Complex numbers of organs. Where are complex numbers used? Differences between complex and scalar quantities

HISTORICAL REFERENCE

Complex numbers were introduced into mathematics to make it possible to take the square root of any real number. This, however, is not a sufficient reason to introduce new numbers into mathematics. It turned out that if you carry out calculations according to the usual rules on expressions in which the square root of a negative number occurs, you can come to a result that no longer contains the square root of a negative number. In the 16th century Cardano found a formula for solving the cubic equation. It turned out that when a cubic equation has three real roots, the Cardano formula contains the square root of a negative number. Therefore, square roots of negative numbers began to be used in mathematics and were called imaginary numbers - thereby, as it were, they acquired the right to illegal existence. Gauss gave full civil rights to imaginary numbers, who called them complex numbers, gave a geometric interpretation and proved the fundamental theorem of algebra, which states that every polynomial has at least one real root.

1. CONCEPT OF COMPLEX NUMBER

Solving many problems in mathematics and physics comes down to solving algebraic equations.

So, to solve equations of the form X+A=B, positive numbers are not enough. For example, the equation X+5=2 has no positive roots. Therefore, you have to enter negative numbers and zero.

Algebraic equations of the first degree are solvable on the set of rational numbers, i.e. equations of the form A· X+B=0 (A0). However, algebraic equations of degree higher than first may not have rational roots. For example, these are the equations X 2 =2, X 3 =5. The need to solve such equations was one of the reasons for the introduction of irrational numbers. Rational and irrational numbers form the set of real numbers.

However, real numbers are not enough to solve any algebraic equation. For example, quadratic equation with real coefficients and a negative discriminant has no real roots. The simplest of them is the equation X 2 +1=0. Therefore, we have to expand the set of real numbers by adding new numbers to it. These new numbers, together with the real numbers, form a set, which is called a set complex numbers.

Let us first find out what form complex numbers should have. We will assume that the equation X 2 +1=0 has a root on the set of complex numbers. Let's denote this root by the letter i Thus, i is a complex number such that i 2 = –1.

As for real numbers, it is necessary to introduce the operations of addition and multiplication of complex numbers so that their sum and product are complex numbers. Then, in particular, for any real numbers A and B the expression A+B+ i can be considered a representation of a complex number in general form. The name “complex” comes from the word “composite”: by the form of the expression A+B· i .

Complex numbers are called expressions of the form A+B i , where A and B are real numbers, and i – some symbol such that i 2 = –1, and is denoted by the letter Z.

The number A is called the real part of the complex number A+B i, and the number B is its imaginary part. Number i called the imaginary unit.

For example, the real part of the complex number 2+3 i is equal to 2, and the imaginary one is equal to 3.

To strictly define a complex number, it is necessary to introduce the concept of equality for these numbers.

Two complex numbers A+B· i and C+D i are called equal if and only if A=C and B=D, i.e. when their real and imaginary parts are equal.

2. GEOMETRIC INTERPRETATION OF COMPLEX NUMBER

Real numbers are represented geometrically by points on the number line. Complex number A+B i can be considered as a pair of real numbers (A;B). Therefore, it is natural to represent a complex number by points on a plane. In a rectangular coordinate system, the complex number Z=A+B· i is represented by a point on the plane with coordinates (A;B), and this point is denoted by the same letter Z (Figure 1). Obviously, the resulting correspondence is one-to-one. It makes it possible to interpret complex numbers as points of the plane on which the coordinate system is chosen. This coordinate plane is called complex plane . The abscissa axis is called real axis , because it contains dots corresponding to real numbers. The ordinate axis is called imaginary axis – it contains points corresponding to imaginary complex numbers.

No less important and convenient is the interpretation of the complex number A+B· i as a vector, i.e. vector with origin at point

O(0;0) and with the end at point M(A;B) (Figure 2).

The correspondence established between the set of complex numbers, on the one hand, and the sets of points or vectors of the plane, on the other, allows complex numbers to be points or vectors.

