Loktionova G.N.
mathematics teacher
GAPOU "Vehicle Transport College"
"Complex numbers and actions
above them"
clause 1 Historical background
The concept of a complex number arose from the practice and theory of solving algebraic equations.
Mathematicians first encountered complex numbers when solving quadratic equations. Until the 16th century, mathematicians around the world, not finding an acceptable interpretation for the complex roots that arose when solving quadratic equations, declared them false and did not take them into account.
Cardano, who worked on solving equations of the 3rd and 4th degrees, was one of the first mathematicians to formally operate with complex numbers, although their meaning remained largely unclear to him.
The meaning of complex numbers was explained by another Italian mathematician R. Bombelli. In his book Algebra (1572), he first set out the rules for operating complex numbers in modern form.
However, until the 18th century, complex numbers were considered “imaginary” and useless. It is interesting to note that even such an outstanding mathematician as Descartes, who identified real numbers with segments of the number line, believed that there could be no real interpretation for complex numbers, and they would forever remain imaginary, imaginary. The great mathematicians Newton and Leibniz held similar views.
Only in the 18th century, many problems of mathematical analysis, geometry, and mechanics required the widespread use of operations on complex numbers, which created the conditions for the development of their geometric interpretation.
In the applied works of d'Alembert and Euler in the mid-18th century, the authors represent arbitrary imaginary quantities in the form z=a+ib, which allows such quantities to be represented by points on the coordinate plane. It was this interpretation that was used by Gauss in his work devoted to the study of solutions to algebraic equations.
And only at the beginning of the 19th century, when the role of complex numbers in various fields of mathematics was already clarified, a very simple and natural geometric interpretation of them was developed, which made it possible to understand the geometric meaning of operations on complex numbers.
P. 2 Basic Concepts
Complex number z called an expression of the form z=a+ib, Where a And b– real numbers, i – imaginary unit, which is determined by the relation:
In this case the number a called real part numbers z
(a = Re z), A b - imaginary part (b = Im z).
If a = Rez =0 , that number z will purely imaginary, If b = Im z =0 , then the number z will valid .
Numbers z=a+ib and are called complex - conjugate .
Two complex numbers z 1 =a 1 +ib 1 And z 2 =a 2 +ib 2 are called equal, if their real and imaginary parts are respectively equal:
a 1 =a 2 ; b 1 =b 2
A complex number is equal to zero if the real and imaginary parts are equal to zero, respectively.
Complex numbers can also be written, for example, in the form z=x+iy , z=u+iv .
P. 3 Geometric representation of complex numbers
Any complex number z=x+iy can be represented by a dot M(x;y) plane xOy such that X = Rez , y = Im z. And, conversely, every point M(x;y) coordinate plane can be considered as the image of a complex number z=x+iy(picture 1).
Picture 1
The plane on which complex numbers are depicted is called complex plane .
The abscissa axis is called real axis, since it contains real numbers z=x+0i=x .
The y-axis is called imaginary axis, it contains imaginary complex numbers z=0+yi=yi .
Often, instead of points on the plane, they take them radius vectors
those. vectors starting with a point O(0;0), end M(x;y) .
Length of the vector representing a complex number z , called module this number is designated | z| or r .
The magnitude of the angle between the positive direction of the real axis and the vector representing a complex number is called argument of this complex number is denoted Arg z or φ .
Complex Number Argument z=0 indefined.
Complex Number Argument z ≠ 0 - the quantity is multi-valued and is determined accurate to the summand 2 π k (k=0,-1,1,-2,2,..) :
Arg z=arg z+2 π k,
Where arg z - main meaning of the argument , concluded in the interim (- π , π ] .
p.4 Forms of writing complex numbers
Writing a number in the form z=x+iy called algebraic form complex number.
From Figure 1 it is clear that x=rcos φ , y=rsin φ , therefore, complex z=x+iy the number can be written as:
This form of recording is called trigonometric notation complex number.
Module r=|z| is uniquely determined by the formula
Argument φ determined from the formulas
When moving from the algebraic form of a complex number to the trigonometric one, it is enough to determine only the main value of the argument of the complex number, i.e. count φ =arg z .
Since from the formula we get that
For interior points I , IV quarters;
For interior points II quarters;
For interior points III quarters.
