Algebraic formula for a complex number. Modulus and argument of a complex number. Trigonometric. Algebraic notation

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).

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.

The expansion of the set of real numbers consists of adding new numbers (imaginary) to the real numbers. The introduction of these numbers is due to the impossibility of extracting the root of a negative number in the set of real numbers.

Introduction to the concept of a complex number.

Imaginary numbers, with which we complement real numbers, are written in the form bi, Where i is an imaginary unit, and i 2 = - 1.

Based on this, we obtain 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 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– real number.

Complex number a+bi counts equal to zero, if its real and imaginary parts are equal to zero: a = b = 0

Complex number a+bi at b = 0 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, differing only in the sign of the imaginary part, are called conjugate.

Operations on complex numbers in algebraic form.

You can perform the following operations on complex numbers in algebraic form.

1) Addition.

Definition. Sum of complex numbers z 1 = a 1 + b 1 i And z 2 = a 2 + b 2 i is 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 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.

Addition of complex numbers has the following properties:

1º. Commutativity: 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 of complex number z, denoted -z. Sum of complex numbers z And -z 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 between complex numbers exists and is unique.

Example 2: Perform a subtraction (4 – 2i) - (-3 + 2i).

(4 – 2i) - (-3 + 2i) = (4 - (-3)) + (-2 - 2) i = 7 – 4i.

3) Multiplication.

Definition. Product of complex numbers z 1 =a 1 +b 1 i And z 2 =a 2 +b 2 i is 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. Commutativity: z 1 z 2 = z 2 z 1.

2º. Associativity: (z 1 z 2)z 3 = z 1 (z 2 z 3)

3º. Distributivity 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- real number.

In practice, multiplication of complex numbers is carried out according to the rule of multiplying a sum by a sum and separating the real and imaginary parts.

In the following example, we will consider multiplying complex numbers in two ways: by rule and by multiplying sum by sum.

Example 3: Do the 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 a complex number z 1 to a complex number z 2, means to 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 z 1 = a 1 + b 1 i, z 2 = a 2 + b 2 i, Then

.

In the following example, we will perform division using the formula and the rule of multiplication by the number conjugate to the denominator.

Example 4. Find the quotient .

5) Raising to a positive whole power.

a) Powers of the imaginary unit.

Taking advantage of 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 degree values i n, Where n– a positive integer, periodically repeated as the indicator increases by 4 .

Therefore, to raise the number i to a positive whole power, we must divide the exponent by 4 and build i to a power whose exponent 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 carried out according to the rule for raising a binomial to the corresponding power, since it is a special case of multiplying identical 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.

§ 1. Complex numbers: definitions, geometric interpretation, operations in algebraic, trigonometric and exponential forms

Definition of a complex number

Complex equalities

Geometric representation of complex numbers

Modulus and argument of a complex number

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. Solution 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

Definition of a complex number ( State 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 - purely 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 separating the real and imaginary parts.

1) ;

2) .

Geometric representation of complex numbers ( What is the geometric representation of complex numbers?)

Complex number z represented by a dot ( 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 a complex plane.

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 polar radius of a 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 it 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;

Multiplying 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 .

Multiplying 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 their arguments are added.

Basic properties of multiplication

1)z 1× z 2 = z 2× z 1 - commutativity;

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 - distributivity with respect to addition;

4)z×0 = 0; z×1 = z ;

Division of complex numbers

Division is the inverse operation of multiplication, so

If z × z 2 = z 1 and z 2 ¹ 0, then .

When performing division 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 performing division in trigonometric form, the modules are divided and the 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 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.

Notes

1. When performing the operations of multiplication and raising to a natural power in trigonometric form, angle values ​​​​beyond one full revolution can be obtained. But they can always be reduced to angles or by dropping an integer number of full revolutions using the periodicity properties of the functions and .

2. Meaning called the principal value of the argument of a complex number;

in this case, the values ​​of all possible angles are denoted by ;

it's obvious that , .

Extracting the root of a natural degree from a complex number

Euler's formulas(16)

according to which trigonometric functions and a real variable are expressed through exponential function(exponent) 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 for the same powers of the variable x, that is

Proof

w Identity (3) is valid for "xО (or "xО)

Þ it is valid for ; substituting , we get an = bn .

Let us mutually cancel the terms in (3) an And bn and divide both parts by x :

This identity is also true for " x, including when x = 0

Þ assuming x= 0, we get an – 1 = bn – 1.

