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 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 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.
Complex numbers are an extension of the set of real numbers, usually denoted by . Any complex number can be represented as a formal sum , where and are real numbers and is the imaginary unit.
Writing a complex number in the form , , is called the algebraic form of a complex number.
Properties of complex numbers. Geometric interpretation of a complex number.
Actions on complex numbers given in algebraic form:
Let's consider the rules by which arithmetic operations are performed on complex numbers.
If two complex numbers α = a + bi and β = c + di are given, then
α + β = (a + bi) + (c + di) = (a + c) + (b + d)i,
α – β = (a + bi) – (c + di) = (a – c) + (b – d)i. (eleven)
This follows from the definition of the operations of addition and subtraction of two ordered pairs of real numbers (see formulas (1) and (3)). We have received the rules for adding and subtracting complex numbers: in order to add two complex numbers, we must separately add their real parts and, accordingly, their imaginary parts; In order to subtract another from one complex number, it is necessary to subtract their real and imaginary parts, respectively.
The number – α = – a – bi is called the opposite of the number α = a + bi. The sum of these two numbers is zero: - α + α = (- a - bi) + (a + bi) = (-a + a) + (-b + b)i = 0.
To obtain the rule for multiplying complex numbers, we use formula (6), i.e., the fact that i2 = -1. Taking this relation into account, we find (a + bi)(c + di) = ac + adi + bci + bdi2 = ac + (ad + bc)i – bd, i.e.
(a + bi)(c + di) = (ac - bd) + (ad + bc)i . (12)
This formula corresponds to formula (2), which determined the multiplication of ordered pairs of real numbers.
Note that the sum and product of two complex conjugate numbers are real numbers. Indeed, if α = a + bi, = a – bi, then α = (a + bi)(a - bi) = a2 – i2b2 = a2 + b2 , α + = (a + bi) + (a - bi) = ( a + a) + (b - b)i= 2a, i.e.
α + = 2a, α = a2 + b2. (13)
When dividing two complex numbers in algebraic form, one should expect that the quotient is also expressed by a number of the same type, i.e. α/β = u + vi, where u, v R. Let us derive the rule for dividing complex numbers. Let the numbers α = a + bi, β = c + di be given, and β ≠ 0, i.e. c2 + d2 ≠ 0. The last inequality means that c and d do not simultaneously vanish (the case is excluded when c = 0, d = 0). Applying formula (12) and the second of equalities (13), we find:
Therefore, the quotient of two complex numbers is determined by the formula:
corresponding to formula (4).
Using the resulting formula for the number β = c + di, you can find its inverse number β-1 = 1/β. Assuming a = 1, b = 0 in formula (14), we obtain
This formula determines the inverse of a given complex number other than zero; this number is also complex.
For example: (3 + 7i) + (4 + 2i) = 7 + 9i;
(6 + 5i) – (3 + 8i) = 3 – 3i;
(5 – 4i)(8 – 9i) = 4 – 77i;
Operations on complex numbers in algebraic form.
55. Argument of a complex number. Trigonometric form of writing a complex number (derivation).
Arg.com.numbers. – between the positive direction of the real X axis and the vector representing the given number.
Trigon formula. Numbers: ,
DEFINITION
The algebraic form of a complex number is to write the complex number \(\z\) in the form \(\z=x+i y\), where \(\x\) and \(\y\) are real numbers, \(\i\ ) - imaginary unit satisfying the relation \(\i^(2)=-1\)
The number \(\ x \) is called the real part of the complex number \(\ z \) and is denoted by \(\ x=\operatorname(Re) z \)
The number \(\y\) is called the imaginary part of the complex number \(\z\) and is denoted by \(\y=\operatorname(Im) z\)
For example:
The complex number \(\ z=3-2 i \) and its adjoint number \(\ \overline(z)=3+2 i \) are written in algebraic form.
The imaginary quantity \(\ z=5 i \) is written in algebraic form.
In addition, depending on the problem you are solving, you can convert a complex number to a trigonometric or exponential number.
Write the number \(\z=\frac(7-i)(4)+13\) in algebraic form, find its real and imaginary parts, as well as its conjugate number.
Using the term division of fractions and the rule of adding fractions, we get:
\(\z=\frac(7-i)(4)+13=\frac(7)(4)+13-\frac(i)(4)=\frac(59)(4)-\frac( 1)(4)i\)
Therefore, the real part of the complex number \(\ z=\frac(5 g)(4)-\frac(1)(4) i \) is the number \(\ x=\operatorname(Re) z=\frac(59) (4) \) , the imaginary part is the number \(\ y=\operatorname(Im) z=-\frac(1)(4) \)
Conjugate number: \(\ \overline(z)=\frac(59)(4)+\frac(1)(4) i \)
\(\ z=\frac(59)(4)-\frac(1)(4) i \), \(\ \operatorname(Re) z=\frac(59)(4) \), \(\ \operatorname(Im) z=-\frac(1)(4) \), \(\ \overline(z)=\frac(59)(4)+\frac(1)(4) i \)
Actions of complex numbers in algebraic form comparison
Two complex numbers \(\ z_(1)=x_(1)+i y_(1) \) are said to be equal if \(\ x_(1)=x_(2) \), \(\ y_(1)= y_(2) \) i.e. Their real and imaginary parts are equal.