Example 1. Represent complex numbers in trigonometric form.
Solution. Complex number z=x+iy in trigonometric form has the form z=r(cos φ +isin φ ) , Where
1) z 1 = 1 +i(number z 1 belongs I quarters), x=1, y=1.
Thus,
2) (number z 2 belongs II quarters)
Since then
Hence,
Answer:
Consider the exponential function w=e z, Where z=x+iy- complex number.
It can be shown that the function w can be written as:
This equality is called Euler's equation.
For complex numbers the following properties will be true:
Where m– an integer.
If in the Euler equation the exponent is taken to be a purely imaginary number ( x=0), then we get:
For a complex conjugate number we get:
From these two equations we get:
These formulas are used to find the values of powers of trigonometric functions through functions of multiple angles.
If you represent a complex number in trigonometric form
z=r(cos φ +isin φ )
and use Euler's formula e i φ =cos φ +isin φ , then the complex number can be written as
z=r e i φ
The resulting equality is called exponential form complex number.
P. 5 Operations on complex numbers
1) Actions on complex numbers given in algebraic form
a) Addition of complex numbers
Amount two complex numbers z 1 =x 1 +y 1 i And z 2 =x 2 +y 2 i
z 1 +z 2 =(x 1 +x 2 )+i(y 1 +y 2 ).
Properties of the addition operation:
1. z 1 +z 2 = z 2 +z 1 ,
2. (z 1 +z 2 )+z 3 =z 1 +(z 2 +z 3 ) ,
3. z+0=z .
b) Subtraction of complex numbers
Subtraction is defined as the inverse of addition.
By difference two complex numbers z 1 =x 1 +y 1 i And z 2 =x 2 +y 2 i such a complex number is called z, which, when added to z 2 , gives the number z 1 and is defined by the equality
z=z 1 – z 2 =(x 1 –x 2 )+i(y 1 -y 2 ).
c) Multiplication of complex numbers
The work complex numbers z 1 =x 1 +y 1 i And z 2 =x 2 +y 2 i, defined by equality
z=z 1 z 2 =(x 1 x 2 –y 1 y 2 )+i(x 1 y 2 –x 2 y 1 ).
From here, in particular, follows the most important relation
i 2 = – 1.
Properties of the multiplication operation:
1. z 1 z 2 = z 2 z 1 ,
2. (z 1 z 2 )z 3 =z 1 (z 2 z 3 ) ,
3. z 1 ( z 2 +z 3 ) =z 1 z 2 +z 1 z 3 ,
4 . z ∙ 1 =z .
d) Division of complex numbers
Division is defined as the inverse of multiplication.
The quotient of two complex numbers z 1 And z 2 ≠ 0 is called a complex number z, which when multiplied by z 2 , gives the number z 1 , i.e. If z 2 z = z 1 .
If you put z 1 =x 1 +y 1 i , z 2 =x 2 +y 2 i ≠ 0, z=x+yi , then from equality (x+yi)(x 2 +iy 2 )= x 1 +y 1 i, should
Solving the system, we find the values x And y :
Thus,
In practice, instead of the resulting formula, the following technique is used: they multiply the numerator and denominator of the fraction by the number conjugate to the denominator (“get rid of the imaginary in the denominator”).
Example 2. Given complex numbers 10+8i , 1+i. Let's find their sum, difference, product and quotient.
Solution.
A) (10+8i)+(1+i)=(10+1)+(8+1)i=11+9i;
b) (10+8i)–(1+i) =(10–1)+(8–1)i= 9 + 7 i;
V) (10+8i)(1+i) = 10+10 i +8 i +8 i 2 =2+18i;
e) Construction of a complex number given in algebraic form in n th degree
Let us write down the integer powers of the imaginary unit:
In general, the result can be written as follows:
Example 3. Calculate i 2 092 .
Solution.
We have: 2092=4 ∙ 523 .
Thus, i 2 092 = i 4 ∙ 523 =(i 4 ) 523 , but since i 4 = 1 , then we finally get i 2 092 = 1 .
Answer: i 2 092 = 1 .