Let us mutually cancel the terms in (3") an– 1 and a n– 1 and divide both sides by x, as a result we get

Continuing the argument similarly, we obtain that an – 2 = bn –2, …, A 0 = b 0.

Thus, it has been proven that the identical equality of 2-x polynomials implies the coincidence of their coefficients at the same degrees x .

The converse statement is rightly obvious, i.e. if two polynomials have the same coefficients, then they are identical functions, therefore, their values ​​coincide for all values ​​of the argument, which means they are identically equal. Property 1 has been completely proven. v

When dividing a polynomial Pn (x) by the difference ( x – X 0) the remainder is equal to Pn (x 0), that is

Bezout's theorem,(4)

Where Qn – 1(x) - whole part from division, is a polynomial of degree ( n – 1).

Proof

w Let's write the division formula with a remainder:

Pn (x) = (x – X 0)∙Qn – 1(x) + A ,

Where Qn – 1(x) - polynomial of degree ( n – 1),

A- the remainder, which is a number due to the well-known algorithm for dividing a polynomial by a binomial “in a column”.

This equality is true for " x, including when x = X 0 Þ

Pn (x 0) = (x 0 – x 0)× Qn – 1(x 0) + A Þ

A = Pn (X 0), etc. v

Corollary to Bezout's theorem. On dividing a polynomial by a binomial without a remainder

If the number X 0 is the zero of the polynomial, then this polynomial is divided by the difference ( x – X 0) without remainder, that is

Þ .(5)

1) , since P 3(1) º 0

2) because P 4(–2) º 0

3) because P 2(–1/2) º 0

Dividing polynomials into binomials “in a column”:

_

_

_

_

_

Every 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 accept the theorem without proof.

Let's work on this theorem and Bezout's theorem with the polynomial Pn (x).

After n-multiple application of these theorems we obtain that

Where a 0 is the coefficient at x n V Pn (x).

Corollary to the fundamental theorem of algebra. On the decomposition of a polynomial into linear factors

Any polynomial of degree on the set of complex numbers can be decomposed into n linear factors, that is

Expansion of a polynomial into linear factors, (6)

where x1, x2, ... xn are the zeros of the polynomial.

Moreover, if k numbers from the set X 1, X 2, … xn coincide with each other and with the number a, then in the product (6) the multiplier ( 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 of the polynomial Pn ( x) .

1)P 4(x) = (x – 2)(x– 4)3 Þ x 1 = 2 - simple zero, x 2 = 4 - triple zero;

2)P 4(x) = (x – i)4 Þ x = i- zero multiplicity 4.

Property 4 (about 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 we count each root 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 a polynomial Pn (x). If all coefficients of this polynomial are real numbers, then is also its zero (by property 5).

Let's 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 quadratic 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. Binomial 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 you need to count each root as many times as its multiplicity. Since all coefficients given equation are real numbers, then the complex roots of the equation, if there are any, will be paired complex conjugates.

By selection we find the first root of the equation, since .

By corollary to Bezout's theorem. We calculate this division “in a column”:

_
_
_

Now representing the polynomial as a product of a linear and a square factor, we get:

.

We find other roots as roots of a quadratic equation:

Answer: , .

5. Construct an algebraic equation of the smallest 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, because the coefficients of the equation must be real.

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 a complex zero?

13. Formulate the meaning of complex equality.

15. What is the modulus and argument of a complex number?

17. What is the argument of a complex number?

18. What is the name or meaning of the formula?

19. Explain the meaning of the notation in this formula:

27. Give definitions and list the main properties of arithmetic operations on complex numbers.

28. What is the name or meaning of the formula?

29. Explain the meaning of the notation in this formula:

31. What is the name or meaning of the formula?

32. Explain the meaning of the notation in this formula:

34. What is the name or meaning of the formula?

35. Explain the meaning of the notation in this formula:

61. List the main properties of polynomials.

63. State the property about dividing a polynomial by the difference (x – x0).

65. What is the name or meaning of the formula?

66. Explain the meaning of the notation in this formula:

67. ⌂ .

