Complex numbers find all values. Complex numbers
FEDERAL EDUCATION AGENCY
STATE EDUCATIONAL INSTITUTION
HIGHER PROFESSIONAL EDUCATION
"VORONEZH STATE PEDAGOGICAL UNIVERSITY"
DEPARTMENT OF AGLEBRA AND GEOMETRY
Complex numbers
(selected tasks)
GRADUATE QUALIFICATION WORK
in the specialty 050201.65 mathematics
(with additional specialty 050202.65 informatics)
Completed: 5th year student
physical and mathematical
faculty
Supervisor:
VORONEZH - 2008
1. Introduction……………………………………………………...…………..…
2. Complex numbers (selected problems)
2.1. Complex numbers in algebraic form…. …… ... ……….….
2.2. Geometric interpretation of complex numbers ………… ..…
2.3. Trigonometric form of complex numbers
2.4. Application of the theory of complex numbers to the solution of equations of the 3rd and 4th degree …………… .. ………………………………………………………
2.5. Complex numbers and parameters ……… ... …………………… ...….
3. Conclusion …………………………………………………… .................
4. References …………………………. ………………… ...............
1. Introduction
In the mathematics curriculum of the school course, number theory is introduced using examples of sets of natural numbers, integers, rational, irrational, i.e. on the set of real numbers, the images of which fill the entire number axis. But already in grade 8, the stock of real numbers is not enough, solving quadratic equations with a negative discriminant. Therefore, it was necessary to replenish the stock of real numbers with complex numbers for which the square root of a negative number makes sense.
The choice of the topic "Complex numbers", as the topic of my final qualifying work, is that the concept of a complex number expands students' knowledge about number systems, about solving a wide class of problems, both algebraic and geometric content, about solving algebraic equations of any degree and about solving problems with parameters.
In this thesis, the solution of 82 problems is considered.
The first part of the main section "Complex numbers" provides solutions to problems with complex numbers in algebraic form, defines the operations of addition, subtraction, multiplication, division, conjugation for complex numbers in algebraic form, the power of the imaginary unit, the modulus of a complex number, and also sets out the rule extracting the square root of a complex number.
In the second part, problems are solved for the geometric interpretation of complex numbers in the form of points or vectors of a complex plane.
The third part deals with actions on complex numbers in trigonometric form. The formulas are used: Moivre and extraction of a root from a complex number.
The fourth part is devoted to solving equations of the 3rd and 4th degrees.
When solving the problems of the last part "Complex numbers and parameters", the information given in the previous parts is used and consolidated. A series of problems in the chapter is devoted to the determination of families of lines in the complex plane, given by equations (inequalities) with a parameter. In part of the exercises, you need to solve equations with a parameter (over the field C). There are tasks where a complex variable simultaneously satisfies a number of conditions. A feature of solving the problems of this section is the reduction of many of them to the solution of equations (inequalities, systems) of the second degree, irrational, trigonometric with a parameter.
A feature of the presentation of the material of each part is the initial introduction of the theoretical foundations, and subsequently their practical application in solving problems.
At the end of the thesis, a list of used literature is presented. In most of them, theoretical material is presented in sufficient detail and in an accessible way, solutions of some problems are considered and practical tasks are given for independent solution. I would like to pay special attention to such sources as:
1. Gordienko N.A., Belyaeva E.S., Firstov V.E., Serebryakova I.V. Complex Numbers and Their Applications: A Study Guide. ... The material of the tutorial is presented in the form of lectures and practical lessons.
2. Shklyarsky DO, Chentsov NN, Yaglom IM Selected problems and theorems of elementary mathematics. Arithmetic and Algebra. The book contains 320 problems related to algebra, arithmetic and number theory. By their nature, these tasks differ significantly from the standard school tasks.
