EntranceThe Diophantus project and its discoveries. Biography of Diophantus
Coefficients whose solutions must be found among integers.
| Diophantus of Alexandria | |
|---|---|
| Διόφαντος ὁ Ἀλεξανδρεύς | |
| Date of Birth | no earlier and no later or |
| Place of Birth |
|
| Date of death | no earlier and no later |
| A country |
|
| Scientific field | number theory |
| Known as | "father of algebra" |
| Diophantus of Alexandria at Wikimedia Commons | |
Biography
Almost nothing is known about the details of his life. On the one hand, Diophantus quotes Hypsicles (2nd century BC); on the other hand, Theon of Alexandria (about 350 AD) writes about Diophantus, from which we can conclude that his life took place within the boundaries of this period. A possible clarification of the life time of Diophantus is based on the fact that he Arithmetic dedicated to “the most venerable Dionysius.” It is believed that this Dionysius is none other than Bishop Dionysius of Alexandria, who lived in the middle of the 3rd century. n. e.
It is equivalent to solving the following equation:
x = x 6 + x 12 + x 7 + 5 + x 2 + 4 (\displaystyle x=(\frac (x)(6))+(\frac (x)(12))+(\frac (x) (7))+5+(\frac (x)(2))+4)This equation gives x = 84 (\displaystyle x=84), that is, the age of Diophantus is equal to 84 years. However, the accuracy of the information cannot be confirmed.
Arithmetic Diophanta
The main work of Diophantus - Arithmetic in 13 books. Unfortunately, only 6 (or 10, see below) of the first 13 books have survived.
The first book is preceded by an extensive introduction, which describes the notation used by Diophantus. Diophantus calls the unknown “number” ( ἀριθμός ) and is denoted by the letter ς , square unknown - symbol Δ Υ (short for δύναμις - “degree”), the cube of the unknown - symbol Κ Υ (short for κύβος - “cube”). Special signs are provided for the following degrees of the unknown, up to the sixth, called cube-cube, and for their opposite degrees, up to minus the sixth.
Diophantus does not have an addition sign: he simply writes positive terms next to each other in descending order of degree, and in each term the degree of the unknown is first written, and then the numerical coefficient. The subtracted terms are also written side by side, and a special sign in the form of an inverted letter Ψ is placed in front of their entire group. The equal sign is represented by two letters ἴσ (short for ἴσος - “equal”).
A rule for bringing similar terms and a rule for adding or subtracting the same number or expression to both sides of an equation were formulated: what al-Khorezmi later began to call “algebra and almukabala.” The rule of signs has been introduced: “minus by plus gives minus”, “minus by minus gives plus”; This rule is used when multiplying two expressions with subtracted terms. All this is formulated in general terms, without reference to geometric interpretations.
Most of the work is a collection of problems with solutions (there are a total of 189 in the six surviving books, together with the four from the Arabic part - 290), skillfully selected to illustrate general methods. Main issues Arithmetic- finding positive rational solutions to uncertain equations. Rational numbers are treated by Diophantus in the same way as natural numbers, which is not typical for ancient mathematicians.
First, Diophantus examines systems of second-order equations in two unknowns; it specifies a method for finding other solutions if one is already known. Then he applies similar methods to equations of higher degrees. Book VI examines problems related to right triangles with rational sides.
Influence Arithmetic for the development of mathematics
In the 10th century Arithmetic was translated into Arabic (see Kusta ibn Luka), after which mathematicians from Islamic countries (Abu Kamil and others) continued some of Diophantus’s research. In Europe, interest in Arithmetic increased after Raphael Bombelli translated and published this work into Latin, and published 143 problems from it in his Algebra(1572). In 1621, a classic, thoroughly commented Latin translation appeared Arithmetic, executed by Bachet de Meziriac.
Diophantus' methods greatly influenced François Viète and Pierre Fermat; however, in modern times, indefinite equations are usually solved in integers, and not in rational numbers, as Diophantus did. When Pierre Fermat read Diophantus's Arithmetic, edited by Bachet de Mezyriac, he came to the conclusion that one of the equations similar to those considered by Diophantus had no solutions in integers, and noted in the margin that he had found "a truly wonderful proof of this theorem ... however, the margins of the book are too narrow to include it.” This statement is now known as Fermat's Last Theorem.