69. State the theorem: the basic theorem of algebra.

70. What is the name or meaning of the formula?

71. Explain the meaning of the notation in this formula:

75. State the property about the number of roots of an algebraic equation.

78. State the property about the decomposition of a polynomial with real coefficients into linear and quadratic factors.

Glossary

The k-fold zero of a polynomial is... (p. 18)

an algebraic polynomial is called... (p. 14)

algebraic equation nth degree called... (page 14)

the algebraic form of a complex number is called... (p. 5)

the argument of a complex number is... (page 4)

the real part of a complex number z is... (page 2)

a complex conjugate number is... (page 2)

complex zero is... (page 2)

a complex number is called... (page 2)

a root of degree n of a complex number is called... (p. 10)

the root of the equation is... (p. 14)

the coefficients of the polynomial are... (p. 14)

the imaginary unit is... (page 2)

the imaginary part of a complex number z is... (page 2)

the modulus of a complex number is called... (p. 4)

the zero of a function is called... (p. 14)

the exponential form of a complex number is called... (p. 11)

a 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)

the trigonometric form of a complex number is called... (p. 5)

Moivre's formula is... (p. 9)

Euler's formulas are... (page 13)

the entire function is called... (p. 14)

a purely imaginary number is... (p. 2)

FEDERAL AGENCY FOR EDUCATION

STATE EDUCATIONAL INSTITUTION

HIGHER PROFESSIONAL EDUCATION

"VORONEZH STATE PEDAGOGICAL UNIVERSITY"

DEPARTMENT OF AGLEBRA AND GEOMETRY

Complex numbers

(selected tasks)

GRADUATE QUALIFYING WORK

specialty 050201.65 mathematics

(With additional specialty 050202.65 computer science)

Completed by: 5th year student

physical and mathematical

faculty

Scientific adviser:

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. List of references………………………….………………………......

1. Introduction

In the mathematics program school course number theory is introduced using examples of sets natural numbers, whole, rational, irrational, i.e. on the set of real numbers, the images of which fill the entire number line. But already in the 8th grade there is not enough supply of real numbers, solving quadratic equations with a negative discriminant. Therefore, it was necessary to replenish the stock of real numbers with the help of 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 qualification work is that the concept of a complex number expands students’ knowledge about number systems, about solving a wide class of problems of both algebraic and geometric content, about solving algebraic equations of any degree and about solving problems with parameters.

This thesis examines the solution to 82 problems.

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, the conjugation operation for complex numbers in algebraic form, the power of an imaginary unit, the modulus of a complex number, and also sets out the rule extraction square root from a complex number.

In the second part, problems on the geometric interpretation of complex numbers in the form of points or vectors of the complex plane are solved.

The third part examines operations on complex numbers in trigonometric form. The formulas used are: Moivre and extracting the root of a complex number.

The fourth part is devoted to solving equations of the 3rd and 4th degrees.

When solving problems in 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 are devoted to defining 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 field C). There are tasks where a complex variable simultaneously satisfies a number of conditions. A special feature of solving problems in 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 in each part is the initial input theoretical foundations, and subsequently their practical application in solving problems.

At the end thesis a list of used literature is presented. Most of them present the theoretical material, solutions to some problems are considered and practical tasks are given for independent decision. Special attention I would like to refer to such sources as:

1. Gordienko N.A., Belyaeva E.S., Firstov V.E., Serebryakova I.V. Complex numbers and their applications: Textbook. . Material teaching aid presented in the form of lectures and practical classes.

2. Shklyarsky D.O., Chentsov N.N., Yaglom I.M. Selected problems and theorems of elementary mathematics. Arithmetic and algebra. The book contains 320 problems related to algebra, arithmetic and number theory. These tasks differ significantly in nature from standard school tasks.

2. Complex numbers (selected problems)

2.1. Complex numbers in algebraic form

The solution of many problems in mathematics and physics comes down 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, it has no real roots quadratic equation with a negative discriminant. 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

.

Let us denote this root by

. Thus, by definition, or,

hence,

. called the 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 because they contained both real and imaginary parts.

So, complex numbers are expressions of the form

, and are real numbers, and is a certain symbol that satisfies the condition . The number is called the real part of a complex number, and the number is its imaginary part. The symbols , are used to denote them.

Complex numbers of the form

are real numbers and, therefore, the set of complex numbers contains the set of real numbers.

Complex numbers of the form

are called purely imaginary. Two complex numbers of the form and are said to be equal if their real and imaginary parts are equal, i.e. if equalities , .

Algebraic notation of complex numbers allows 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 .

Product of two complex numbers