When studying the properties quadratic equation a restriction was set - for a discriminant less than zero, there is no solution. It was immediately stated that we're talking about about the set of real numbers. The inquisitive mind of a mathematician will be interested in what secret is contained in the clause about real values?

Over time, mathematicians introduced the concept of complex numbers, where one is taken as conditional meaning second degree root of minus one.

Historical reference

Mathematical theory develops sequentially, from simple to complex. Let's figure out how the concept called “complex number” arose and why it is needed.

Since time immemorial, the basis of mathematics has been ordinary counting. The researchers knew only the natural set of values. Addition and subtraction were simple. As economic relations become more complex, instead of adding identical values started using multiplication. The inverse operation to multiplication appeared - division.

The concept of a natural number limited the use arithmetic operations. It is impossible to solve all division problems on a set of integer values. led first to the concept rational values, and then to irrational values. If for rational it is possible to indicate the exact location of a point on a line, then for irrational it is impossible to indicate such a point. You can only approximately indicate the location interval. Combining rational and irrational numbers formed a real set, which can be represented as a certain line with a given scale. Each step along the line is a natural number, and between them are rational and irrational values.

An era has begun theoretical mathematics. The development of astronomy, mechanics, and physics required solutions to increasingly complex equations. In general form, the roots of the quadratic equation were found. When solving more complex cubic polynomial scientists are faced with a contradiction. Concept cube root from the negative it makes sense, but for the square it results in uncertainty. In this case, the quadratic equation is only special case cubic.

In 1545, the Italian G. Cardano proposed introducing the concept of an imaginary number.

This number became the second root of minus one. The term complex number was finally formed only three hundred years later, in the works famous mathematician Gauss. He proposed to formally extend all the laws of algebra to an imaginary number. The real line has expanded to a plane. The world has become bigger.

Basic Concepts

Let us recall a number of functions that have restrictions on the real set:

• y = arcsin(x), defined in the range of values ​​between negative and positive unity.
• y = ln(x), makes sense for positive arguments.
• Square root y = √x, calculated only for x ≥ 0.

By denoting i = √(-1), we introduce such a concept as an imaginary number, this will allow us to remove all restrictions from the domain of definition of the above functions. Expressions like y = arcsin(2), y = ln(-4), y = √(-5) take on meaning in a certain space of complex numbers.

The algebraic form can be written as z = x + i×y on the set of real values ​​x and y, and i 2 = -1.

The new concept removes all restrictions on the use of any algebraic function and its appearance resembles a graph of a straight line in the coordinates of real and imaginary values.

Complex plane

Geometric shape complex numbers makes it possible to visualize many of their properties. Along the Re(z) axis we mark the real values ​​of x, along the Im(z) - imaginary values ​​of y, then the point z on the plane will display the required complex value.

Definitions:

• Re(z) - real axis.
• Im(z) - means the imaginary axis.
• z is the conditional point of a complex number.
• Numerical value vector length from zero point to z is called modulus.
• The real and imaginary axes divide the plane into quarters. At positive value coordinates - I quarter. When the argument of the real axis is less than 0, and the imaginary axis is greater than 0 - the second quarter. When the coordinates are negative - III quarter. The last, IV quarter contains many positive real values ​​and negative imaginary values.

Thus, on a plane with coordinates x and y, you can always visually depict a point of a complex number. The symbol i is introduced to separate the real part from the imaginary part.

Properties

1. With a zero value of the imaginary argument, we simply obtain a number (z = x), which is located on the real axis and belongs to the real set.
2. A special case, when the value of the real argument becomes zero, the expression z = i×y corresponds to the location of the point on the imaginary axis.
3. The general form z = x + i×y will be for non-zero values ​​of the arguments. Indicates the location of the point characterizing a complex number in one of the quarters.

Trigonometric notation

Let us recall the polar coordinate system and definition of sin and cos. Obviously, using these functions you can describe the location of any point on the plane. To do this, it is enough to know the length of the polar ray and the angle of inclination to the real axis.

Definition. Notation of the form ∣z ∣, multiplied by the sum of trigonometric cos functions(ϴ) and the imaginary part i ×sin(ϴ), is called a trigonometric complex number. Here we use the notation angle of inclination to the real axis

ϴ = arg(z), and r = ∣z∣, the beam length.

From the definition and properties of trigonometric functions, it follows very important formula Moivre:

z n = r n × (cos(n × ϴ) + i × sin(n × ϴ)).