2. Complex numbers (selected problems)
2.1. Complex numbers in algebraic form
The solution of many problems in mathematics and physics is reduced to solving algebraic equations, i.e. equations of the form
,where a0, a1,…, an are real numbers. Therefore, the study of algebraic equations is one of the most important issues in mathematics. For example, a quadratic equation with negative discriminant does not have real roots. The simplest such equation is the equation
.In order for this equation to have a solution, it is necessary to expand the set of real numbers by adding to it the root of the equation
.We denote this root by
... Thus, by definition, or,hence,
... is called an imaginary unit. With its help and with the help of a pair of real numbers, an expression of the form is compiled.The resulting expression was called complex numbers, since they contained both real and imaginary parts.
So, complex numbers are expressions of the form
, and are real numbers, and is some symbol that satisfies the condition. A number is called the real part of a complex number, and a number is called its imaginary part. Symbols are used to denote them,.Complex numbers of the form
are real numbers and, therefore, the set of complex numbers contains a set of real numbers.Complex numbers of the form
are called purely imaginary. Two complex numbers of the form and are called equal if their real and imaginary parts are equal, i.e. if the equalities hold,.Algebraic notation of complex numbers allows you to perform operations on them according to the usual rules of algebra.
The sum of two complex numbers
and is called a complex number of the form.The product of two complex numbers
§ 1. Complex numbers: definitions, geometric interpretation, actions in algebraic, trigonometric and exponential forms
Definition of a complex number
Complex equalities
Geometric representation of complex numbers
Complex number modulus and argument
Algebraic and trigonometric forms of a complex number
Exponential form of a complex number
Euler's formulas
§ 2. Entire functions (polynomials) and their basic properties. Solving algebraic equations on the set of complex numbers
Definition of an algebraic equation of the th degree
Basic properties of polynomials
Examples of solving algebraic equations on the set of complex numbers
Self-test questions
Glossary
§ 1. Complex numbers: definitions, geometric interpretation, actions in algebraic, trigonometric and exponential forms
The definition of a complex number ( Formulate the definition of a complex number)
A complex number z is an expression of the following form:
Complex number in algebraic form, (1)
Where x, y Î;
- complex conjugate number number z ;
- opposite number number z ;
- complex zero ;
- this is how the set of complex numbers is denoted.
1)z = 1 + iÞ Re z= 1, Im z = 1, = 1 – i, = –1 – i ;
2)z = –1 + iÞ Re z= –1, Im z = , = –1 – i, = –1 –i ;
3)z = 5 + 0i= 5 Þ Re z= 5, Im z = 0, = 5 – 0i = 5, = –5 – 0i = –5
Þ if Im z= 0, then z = x- real number;
4)z = 0 + 3i = 3iÞ Re z= 0, Im z = 3, = 0 – 3i = –3i , = –0 – 3i = – 3i
Þ if Re z= 0, then z = iy - pure imaginary number.
Complex equalities (Formulate the meaning of complex equality)
1) 
;
2)
.
One complex equality is equivalent to a system of two real equalities. These real equalities are obtained from complex equality by dividing the real and imaginary parts.
1) 
;
2) 
.
Geometric representation of complex numbers ( What is the geometric representation of complex numbers?)
Complex number z is represented by a point ( x , y) on the complex plane or the radius vector of this point.
Sign z in the second quarter means that the Cartesian coordinate system will be used as the complex plane.
The modulus and argument of a complex number ( What is the modulus and argument of a complex number?)
The modulus of a complex number is a non-negative real number
.(2)
Geometrically, the modulus of a complex number is the length of the vector representing the number z, or the polar radius of the point ( x , y).
Draw the following numbers on the complex plane and write them in trigonometric form.
1)z = 1 + i Þ
,
Þ 
Þ 
;
,
Þ 
Þ 
;
,
5)
,
that is, for z = 0 there will be
, j indefined.
Arithmetic operations on complex numbers (Give definitions and list the main properties of arithmetic operations on complex numbers.)
