Find the set of all complex numbers. Complex numbers

Topic Complex numbers and polynomials

Lecture22

§1. Complex numbers: basic definitions

Symbol introduced by the ratio
and is called an imaginary unit. In other words,
.

Definition. Expression of the form
where
, is called a complex number, and the number called the real part of the complex number and denote
, number - imaginary part and denote
.

From this definition it follows that real numbers are those complex numbers, the imaginary part of which is equal to zero.

It is convenient to represent complex numbers by points of a plane on which a Cartesian rectangular coordinate system is given, namely: a complex number
match point
and vice versa. On axis
real numbers are depicted and it is called the real axis. Complex numbers of the form

are called purely imaginary. They are represented by dots on the axis
, which is called the imaginary axis. This plane, which serves to represent complex numbers, is called a complex plane. A complex number that is not valid, i.e. such that
are sometimes called imaginary.

Two complex numbers are called equal if and only if their real and imaginary parts coincide.

Addition, subtraction and multiplication of complex numbers is carried out according to the usual rules of the algebra of polynomials, taking into account that

... The division operation can be defined as the inverse of the multiplication operation and prove the uniqueness of the result (if the divisor is nonzero). However, in practice, a different approach is used.

Complex numbers
and
are called conjugate, on the complex plane they are depicted by points symmetric about the real axis. It's obvious that:

1)

;

2)
;

3)
.

Now split on as follows:

It's not hard to show that

where the symbol denotes any arithmetic operation.

Let be
some imaginary number, and Is a real variable. Product of two binomials

is a square trinomial with real coefficients.

Now, with complex numbers at our disposal, we can solve any quadratic equation
.If, then

and the equation has two complex conjugate roots

If
, then the equation has two different real roots. If
, then the equation has two identical roots.

§2. Trigonometric form of a complex number

As mentioned above, the complex number
it is convenient to represent as a point
... You can also identify such a number with the radius vector of this point
... With this interpretation, the addition and subtraction of complex numbers is performed according to the rules of addition and subtraction of vectors. Another form is more convenient for multiplying and dividing complex numbers.

Introduce on the complex plane
polar coordinate system. Then where
,
and complex number
can be written as:

This form of notation is called trigonometric (in contrast to the algebraic form
). In this form, the number is called a module, and - the argument of a complex number ... They are designated:
,

... For the module we have the formula

The argument of the number is not uniquely defined, but up to a term
,
... The value of the argument satisfying the inequalities
, called the main and denoted
... Then,
... For the main value of the argument, you can get the following expressions:

number argument
is considered undefined.

The condition of equality of two complex numbers in trigonometric form is as follows: the absolute values \u200b\u200bof the numbers are equal, and the arguments differ by a multiple
.

Let's find the product of two complex numbers in trigonometric form:

So, when multiplying numbers, their modules are multiplied, and the arguments are added.

Similarly, you can establish that when dividing, the absolute values \u200b\u200bof numbers are divided, and the arguments are subtracted.

Understanding exponentiation as multiple multiplication, you can get the formula for raising a complex number to a power:

Let us derive a formula for
- root -th power of a complex number (not to be confused with the arithmetic root of a real number!). The operation of extracting a root is the reverse of the operation of exponentiation. therefore
Is a complex number such that
.

Let be
known, but
needs to be found. Then

It follows from the equality of two complex numbers in trigonometric form that

,
,
.

From here
(this is the arithmetic root!),

,
.

It is not hard to see that can only accept essentially different values, for example, for
... Finally, we have the formula:

,
.

So the root -th degree of a complex number has different values. On the complex plane, these values \u200b\u200bare located at the vertices correctly -gon inscribed in a circle of radius
centered at the origin. The “first” root has an argument
, the arguments of the two “neighboring” roots differ by
.

Example. Let's extract the cube root of the imaginary unit:
,
,
... Then:

Let us recall the necessary information about complex numbers.