Using this formula, it is convenient to solve many systems of equations containing trigonometric functions. Especially when the problem of exponentiation arises.

Module and phase

To complete the description complex set we'll offer two important definitions.

Knowing the Pythagorean theorem, it is easy to calculate the length of the ray in polar system coordinates

r = ∣z∣ = √(x 2 + y 2), such a notation in complex space is called “modulus” and characterizes the distance from 0 to a point on the plane.

The angle of inclination of the complex ray to the real line ϴ is usually called the phase.

From the definition it is clear that the real and imaginary parts are described using cyclic functions. Namely:

• x = r × cos(ϴ);
• y = r × sin(ϴ);

Conversely, the phase has a connection with algebraic values through the formula:

ϴ = arctan(x / y) + µ, the correction µ is introduced to take into account the periodicity geometric functions.

Euler's formula

Mathematicians often use exponential form. The numbers of the complex plane are written as the expression

z = r × e i × ϴ, which follows from Euler’s formula.

I received this entry wide use for practical calculation physical quantities. The form of representation in the form of exponential complex numbers is especially convenient for engineering calculations, where there is a need to calculate circuits with sinusoidal currents and it is necessary to know the value of the integrals of functions with a given period. The calculations themselves serve as a tool in the design of various machines and mechanisms.

Defining Operations

As already noted, all algebraic laws of working with basic mathematical functions apply to complex numbers.

Sum operation

When adding complex values, their real and imaginary parts also add up.

z = z 1 + z 2, where z 1 and z 2 are complex numbers general view. Transforming the expression, after opening the brackets and simplifying the notation, we get real argument x=(x 1 + x 2), imaginary argument y = (y 1 + y 2).

On the graph it looks like the addition of two vectors, according to the well-known parallelogram rule.

Subtraction operation

It is considered as a special case of addition, when one number is positive, the other is negative, that is, located in the mirror quarter. The algebraic notation looks like the difference between the real and imaginary parts.

z = z 1 - z 2 , or, taking into account the values ​​of the arguments, similar to the addition operation, we obtain for real values ​​x = (x 1 - x 2) and imaginary values ​​y = (y 1 - y 2).

Multiplication in the complex plane

Using the rules for working with polynomials, we will derive a formula for solving complex numbers.

Following the general algebraic rules z=z 1 ×z 2, we describe each argument and present similar ones. The real and imaginary parts can be written as follows:

• x = x 1 × x 2 - y 1 × y 2,
• y = x 1 × y 2 + x 2 × y 1.

It looks more beautiful if we use exponential complex numbers.

The expression looks like this: z = z 1 × z 2 = r 1 × e i ϴ 1 × r 2 × e i ϴ 2 = r 1 × r 2 × e i(ϴ 1+ ϴ 2) .

Division

When considering the division operation as the inverse of the multiplication operation, in exponential notation we obtain a simple expression. Dividing the value of z 1 by z 2 is the result of dividing their modules and the phase difference. Formally, when using the exponential form of complex numbers, it looks like this:

z = z 1 / z 2 = r 1 × e i ϴ 1 / r 2 × e i ϴ 2 = r 1 / r 2 × e i(ϴ 1- ϴ 2) .

In the form of algebraic notation, the operation of dividing numbers in the complex plane is written a little more complicated:

By describing the arguments and carrying out transformations of polynomials, it is easy to obtain the values ​​x = x 1 × x 2 + y 1 × y 2 , respectively y = x 2 × y 1 - x 1 × y 2 , however, within the framework of the described space this expression makes sense, if z 2 ≠ 0.

Extracting the root

All of the above can be used to define more complex algebraic functions - raising to any power and its inverse - extracting the root.

Taking advantage general concept raising to the power n, we get the definition:

z n = (r × e i ϴ) n .

Using general properties, we rewrite it in the form:

z n = r n × e i ϴ n .

Got simple formula raising a complex number to a power.

From the definition of degree we obtain a very important corollary. An even power of the imaginary unit is always equal to 1. Any odd power of the imaginary unit is always equal to -1.

Now let's study inverse function- root extraction.

For simplicity of notation, we take n = 2. The square root w of a complex value z on the complex plane C is usually considered to be the expression z = ±, valid for any real argument greater than or equal to zero. For w ≤ 0 there is no solution.