Addition (subtraction) of complex numbers
z 1 ± z 2 = (x 1 + iy 1) ± ( x 2 + iy 2) = (x 1 ± x 2) + i (y 1 ± y 2),(5)
that is, when adding (subtracting) complex numbers, their real and imaginary parts are added (subtracted).
1)(1 + i) + (2 – 3i) = 1 + i + 2 –3i = 3 – 2i ;
2)(1 + 2i) – (2 – 5i) = 1 + 2i – 2 + 5i = –1 + 7i .
Basic properties of addition
1)z 1 + z 2 = z 2 + z 1;
2)z 1 + z 2 + z 3 = (z 1 + z 2) + z 3 = z 1 + (z 2 + z 3);
3)z 1 – z 2 = z 1 + (– z 2);
4)z + (–z) = 0;
Multiplication of complex numbers in algebraic form
z 1∙z 2 = (x 1 + iy 1)∙(x 2 + iy 2) = x 1x 2 + x 1iy 2 + iy 1x 2 + i 2y 1y 2 = (6)
= (x 1x 2 – y 1y 2) + i (x 1y 2 + y 1x 2),
that is, the multiplication of complex numbers in algebraic form is carried out according to the rule of algebraic multiplication of a binomial by a binomial, followed by replacement and reduction of similar ones in real and imaginary terms.
1)(1 + i)∙(2 – 3i) = 2 – 3i + 2i – 3i 2 = 2 – 3i + 2i + 3 = 5 – i ;
2)(1 + 4i)∙(1 – 4i) = 1 – 42 i 2 = 1 + 16 = 17;
3)(2 + i)2 = 22 + 4i + i 2 = 3 + 4i .
Multiplication of complex numbers in trigonometric form
z 1∙z 2 = r 1 (cos j 1 + i sin j 1) × r 2 (cos j 2 + i sin j 2) =
= r 1r 2 (cos j 1cos j 2 + i cos j 1sin j 2 + i sin j 1cos j 2 + i 2 sin j 1sin j 2) =
= r 1r 2 ((cos j 1cos j 2 - sin j 1sin j 2) + i(cos j 1sin j 2 + sin j 1cos j 2))
The product of complex numbers in trigonometric form, that is, when multiplying complex numbers in trigonometric form, their modules are multiplied, and the arguments are added.
Basic properties of multiplication
1)z 1 × z 2 = z 2 × z 1 - commutability;
2)z 1 × z 2 × z 3 = (z 1 × z 2) × z 3 = z 1 × ( z 2 × z 3) - associativity;
3)z 1 × ( z 2 + z 3) = z 1 × z 2 + z 1 × z 3 - distributiveness with respect to addition;
4)z× 0 = 0; z× 1 = z ;
Division of complex numbers
Division is the inverse of multiplication, so
if z × z 2 = z 1 and z 2 ¹ 0, then.
When division is performed in algebraic form, the numerator and denominator of the fraction are multiplied by the complex conjugate of the denominator:
Division of complex numbers in algebraic form. (7)
When doing division in trigonometric form, modules are divided and arguments are subtracted:
Division of complex numbers in trigonometric form. (8)
2)
.
Raising a complex number to a natural power
It is more convenient to perform natural exponentiation in trigonometric form:
Moivre's formula, (9)
that is, when a complex number is raised to a natural power, its modulus is raised to this power, and the argument is multiplied by the exponent.
Calculate (1 + i)10.
Remarks
1. When performing operations of multiplication and raising to a natural power in trigonometric form, angles can be obtained outside the limits of one full revolution. But they can always be reduced to angles or by dropping an integer number of complete revolutions according to the properties of the periodicity of the functions and.
2. Value 
called the main value of the argument of a complex number;
the values of all possible angles mean;
it's obvious that , .
Extracting the natural root of a complex number
Euler's formulas (16)
by which trigonometric functions and a real variable are expressed through an exponential function (exponential) with a purely imaginary exponent.