3. COMPLEX NUMBER MODULE

Let a complex number Z=A+B· be given i . Conjugate With Z is called a complex number A – B i , which is denoted, i.e.

A – B i .

Note that = A+B· i , therefore for any complex number Z the equality =Z holds.

Module complex number Z=A+B· i called number and is denoted by , i.e.

From formula (1) it follows that for any complex number Z, and =0 if and only if Z=0, i.e. when A=0 and B=0. Let us prove that for any complex number Z the following formulas are valid:

4.ADDING AND MULTIPLYING COMPLEX NUMBERS

Amount two complex numbers A+B i and C+D i is called a complex number (A+C ) + ( B+D) i , i.e. ( A+B i) + ( C+D i)=( A+C) + (B+D) i

The work two complex numbers A+B i and C+D i is called a complex number (A· C – B· D)+(A· D+B· C) · i , i.e.

(A + B i ) (C + D) i )=(A·C – B·D) + (A·D + B·C)· i

It follows from the formulas that addition and multiplication can be performed according to the rules of operations with polynomials, considering i 2 = –1. The operations of addition and multiplication of complex numbers have the properties of real numbers. Basic properties:

Displacement property:

Z 1 +Z 2 =Z 2 +Z 1, Z 1· Z 2 =Z 2· Z 1

Matching property:

(Z 1 +Z 2)+Z 3 =Z 1 +(Z 2 +Z 3), (Z 1 Z 2) Z 3 =Z 1 (Z 2 Z 3)

Distribution property:

Z 1 (Z 2 +Z 3)=Z 1 Z 2 +Z 1 Z 3

Geometric representation of the sum of complex numbers

According to the definition of adding two complex numbers, the real part of the sum is equal to the sum of the real parts of the terms, the imaginary part of the sum is equal to the sum of the imaginary parts of the terms. The coordinates of the sum of vectors are determined in the same way:

The sum of two vectors with coordinates (A 1 ;B 1) and (A 2 ;B 2) is a vector with coordinates (A 1 +A 2 ;B 1 +B 2). Therefore, to find the vector corresponding to the sum of complex numbers Z 1 and Z 2, you need to add the vectors corresponding to the complex numbers Z 1 and Z 2.

Example 1: Find the sum and product of complex numbers Z 1 =2 – 3× i And

Z 2 = –7 + 8× i .

Z 1 + Z 2 = 2 – 7 + (–3 + 8)× i = – 5 + 5× i

Z 1× Z 2 = (2 – 3× i )× (–7 + 8× i ) = –14 + 16× i + 21× i + 24 = 10 + 37× i

5.SUBTRACT AND DIVISION OF COMPLEX NUMBERS

Subtraction of complex numbers is the inverse operation of addition: for any complex numbers Z 1 and Z 2 there is, and only one, the number Z, such that:

If we add (–Z 2) the opposite of the number Z 2 to both sides of the equality:

Z+Z 2 +(–Z 2)=Z 1 +(–Z 2), whence

The number Z=Z 1 +Z 2 is called difference in numbers Z 1 and Z 2.

Division is introduced as the inverse operation of multiplication:

Z×Z 2 =Z 1

Dividing both sides by Z 2 we get:

From this equation it is clear that Z 2 0

Geometric representation of the difference of complex numbers

The difference Z 2 – Z 1 of the complex numbers Z 1 and Z 2 corresponds to the difference of the vectors corresponding to the numbers Z 1 and Z 2. The modulus of the difference between two complex numbers Z 2 and Z 1, by definition of the modulus, is the length of the vector Z 2 – Z 1. Let's construct this vector as the sum of the vectors Z 2 and (–Z 1) (Figure 4). Thus, the modulus of the difference of two complex numbers is the distance between the points of the complex plane that correspond to these numbers.

This important geometric interpretation of the modulus of the difference of two complex numbers makes simple geometric facts useful.