Let's look at the simplest quadratic equation z 2 = 1. Using the formulas for complex numbers, we rewrite r 2 × e i 2ϴ = r 2 × e i 2ϴ = e i 0. From the record it is clear that r 2 = 1 and ϴ = 0, therefore, we have only decision, equal to 1. But this contradicts the concept that z = -1, also consistent with the definition of a square root.

Let's figure out what we don't take into account. If we remember trigonometric notation, then we restore the statement - when periodic change phase ϴ the complex number does not change. Let us denote the value of the period by the symbol p, then the following is valid: r 2 × e i 2ϴ = e i (0+ p), from which 2ϴ = 0 + p, or ϴ = p / 2. Therefore, e i 0 = 1 and e i p /2 = -1 . We received the second solution, which corresponds to common understanding square root.

So, to find an arbitrary root of a complex number, we will follow the procedure.

• Let's write the exponential form w= ∣w∣ × e i (arg (w) + pk), k is an arbitrary integer.
• We can also represent the required number using the Euler form z = r × e i ϴ .
• Let's take advantage general definition root extraction functions r n *e i n ϴ = ∣w∣ × e i (arg (w) + pk) .
• From general properties equality of modules and arguments, we write r n = ∣w∣ and nϴ = arg (w) + p×k.
• The final notation for the root of a complex number is described by the formula z = √∣w∣ × e i (arg (w) + pk) / n.
• Comment. The value ∣w∣, by definition, is a positive real number, which means that the root of any power makes sense.

Field and mate

In conclusion, we give two important definitions that have little significance for the solution applied problems with complex numbers, but are significant for further development mathematical theory.

Expressions for addition and multiplication are said to form a field if they satisfy the axioms for any elements of the complex plane z:

1. Changing the places of complex terms does not change the complex sum.
2. The statement is true - in complex expression any sum of two numbers can be replaced by their value.
3. There is a neutral value 0 for which z + 0 = 0 + z = z is true.
4. For any z there is an opposite - z, the addition of which gives zero.
5. When changing the places of complex factors, the complex product does not change.
6. The multiplication of any two numbers can be replaced by their value.
7. There is a neutral value 1, multiplying by which does not change the complex number.
8. For every z ≠ 0, there is an inverse value z -1, multiplying by which results in 1.
9. Multiplying the sum of two numbers by a third is equivalent to the operation of multiplying each of them by this number and adding the results.
10. 0 ≠ 1.

The numbers z 1 = x + i×y and z 2 = x - i×y are called conjugate.

Theorem. For pairing, the following statement is true:

• The conjugate of a sum is equal to the sum of the conjugate elements.
• The conjugate of a product is equal to the product of conjugates.
• equal to the number itself.

In general algebra, such properties are usually called field automorphisms.

Examples

Following the given rules and formulas for complex numbers, you can easily operate with them.

Let's look at the simplest examples.

Task 1. Using the equation 3y +5 x i= 15 - 7i, determine x and y.

Solution. Let us recall the definition of complex equalities, then 3y = 15, 5x = -7. Therefore x = -7 / 5, y = 5.

Task 2. Calculate the values ​​of 2 + i 28 and 1 + i 135.

Solution. Obviously 28 - even number, from the corollary of the definition of a complex number to the power we have i 28 = 1, which means the expression is 2 + i 28 = 3. The second value, i 135 = -1, then 1 + i 135 = 0.

Task 3. Calculate the product of the values ​​2 + 5i and 4 + 3i.

Solution. From the general properties of multiplication of complex numbers we obtain (2 + 5i)X(4 + 3i) = 8 - 15 + i(6 + 20). The new value will be -7 + 26i.

Task 4. Calculate the roots of the equation z 3 = -i.

Solution. There may be several options for finding a complex number. Let's consider one of the possible ones. By definition, ∣ - i∣ = 1, the phase for -i is -p / 4. The original equation can be rewritten as r 3 *e i 3ϴ = e - p/4+ pk, from where z = e - p / 12 + pk /3 , for any integer k.

The set of solutions has the form (e - ip/12, e ip /4, e i 2 p/3).

Why are complex numbers needed?

History knows many examples when scientists, working on a theory, do not even think about the practical application of their results. Mathematics is, first of all, a game of the mind, a strict adherence to cause-and-effect relationships. Almost all mathematical constructions are reduced to solving integral and differential equations, and those, in turn, with some approximation, are solved by finding the roots of the polynomials. Here we first encounter the paradox of imaginary numbers.