§ 2. Entire functions (polynomials) and their basic properties. Solving algebraic equations on the set of complex numbers
Two polynomials of the same degree n are identically equal to each other if and only if their coefficients coincide at the same powers of the variable x, that is
Proof
w Identity (3) is valid for "xÎ (or" xÎ)
Þ it is valid for; substituting, we get an = bn .
We mutually annihilate in (3) the terms an and bn and divide both parts into x :
This identity is also true for " x, including at x = 0
Þ assuming x= 0, we get an – 1 = bn – 1.
We mutually annihilate in (3 ") the terms an- 1 and a n- 1 and divide both parts by x, as a result we get
Continuing the reasoning in a similar way, we find that an – 2 = bn –2, …, a 0 = b 0.
Thus, it has been proved that the identity of 2-x polynomials implies the coincidence of their coefficients at the same degrees x .
The converse statement is true and obvious, i.e. if two polynomials have the same all coefficients, then they are the same functions, therefore, their values coincide for all values of the argument, which means their identical equality. Property 1 is completely proved. v
When dividing a polynomial Pn (x) by the difference ( x – NS 0), the remainder is equal to Pn (x 0), that is
Bezout's theorem, (4)
where Qn – 1(x) is the integer part of division, is a polynomial of degree ( n – 1).
Proof
w Let's write the formula for division with remainder:
Pn (x) = (x – NS 0)∙Qn – 1(x) + A ,
where Qn – 1(x) is a polynomial of degree ( n – 1),
A- the remainder, which is a number due to the well-known algorithm for dividing a polynomial by a two-term "column".
This equality is true for " x, including at x = NS 0 Þ
Pn (x 0) = (x 0 – x 0)× Qn – 1(x 0) + A Þ
A = Pn (NS 0), p.t.d. v
Corollary from Bezout's theorem. On dividing a polynomial by a binomial without remainder
If the number NS 0 is the zero of the polynomial, then this polynomial is divisible by the difference ( x – NS 0) without a remainder, that is
Þ 
.(5)
1), since P 3 (1) º 0
2), since P 4 (–2) º 0
3), since P 2 (–1/2) º 0
Division of polynomials into binomials "in a column":
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Any polynomial of degree n ³ 1 has at least one zero, real or complex
The proof of this theorem is beyond the scope of our course. Therefore, we will accept the theorem without proof.
Let us work on this theorem and on the Bezout theorem with the polynomial Pn (x).
After n-fold application of these theorems, we obtain
where a 0 is the coefficient at x n v Pn (x).
Corollary from the main theorem of algebra. Decomposition of a polynomial into linear factors
Any polynomial of degree on the set of complex numbers is decomposed into n linear factors, that is
Decomposition of a polynomial into linear factors, (6)
where x1, x2, ... xn are the zeros of the polynomial.
Moreover, if k numbers from the set NS 1, NS 2, … xn coincide with each other and with the number a, then the factor ( x- a) k... Then the number x= a is called k-fold zero of the polynomial Pn ( x) ... If k= 1, then zero is called simple zero polynomial Pn ( x) .
1)P 4(x) = (x – 2)(x- 4) 3 Þ x 1 = 2 - simple zero, x 2 = 4 - three-fold zero;
2)P 4(x) = (x – i) 4 Þ x = i- zero of multiplicity 4.
Property 4 (on the number of roots of an algebraic equation)
Any algebraic equation Pn (x) = 0 of degree n has exactly n roots on the set of complex numbers, if each root is counted as many times as its multiplicity.
1)x 2 – 4x+ 5 = 0 - algebraic equation of the second degree
Þ x 1.2 = 2 ± = 2 ± i- two roots;
2)x 3 + 1 = 0 - algebraic equation of the third degree
Þ x
1,2,3 = 
- three roots;
3)P 3(x) = x 3 + x 2 – x- 1 = 0 Þ x 1 = 1, because P 3(1) = 0.