Example 2: Given complex numbers Z 1 = 4 + 5 i and Z 2 = 3 + 4 i . Find the difference Z 2 – Z 1 and the quotient

Z 2 – Z 1 = (3 + 4 i) – (4 + 5· i) = –1 – i

==

6. TRIGONOMETRIC FORM OF COMPLEX NUMBER

Writing a complex number Z as A+B· i called algebraic form complex number. In addition to the algebraic form, other forms of writing complex numbers are also used.

Let's consider trigonometric form writing a complex number. Real and imaginary parts of a complex number Z=A+B· i are expressed through its modulus = r and argument j as follows:

A= r cosj ; B= r sinj .

The number Z can be written like this:

Z= r cosj + i sinj = r (cosj + i sinj)

Z = r (cosj + i sinj) (2)

This entry is called trigonometric form of a complex number .

r =– modulus of a complex number.

The number j is called argument of a complex number.

The argument of the complex number Z0 is the magnitude of the angle between the positive direction of the real axis and the vector Z, and the angle is considered positive if the count is counterclockwise, and negative if it is counted clockwise.

For the number Z=0, the argument is not defined, and only in this case the number is specified only by its modulus.

As mentioned above = r =, equality (2) can be written in the form

A+B i =· cosj + i · sinj, from where, equating the real and imaginary parts, we get:

cosj =, sinj = (3)

If sinj divide by cosj we get:

tgj= (4)

This formula is more convenient to use to find the argument j than formula (3). However, not all values ​​of j that satisfy equality (4) are arguments of the number A + B i . Therefore, when finding the argument, you need to take into account in which quarter the point A+B is located i .

7. PROPERTIES OF THE MODULE AND ARGUMENT OF A COMPLEX NUMBER

Using the trigonometric form it is convenient to find the product and quotient of complex numbers.

Let Z 1 = r 1 ( cosj 1 +i sinj 1), Z 2 = r 2 ( cosj 2 +i sinj 2). Then:

Z 1 Z 2 = r 1 · r 2 =

= r 1 r 2 .

Thus, the product of complex numbers written in trigonometric form can be found using the formula:

Z 1 Z 2 = r 1 · r 2 (5)

From formula (5) it follows that When multiplying complex numbers, their modules are multiplied and their arguments are added.

If Z 1 =Z 2 then we get:

Z 2 = 2 = r 2 (cos2j +i sin2j)

Z 3 =Z 2 Z= r 2 ( cos2j +i sin2j ) r (cosj + i sinj )=

= r 3 ( cos3j +i sin3j)

In general, for any complex number Z=r (cosj + i sinj )0 and any natural number n the formula is valid:

Zn=[ r (cosj + i sinj )] n = r n (cosnj + i sinnj),(6)

which is called Moivre's formula.

The quotient of two complex numbers written in trigonometric form can be found using the formula:

[cos(j 1 – j 2) + i sin(j 1 – j 2)].(7)

= = cos(–j 2) + i sin(–j 2)

Using formula 5

(cosj 1 + i sinj 1)× (cos(–j 2) + i sin(–j 2)) =

cos(j 1 – j 2) + i sin(j 1 – j 2).

Example 3:

We write the number –8 in trigonometric form

8 = 8 (cos(p + 2p k ) + i·sin(p + 2p k )), k О Z

Let Z = r×(cosj + i×

r 3× (cos3j + i× sin3j ) = 8 (cos(p + 2p k ) + i·sin(p + 2p k )), k О Z

Then 3j =p + 2p k , k О Z

j= , k О Z

Hence:

Z = 2 (cos() + i·sin()), k О Z

k = 0,1,2...

k = 0

Z 1 = 2 (cos + i sin) = 2 ( i) = 1+× i

k = 1

Z 2 = 2 (cos( ​​+ ) + i sin( + )) = 2 (cosp + i sinp ) = –2

k = 2

Z 3 = 2 (cos( ​​+ ) + i sin( + )) = 2 (cos + i sin) = 1–× i

Answer: Z 13 = ; Z2 = –2

Example 4:

We write the number 1 in trigonometric form

1 = 1· (cos(2p k ) + i·sin(2p k )), k О Z

Let Z = r×(cosj + i× sinj ), then this equation will be written as:

r 4× (cos4j + i× sin4j ) = cos(2p k ) + i·sin(2p k )), k О Z

4j = 2p k , k О Z

j = , k О Z

Z = cos+ i× sin

k = 0,1,2,3...

k = 0

Z 1 = cos0+ i× sin0 = 1 + 0 = 1

k = 1

Z 2 = cos+ i× sin = 0 + i = i

k = 2

Z 3 = cosp + i sinp = –1 + 0 = –1

k = 3

Z 4 = cos+ i× sin

Answer: Z 13 = 1

Z 24 = i

8. RAISING TO A POWER AND EXTRACTING THE ROOT

From formula 6 it is clear that raising the complex number r· (cosj + i sinj ) to a positive integer power with a natural exponent, its module is raised to a power with the same exponent, and the argument is multiplied by the exponent.

[ r (cosj + i sinj )] n = r n (cos nj + i sinnj)

Number Z called root of the degree n from the number w (denoted) if Z n =w.

From this definition it follows that every solution of the equation Zn=w is the root of the degree n from the number w. In other words, in order to extract the root of the power n from the number w, it is enough to solve the equation Z n =w. If w =0, then for any n the equation Zn=w has only one solution Z= 0. If w 0, then Z0 , and, therefore, Z and w can be represented in trigonometric form

Z = r (cosj + i sinj ), w = p (cozy + i siny)

The equation Z n = w will take the form:

r n (cos nj + i sin nj ) = p (cosy + i siny)

Two complex numbers are equal if and only if their moduli are equal and the arguments differ by terms that are multiples of 2p. Therefore, r n = p and nj = y + 2p k , wherekО Z or r = and j= , where kО Z .

So, all solutions can be written as follows:

Z K =, kО Z (8)

Formula 8 is called Moivre's second formula.

Thus, if w 0, then there are exactly n roots of degree n from the number w: they are all contained in formula 8. All roots of degree n from the number w have the same module , but different arguments, differing by a term that is a multiple of the number . It follows that complex numbers, which are roots of degree n from a complex number w, correspond to points of the complex plane located at the vertices of a regular n-gon inscribed in a circle of radius centered at the point Z = 0.

The symbol does not have a clear meaning. Therefore, when using it, you should clearly understand what is meant by this symbol. For example, when using the notation , you should consider making it clear whether this symbol means a pair of complex numbers i And –i , or one thing, which one exactly.

Equations of higher degrees

Formula 8 determines all the roots of a binomial equation of degree n. The situation is immeasurably more complicated in the case of a general algebraic equation of degree n:

a n× Z n+ a n–1× Z n–1 +...+ a 1× Z 1 + a 0 = 0(9)

Where a n ,..., a 0 are given complex numbers.

In the course of higher mathematics, Gauss's theorem is proven: every algebraic equation has at least one root in the set of complex numbers. This theorem was proven by the German mathematician Carl Gauss in 1779.

Based on Gauss's theorem, it can be proven that the left side of equation 9 can always be represented as a product:

,

Where Z 1, Z 2,..., Z K are some different complex numbers,

and a 1 ,a 2 ,...,a k are natural numbers, and:

a 1 + a 2 + ... + a k = n

It follows that the numbers Z 1, Z 2,..., Z K are the roots of equation 9. In this case, they say that Z 1 is a root of multiplicity a 1, Z 2 is a root of multiplicity a 2, and so on.

