Presentation on the topic geometric solution of a complex number. Complex numbers The history of the emergence of complex numbers

1.85  -2  0.8 The world of numbers is endless.  The first ideas about the number arose from the counting of objects (1, 2, 3, etc.) - NATURAL NUMBERS.  Subsequently, FRACTIONS appeared as a result of measuring length, weight, etc. (, etc.)  NEGATIVE NUMBERS, appeared with the development of algebra Integer numbers (i.e. natural 1, 2, 3, etc. .), negative numbers (-1, -2, -3, etc. and zero), fractions are called RATIONAL NUMBERS. ,  Rational numbers cannot accurately express the length of the diagonal of a square if the length of the side is equal to the unit of measurement. To accurately express the relationship of incommensurable segments, you need to introduce a new number:  IRRATIONAL (etc.) Rational and irrational - form a set: Real numbers. When considering real numbers, it was noted that in the set of real numbers it is impossible, for example, to find a number whose square is equal to. When considering quadratic equations with negative discriminants, it was also noted that such equations do not have roots that would be real numbers. To make such problems solvable, new numbers are introduced - Complex numbers Complex numbers 2=-1 3=- = 4 =1 b - Imaginary numbers a + b - Complex numbers a, b - Any real numbers Past and present complex numbers. Complex numbers appeared in mathematics more than 400 years ago. For the first time, we encountered the square roots of negative numbers.  What is, what meaning should be given to this expression, no one knew. The square root of any negative number has no meaning in the set of real numbers. This is encountered when solving quadratic, cubic equations, equations of the fourth degree. MATHEMATICS CONSIDERED: LEONHARD EULER The square roots of negative numbers - because they are not greater, not less, and not equal to zero - cannot be counted among the possible numbers. Gottfried Wilim Leibnetz Gottfried Leibnetz called complex numbers "an elegant and wonderful refuge of the divine spirit", a degenerate of the world of ideas, an almost dual being, between being and not being." He even bequeathed to draw a sign on his grave as a symbol of the other world. K. Gauss at the beginning of the 19th century suggested calling them "complex numbers". K. F. Gauss Forms of complex numbers: Z=a+bi - algebraic form Z=r() - trigonometric Z=rE - exponential Complex numbers are used:  When compiling geographical maps In the theory of aircraft construction  Used in various studies on number theory  In electromechanics  When studying the movement of natural and artificial celestial bodies, etc. And at the end of the presentation, offering Solve the crossword “Test yourself” 8 1 3 2 7 5 6 4 1. What is the name of a number like Z=a+bc? 2. To what extent of the imaginary unit is one obtained? 3. What are the names of numbers that differ only in sign at the imaginary part? 4. The length of the vector. 5. The angle under which the vector is located. 6. What is the form of a complex number: Z=r(cos +sin)? 7. What is the form of the complex number Z=re? 8. View D \u003d b -4ac, what is D?

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N C Z C Q C R C C N- ”natural” R- “real” C - “complex” Z – exclusive role of zero “zero” Q – “quotient” relation (since rational numbers are m/n) C R Q Z N

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Minimum conditions for a complex number 1) There is a number whose square = -1. 2) The set of complex numbers contains all real numbers. 3) The operations of addition, subtraction, multiplication and division of complex numbers satisfy the usual law of arithmetic operations.

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An element whose square is -1 is called an imaginary unit. Denoted by i (translated as "imaginary", "imaginary") "Mathematicians used complex numbers and functions of a complex variable in their studies already in the 18th century. The merits of the largest mathematician of the 18th century Leonard Euler (1707-1783), who is rightfully considered one of the creators of the theory of functions of a complex variable. In the remarkable works of Euler, the elementary functions of a complex variable were studied in detail. After Euler, the results and methods discovered by him were developed, improved and systematized, and in the first half of the 19th century the theory of functions of a complex variable took shape as the most important branch of mathematical analysis. " The first mention of "imaginary" numbers as the roots of square and negative numbers dates back to the 16th century. (J. Cardano, 1545). Until the middle of the XVIII century. complex numbers appear only sporadically in the works of individual mathematicians (I. Newton, N. Bernoulli, A. Clairaut). The first presentation of the theory of complex numbers in Russian belongs to L. Euler ("Algebra", St. Petersburg, 1763, later the book was translated into foreign languages and reprinted many times): the symbol "i" was also introduced by L. Euler. The geometric interpretation of complex numbers dates back to the end of the 18th century. (Dane Kaspar Wessel, 1799)."