Divide the polynomial P 3(x) on ( x – 1):
| x 3 | + | x 2 | – | x | – | 1 | x – 1 |
| x 3 | – | x 2 | x 2 + 2x +1 | ||||
| 2x 2 | – | x | |||||
| 2x 2 | – | 2x | |||||
| x | – | 1 | |||||
| x | – | 1 | |||||
| 0 |
Original equation
P 3(x) = x 3 + x 2 – x- 1 = 0 Û ( x – 1)(x 2 + 2x+ 1) = 0 Û ( x – 1)(x + 1)2 = 0
Þ x 1 = 1 - simple root, x 2 = –1 - double root.
1) - paired complex conjugate roots;
Any polynomial with real coefficients is decomposed into the product of linear and quadratic functions with real coefficients.
Proof
w Let x 0 = a + bi- zero of the polynomial Pn (x). If all the coefficients of this polynomial are real numbers, then it is also its zero (by property 5).
We calculate the product of binomials 
:
complex number polynomial equation
Got ( x – a)2 + b 2 - square trinomial with real coefficients.
Thus, any pair of binomials with complex conjugate roots in formula (6) leads to a square trinomial with real coefficients. v
1)P 3(x) = x 3 + 1 = (x + 1)(x 2 – x + 1);
2)P 4(x) = x 4 – x 3 + 4x 2 – 4x = x (x –1)(x 2 + 4).
Examples of solving algebraic equations on the set of complex numbers ( Give examples of solving algebraic equations on the set of complex numbers)
1. Algebraic equations of the first degree:
, Is the only simple root.
2. Quadratic equations:
, 
- always has two roots (different or equal).
1) 
.
3. Two-term equations of degree:
, - always has different roots.
,
Answer: , 
.
4. Solve the cubic equation.
An equation of the third degree has three roots (real or complex), and each root must be counted as many times as its multiplicity. Since all the coefficients of this equation are real numbers, the complex roots of the equation, if any, will be paired complex conjugate.
By selection, we find the first root of the equation, since.
By the corollary to Bezout's theorem. We calculate this division "in a column":
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Representing now the polynomial as a product of a linear and a square factor, we get:
.
We find other roots as the roots of the quadratic equation: 
Answer: , 
.
5. Write the algebraic equation of the least degree with real coefficients, if it is known that the numbers x 1 = 3 and x 2 = 1 + i are its roots, and x 1 is a double root, and x 2 - simple.
The number is also the root of the equation, since the coefficients of the equation must be valid.
In total, the required equation has 4 roots: x 1, x 1,x 2,. Therefore, its degree is 4. We compose a polynomial of the 4th degree with zeros x
11. What is complex zero?
13. Formulate the meaning of complex equality.
15. What is the modulus and argument of a complex number?
17. What is a complex number argument?
18. What name or meaning does the formula have?
19. Explain the meaning of the notation in this formula:
27. Give definitions and list the basic properties of arithmetic operations on complex numbers.
28. What name or meaning does the formula have?
29. Explain the meaning of the designations in this formula:
31. What name or meaning does the formula have?
32. Explain the meaning of the notation in this formula:
34. What name or meaning does the formula have?
35. Explain the meaning of the designations in this formula:
61. List the basic properties of polynomials.
63. Formulate the property of dividing the polynomial by the difference (x - x0).
65. What name or meaning does the formula have?
66. Explain the meaning of the designations in this formula:
67. ⌂ 
.
69. Formulate the theorem. The main theorem of algebra.
70. What name or meaning does the formula have?
71. Explain the meaning of the designations in this formula:
75. Formulate the property about the number of roots of an algebraic equation.
78. Formulate the property on the decomposition of a polynomial with real coefficients into linear and quadratic factors.