Gauss's theorem and the theorem just stated give solutions to the existence of roots, but say nothing about how to find these roots. If the roots of the first and second degrees can be easily found, then for equations of the third and fourth degrees the formulas are cumbersome, and for equations of degree above the fourth such formulas do not exist at all. The lack of a general method does not prevent us from finding all the roots of the equation. To solve an equation with integer coefficients, the following theorem is often useful: the integer roots of any algebraic equation with integer coefficients are divisors of the free term.

Let's prove this theorem:

Let Z = k be the integer root of the equation

a n× Z n + a n–1× Z n–1 +...+ a 1× Z 1 + a 0 = 0

with integer coefficients. Then

a n× k n + a n–1× k n–1 +...+ a 1× k 1 + a 0 = 0

a 0 = – k(a n× k n–1 + a n–1× k n–2 +...+ a 1)

The number in brackets, under the assumptions made, is obviously an integer, which means k is a divisor of the number a 0 .

9.QUADRATE EQUATION WITH COMPLEX UNKNOWN

Consider the equation Z 2 = a, where a is a given real number, Z is an unknown number.

This is the equation:

Let's write the number a in the form a = (– 1)× (– a) = i 2× = i 2× () 2 . Then the equation Z 2 = a will be written in the form: Z 2 – i 2× () 2 = 0

those. (Z – i× )(Z+ i× ) = 0

Therefore, the equation has two roots: Z 1.2 = i×

The introduced concept of a root of a negative number allows us to write down the roots of any quadratic equation with real coefficients

a× Z 2 + b× Z + c = 0

According to the well-known general formula

Z 1.2 = (10)

So, for any real a(a0), b, c, the roots of the equation can be found using formula 10. Moreover, if the discriminant, i.e. radical expression in formula 10

D = b 2 – 4× a× c

is positive, then the equation a× Z 2 + b× Z + c = 0 are two real distinct roots. If D = 0, then the equation a× Z 2 + b× Z + c = 0 has one root. If D< 0, то уравнение a× Z 2 + b× Z + c = 0 имеет два различных комплексных корня.

Complex roots of a quadratic equation have the same properties as the known properties of real roots.

Let us formulate the main ones:

Let Z 1 ,Z 2 be the roots of the quadratic equation a× Z 2 + b× Z + c = 0, a0. Then the following properties hold:

Z 1× Z 2 =

1. For all complex Z the formula is valid

a× Z 2 + b× Z + c = a× (Z – Z 1)× (Z – Z 2)

Example 5:

Z 2 – 6 Z + 10 = 0

D = b 2 – 4 a c

D = 6 2 – 4 10 = – 4

– 4 = i 2 ·4

Z 1.2 =

Answer: Z 1 = Z 2 = 3 + i

Example 6:

3·Z 2 +2·Z + 1 = 0

D = b 2 – 4 a c

D = 4 – 12 = – 8

D = –1·8 = 8· i 2

Z 1.2 = =

Answer: Z 1 = Z 2 = –

Example 7:

Z 4 – 8 Z 2 – 9 = 0

t 2 – 8 t – 9 = 0

D = b 2 – 4 a c = 64 + 36 = 100

t 1 = 9 t 2 = – 1

Z 2 = 9 Z 2 = – 1

Z 3.4 = i

Answer: Z 1.2 =3, Z 3.4 = i

Example 8:

Z 4 + 2 Z 2 – 15 = 0

t 2 + 2 t – 15 = 0

D = b 2 – 4 a c = 4 + 60 = 64

t 1.2 = = = –14

t 1 = – 5 t 2 = 3

Z 2 = – 5 Z 2 = 3

Z 2 = – 1·5 Z 3.4 =

Z 2 = i 2 ·5

Z 1.2 = i

Answer: Z 1.2 = i , Z 3.4 =

Example 9:

Z 2 = 24 10 i

Let Z = X + Y i

(X + Y i ) 2 = X 2 + 2· X· Y· i – Y2

X 2 + 2 X Y i – Y 2 = 24 10 i

(X 2 Y 2) + 2· X· Y· i = 24 10· i

multiply by X 2 0

X 4 – 24 X 2 – 25 = 0

t 2 – 24 t – 25 = 0

t 1 t 2 = – 25

t 1 = 25 t 2 = – 1

X 2 = 25 X 2 = – 1 - no solutions

X 1 = 5 X 2 = – 5

Y 1 = – Y 2 =

Y 1 = – 1 Y 2 = 1

Z 1.2 =(5 – i )

Answer: Z 1.2 =(5 – i )

TASKS:

(2 – Y) 2 + 3 (2 – Y) Y + Y 2 = 6

4 – 4·Y + Y 2 + 6·Y – 3·Y 2 + Y 2 = 6

–Y 2 + 2Y – 2 = 0 / –1

Y 2 – 2Y + 2 = 0

D = b 2 – 4 a c = 4 – 8 = – 4

– 4 = – 1·4 = 4· i 2

Y 1.2 = = = 1 i

Y 1 = 1– i Y 2 = 1 + i

X 1 = 1 + i X 2 = 1– i

Answer: (1 + i ; 1–i }

{1–i ; 1 + i }

Let's square it

§1. Complex numbers

1°. Definition. Algebraic notation.

Definition 1. Complex numbers ordered pairs of real numbers are called And , if for them the concept of equality, addition and multiplication operations are defined, satisfying the following axioms:

1) Two numbers
And
equal if and only if
,
, i.e.

 , .

2) The sum of complex numbers
And

and equal
, i.e.

+ = .

3) Product of complex numbers
And
is the number denoted by
and equal, i.e.

∙=.

The set of complex numbers is denoted C.

Formulas (2), (3) for numbers of the form
take the form

whence it follows that the operations of addition and multiplication for numbers of the form
coincide with addition and multiplication for real numbers  complex number of the form
identified with a real number .

Complex number
called imaginary unit and is designated , i.e.
Then from (3) 

From (2), (3)  which means

Expression (4) is called algebraic notation complex number.

In algebraic notation, the operations of addition and multiplication take the form:

A complex number is denoted by
, – real part, – imaginary part, is a purely imaginary number. Designation:
,
.

Definition 2. Complex number
called conjugate with a complex number
.

Properties of complex conjugation.

1)

2)
.

3) If
, That
.

4)
.

5)
– real number.

The proof is carried out by direct calculation.

Definition 3. Number
called module complex number
and is designated
.

It's obvious that
, and


. The formulas are also obvious:
And
.

2°. Properties of addition and multiplication operations.

1) Commutativity:
,
.

2) Associativity:,
.

3) Distributivity: .

Proof 1) – 3) is carried out by direct calculations based on similar properties for real numbers.

4)
,
.

5) , C ! , satisfying the equation
. This

6) ,C, 0, ! :
. This is found by multiplying the equation by



.

Example. Let's imagine a complex number
in algebraic form. To do this, multiply the numerator and denominator of the fraction by the conjugate number of the denominator. We have:

3°. Geometric interpretation of complex numbers. Trigonometric and exponential form of writing a complex number.

Let a rectangular coordinate system be specified on the plane. Then
C you can match a point on the plane with the coordinates
.(see Fig. 1). Obviously, such a correspondence is one-to-one. In this case, real numbers lie on the abscissa axis, and purely imaginary numbers lie on the ordinate axis. Therefore, the abscissa axis is called real axis, and the ordinate axis − imaginary axis. The plane on which complex numbers lie is called complex plane.

Note that And
are symmetrical about the origin, and And symmetrical with respect to Ox.

Each complex number (i.e., each point on the plane) can be associated with a vector with the beginning at the point O and the end at the point
. The correspondence between vectors and complex numbers is one-to-one. Therefore, the vector corresponding to a complex number , denoted by the same letter

D vector line
corresponding to a complex number
, is equal
, and
,
.