In the 20th century, the Arabic text of four more books was discovered under the name of Diophantus. Arithmetic. I. G. Bashmakova and E. I. Slavutin, having analyzed this text, put forward a hypothesis that its author was not Diophantus, but a commentator well versed in Diophantus’ methods, most likely Hypatia. However, the significant gap in the methodology for solving problems in the first three and last three books is well filled by four books of Arabic translation. This forces us to reconsider the results of previous studies. . [ ]
Other works of Diophantus
Treatise of Diophantus About polygonal numbers (Περὶ πολυγώνων ἀριθμῶν ) not completely preserved; in the preserved part, a number of auxiliary theorems are derived using geometric algebra methods.
From the works of Diophantus About measuring surfaces (ἐπιπεδομετρικά ) And About multiplication (Περὶ πολλαπλασιασμοῦ ) also only fragments have survived.
Book of Diophantus Porisms known only from a few theorems used in Arithmetic.
see also
Collection Budé" (2 volumes published: Books 4 - 7).
Research:
- Bashmakova I. G., Slavutin E. I., Rosenfeld B. A. Arabic version of “Arithmetic” of Diophantus // Historical and mathematical studies. - M., 1978. - Issue. XXIII. - P. 192 - 225.
- Bashmakova I. G. Arithmetic of algebraic curves: (From Diophantus to Poincaré) // Historical and mathematical studies. - 1975. - Issue. 20. - pp. 104 - 124.
- Bashmakova I. G. Diophantus and Diophantine equations. - M.: Nauka, 1972 (Reprint: M.: LKI, 2007). Per. On him. language: Diophant und diophantische Gleichungen. - Basel; Stuttgart: Birkhauser, 1974. Trans. in English. language: Diophantus and Diophantine Equations/ Transl. by A. Shenitzer with the editorial assistance of H. Grant and updated by J. Silverman // The Dolciani Mathematical Expositions. - No. 20. - Washington, DC: Mathematical Association of America, 1997.
- Bashmakova I. G. Diophantus and Fermat: (On the history of the method of tangents and extrema) // Historical and mathematical studies. - M., 1967. - Issue. VII. - P. 185 - 204.
- Bashmakova I. G., Slavutin E. I. History of Diophantine analysis from Diophantus to Fermat. - M.: Nauka, 1984.
- History of mathematics from ancient times to the beginning of the 19th century. - T. I: From the most ancient. times before the beginning of the New Age. time / Ed. A. P. Yushkevich. - M., Nauka, 1970.
- Slavutin E. I. Diophantus’ algebra and its origins // Historical and mathematical studies. - M., 1975. - Issue. 20. - pp. 63 - 103.
- Shchetnikov A. I. Can Diophantus of Alexandria’s book “On Polygonal Numbers” be called purely algebraic? // Historical and mathematical research. - M., 2003. - Issue. 8 (43). - pp. 267 - 277.
- Heath Th. L. Diophantus of Alexandria, A Study in the History of Greek Algebra. - Cambridge, 1910 (Repr.: NY, 1964).
- Knorr W. R. Arithmktikê stoicheiôsis: On Diophantus and Hero of Alexandria // Historia Mathematica. - 20. - 1993. - P. 180 - 192.
- Christianidis J. The way of Diophantus: Some clarifications on Diophantus’ method of solution // Historia Mathematica. - 34. - 2007. - P. 289 - 305.
- Rashed R., Houzel C. Les Arithmétiques de Diophante. Lecture historique et mathématique . - De Gruyter, 2013.
Two works of Diophantus have survived to this day, both incompletely. These are “Arithmetic” (six books out of thirteen) and excerpts from the treatise “On Polygonal Numbers”. But almost nothing is known about the author himself. His Arithmetic was a turning point in the development of algebra and number theory. It was here that the final abandonment of geometric algebra took place. At the beginning of his work, Diophantus included a brief introduction, which became the first presentation of the foundations of algebra. It constructs a field of rational numbers and introduces alphabetic symbolism. The rules for dealing with polynomials and equations are also formulated there. The works of Diophantus were of fundamental importance for the development of algebra and number theory. The name of this scientist is associated with the emergence and development of algebraic geometry, the problems of which were subsequently studied by Leonard Euler, Carl Jacobi and other authors.
“Arithmetic” by Diophantus is a collection of problems (there are 189 in total), each of which is equipped with a solution (or several methods of solution) and the necessary explanations. Therefore, at first glance, it seems that it is not a theoretical work. However, a careful reading shows that the problems are carefully selected and serve to illustrate very specific, strictly thought out methods. As was customary in ancient times, methods are not formulated in a general form, but are repeated to solve similar problems.