When constructing a complex number a+bi to the second and third powers, use the formula for the square and cube of the sum of two numbers, and when raising to a power n (n- natural number, n ≥ 4 ) – Newton’s binomial formula:
To find the coefficients in this formula, it is convenient to use Pascal's triangle.
e) Extracting the square root of a complex number
Square root A complex number is called a complex number whose square is equal to the given one.
Let us denote the square root of a complex number x+yi through u+vi, then by definition
Formulas for finding u And v look like
Signs u And v are chosen so that the resulting u And v satisfied equality 2uv=y .
0, then u and v are one complex number of identical signs.) Answer: content" width="640" Example 4. Finding the square root of a complex number z=5+12i .
Solution.
Let us denote the square root of the number z through u+vi, Then (u+vi) 2 =5+12i .
Because in this case x=5 , y=12, then using formulas (1) we obtain:
u 2 =9; u 1 =3; u 2 = – 3; v 2 =4; v 1 =2; v 2 = – 2.
Thus, two values of the square root are found: u 1 +v 1 i=3+2i , u 2 +v 2 i= –3 –2i, . (The signs were chosen according to the equality 2uv=y, i.e. because the y=120, That u And v one complex number of identical signs.)
Answer:
2) Operations on complex numbers given in trigonometric form
Consider two complex numbers z 1 And z 2 , given in trigonometric form
a) Product of complex numbers
Doing number multiplication z 1 And z 2 , we get
b) The quotient of two complex numbers
Let complex numbers be given z 1 And z 2 ≠ 0 .
Let's consider the quotient we have
Example 5. Given two complex numbers
Solution.
1) Using the formula. we get
Hence,
2) Using the formula. we get
Hence,
Answer:
V) Construction of a complex number given in trigonometric form in n th degree
From the operation of multiplying complex numbers it follows that
In the general case we get:
Where n – positive integer.
Hence , when raising a complex number to a power, the modulus is raised to the same power, and the argument is multiplied by the exponent .
Expression (2) is called Moivre's formula .
Abraham de Moivre (1667 - 1754) - English mathematician of French origin.
Merits of Moivre:
Moivre's formula can be used to find trigonometric functions of double, triple, etc. corners
Example 6. Find formulas sin 2 And cos 2 .
Solution.
Consider some complex number
Then on the one hand
According to Moivre's formula:
Equating, we get
Because two complex numbers are equal if their real and imaginary parts are equal, then
We obtained the well-known double angle formulas.
d) Root extraction P
Root P -th power of a complex number z is called a complex number w, satisfying the equality w n =z, i.e. If w n =z .
If we put and then, by the definition of a root and Moivre’s formula, we get
From here we have
Therefore the equality takes the form
where (i.e. from 0 to n-1).
Thus, root extraction n -th power of a complex number z is always possible and gives n different meanings. All root meanings n th degree located on a circle of radius with center at zero and divide this circle by n equal parts.
Example 7. Find all values
Solution.
First, let's represent the number in trigonometric form.
In this case x=1 , , Thus,
Hence,
Using formula
Where k=0,1,2,…,(n-1), we have:
Let's write down all the values:
Answer:
Questions for self-control
1 . Formulate the definition of a complex number.
2. What complex number is called purely imaginary?
3. What two complex numbers are called conjugate?
4. Explain what it means to add complex numbers given in algebraic form; multiply a complex number by a real number.
5. Explain the principle of dividing complex numbers given in algebraic form.
6. Write in general terms the integer powers of the imaginary unit.
7. What does it mean to raise a complex number given by an algebraic form to a power (n is a natural number)?
8. Tell us how complex numbers are depicted on a plane.
9 . What form of notation is called the trigonometric form of complex numbers?
10. Formulate the definition of the modulus and argument of a complex number.
11. Formulate the rule for multiplying complex numbers written in trigonometric form.
12. Formulate a rule for finding the quotient of two complex numbers given in trigonometric form.
13. Formulate the rule for raising complex numbers given in trigonometric form to powers.
14. Formulate a rule for extracting the nth root of a complex number given in trigonometric form.
15. Tell us about the meaning of the nth root of unity and the scope of its application.
1. History of the development of numbers.
Speaker: Do you know that in ancient times you and I were most likely considered sorcerers? In ancient times, a person who could count was considered a sorcerer. Not all literate people possessed such “witchcraft”. It was mainly scribes who knew how to count, and also, of course, merchants.