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Conditions about the operations of complex numbers allow you to multiply complex numbers by an imaginary unit (i). Such a product is called purely imaginary numbers. For example: i, 2i, -0,3i are purely imaginary numbers. 3i +13i=(3+13)i = 16i 3i 13i = (3 13) (i i)=39i2=-39 RULES OF ARITHMETIC OPERATIONS 10 ai+bi=(a+b)i 20 a(bi) =(ab)i 30 (ai)(bi)=abi2= -ab 40 0i =0

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The sum a+bi (a and b are real numbers) a = 0, then a+bi =0+bi=bi (imaginary) b = 0, then a+bi =a+0=a (real) and not equal to zero, then a+bi is neither real nor imaginary. It is a more complex composite number. COMPLEX NUMBERS ARE THE SUM OF A REAL NUMBER AND A PURELY IMAGINARY NUMBER Z=a + bi

1. Development of the concept of number 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 extracted from negative numbers.

This formula works flawlessly in the case when the equation has one real root, and if it has three real roots, then under the square root sign turned out to be a negative number. It turned out that the path to these roots leads through the impossible operation of extracting the square root of a negative number.

3. Statement of complex numbers in mathematics 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.

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. 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.

The word complex (from the Latin complexus) means a connection, a combination, a set of concepts, objects, phenomena, etc. that form a single whole. The word complex (from the Latin complexus) means a connection, a combination, a set of concepts, objects, phenomena, etc. that form a single whole.

Which connected together the exponential function with the trigonometric function. With the help of L. Euler's formula, it was possible to raise the number e to any complex power. which linked together the exponential function with the trigonometric function. With the help of L. Euler's formula, it was possible to raise the number e to any complex power.

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. 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

Such a plane is called complex. The real numbers on it occupy the horizontal axis, the imaginary unit is represented by the unit 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 the ordinate b of a complex number a + bi are expressed in terms of the modulus r and the argument q. The formulas The abscissa a and the 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 a = r cos q, r=a/cos q b = r sin q, r=b/sin q b = r sin q, r=b/sin q r is the length of the vector ( a + bi), q - the angle that it forms with the positive direction of the x-axis

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.

0 i.e. z=a+bi or z=r*cos q + r*sin q where r > 0 i.e. z=a+bi or z=r*cos q + r*sin q Et" title="(!LANG: Therefore, any complex number can be represented as Therefore, any complex number can be represented as r(cos q + i sin q ), r(cos q + i sin q), where r > 0 i.e. z=a+bi or z=r*cos q + r*sin q where r > 0 i.e. z=a+bi or z=r*cos q + r*sin q Et" class="link_thumb"> 25 !} Therefore, any complex number can be represented as Therefore, any complex number can be represented as r(cos q + i sin q), r(cos q + i sin q), where r > 0 i. z=a+bi or z=r*cos q + r*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. This expression is called the normal trigonometric form, or, in short, the trigonometric form of a complex number. 0 i.e. z=a+bi or z=r*cos q + r*sin q where r > 0 i.e. z=a+bi or z=r*cos q + r*sin q Et "> 0 i.e. z=a+bi or z=r*cos q + r*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. This expression is called the normal trigonometric form, or, in short, the trigonometric form of a complex number."> 0 those. z=a+bi or z=r*cos q + r*sin q where r > 0 i.e. z=a+bi or z=r*cos q + r*sin q Et" title="(!LANG: Therefore, any complex number can be represented as Therefore, any complex number can be represented as r(cos q + i sin q ), r(cos q + i sin q), where r > 0 i.e. z=a+bi or z=r*cos q + r*sin q where r > 0 i.e. z=a+bi or z=r*cos q + r*sin q Et"> title="Therefore, any complex number can be represented as Therefore, any complex number can be represented as r(cos q + i sin q), r(cos q + i sin q), where r > 0 i. z=a+bi or z=r*cos q + r*sin q where r > 0 i.e. z=a+bi or z=r*cos q + r*sin q"> !}

Complex numbers Complex numbers and operations on them.

Number system Allowed algebraic operations Partially allowed algebraic operations. Natural numbers, N Addition, multiplication Subtraction, division, extraction of roots. But on the other hand, the equation has no roots in N Integers, Z Addition, subtraction, multiplication. Division, extraction of roots. But on the other hand, the equation has no roots in Z Rational numbers, Q Addition, subtraction, multiplication, division. Extracting roots from non-negative numbers. But on the other hand, the equation has no roots in Q Real numbers, R Addition, subtraction, multiplication, division, taking roots from non-negative numbers. Extracting roots from arbitrary numbers. But on the other hand, the equation has no roots in R Complex numbers, C All operations

CONDITIONS that complex numbers must satisfy ... 1. There is a complex number whose square is -1 2. The set of complex numbers contains all real numbers. 3. Operations of addition, subtraction, multiplication and division of complex numbers satisfy the usual law of arithmetic operations (associative, commutative, distributive)

Type of complex number B general view the rules for arithmetic operations with purely imaginary numbers are as follows: ai+bi =(a+b) i ; ai-bi=(a-b) i ; a(bi)=(ab) i ; (ai)(bi)=abi²=- ab (a and b are real numbers) i²= -1, i - imaginary unit

Definitions Definition #1 A complex number is the sum of a real number and a purely imaginary number. Z= a+bi c C ↔ a c R , b c R, i is the imaginary unit. In the notation z \u003d a + bi, the number a is called the real part of the complex number z, and the number b is called the imaginary part of the complex number z. Definition №2 Two complex numbers are called equal if their real parts are equal and their imaginary parts are equal. a+bi = c+di ↔ a=c, b=d.