The main problem of "Arithmetic" is finding positive rational solutions to uncertain equations. Rational numbers are interpreted by Diophantus in the same way as natural numbers, which is not typical for ancient mathematicians. First, Diophantus examines systems of second-order equations in two unknowns. It specifies a method for finding other solutions if one is already known. Then he applies similar methods to equations of higher degrees.
In the 10th century, Arithmetic was translated into Arabic, after which mathematicians from Islamic countries (Abu Kamil and others) continued some of Diophantus’s research. In Europe, interest in Arithmetic increased after Raphael Bombelli discovered this work in the Vatican Library and published 143 problems from it in his Algebra (1572). In 1621, a classic, thoroughly commented Latin translation of “Arithmetic” appeared, carried out by Bachet de Meziriak. Diophantus's methods had a huge influence on François Viète and Pierre Fermat, however, in modern times, indefinite equations are usually solved in whole numbers, and not in rational numbers, as Diophantus did.
Other works of Diophantus are also known. The treatise “On Polygonal Numbers” has not been completely preserved. In the surviving part, a number of auxiliary theorems are derived using geometric algebra methods. From the works of Diophantus “On the Measurement of Surfaces” and “On Multiplication” only fragments have also survived. Diophantus' book "Porisms" is known only from a few theorems used in Arithmetic.
The Palatine Anthology contains an epigram - a task from which we can conclude that Diophantus lived 84 years: Diophantus is buried here, and the gravestone, when counted, will tell us how long his life was. By God's decree he was a boy for a sixth of his life; In the twelfth part, his bright youth then passed. Let's add the seventh part of life - before us is the hearth of Hymen. Five years have passed; and Hymen sent him a son. But woe to the child! He had barely lived half of the years that his father had died when the unfortunate man died. Diophantus suffered for four years from such a severe loss and died, having lived for science. Tell me, how old was Diophantus when he reached death?
Diophantine equations Diophantine equations are algebraic equations or systems of algebraic equations with integer coefficients for which integer or rational solutions must be found. In this case, the number of unknowns in the equations must be at least two (if you are not limited to only integers). Diophantine equations, as a rule, have many solutions, which is why they are called indeterminate equations. These are, for example, the equations: 3x+5y=7; x²+y²= z²; 3х³+4у³= 5z³
Problem 1 There are rabbits and pheasants in a cage; they have 18 legs in total. Find out how many of both are in the cage. Solution. An equation is created with two unknown variables, in which x is the number of rabbits. y – number of pheasants: 4x + 2y = 18, or 2x + y = 9. Let’s express y through x: y = 9 – 2x. Next, we will use the brute force method: x1234 y7531 Thus, the problem has four solutions. Answer: (1; 7), (2; 5), (3; 3), (4; 1).
Task 2 The subjects brought 300 precious stones as gifts to the Shah: in small boxes of 15 pieces each and in large boxes of 40 pieces. How many of these and other boxes were there, if it is known that there were fewer small ones than large ones? Solution: Let's denote by X the number of small boxes, and by Y the number of large ones. Moreover X 
Municipal educational institution
"Lyceum No. 10" Perm
Diophantus. Diophantine equations
Done the job
Ilyina Yana,
11th grade student
Supervisor
Zolotukhina L. V.
mathematic teacher
Perm, 2010
Introduction……………………………………………………………………………….3
1. Diophantus………………………………………………………………………………..…4
2. Numbers and symbols………………………………………………………6
3. Diophantine equation……………………………………………..…8
4. Solutions………………………………………………………..12
Conclusion…………………………………………………………………………………15
References………………………………………………………16
Introduction
Today's schoolchildren solve various equations. In Part C of the Unified State Examination tasks there is an interesting equation called the Diophantine equation. In his works, Diophantus not only posed the problem of solving indefinite equations in rational numbers, but also gave some general methods for solving them. These methods will be very useful for today's eleventh graders who are about to take the math exam.
Diophantus made as great a contribution to the development of mathematics as Archimedes. This is what Archimedes did, for example: when determining the areas of an ellipse, a segment of a parabola, the surface of a sphere, the volumes of a sphere and other bodies, he used the method of integral sums and the method of passage to the limit, but nowhere did he give a general abstract description of these methods. Scientists of the 16th and 17th centuries had to carefully study and rearrange his works in a new way in order to isolate the methods of Archimedes from there. The situation is similar with Diophantus. His methods were understood and applied to new problems by Viethe and Fermat, i.e. at the same time when Archimedes was solved.