Merchants appear.
Merchants. Addition, the simplest arithmetic operation, can be mastered with a certain amount of imagination. All you had to do was imagine identical sticks, pebbles, and shells.
Speaker: This is roughly how we were taught counting in first grade. In the fifth grade we LEARNED the name of these numbers. What are they called and designated? ? (Natural "
N
» -
natural
,
Slide No. 1) What operations are allowed on the set of natural numbers? (addition, multiplication)
But problems were already beginning with subtraction. It was not always possible to subtract one number from another. Sometimes you take away, take away, and lo and behold, there’s nothing left. Nothing more to take away! So subtraction was considered a tricky action and it was not always possible to perform it.
But then the merchants came to the rescue.
“Two black sticks are, let’s say, two sheep that you have to give away, but haven’t given up yet. This is a duty!
Speaker: In general, humanity needs to interpret negative numbers, and at the same time to define the concept of integers Z -« zero » it took more than a thousand years. But operations have become permissible...( addition, subtraction and multiplication).
In general, problems similar to those described above with negative numbers arose with all “reverse” arithmetic operations. Two integers could be multiplied to produce a whole number. But the result of dividing two integers by an integer was not always the same. This also led to confusion.
Merchants: chocolate sharing scene. Look, we earned some sweets. Let's share!!!
But as? she is alone, and there are two of us, and also guests... I came up with fractions of her into parts...
Speaker: That is, in order for the result of division to always exist, it was necessary to introduce, master and understand, so to speak, the “physical meaning” of fractional numbers. This is how rational numbers came into play - Q - “quotient” - “ratio”.
Many operations have become permissible in the system of rational numbers. But what didn't always work out ? (extracting roots from non-negative numbers was partially permissible. For example, “root of 81” and “root of 2.”)
This need led to the introduction of the set of real numbers (R – real), for which the extraction of roots from non-negative numbers was an admissible algebraic operation. And yet there was one drawback - this...? ( taking the root of negative numbers.)
2. New material.
In the 18th century, mathematicians came up with special numbers to perform another “inverse” operation, taking the square root of negative numbers. These are the so-called “complex” numbers (C-complex). It’s difficult to imagine them, but it’s possible to get used to them. It is believed that all algebraic operations are permissible on the set of complex numbers. And the benefits of using complex numbers are great. The existence of these “strange” numbers greatly facilitated the calculation of complex AC electrical circuits, and also made it possible to calculate the profile of an aircraft wing. Let's get to know them better.
Let us list the minimum conditions that complex numbers must satisfy:
C2 The set of complex numbers contains all real numbers.
C3 The operations of addition, subtraction, multiplication and division satisfy the laws of arithmetic operations (combinative, commutative, distributive)
i 2 =-1, i-imaginary unit
Definition 1:
Numbers of the form bi, where i is the imaginary unit, are called purely imaginary.
For example 2i, -3i, 0.5i
Definition 2:
A complex number is the sum of a real number and a purely imaginary number.
A complex number is written as z = a + bi.
Number a is called the real part of the number z,
number bi is the imaginary part of the number z.
They are designated accordingly: a = Re z, b = Im z.
Arithmetic operations:
Comparison
a + bi = c + di means that a = c and b = d (two complex numbers are equal if and only if their real and imaginary parts are equal)
Addition
(a + bi) + (c + di) = (a + c) + (b + d)i
Subtraction
(a + bi) − (c + di) = (a − c) + (b − d)i
Multiplication
(a + bi)
× (c + di) = ac + bci + adi + bdi 2 = (ac − bd) + (bc + ad)iDivision
3. Practice.
Textbook Mordkovich A.G. Profile level. Grade 11. Let's look at the simplest examples of working on the set of complex numbers.
Consider example No. 1,2 - two ways. (p.245).
Working with the textbook. No. 32.7, 32.10, 32.12
4.Test(Application)
D/Z No. 32.5, 32.8, 32.11 a, b
Slide 2
Ancient Greek mathematicians considered only natural numbers to be “real”. Along with natural numbers, fractions were used - numbers made up of a whole number of fractions of a unit.