Using vector interpretation, we can see that the vector
− sum of vectors And , A
− sum of vectors And
.(see Fig. 2). Therefore, the following inequalities are valid: ,

Along with the length vector let's introduce the angle between vector and the Ox axis, counted from the positive direction of the Ox axis: if the counting is counterclockwise, then the sign of the angle is considered positive, if clockwise, then it is negative. This angle is called complex number argument and is designated
. Corner is not determined unambiguously, but with precision
… . For
the argument is not defined.

Formulas (6) define the so-called trigonometric notation complex number.

From (5) it follows that if
And
That

, .

From (5)
what about And a complex number is uniquely determined. The converse is not true: namely, over a complex number its module is uniquely found, and the argument , by virtue of (7), − with accuracy
. It also follows from (7) that the argument can be found as a solution to the equation

However, not all solutions to this equation are solutions to (7).

Among all the values ​​of the argument of a complex number, one is selected, which is called the main value of the argument and is denoted
. Usually the main value of the argument is chosen either in the range
, or in the interval

It is convenient to perform multiplication and division operations in trigonometric form.

Theorem 1. Modulus of product of complex numbers And is equal to the product of the modules, and the argument is the sum of the arguments, i.e.

, A .

Likewise

,

Proof. Let , . Then by direct multiplication we get:

Likewise

.■

Consequence(Moivre's formula). For
Moivre's formula is valid

P example. Let us find the geometric location of the point
. From Theorem 1 it follows that .

Therefore, to construct it, you must first construct a point , which is the inversion relative to the unit circle, and then find a point symmetrical to it relative to the Ox axis.

Let
, i.e.
Complex number
denoted by
, i.e. R Euler's formula is valid

Because
, That
,
. From Theorem 1
what's with the function
you can work as with a regular exponential function, i.e. equalities are valid

,
,
.

From (8)
demonstrative notation complex number

, Where
,

Example. .

4°. Roots -th power of a complex number.

Consider the equation

, WITH , N .

Let
, and the solution to equation (9) is sought in the form
. Then (9) takes the form
, from where we find that
,
, i.e.

,
,
.

Thus, equation (9) has roots

, .

Let us show that among (10) there is exactly different roots. Really,

are different, because their arguments are different and differ less than
. Further,
, because
. Likewise
.

Thus, equation (9) at
has exactly roots
, located at the vertices of the regular -a triangle inscribed in a circle of radius with center at point O.

Thus it is proven

Theorem 2. Root extraction -th power of a complex number
It's always possible. All root meanings th degree of located at the vertices of the correct -gon inscribed in a circle with center at zero and radius
. Wherein,

Consequence. Roots -th power of 1 are expressed by the formula

.

The product of two roots of 1 is a root, 1 is a root -th power of unity, root
:
.

If you need to name the distance between two cities, you can give an answer consisting of a single number in miles, kilometers, or other units of linear distance. However, if you must describe how to get from one city to another, then you need to provide more information than just the distance between two points on the map. In this case, it is worth talking about the direction in which you need to move and about.

The type of information that expresses a one-dimensional measurement is called a scalar quantity in science. Scalars are numbers used in most mathematical calculations. For example, the mass and speed that an object has are scalar quantities.

In order to successfully analyze natural phenomena, we must work with abstract objects and methods that can represent multidimensional quantities. Here it is necessary to abandon scalar numbers in favor of complex ones. They make it possible to express two dimensions simultaneously.

Complex numbers are easier to understand when they are represented in graphical form. If a line has a certain length and direction, then this will be a graphical representation. It is also commonly known as vector.

Differences between complex and scalar quantities

Such types of numbers as integers, rationals, and reals are familiar to children from school. They all have a one-dimensional quality. The straightness of the number line illustrates this graphically. You can move up or down on it, but all "movement" along that line will be limited to the horizontal axis. One-dimensional, scalar numbers are sufficient for counting the number of objects, expressing weight, or measuring the DC voltage of a battery. But they cannot mean anything more complex. It is impossible to simultaneously express distance and direction between two cities, or amplitude with phase, using scalars. These types of numbers must be represented in the form of a multidimensional range of values. In other words, we need vector quantities that can have not only a magnitude, but also a direction of propagation.