Definition No. 3 If we keep the real part of a complex number and change the sign of the imaginary part, then we get a complex number conjugate to this one. Z=X+YI X - YI

Formulas Sum of complex numbers: z 1+ z 2 = (a+bi)+(c+di)=(a+c)+(bi+di)=(a+c)+ i (b+d) Difference of complex numbers : z 1 - z 2 = (a+bi)-(c+di)=(a-c)+ i (b-d) Product of complex numbers: (a+bi)(c+di)= i (ac-bd)+( bc+ad) Formula for the quotient of two complex numbers: a+bi = ac+bd + bc -ad c+di c²+d² c²+d² i

z 2 Properties Property 1 If z = x + yi , then z*z = x ² + y ² z 1 Both the numerator and denominator of the fraction should be multiplied by the conjugate of the denominator. Property 2 Z1+ Z2=Z1+Z2 i.e. the conjugate of the sum of two complex numbers is equal to the sum of the conjugates of the given numbers. Property 3 Z 1- Z 2= Z 1- Z 2, i.e. the conjugate of the difference of two complex numbers is equal to the difference of the conjugates of the given numbers.

Property 4 Z 1 Z 2= Z 1 Z 2 i.e. the number conjugate to the product of two complex numbers is equal to the product of the conjugate given numbers. On the other hand, Z 1= a-bi, c- di , so Z 1 Z 2 = (ac – bd)- i (bc+ad) Property 5 Property 6

Geometric interpretation of a complex number. Y 0 X Bi A Z= A+Bl Y Bi 0 A M(A ; B) X

Addition and multiplication of complex numbers. Algebraic form Geometric form Product Z 1 = r 1 (cos φ 1 + i sin φ 1) Z 2 = r 2 (cos φ 2 + i sin φ 2) Z 1 Z 2 = r 1 r 2 [ cos (φ 1 + φ 2)+ isin (φ 1 + φ 2)] Product (A+iB) (C+iD)= (AC-BD)+(AD+BC) i Sum (A+iB) + (C+iD )= (A+C)+(B+D)I

Moivre formula For any Z= r (cos φ + i sin φ)≠0 and any natural number n

Gauss' theorem: every algebraic equation has at least one root in the set of complex numbers. Every algebraic equation of degree n has exactly n roots in the set of complex numbers. The second formula of De Moivre determines all the roots of a two-term equation of degree n

Thank you for your attention! The presentation was made by: a student of 10 "a" class MOAU "Gymnasium No. 7" of Orenburg Elimova Maria.

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1. Development of the concept of number

Ancient Greek mathematicians considered "real" only natural numbers. Along with natural numbers, fractions were used - numbers made up of a whole number of fractions of a unit.

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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 extracted from negative numbers.

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2. On the way to complex numbers

In the 16th century, in connection with the study of cubic equations, it became necessary to extract square roots from negative numbers.

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In the formula for solving cubic equations of the form:

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cube and square roots:

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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 turned out to be 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.

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In addition to x=1, there are two more roots

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The Italian algebraist G. Cardano in 1545 suggested introducing numbers of a new nature. He showed that the system of equations

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which has no solutions in the set of real numbers, has solutions of the form

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we only need to agree to act on such expressions according to the rules of ordinary algebra and assume that

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3. Statement of complex numbers in mathematics

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.

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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.

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The word complex (from the Latin complexus) means a connection, a combination, a set of concepts, objects, phenomena, etc. that form a single whole.

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L. Euler derived in 1748 a wonderful formula

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which linked together the exponential function with the trigonometric function. With the help of L. Euler's formula, it was possible to raise the number e to any complex power.

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At the end of the 18th century, the French mathematician J. Lagrange was able to say that imaginary quantities no longer complicate mathematical analysis.

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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”

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4.Geometric representation of a complex number

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5. Trigonometric form of a complex number.

The abscissa a and the ordinate b of a complex number a + bi are expressed in terms of the modulus r and the argument q. Formulas a \u003d r cos q, r \u003d a / cos q b \u003d r sin q, r \u003d b / sin q r is the length of the vector (a + bi), q is the angle that it forms with the positive direction of the x-axis

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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.

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Therefore, any complex number can be represented as 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.

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Thank you for your attention!

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