Glossary
k-fold zero of a polynomial is called ... (p. 18)
an algebraic polynomial is called ... (p. 14)
an algebraic equation of the nth degree is called ... (p. 14)
the algebraic form of a complex number is called ... (p. 5)
complex number argument is ... (p. 4)
the real part of a complex number z is ... (p. 2)
a complex conjugate number is ... (p. 2)
complex zero is ... (page 2)
a complex number is called ... (p. 2)
nth root of a complex number is called ... (p. 10)
the root of the equation is called ... (p. 14)
the coefficients of the polynomial are ... (p. 14)
the imaginary unit is ... (p. 2)
the imaginary part of a complex number z is ... (p. 2)
the modulus of a complex number is called ... (p. 4)
the function zero is called ... (p. 14)
the exponential form of a complex number is called ... (p. 11)
polynomial is called ... (p. 14)
a simple zero of a polynomial is called ... (p. 18)
the opposite number is ... (page 2)
the degree of a polynomial is ... (p. 14)
trigonometric form of a complex number is called ... (p. 5)
Moivre's formula is ... (p. 9)
Euler's formulas are ... (p. 13)
the whole function is called ... (p. 14)
a purely imaginary number is ... (p. 2)
Lesson plan.
1. Organizational moment.
2. Presentation of the material.
3. Homework.
4. Summing up the lesson.
During the classes
I. Organizational moment.
II. Presentation of the material.
Motivation.
Expansion of the set of real numbers is that new numbers (imaginary) are added to the real numbers. The introduction of these numbers is associated with the impossibility of extracting a root from a negative number in the set of real numbers.
Introduction of the concept of a complex number.
The imaginary numbers with which we supplement the real numbers are written as bi, where i Is an imaginary unit, and i 2 = - 1.
Based on this, we get the following definition of a complex number.
Definition... A complex number is an expression of the form a + bi, where a and b- real numbers. In this case, the following conditions are met:
a) Two complex numbers a 1 + b 1 i and a 2 + b 2 i are equal if and only if a 1 = a 2, b 1 = b 2.
b) The addition of complex numbers is determined by the rule:
(a 1 + b 1 i) + (a 2 + b 2 i) = (a 1 + a 2) + (b 1 + b 2) i.
c) Multiplication of complex numbers is determined by the rule:
(a 1 + b 1 i) (a 2 + b 2 i) = (a 1 a 2 - b 1 b 2) + (a 1 b 2 - a 2 b 1) i.
Algebraic form of a complex number.
Writing a complex number in the form a + bi is called the algebraic form of a complex number, where a- real part, bi Is the imaginary part, and b Is a real number.
Complex number a + bi is considered equal to zero if its real and imaginary parts are equal to zero: a = b = 0
Complex number a + bi at b = 0 is considered to be the same as a real number a: a + 0i = a.
Complex number a + bi at a = 0 is called purely imaginary and is denoted bi: 0 + bi = bi.
Two complex numbers z = a + bi and = a - bi that differ only in the sign of the imaginary part are called conjugate.
Actions on complex numbers in algebraic form.
You can do the following on complex numbers in algebraic form.
1) Addition.
Definition... The sum of complex numbers z 1 = a 1 + b 1 i and z 2 = a 2 + b 2 i called a complex number z, the real part of which is equal to the sum of the real parts z 1 and z 2, and the imaginary part is the sum of the imaginary parts of the numbers z 1 and z 2, that is z = (a 1 + a 2) + (b 1 + b 2) i.
Numbers z 1 and z 2 are called terms.
The addition of complex numbers has the following properties:
1º. Commutability: z 1 + z 2 = z 2 + z 1.
2º. Associativity: (z 1 + z 2) + z 3 = z 1 + (z 2 + z 3).
3º. Complex number –A –bi called the opposite of a complex number z = a + bi... Complex number opposite to complex number z, denoted -z... Sum of complex numbers z and -z is equal to zero: z + (-z) = 0
Example 1. Perform addition (3 - i) + (-1 + 2i).