Determine for which x and y the two complex numbers \(\ z_(1)=13+y i \) and \(\ z_(2)=x+5 i \) are equal.
By definition, two complex numbers are equal if their real and imaginary parts are equal, i.e. \(\x=13\), \(\y=5\).
addition
Adding complex numbers \(\z_(1)=x_(1)+i y_(1)\) is done by directly summing the real and imaginary parts:
\(\ z_(1)+z_(2)=x_(1)+i y_(1)+x_(2)+i y_(2)=\left(x_(1)+x_(2)\right) +i\left(y_(1)+y_(2)\right) \)
Find the sum of complex numbers \(\ z_(1)=-7+5 i \), \(\ z_(2)=13-4 i \)
The real part of a complex number \(\ z_(1)=-7+5 i \) is the number \(\ x_(1)=\operatorname(Re) z_(1)=-7 \) , the imaginary part is the number \( \ y_(1)=\mathrm(Im) \), \(\ z_(1)=5 \) . The real and imaginary parts of the complex number \(\ z_(2)=13-4 i \) are equal to \(\ x_(2)=\operatorname(Re) z_(2)=13 \) and \(\ y_(2) respectively )=\operatorname(Im) z_(2)=-4 \) .
Therefore, the sum of complex numbers is:
\(\z_(1)+z_(2)=\left(x_(1)+x_(2)\right)+i\left(y_(1)+y_(2)\right)=(-7+ 13)+i(5-4)=6+i \)
\(\ z_(1)+z_(2)=6+i \)
Read more about adding complex numbers in a separate article: Adding complex numbers.
Subtraction
Subtraction of complex numbers \(\z_(1)=x_(1)+i y_(1)\) and \(\z_(2)=x_(2)+i y_(2)\) is performed by directly subtracting the real and imaginary parts:
\(\ z_(1)-z_(2)=x_(1)+i y_(1)-\left(x_(2)+i y_(2)\right)=x_(1)-x_(2) +\left(i y_(1)-i y_(2)\right)=\left(x_(1)-x_(2)\right)+i\left(y_(1)-y_(2)\right ) \)
find the difference of complex numbers \(\ z_(1)=17-35 i \), \(\ z_(2)=15+5 i \)
Find the real and imaginary parts of complex numbers \(\ z_(1)=17-35 i \), \(\ z_(2)=15+5 i \) :
\(\ x_(1)=\operatorname(Re) z_(1)=17, x_(2)=\operatorname(Re) z_(2)=15 \)
\(\ y_(1)=\operatorname(Im) z_(1)=-35, y_(2)=\operatorname(Im) z_(2)=5 \)
Therefore, the difference of complex numbers is:
\(\ z_(1)-z_(2)=\left(x_(1)-x_(2)\right)+i\left(y_(1)-y_(2)\right)=(17-15 )+i(-35-5)=2-40 i \)
\(\ z_(1)-z_(2)=2-40 i \) multiplication
Multiplication of complex numbers \(\ z_(1)=x_(1)+i y_(1) \) and \(\ z_(2)=x_(2)+i y_(2) \) is performed by directly creating numbers in algebraic form taking into account the property of the imaginary unit \(\i^(2)=-1\) :
\(\ z_(1) \cdot z_(2)=\left(x_(1)+i y_(1)\right) \cdot\left(x_(2)+i y_(2)\right)=x_ (1) \cdot x_(2)+i^(2) \cdot y_(1) \cdot y_(2)+\left(x_(1) \cdot i y_(2)+x_(2) \cdot i y_(1)\right)=\)
\(\ =\left(x_(1) \cdot x_(2)-y_(1) \cdot y_(2)\right)+i\left(x_(1) \cdot y_(2)+x_(2 ) \cdot y_(1)\right) \)
Find the product of complex numbers \(\ z_(1)=1-5 i \)
Complex of complex numbers:
\(\ z_(1) \cdot z_(2)=\left(x_(1) \cdot x_(2)-y_(1) \cdot y_(2)\right)+i\left(x_(1) \cdot y_(2)+x_(2) \cdot y_(1)\right)=(1 \cdot 5-(-5) \cdot 2)+i(1 \cdot 2+(-5) \cdot 5 )=15-23 i\)
\(\ z_(1) \cdot z_(2)=15-23 i \) division
The factor of complex numbers \(\z_(1)=x_(1)+i y_(1)\) and \(\z_(2)=x_(2)+i y_(2)\) is determined by multiplying the numerator and denominator to the conjugate number with the denominator:
\(\ \frac(z_(1))(z_(2))=\frac(x_(1)+i y_(1))(x_(2)+i y_(2))=\frac(\left (x_(1)+i y_(1)\right)\left(x_(2)-i y_(2)\right))(\left(x_(2)+i y_(2)\right)\left (x_(2)-i y_(2)\right))=\frac(x_(1) \cdot x_(2)+y_(1) \cdot y_(2))(x_(2)^(2) +y_(2)^(2))+i \frac(x_(2) \cdot y_(1)-x_(1) \cdot y_(2))(x_(2)^(2)+y_(2 )^(2)) \)
To divide the number 1 by the complex number \(\z=1+2i\).
