i – imaginary unit, i.e. e. i 2 = –1. Number a called abscissa,a b – ordinatecomplex numbera + bi.Two complex numbersa+bi And a–bi are called conjugate complex numbers.

Main agreements:

1. Real numberAcan also be written in the formcomplex number:a+ 0 i or a – 0 i. For example, records 5 + 0i and 5 – 0 imean the same number 5 .

2. Complex number 0 + bicalled purely imaginary number. Recordbimeans the same as 0 + bi.

3. Two complex numbersa+bi Andc + diare considered equal ifa = c And b = d. Otherwise complex numbers are not equal.

Addition. Sum of complex numbersa+bi And c + diis called a complex number (a+c ) + (b+d ) i.Thus, when adding complex numbers, their abscissas and ordinates are added separately.

This definition corresponds to the rules for operations with ordinary polynomials.

Subtraction. The difference of two complex numbersa+bi(diminished) and c + di(subtrahend) is called a complex number (a–c ) + (b–d ) i.

Thus, When subtracting two complex numbers, their abscissas and ordinates are subtracted separately.

Multiplication. Product of complex numbersa+bi And c + di is called a complex number:

(ac–bd ) + (ad+bc ) i.This definition follows from two requirements:

1) numbers a+bi And c + dimust be multiplied like algebraic binomials,

2) number ihas the main property:i 2 = – 1.

EXAMPLE ( a+ bi )(a–bi) = a 2 + b 2 . Hence, work

two conjugate complex numbers is equal to the real

a positive number.

Division. Divide a complex numbera+bi (divisible) by anotherc + di(divider) - means to find the third numbere + f i(chat), which when multiplied by a divisorc + di, results in the dividenda + bi.

If the divisor is not zero, division is always possible.

EXAMPLE Find (8 +i ) : (2 – 3 i) .

Solution. Let's rewrite this ratio as a fraction:

Multiplying its numerator and denominator by 2 + 3i

AND Having performed all the transformations, we get:

Geometric representation of complex numbers. Real numbers are represented by points on the number line:

Here is the point Ameans the number –3, dotB– number 2, and O- zero. In contrast, complex numbers are represented by dots on coordinate plane. For this purpose, we choose rectangular (Cartesian) coordinates with the same scales on both axes. Then the complex numbera+bi will be represented by a dot P with abscissa a and ordinate b (see picture). This coordinate system is called complex plane .

Module complex number is the length of the vectorOP, representing a complex number on the coordinate ( comprehensive) plane. Modulus of a complex numbera+bi denoted | a+bi| or letter r

Go) numbers.

2. Algebraic form of representation of complex numbers

Complex number or complex, is a number consisting of two numbers (parts) – real and imaginary.

Real is called any positive or a negative number, for example, + 5, - 28, etc. Let's denote a real number by the letter “L”.

Imaginary is a number equal to the product of a real number and Square root from a negative unit, for example, 8, - 20, etc.

A negative unit is called imaginary and is denoted by the letter “yot”:

Let us denote the real number in the imaginary number by the letter “M”.

Then the imaginary number can be written like this: j M. In this case, the complex number A can be written like this:

A = L + j M (2).

This form of writing a complex number (complex), which is algebraic sum real and imaginary parts is called algebraic.

Example 1. Represent in algebraic form a complex whose real part is 6 and whose imaginary part is 15.

Solution. A = 6 +j 15.

In addition to the algebraic form, a complex number can be represented by three more:

1. graphic;

2. trigonometric;

3. indicative.

Such a variety of forms is dramatically simplifies calculations sinusoidal quantities and their graphical representation.

Let's look at the graphical, trigonometric and exponent in turn.

new forms of representing complex numbers.

Graphical form of representing complex numbers

For graphical representation of complex numbers, direct

carbon coordinate system. In a regular (school) coordinate system, positive or negative values ​​are plotted along the “x” (abscissa) and “y” (ordinate) axes. real numbers.

In the coordinate system adopted in the symbolic method, along the “x” axis

real numbers are plotted in the form of segments, and imaginary numbers are plotted along the “y” axis

Rice. 1. Coordinate system for graphical representation of complex numbers

Therefore, the x-axis is called the axis of real quantities or, for short, real axis.

The ordinate axis is called the axis of imaginary quantities or imaginary axis.

The plane itself (i.e., the plane of the drawing), on which complex numbers or quantities are depicted, is called comprehensive flat.

In this plane, the complex number A = L + j M is represented by the vector A

(Fig. 2), the projection of which onto the real axis is equal to its real part Re A = A" = L, and the projection onto the imaginary axis is equal to the imaginary part Im A = A" = M.

(Re - from the English real - real, real, real, Im - from the English imaginary - unreal, imaginary).

Rice. 2. Graphical representation of a complex number

In this case, the number A can be written as follows

A = A" + A" = Re A + j Im A (3).

Using a graphical representation of the number A in the complex plane, we introduce new definitions and obtain some important relationships:

1. the length of vector A is called module vector and is denoted by |A|.

According to the Pythagorean theorem

|A| = (4) .