Slide 3
The introduction of negative numbers - this was done by Chinese mathematicians two centuries BC. e. Already in the 8th century, it was established that the square root of a positive number has two meanings - positive and negative, and the square root cannot be taken from negative numbers.
Slide 4
In the 16th century, in connection with the study of cubic equations, it became necessary to extract square roots from negative numbers.
Slide 5
Slide 6
Slide 7
This formula works flawlessly in the case when the equation has one real root, and if it has three real roots, then a negative number appears under the square root sign. It turned out that the path to these roots leads through the impossible operation of extracting the square root of a negative number.
Slide 8
Slide 9
Besides x=1, there are two more roots
Slide 10
The Italian algebraist G. Cardano in 1545 proposed introducing numbers of a new nature. He showed that the system of equations
Slide 11
which has no solutions in the set of real numbers, has solutions of the form
Slide 12
you just need to agree to act on such expressions according to the rules of ordinary algebra and assume that
Slide 13
Cardano called such quantities “purely negative” and even “sophistically negative,” considered them useless and tried not to use them. But already in 1572, a book by the Italian algebraist R. Bombelli was published, in which the first rules for arithmetic operations on such numbers were established, up to the extraction of cube roots from them.
Slide 14
The name “imaginary numbers” was introduced in 1637 by the French mathematician and philosopher R. Descartes.
In 1777, one of the greatest mathematicians of the 18th century, L. Euler, proposed using the first letter of the French word imaginaire (imaginary) to denote a number (an imaginary unit). This symbol came into general use thanks to K. Gauss. The term “complex numbers” was also introduced by Gauss in 1831.
Slide 15
The word complex (from the Latin complexus) means a connection, combination, a set of concepts, objects, phenomena, etc. that form a single whole.
L. Euler derived a remarkable formula in 1748
Slide 17
which linked together the exponential function with the trigonometric one. Using L. Euler's formula, it was possible to raise the number e to any complex power.
Slide 18
At the end of the 18th century, the French mathematician J. Lagrange was able to say that mathematical analysis was no longer complicated by imaginary quantities.
Slide 19
After the creation of the theory of complex numbers, the question arose about the existence of “hypercomplex” numbers - numbers with several “imaginary” units. Such a system was built in 1843 by the Irish mathematician W. Hamilton, who called them “quaternions”
Slide 20
4. Geometric representation of a complex number
Slide 22
Such a plane is called complex. The real numbers on it occupy the horizontal axis, the imaginary unit is depicted as one on the vertical axis; for this reason, the horizontal and vertical axes are called the real and imaginary axes, respectively.
5. Trigonometric form of a complex number.
The abscissa a and ordinate b of a complex number a + bi are expressed in terms of the modulus r and the argument q. Formulas a = r cos q , r=a/cos q b = r sin q , r=b/sin q r is the length of the vector (a+bi), q is the angle it forms with the positive direction of the abscissa axis
Slide 24
Complex numbers, despite their “falsity” and invalidity, have a very wide application. They play a significant role not only in mathematics, but also in such sciences as physics and chemistry. Currently, complex numbers are actively used in electromechanics, computer and space industries
Therefore, any complex number can be represented in the form r(cos q + i sin q), where r > 0 i.e. z=a+bi or z=r*cos q + r*sin q This expression is called the normal trigonometric form or, in short, the trigonometric form of a complex number.
Slide 26
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Complex numbers
After studying the topic “Complex numbers”, students should: Know: algebraic, geometric and trigonometric forms of a complex number. Be able to: perform addition, multiplication, subtraction, division, exponentiation operations on complex numbers, extracting the root of a complex number; convert complex numbers from algebraic to geometric and trigonometric forms; use the geometric interpretation of complex numbers; in the simplest cases, find complex roots of equations with real coefficients.
What number sets are you familiar with? N Z Q R I . Preparing to study new material
Number system Valid algebraic operations Partially valid algebraic operations Natural numbers, N Integers, Z Rational numbers, Q Real numbers, R Addition, multiplication Subtraction, division, rooting Addition, subtraction, multiplication Division, rooting Addition, subtraction, multiplication, division Extracting roots from non-negative numbers Addition, subtraction, multiplication, division, taking roots from non-negative numbers Extracting roots from arbitrary numbers Complex numbers, C All operations
The minimum conditions that complex numbers must satisfy: C 1) There is a square root of, i.e. there is a complex number whose square is equal to. C 2) The set of complex numbers contains all real numbers. C 3) The operations of addition, subtraction, multiplication and division of complex numbers satisfy the usual laws of arithmetic operations (combinative, commutative, distributive). The fulfillment of these minimal conditions allows us to determine the entire set C of complex numbers.
