TopicComplex numbers and polynomials

Lecture 22

§one. Complex numbers: basic definitions

Symbol enter the ratio
and is called the imaginary unit. In other words,
.

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

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

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

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

Two complex numbers are said to be equal if and only if they have the same real and imaginary parts.

Addition, subtraction and multiplication of complex numbers are performed according to the usual rules of polynomial algebra, taking into account the fact that

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

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

1)

;

2)
;

3)
.

Now split on the can be done as follows:

.

It is not difficult to show that

,

where symbol stands for any arithmetic operation.

Let
some imaginary number, and is a real variable. The product of two binomials

is a square trinomial with real coefficients.

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

and the equation has two complex conjugate roots

.

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

§2. Trigonometric form of a complex number

As mentioned above, the complex number
convenient to represent with a dot
. One 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. For multiplication and division of complex numbers, another form is more convenient.

We 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 - complex number argument . They are marked:
,

. For the module, we have the formula

The number argument is defined ambiguously, but up to a term
,
. The value of the argument that satisfies the inequalities
, is called principal and denoted
. Then,
. For the main value of the argument, you can get the following expressions:

,

number argument
considered to be undefined.

The condition for the equality of two complex numbers in trigonometric form has the form: the modules of the numbers are equal, and the arguments differ by a multiple
.

Find the product of two complex numbers in trigonometric form:

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

Similarly, it can be established that when dividing, the modules of numbers are divided, and the arguments are subtracted.

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

We derive a formula for
- root th power of a complex number (not to be confused with the arithmetic root of a real number!). The root extraction operation is the inverse of the exponentiation operation. That's why
is a complex number such that
.

Let
known, and
required to be found. Then

From the equality of two complex numbers in trigonometric form, it follows that

,
,
.

From here
(it's an arithmetic root!),

,
.

It is easy to verify that can only accept essentially different values, for example, when
. Finally we have the formula:

,
.

So the root th degree from 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 two “neighboring” roots differ by
.

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

,

The topic "Complex numbers" often causes difficulties for students, but in fact there is nothing terrible in them, as it might seem at first glance.

So, now we will analyze and consider with simple examples what a complex number is, how it is denoted and what it consists of. Expression z = a + bi is called a complex number. It's a single number, not an addition.

Example 1 : z = 6 + 4i

What is a complex number?

A complex number has a real and an imaginary part in its composition.

The number a is called the real part of the complex number and is denoted a = Re(z). And here is what stands with the letter i- i.e. number b is called the coefficient of the imaginary part of a complex number and is denoted b = Im(z). Together bi form the imaginary part of a complex number.

It is easy to guess and easy to remember that the abbreviation "Re" comes from the word Real- real, real part. Respectively, "im" is an abbreviation of the word «Imaginary» imaginary part.

Example 2 : z = 0.5 + 9i. Here is the real part a=Re(z)=0.5, and the imaginary part b = Im(z) = 9i

Example 3 : z = -5 + 19i. Here is the real part a=Re(z)=-5, and the imaginary part b=Im(z)=19.

Purely imaginary complex number

A complex number that does not have a real part, i.e. Re(z) = 0, is called purely imaginary.

Example 4 : z = 2i. The real part is missing a = Re(z) = 0, and the imaginary part b = Im(z) = 2.

Example 5 . z=-8i. Here is the imaginary part b=Im(z)=-8, real part a = Re(z) = 0.

Conjugate complex numbers

The complex conjugate number is denoted "z" with a bar and is used, for example, to find the quotient of two complex numbers, in other words, to implement the division of numbers. Those who are now thinking, you are here - read about the division of complex numbers.

The numbers are called complex conjugate, they have the same real parts and differ only in the sign of the imaginary parts. Consider an example:

Example 6 . Complex conjugate of a number z = 7 + 13i is a number.

Imaginary unit of a complex number

And finally, let's talk about the letter i. The same letter that forms the imaginary component in a complex number. Even if we have an expression z=5, it simply means that the imaginary part of the given number is zero and the real part is five.

Value i called imaginary unit.

The imaginary unit is useful when solving quadratic equations in the case when the discriminant is less than zero. We are accustomed to believing that if it is negative, there is no solution, no roots. This is not entirely correct. Roots exist, they're just complex. But more on that later. And now, let's move on to the next article on the study of complex numbers, we will learn how to calculate

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- so-called imaginary unit, the symbol whose square is -1, i.e. i 2 = -1. Number a called real part, and the number b - imaginary part complex number z = a + bi. If a b= 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 proceed 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(here it is just 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: the number z = a + bi can be represented as 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 to . This value 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 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 turning through 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 ). 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. Multiplication of complex numbers in trigonometric form looks very simple: z one · 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 moduli are multiplied and the arguments are added). From here follow De Moivre formulas: z n = |z|n(cos( n(Arg z)) + i sin( n(Arg z))). With the help of 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- one). This means that there is always exactly n roots n th degree from a complex number (on the plane they are located at the vertices of a regular n-gon).