(3 - i) + (-1 + 2i) = (3 + (-1)) + (-1 + 2) i = 2 + 1i.
2) Subtraction.
Definition. Subtract from a complex number z 1 complex number z 2 z, what z + z 2 = z 1.
Theorem... The difference of complex numbers exists and, moreover, is unique.
Example 2. Perform subtraction (4 - 2i) - (-3 + 2i).
(4 - 2i) - (-3 + 2i) = (4 - (-3)) + (-2 - 2) i = 7 - 4i.
3) Multiplication.
Definition... The product of complex numbers z 1 = a 1 + b 1 i and z 2 = a 2 + b 2 i called a complex number z defined by the equality: z = (a 1 a 2 - b 1 b 2) + (a 1 b 2 + a 2 b 1) i.
Numbers z 1 and z 2 are called factors.
Multiplication of complex numbers has the following properties:
1º. Commutability: z 1 z 2 = z 2 z 1.
2º. Associativity: (z 1 z 2) z 3 = z 1 (z 2 z 3)
3º. Distributiveness of multiplication relative to addition:
(z 1 + z 2) z 3 = z 1 z 3 + z 2 z 3.
4º. z = (a + bi) (a - bi) = a 2 + b 2 is a real number.
In practice, the multiplication of complex numbers is carried out according to the rule of multiplying the sum by the sum and separating the real and imaginary parts.
In the following example, we will consider multiplication of complex numbers in two ways: by rule and multiplication of the sum by the sum.
Example 3. Perform multiplication (2 + 3i) (5 - 7i).
1 way. (2 + 3i) (5 - 7i) = (2 × 5 - 3 × (- 7)) + (2 × (- 7) + 3 × 5) i = = (10 + 21) + (- 14 + 15 ) i = 31 + i.
Method 2. (2 + 3i) (5 - 7i) = 2 × 5 + 2 × (- 7i) + 3i × 5 + 3i × (- 7i) = = 10 - 14i + 15i + 21 = 31 + i.
4) Division.
Definition... Divide complex number z 1 on a complex number z 2, then find such a complex number z, what z z 2 = z 1.
Theorem. The quotient of complex numbers exists and is unique if z 2 ≠ 0 + 0i.
In practice, the quotient of complex numbers is found by multiplying the numerator and denominator by the conjugate of the denominator.
Let be z 1 = a 1 + b 1 i, z 2 = a 2 + b 2 i, then
.
In the following example, we will divide by the formula and the rule of multiplication by the conjugate of the denominator.
Example 4. Find the quotient 
.
5) Raising to a positive integer.
a) The powers of the imaginary unit.
Using the equality i 2 = -1, it is easy to define any positive integer power of the imaginary unit. We have:
i 3 = i 2 i = -i,
i 4 = i 2 i 2 = 1,
i 5 = i 4 i = i,
i 6 = i 4 i 2 = -1,
i 7 = i 5 i 2 = -i,
i 8 = i 6 i 2 = 1 etc.
This shows that the values of the degree i n, where n- a positive integer, periodically repeated when the indicator increases by 4 .
Therefore, to raise the number i to a whole positive degree, the exponent must be divided by 4 and erect i to the power, the exponent of which is equal to the remainder of the division.
Example 5. Calculate: (i 36 + i 17) i 23.
i 36 = (i 4) 9 = 1 9 = 1,
i 17 = i 4 × 4 + 1 = (i 4) 4 × i = 1 i = i.
i 23 = i 4 × 5 + 3 = (i 4) 5 × i 3 = 1 · i 3 = - i.
(i 36 + i 17) i 23 = (1 + i) (- i) = - i + 1 = 1 - i.
b) Raising a complex number to a positive integer power is performed according to the rule of raising a binomial to the appropriate power, since it is a special case of multiplying the same complex factors.