Conclusion

A scalar number is a type of mathematical object that people are accustomed to using in Everyday life- this is temperature, length, weight, etc. A complex number is a value that includes two types of data.

A vector is a graphical representation of a complex number. It looks like an arrow with a starting point, a specific length and direction. Sometimes the word "vector" is used in radio engineering, where it expresses the phase shift between signals.

Let us 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, a symbol whose square is equal to –1, that is i 2 = –1. Number a called real part, and the number b - imaginary part complex number z = a + bi. If b= 0, then instead a + 0i they simply write 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 numbers: they can be added, subtracted, multiplied and divided by each other. Addition and subtraction occur according to the rule ( a + bi) ± ( c + di) = (a ± c) + (b ± d)i, and multiplication follows the rule ( a + bi) · ( c + di) = (ac – bd) + (ad + bc)i(here it is 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: number z = a + bi can be represented by a vector with coordinates ( a; b) on the Cartesian plane (or, which is almost the same thing, a point - the end of a 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 using the parallelogram rule). According to the Pythagorean theorem, the length of the vector with coordinates ( a; b) is equal to . This quantity 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 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 counted in degrees) - after all, it is clear that a 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 the positive direction of the x-axis, then its coordinates are equal to ( r cos φ ; r sin φ ). From here 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 the calculations. Multiplying complex numbers in trigonometric form is 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 their arguments are added). From here follow Moivre's 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 powers from number z- this is 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 regular n-gon).

When studying the properties of a 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 the conditional value of the second root of minus one is taken as 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 performed simply. 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 of arithmetic operations. It is impossible to solve all division problems on a set of integer values. led first to the concept of 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. The combination of rational and irrational numbers formed a real set, which can be represented as a certain line with a given scale. Every step along the line is natural number, and between them are rational and irrational values.

The era of theoretical mathematics began. The development of astronomy, mechanics, and physics required the solution of increasingly complex equations. In general form, the roots of the quadratic equation were found. When solving a more complex cubic polynomial, scientists encountered a contradiction. The concept of a negative cube root makes sense, but for a square root it results in uncertainty. Moreover, the quadratic equation is only a special case of the cubic one.

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 of the 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 a 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

The geometric form of 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.
• The numerical value of the length of the vector from the zero point to z is called the module.
• 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's remember the polar coordinate system and the 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. A notation of the form ∣z ∣ multiplied by the sum of the trigonometric functions cos(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, a very important Moivre formula follows:

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 of a complex set, we propose two important definitions.

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

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

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

This notation has become widespread for practical calculations. 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 the 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 .

We have obtained a simple formula for 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 the inverse function - extracting the root.

For simplicity of notation, we take n = 2. The square root w of the 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 a unique solution equal to 1. But this contradicts the concept that z = -1, also corresponds to the definition of a square root.

Let's figure out what we don't take into account. If we remember the trigonometric notation, we will restore the statement - with a periodic change in the phase ϴ, the complex number does not change. Let us denote the value of the period by the symbol p, then the following is true: 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 obtained the second solution, which corresponds to the general understanding of the 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 in the Euler form z = r × e i ϴ .
• Let's use the general definition of the root extraction function r n *e i n ϴ = ∣w∣ × e i (arg (w) + pk) .
• From the general properties of 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 are of little importance for solving applied problems with complex numbers, but are essential for the further development of 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 a 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 is an 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 polynomials. Here we first encounter the paradox of imaginary numbers.

Scientific natural scientists, solving completely practical problems, resorting to solutions of various equations, discover mathematical paradoxes. The interpretation of these paradoxes leads to completely surprising discoveries. The dual nature of electromagnetic waves is 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 more difficult 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 are made in living nature, during the synthesis of macromolecules, and 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.