Natural scientists, deciding completely practical problems, resorting to solutions different equations, discover mathematical paradoxes. The interpretation of these paradoxes leads to completely surprising discoveries. Dual nature electromagnetic waves one such example. Complex numbers play a decisive role in understanding their properties.

This, in turn, found practical use in optics, radio electronics, energy and many other technological fields. Another example, much harder to understand physical phenomena. Antimatter was predicted at the tip of the pen. And only many years later attempts to physically synthesize it begin.

One should not think that such situations exist only in physics. No less interesting discoveries occur in living nature, during the synthesis of macromolecules, during the study of artificial intelligence. And all this thanks to the expansion of our consciousness, moving away from simple addition and subtraction of natural quantities.

IN modern mathematics complex number is one of the most fundamental concepts, finding application in “ pure science", and in applied areas. It is clear that this was not always the case. In ancient times, when even ordinary negative numbers seemed a strange and dubious innovation, the need to extend the square root operation to them was not at all obvious. However, in mid-16th century century mathematician Raphael Bombelli introduces complex (in in this case more precisely, imaginary) numbers in circulation. Actually, I propose to look at what was the essence of the difficulties that ultimately brought the respectable Italian to such extremes.

There is a common misconception that complex numbers were required in order to solve quadratic equations. In fact, this is completely wrong: the task of finding the roots of a quadratic equation in no way motivates the introduction of complex numbers. That's perfect.

Let's see for ourselves. Any quadratic equation can be represented as:
.
Geometrically, this means that we want to find the intersection points of a certain line and a parabola
I even made a picture here for illustration.

As we all know well from school, the roots of a quadratic equation (in the above notations) are found by the following formula:

There are 3 possible options:
1. The radical expression is positive.
2. The radical expression is equal to zero.
3. The radical expression is negative.

In the first case there are 2 various roots, in the second there are two coinciding ones, in the third the equation is “not solved”. All these cases have a very clear geometric interpretation:
1. A straight line intersects a parabola (blue line in the figure).
2. A straight line touches a parabola.
3. A straight line has no relation to a parabola common points(lilac line in the picture).

The situation is simple, logical, and consistent. There is absolutely no reason to try to take the square root of a negative number. Nobody even tried.

The situation changed significantly when inquisitive mathematical thought reached cubic equations. A little less obvious, using some simple substitution , any cubic equation can be reduced to the form: . From a geometric point of view, the situation is similar to the previous one: we are looking for the intersection point of a straight line and a cubic parabola.
Take a look at the picture:

The significant difference from the case of a quadratic equation is that no matter what line we take, it will always intersect the parabola. That is, from purely geometric considerations, a cubic equation always has at least one solution.
You can find it using the Cardano formula:

Where
.
A little bulky, but so far everything seems to be in order. Or not?

In general, the Cardano formula is shining example"Arnold's principle" in action. And what is characteristic is that Cardano never claimed authorship of the formula.

Let us return, however, to our sheep. The formula is remarkable, without exaggeration, a great achievement of mathematics in the early to mid-16th century. But she has one nuance.
Let's take classic example, which was also considered by Bombelli:
.
Suddenly,
,
and correspondingly,
.
We've arrived. It’s a pity for the formula, but the formula is good. Dead end. Despite the fact that the equation certainly has a solution.

Rafael Bombelli's idea was the following: let's pretend to be a hose and pretend that the root of a negative is some kind of number. We, of course, know that there are no such numbers, but nevertheless, let's imagine that it exists and how regular numbers, can be added with others, multiplied, raised to a power, etc.

Using a similar approach, Bombelli found, in particular, that
,
And
.
Let's check:
.
Please note that in the calculations no assumptions were made about the properties of the square roots of negative numbers, except for the assumption mentioned above that they behave like “ordinary” numbers.

In total we get . Which is quite the correct answer, which can be easily verified by direct substitution. It was a real breakthrough. Breakthrough into the complex plane.

Nevertheless, such calculations look like some kind of magic, a mathematical trick. The attitude towards them as some kind of trick persisted among mathematicians for a very long time. Actually, the name “imaginary numbers”, invented by Rene Descartes for roots of negative numbers, fully reflects the attitude of mathematicians of those times to such entertainment.