Example 6. Calculate: (4 + 2i) 3
(4 + 2i) 3 = 4 3 + 3 × 4 2 × 2i + 3 × 4 × (2i) 2 + (2i) 3 = 64 + 96i - 48 - 8i = 16 + 88i.
Complex numbers are the minimum extension of the set of real numbers we are used to. Their fundamental difference is that an element appears that gives -1 in the square, i.e. i, or.
Any complex number has two parts: real and imaginary:
Thus, it can be seen that the set of real numbers coincides with the set of complex numbers with a zero imaginary part.
The most popular model for the set of complex numbers is the Plane. The first coordinate of each point will be its real part, and the second one will be imaginary. Then vectors with the origin at the point (0,0) will act as the complex numbers themselves.
Operations on complex numbers.
In fact, if we take into account the model of a set of complex numbers, it is intuitively clear that addition (subtraction) and multiplication of two complex numbers are performed in the same way as the corresponding operations on vectors. And we mean the vector product of vectors, because the result of this operation is again a vector.
1.1 Addition.
(As you can see, this operation exactly matches)
1.2 Subtraction, similarly, is performed according to the following rule:
2. Multiplication.
3. Division.
Defined simply as the inverse of multiplication.
Trigonometric form.
The modulus of a complex number z is the following quantity:
,
obviously this is, again, just the modulus (length) of the vector (a, b).
Most often, the modulus of a complex number is denoted as ρ.
It turns out that
z = ρ (cosφ + isinφ).
The following immediately follow from the trigonometric form of notation for a complex number. formulas :
The last formula is called Moivre formula. The formula is derived directly from it nth root of a complex number:
thus, there are n roots of the nth degree of the complex number z.
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 character whose square is -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 of a + 0i write simply a... It can be seen that real numbers are a special case of complex numbers.
Arithmetic operations on complex numbers are the same as on real ones: they can be added, subtracted, multiplied and divided by each other. Addition and subtraction occurs according to the rule ( a + bi) ± ( c + di) = (a ± c) + (b ± d)i, and multiplication - according to the rule ( a + bi) · ( c + di) = (ac – bd) + (ad + bc)i(it is just used here that i 2 = –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 intuitive geometric representation: the number z = a + bi can be represented by a vector with coordinates ( a; b) on the Cartesian plane (or, which is almost the same, a point - the end of the vector with these coordinates). In this case, the sum of two complex numbers is depicted as the sum of the corresponding vectors (which can be found by the parallelogram rule). By the Pythagorean theorem, the length of the vector with coordinates ( a; b) is equal. This quantity is called module complex number z = a + bi and denoted by | z|. The angle that this vector makes with the positive direction of the abscissa axis (counted counterclockwise) is called argument complex number z and is denoted by Arg z... The argument is not uniquely defined, but only up to the addition of a multiple of 2 π
radians (or 360 °, if you count in degrees) - after all, it is clear that rotation by such an angle around the origin will not change the vector. But if the vector of length r forms an angle φ
with a positive direction of the abscissa axis, then its coordinates are ( r Cos φ
; r Sin φ
). Hence it turns out trigonometric notation complex number: z = |z| (Cos (Arg z) + i sin (Arg z)). It is often convenient to write complex numbers in this form, because it greatly simplifies calculations. Multiplying complex numbers in trigonometric form looks very simple: z 1 · z 2 = |z 1 | · | z 2 | (Cos (Arg z 1 + Arg z 2) + i sin (Arg z 1 + Arg z 2)) (when multiplying two complex numbers, their modules are multiplied, and the arguments are added). Hence follow Moivre formulas: z n = |z|n(Cos ( n(Arg z)) + i sin ( n(Arg z))). Using these formulas, it is easy to learn how to extract roots of any degree from complex numbers. Nth root of z is such a complex number w, what w n = z... It's clear that 
, And where k can take any value from the set (0, 1, ..., n- 1). This means that there is always exactly n roots n-th degree of a complex number (on the plane, they are located at the vertices of the correct n-gon).
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