Complex number is an expression like a + biwhere a, b are real numbers, and i - so-called imaginary unit, a symbol whose square is -1, that is i 2 \u003d -1. Number a called real partand the number b - imaginary part complex number z = a + bi... If b \u003d 0, then instead of a + 0i 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 ones: they can be added, subtracted, multiplied and divided by each other. Addition and subtraction occurs according to the rule ( a + bi) ± ( c + di) = (a ± c) + (b ± d)i, and multiplication according to the rule ( a + bi) · ( c + di) = (ac – bd) + (ad + bc)i (it is just used here that i 2 \u003d –1). Number \u003d 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 instance, .)

Complex numbers have a convenient and intuitive geometric representation: the number z = a + bi can be represented by a vector with coordinates ( a; b) on the Cartesian plane (or, which is almost the same, a point - the end of the 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 by the parallelogram rule). By the Pythagorean theorem, the length of the vector with coordinates ( a; b) is equal. This quantity is called module complex number z = a + bi and denoted by | z|. The angle that this vector makes with the positive direction of the abscissa 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 you count in degrees) - after all, it is clear that 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 a positive direction of the abscissa axis, then its coordinates are ( r Cos φ ; r Sin φ ). Hence 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 calculations. Multiplying complex numbers in trigonometric form looks 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 the arguments are added). Hence follow moivre 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 is such a complex number w, what w n = z... It's 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 correct n-gon).

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.

The modulus and argument of a complex number. Trigonometric

complex number form. Operations with complex

numbers in trigonometric form. Moivre's formula.

Initial 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 Is the discriminant of the quadratic equation). For a long time these numbers did not find physical application, therefore they were called "imaginary" numbers. However, now they are very widely used in various fields of physics.

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

Complex numbers are written as: a + bi... Here aand b – real numbers , and i – imaginary unit, i.e.e. i 2 = –1. Number acalled abscissa, a b - ordinate complex numbera + bi.Two complex numbersa + bi and a - bi are called associated complex numbers.

Basic agreements:

1. Real number andcan also be written in the form complex number:a +0 ior a -0 i. For example, records 5 + 0 i and 5 - 0 imean the same number5 .

2. Complex number 0 + bi called purely imaginary number. Recordingbimeans the same as 0 + bi.

3. Two complex numbers a + bi andc + diare considered equal ifa \u003d cand b \u003d d... Otherwise complex numbers are not equal.

Addition. The sum of complex numbersa + bi and c + diis called a complex number (a + c ) + (b + d ) i.In this way, when adding complex numbers, their abscissas and ordinates are added separately.

This definition follows the rules for dealing with ordinary polynomials.

Subtraction. Difference of two complex numbersa + bi (decreasing) and c + di (subtracted) is called a complex number ( a - c ) + (b - d ) i.

In this way, when subtracting two complex numbers, their abscissas and ordinates are subtracted separately.

Multiplication. The product of complex numbersa + bi and c + di 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 binomial,

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

PRI me r. ( a + bi )( a - bi) \u003d a 2 + b 2 . Consequently, composition

two conjugate complex numbers is equal to the real

a positive number.

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

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

PRI me r. Find (8 + i ) : (2 – 3 i) .

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

Multiplying its numerator and denominator by 2 + 3i

AND after performing all the transformations, we get:

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

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

Module complex number is the length of the vector OPrepresenting a complex number on the coordinate ( an integrated) plane. Complex number module a + bi denoted by | a + bi | or letter r

Complex numbers are the minimum extension of the set of real numbers we are used to. Their fundamental difference is that an element appears that gives -1 in the square, i.e. i, or.

Any complex number has two parts: real and imaginary:

Thus, it can be seen that the set of real numbers coincides with the set of complex numbers with zero imaginary part.

The most popular model for a set of complex numbers is the regular plane. The first coordinate of each point will be its real part, and the second one will be imaginary. Then vectors with the origin at point (0,0) will act as the complex numbers themselves.

Operations on complex numbers.

In fact, if we take into account the model of a set of complex numbers, it is intuitively clear that addition (subtraction) and multiplication of two complex numbers are performed in the same way as the corresponding operations on vectors. And we mean the vector product of vectors, because the result of this operation is again a vector.

1.1 Addition.

(As you can see, this operation exactly matches)

1.2 Subtraction, similarly, is performed according to the following rule:

2. Multiplication.