2. angle α formed by vector A and real positive half-

the axis is called argument vector A and is determined through its tangent:

tg α = A" / A" = Im A / Re A (5).

Thus, for a graphical representation of a complex number

A = A" + A" in the form of a vector you need:

1. find the modulus of the vector |A| according to formula (4);

2. find the argument of the vector tan α using formula (5);

3. find the angle α from the relation α = arc tan α;

4. in the coordinate system j (x) draw an auxiliary

straight line and on it, on a certain scale, plot a segment equal to the absolute value of the vector |A|.

Example 2. Present the complex number A = 3 + j 4 in graphical form.

Complex numbers and
coordinate
plane

The geometric model of the set R of real numbers is the number line. Any real number corresponds to a single point

on
number line and any point on the line
only one matches
real number!

By adding one more dimension to the number line corresponding to the set of all real numbers - the line containing the set of pure numbers

By adding to the number line corresponding to the set
everyone real numbers one more dimension -
a straight line containing a set of purely imaginary numbers –
we obtain a coordinate plane in which each
the complex number a+bi can be associated
point (a; b) of the coordinate plane.
i=0+1i corresponds to point (0;1)
2+3i corresponds to point (2;3)
-i-4 corresponds to point (-4;-1)
5=5+1i corresponds to melancholy (5;0)

Geometric meaning of the conjugation operation

! The mating operation is axial
symmetry about the abscissa axis.
!! Conjugated to each other
complex numbers are equidistant from
origin.
!!! Vectors depicting
conjugate numbers, inclined to the axis
abscissa at the same angle, but
located on opposite sides of
this axis.

Image of real numbers

Picture of complex numbers

Algebraic
way
Images:
Complex number
a+bi is depicted
plane point
with coordinates
(a;b)

Examples of depicting complex numbers on the coordinate plane

(We are interested
complex numbers
z=x+yi , for which
x=-4. This is the equation
straight,
parallel axis
ordinate)
at
X= - 4
Valid
part is -4
0
X

Draw on the coordinate plane the set of all complex numbers for which:

Imaginary part
is even
unambiguous
natural
number
(We are interested
complex numbers
z=x+yi, for which
y=2,4,6,8.
Geometric image
consists of four
straight, parallel
x-axis)
at
8
6
4
2
0
X

Specifying a complex number is equivalent to specifying two real numbers a, b - the real and imaginary parts of a given complex number. But an ordered pair of numbers is depicted in Cartesian rectangular system coordinates by a point with coordinates. Thus, this point can serve as an image for the complex number z: a one-to-one correspondence is established between complex numbers and points of the coordinate plane. When using the coordinate plane to depict complex numbers, the Ox axis is usually called the real axis (since the real part of the number is taken as the abscissa of the point), and the Oy axis is the imaginary axis (since the imaginary part of the number is taken as the ordinate of the point). The complex number z represented by the point (a, b) is called the affix of this point. In this case, real numbers are represented by points lying on the real axis, and all purely imaginary numbers (for a = 0) are represented by points lying on the imaginary axis. The number zero is represented by the point O.

In Fig. 8 images of numbers are constructed.

Two complex conjugate numbers are represented by points symmetrical about the Ox axis (points in Fig. 8).

Often associated with a complex number is not only the point M, representing this number, but also the vector OM (see paragraph 93), leading from O to M; The representation of a number as a vector is convenient from the point of view of the geometric interpretation of the action of addition and subtraction of complex numbers.

In Fig. 9, a it is shown that the vector representing the sum of complex numbers is obtained as the diagonal of a parallelogram constructed on vectors representing the terms.

This rule for adding vectors is known as the parallelogram rule (for example, for adding forces or velocities in a physics course). Subtraction can be reduced to addition with the opposite vector (Fig. 9, b).

As is known (item 8), the position of a point on the plane can also be specified by its polar coordinates. Thus, the complex number - the affix of the point will also be determined by the task From Fig. 10 it is clear that at the same time the modulus of a complex number is: the polar radius of the point representing the number is equal to the modulus of this number.

The polar angle of a point M is called the argument of the number represented by this point. The argument of a complex number (like the polar angle of a point) is not defined ambiguously; if is one of its values, then all its values ​​are expressed by the formula

All values ​​of the argument are collectively denoted by the symbol.

So, any complex number can be associated with a pair of real numbers: the modulus and the argument of the given number, and the argument is determined ambiguously. On the contrary, it corresponds to the given module and argument singular, having the given module and argument. The number zero has special properties: its modulus is zero, and no specific value is assigned to its argument.

To achieve unambiguity in the definition of the argument of a complex number, one can agree to call one of the values ​​of the argument the main one. It is designated by the symbol. Typically, the main value of the argument is chosen to be a value that satisfies the inequalities

(in other cases inequalities).

Let us also pay attention to the values ​​of the argument of real and purely imaginary numbers:

The real and imaginary parts of a complex number (as the Cartesian coordinates of a point) are expressed through its modulus and argument (polar coordinates of the point) using formulas (8.3):

and a complex number can be written in the following trigonometric form.