Imaginary numbers i = - 1, i – imaginary unit i, 2 i, -0.3 i – purely imaginary numbers Arithmetic operations on purely imaginary numbers are performed in accordance with condition C3. where a and b are real numbers. In general, the rules for arithmetic operations with purely imaginary numbers are as follows:
Complex numbers Definition 1. A complex number is the sum of a real number and a purely imaginary number. Definition 2. Two complex numbers are called equal if their real parts are equal and their imaginary parts are equal:
Classification of complex numbers Complex numbers a + bi Real numbers b = o Imaginary numbers b ≠ o Rational numbers Irrational numbers Imaginary numbers with non-zero real part a ≠ 0, b ≠ 0. Pure imaginary numbers a = 0, b ≠ 0.
Arithmetic operations on complex numbers (a + bi) + (c + di) = (a + c) + (b + d) i (a + bi) - (c + di) = (a - c) + (b - d) i (a + bi) (c + di) = (ac - bd) + (ad + bc)i
Conjugate complex numbers Definition: If you keep the real part of a complex number and change the sign of the imaginary part, you get a complex number conjugate to the given one. If a given complex number is denoted by the letter z, then the conjugate number is denoted: :. Of all complex numbers, real numbers (and only they) are equal to their conjugate numbers. The numbers a + bi and a - bi are called mutually conjugate complex numbers.
Properties of conjugate numbers The sum and product of two conjugate numbers is a real number. The conjugate of the sum of two complex numbers is equal to the sum of the conjugates of these numbers. The conjugate of the difference of two complex numbers is equal to the difference of the conjugates of these numbers. The conjugate of the product of two complex numbers is equal to the product of the conjugates of these numbers.
Properties of conjugate numbers The number conjugate to the nth power of a complex number z is equal to the pth power of the number conjugate to the number z, i.e. The conjugate number of the quotient of two complex numbers, of which the divisor is non-zero, is equal to the quotient of the conjugate numbers, i.e.
Powers of an imaginary unit By definition, the first power of the number i is the number i itself, and the second power is the number -1: . Higher powers of the number i are found as follows: i 4 = i 3 ∙ i = -∙ i 2 = 1; i 5 = i 4 ∙ i = i ; i 6 = i 5 ∙ i = i 2 = - 1, etc. i 1 = i, i 2 = -1 Obviously, for any natural number n i 4n = 1; i 4n+1 = i ; i 4n +2 = - 1 i 4n+3 = - i .
Extracting square roots of complex numbers in algebraic form. Definition. A number w is called the square root of a complex number z if its square is equal to z: Theorem. Let z=a+bi be a non-zero complex number. Then there are two mutually opposite complex numbers whose squares are equal to z. If b ≠0, then these two numbers are expressed by the formula:
Geometric representation of complex numbers. The complex number z on the coordinate plane corresponds to the point M(a, b). Often, instead of points on the plane, their radius vectors are taken Definition: The modulus of a complex number z = a + bi is a non-negative number equal to the distance from the point M to the origin b a M (a, b) y x O φ
Trigonometric form of a complex number where φ is the argument of the complex number, r = is the modulus of the complex number,
Multiplication and division of complex numbers given in trigonometric form Theorem 1. If and then: b) a) Theorem 2 (Moivre’s formula). Let z be any non-zero complex number, n be any integer. Then
Extracting the root of a complex number. Theorem. For any natural number n and non-zero complex number z, there are n different values of the n-degree root. If
After studying the topic “Complex numbers
students must:
Know:
algebraic, geometric and trigonometric forms
complex number.
Be able to:
perform addition operations on complex numbers,
multiplication, subtraction, division, exponentiation, extraction
root of a complex number;
convert complex numbers from algebraic form to
geometric and trigonometric;
use the geometric interpretation of complex numbers;
in the simplest cases, find complex roots of equations with
real coefficients.
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