However, as time went on, the “trick” was used with continued success, the authority of “imaginary numbers” in the eyes of the mathematical community grew, restrained, however, by the inconvenience of their use. Only the receipt by Leonhard Euler (by the way, it was he who introduced the now commonly used designation for the imaginary unit) of the famous formula

opened the way for complex numbers to various areas of mathematics and its applications. But that's a completely different story.

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.

Initial information O 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 of a 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. IN 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 equal to 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

New Page 1

Complex numbers for dummies. Lesson 1. What are they and what are they eaten with. Imaginary unit.

In order to understand what complex numbers are, let's remember about ordinary numbers and take a comprehensive look at them. And so, the simplest thing is natural numbers. They are called natural because through them something can be expressed “in kind,” that is, something can be counted. Here are two apples. They can be counted. There are five boxes of chocolates. We can count them. In other words, integers- these are numbers with which we can count specific items. You know very well that these numbers can be added, subtracted, multiplied and divided. Everything is clear with addition and multiplication. There were two apples, they added three, it became five. We took three boxes of chocolates, 10 pieces each, which means a total of thirty sweets. Now let's move on to whole numbers. If natural numbers denote a specific number of objects, then abstractions are introduced into the set of integers. This zero And negative numbers. Why are these abstractions? Zero is the absence of something. But can we touch, feel what is not there? We can touch two apples, here they are. We can even eat them. What does zero apples mean? Can we touch, feel this zero? No we can not. So this is abstraction. You have to somehow indicate the absence of something. So we designated zero as a number. But why signify this somehow? Let's imagine that we had two apples. We ate two. How much do we have left? That's right, not at all. We will write this operation (we ate two apples) as subtraction 2-2. And what did we end up with? How should we label the result? Only by introducing a new abstraction (zero), which will indicate that as a result of subtraction (eating) it turns out that we do not have a single apple left. But we can subtract not 2, but 3 from two. It would seem that this operation is meaningless. If we only have two apples, how can we eat three?

Let's look at another example. We go to the store for beer. We have 100 rubles with us. Beer costs 60 rubles per bottle. We want to buy two bottles, but we don’t have enough money. We need 120 rubles. And then we meet our old friend and borrow twenty from him. We buy beer. Question. How much money do we have left? Common sense suggests that not at all. But from a mathematical point of view this would be absurd. Why? Because in order to get zero as a result, you need to subtract 100 from 100. And we do 100-120. Here we should get something else. What did we get? And the fact that we still owe our friend 20 rubles. The next time we have 140 rubles with us, we will come to the store for beer, meet a friend, pay off our debts with him and be able to buy two more bottles of beer. As a result, we get 140-120-20=0. Note the -20. This is another abstraction - a negative number . That is, our debt to a friend is a number with a minus sign, because when we repay the debt, we subtract this amount. I will say more, this is an even greater abstraction than zero. Zero means something that doesn't exist. And a negative number is like something that will be taken away from us in the future.

And so, using an example, I showed how abstractions are born in mathematics. And that, it would seem, despite all the absurdity of such abstractions (like taking away more than was), they find application in real life. In the case of dividing integers, another abstraction arises - fractional numbers. I will not dwell on them in detail, and it is clear that they are needed in the case when we have integers that are not divisible by an integer. For example, we have four apples, but we need to divide them among three people. It’s clear here that we divide the one remaining apple into three parts and get fractions.

Now let’s very smoothly get to the complex numbers themselves. But first, remember that when you multiply two negative numbers, you get a positive number. Someone ask - why is this so? Let's first understand multiplying a negative number by a positive one. Let's say we multiply -20 by 2. That is, we need to add -20+-20. The result is -40, since adding a negative number is a subtraction. Why subtraction - see above, a negative number is a debt; when we take it away, something is taken away from us. There is another everyday meaning. What happens if the debt increases? For example, in the case when we were given a loan at interest? As a result, the same number with a minus sign remained, the one that became larger after the minus. What does it mean to multiply by a negative number? What does 3*-2 mean? This means that the number three must be taken minus two times. That is, put a minus before the result of multiplication. By the way, this is the same as -3*2, since rearranging the factors does not change the product. Now pay attention. Multiply -3 by -2. We take the number -3 minus two times. If we take the number -3 twice, then the result will be -6, you understand that. What if we take minus two times? But what does it mean to take minus times? If you take positive number minus times, then the result will be negative, its sign changes. If we take a negative number minus times, then its sign changes and it becomes positive.