Since the imaginary part of the real number 1 is zero, the factor is:
\(\ \frac(1)(1+2 i)=\frac(1 \cdot 1)(1^(2)+2^(2))-i \frac(1 \cdot 2)(1^( 2)+2^(2))=\frac(1)(5)-i \frac(2)(5)\)
\(\ \frac(1)(1+2 i)=\frac(1)(5)-i \frac(2)(5) \)
Complex numbers
Imaginary And complex numbers. Abscissa and ordinate
complex number. Conjugate complex numbers.
Operations with complex numbers. Geometric
representation of complex numbers. Complex plane.
Modulus and argument of a complex number. Trigonometric
complex number form. Operations with complex
numbers in trigonometric form. Moivre's formula.
Basic information about imaginary And complex numbers are given in the section “Imaginary and complex numbers”. The need for these numbers of a new type arose when solving quadratic equations for the case
D< 0 (здесь D– discriminant of a quadratic equation). For a long time, these numbers did not find physical application, which is why they were called “imaginary” numbers. However, now they are very widely used in various fields of physics.and technology: electrical engineering, hydro- and aerodynamics, elasticity theory, etc.
Complex numbers are written in the form:a+bi. Here a And b – real numbers , A i – imaginary unit, i.e. e. i 2 = –1. Number a called abscissa, a b – ordinatecomplex numbera + bi.Two complex numbersa+bi And a–bi are called conjugate complex numbers.
Main agreements:
1. Real number
Acan also be written in the formcomplex number:a+ 0 i or a – 0 i. For example, records 5 + 0i and 5 – 0 imean the same number 5 .2. Complex number 0 + bicalled purely imaginary number. Recordbimeans the same as 0 + bi.
3. Two complex numbersa+bi Andc + diare considered equal ifa = c And b = d. Otherwise complex numbers are not equal.
Addition. Sum of complex numbersa+bi And c + diis called a complex number (a+c ) + (b+d ) i.Thus, when adding complex numbers, their abscissas and ordinates are added separately.
This definition corresponds to the rules for operations with ordinary polynomials.
Subtraction. The difference of two complex numbersa+bi(diminished) and c + di(subtrahend) is called a complex number (a–c ) + (b–d ) i.
Thus, When subtracting two complex numbers, their abscissas and ordinates are subtracted separately.
Multiplication. Product of complex numbersa+bi And c + di is called a complex number:
(ac–bd ) + (ad+bc ) i.This definition follows from two requirements:
1) numbers a+bi And c + dimust be multiplied like algebraic binomials,
2) number ihas the main property:i 2 = – 1.
EXAMPLE ( a+ bi )(a–bi) =a 2 + b 2 . Hence, work
two conjugate complex numbers is equal to the real
a positive number.
Division. Divide a complex numbera+bi (divisible) by anotherc + di(divider) - means to find the third numbere + f i(chat), which when multiplied by a divisorc + di, results in the dividenda + bi.
If the divisor is not zero, division is always possible.
EXAMPLE Find (8 +i ) : (2 – 3 i) .
Solution. Let's rewrite this ratio as a fraction:
Multiplying its numerator and denominator by 2 + 3i
AND Having performed all the transformations, we get:
Geometric representation of complex numbers. Real numbers are represented by points on the number line:
Here is the point Ameans the number –3, dotB– number 2, and O- zero. In contrast, complex numbers are represented by points on the coordinate plane. For this purpose, we choose rectangular (Cartesian) coordinates with the same scales on both axes. Then the complex numbera+bi will be represented by a dot P with abscissa a and ordinate b (see picture). This coordinate system is called complex plane .
Module complex number is the length of the vectorOP, representing a complex number on the coordinate ( comprehensive) plane. Modulus of a complex numbera+bi denoted | a+bi| or letter r