Why did we talk about multiplying minus by minus? And in order to consider another abstraction, this time it is directly related to complex numbers. This imaginary unit. The imaginary unit is equal to the square root of minus 1:

Let me remind you what a square root is. This is the inverse operation of squaring. And squaring is multiplying a number by itself. So the square root of 4 is 2 because 2*2=4. The square root of 9 is 3, since 3*3=9. The square root of one also turns out to be one, and the square root of zero is zero. But how do we take the square root of minus one? What number must be multiplied by itself to get -1? But there is no such number! If we multiply -1 by itself, we will ultimately get 1. If we multiply 1 by 1, we will get 1. But we will not get minus -1 in this way. But, nevertheless, we may encounter a situation where there is a negative number under the root. What to do? You can, of course, say that there is no solution. It's like dividing by zero. Until some time, we all believed that it was impossible to divide by zero. But then we learned about such an abstraction as infinity, and it turned out that dividing by zero is still possible. Moreover, abstractions such as division by zero, or the uncertainty obtained by dividing zero by zero or infinity by infinity, as well as other similar operations, are widely used in higher mathematics (), and higher mathematics- this is the basis of many exact sciences, which move forward technical progress. So maybe in the imaginary unit there is some kind of secret meaning? Eat. And you will understand it by reading my further lessons on complex numbers. In the meantime, I will talk about some areas where complex numbers (numbers that contain an imaginary unit) are used.

And so, here is a list of areas where complex numbers are used:

Electrical engineering. Calculation of alternating current circuits. The use of complex numbers in this case greatly simplifies the calculation; without them, differential and integral equations would have to be used.

Quantum mechanics.In short - in quantum mechanics there is such a thing as wave function, which itself is complex-valued and whose square (already a real number) is equal to the probability density of finding a particle at a given point. See also the series of lessons

Digital signal processing. Theory digital processing signals includes such a concept as the z-transform, which greatly facilitates various calculations related to the calculation of the characteristics of various signals, such as frequency and amplitude characteristics, etc.

Description of processes of plane flow of liquids.

Liquid flow around profiles.

Wave movements of liquid.

And this is far from an exhaustive list of where complex numbers are used. This completes the first acquaintance with complex numbers, until we meet again.

Complex or imaginary numbers first appeared in Cardano's famous work, The Great Art, or algebraic rules» 1545. In the opinion of the author, these numbers were not suitable for use. However, this claim was later refuted. In particular, Bombelli in 1572, when deciding cubic equation justified the use of imaginary numbers. He compiled the basic rules for operations with complex numbers.

But still for a long time V mathematical world there was no common idea about the essence of complex numbers.

The symbol for imaginary numbers was first proposed outstanding mathematician Euler. The proposed symbolism looked like in the following way: i = sqr -1, where i is imaginarius, which means fictitious. Euler's merit also includes the idea of ​​the algebraic closedness of the field of complex numbers.

So, the need for numbers of a new type arose when solving quadratic equations for the case D< 0 (где D - дискриминант квадратного уравнения). В настоящее время комплексные числа нашли широкое применение в физике и технике, гидро- и аэродинамике, теории упругости и т.п.

The graphical representation of complex numbers has the form: a + bi, where a and b are real numbers, and i is an imaginary unit, i.e. i 2 = -1. The number a is called the abscissa, and b is the ordinate of the complex number a + bi. Two complex numbers a + bi and a - bi are called conjugate complex numbers.

There are a number of rules associated with complex numbers:

• Firstly, real number and can be written in the form of a complex number: a+ 0 i or a - 0 i. For example, 5 + 0 i and 5 - 0 i mean the same number 5.
• Secondly, the complex number 0+ bi is called a purely imaginary number. The notation bi means the same as 0+ bi .
• Third, two complex numbers a + bi and c + di are considered equal if a = c and b = d. Otherwise, complex numbers are not equal.

Basic operations on complex numbers include:

In geometric representation, complex numbers, unlike real numbers, which are represented on the number line by points, are marked by points on the coordinate plane. For this we take rectangular (Cartesian) coordinates with identical scales on the axes. In this case, the complex number a + bi will be represented by a point P with abscissa a and ordinate b. This coordinate system is called complex plane.

Module complex number is the length of the vector OP representing the complex number of the complex plane. The modulus of a complex number a + bi is written as |a + bi| or the letter r and is equal to: r = |a + ib| = sqr a 2 + b 2 .

Conjugate complex numbers have the same modulus.