1. Diophantus
Diophantus presents one of the most difficult mysteries in the history of science. We do not know the time when he lived, nor his predecessors who would have worked in the same field. His works are like a sparkling fire in the midst of complete impenetrable darkness. The period of time when Diophantus could have lived is half a millennium! The lower bound of this interval is determined without difficulty: in his book on polygonal numbers, Diophantus repeatedly mentions the mathematician Hypsicles of Alexandria, who lived in the middle of the 2nd century BC. e. On the other hand, in the comments of Theon of Alexandria to the “Almagest” of the famous astronomer Ptolemy, an excerpt from the work of Diophantus is placed. Theon lived in the middle of the 4th century AD. e. This determines the upper bound of this interval. So, 500 years!
But the place of residence of Diophantus is well known - this is the famous Alexandria, the center of scientific thought of the Hellenistic world.
To exhaust everything known about the personality of Diophantus, we present a riddle poem that has come down to us:
The ashes of Diophantus rest in the tomb; marvel at her - and the stone
The age of the deceased will speak through his wise art.
By the will of the gods, he lived a sixth of his life as a child.
And I met half past five with fluff on my cheeks.
It was only the seventh day when he became engaged to his girlfriend.
After spending five years with her, the sage waited for his son;
His father's beloved son lived only half his life.
He was taken from his father by his early grave.
Twice two years the parent mourned a heavy grief,
Here I saw the limit of my sad life.
From here it is easy to calculate that Diophantus lived 84 years. However, for this you do not need to master the art of Diophantus! It is enough to be able to solve an equation of the 1st degree with one unknown, and Egyptian scribes were able to do this back 2 thousand years BC. e.
But the most mysterious is the work of Diophantus. Six books out of 13 have reached us, which were combined into “Arithmetic”. The style and content of these books differ sharply from classical ancient works on number theory and algebra, examples of which we know from Euclid’s Elements, his Data, and lemmas from the works of Archimedes and Apollonius. "Arithmetic" was undoubtedly the result of numerous studies that remained completely unknown to us. We can only guess about its roots and marvel at the richness and beauty of its methods and results.
“Arithmetic” by Diophantus is a collection of problems (there are 189 in total), each of which is equipped with a solution (or several methods of solution) and the necessary explanations. Therefore, at first glance it seems that it is not a theoretical work. However, a careful reading shows that the problems are carefully selected and serve to illustrate very specific, strictly thought-out methods. As was customary in ancient times, methods are not formulated in a general form, but are repeated to solve similar problems.
2. Numbers and symbols
Diophantus begins with basic definitions and a description of the letter symbols he will use.
In classical Greek mathematics, which found its completion in Euclid’s Elements, under the number άριJμός - “ arrhythmos" or " arithmos"; hence the name “arithmetic” for the science of numbers) was understood as a set of units, i.e. integer. Neither fractions nor irrationality were called numbers. Strictly speaking, there are no fractions in the Principia. The unit is considered indivisible and instead of fractions of a unit, ratios of integers are considered; irrationalities appear as ratios of incommensurable segments, for example, the number we now denote √2 was for the classical Greeks the ratio of the diagonal of a square to its side. There was no talk of negative numbers. There weren't even any equivalents for them. We find a completely different picture in Diophantus.
Diophantus gives the traditional definition of number as a set of units, but later seeks for his problems positive rational solutions, and calls each such solution a number (άριJμός - “ arrhythmos »).
But the matter does not stop there. Diophantus introduces negative numbers: he calls them the special term λει̃ψις - “ leipsis" - derived from the verb λει̃πω - " leipo”, which means to lack, to lack, so that the term itself could be translated by the word “lack”. By the way, this is what the famous Russian historian of science I. Timchenko does. Diophantus calls a positive number the word ΰπαρξις - “ iparxis”, which means existence, being, and in the plural this word can mean property or property. Thus, Diophantus's terminology for relative numbers is close to that used in the Middle Ages in the East and Europe. Most likely, it was simply a translation from Greek into Arabic, Sanskrit, Latin, and then into various languages of Europe.
Note that the term λει̃ψις is “ leipsis" - cannot be translated as "subtracted", as many translators of Diophantus do, because for the operation of subtraction Diophantus uses completely different terms, namely άφελει̃ν - " afelein"or άφαιρει̃ν - " afirerain", which are derived from the verb άφαιρεω - " afireo"- take away. When transforming equations, Diophantus himself often uses the standard expression “add λει̃ψις to both sides.”
We have dwelled in such detail on the philological analysis of Diophantus’s text in order to convince the reader that we will not deviate from the truth if we translate Diophantus’s terms as “positive” and “negative”.
Diophantus formulates the rule of signs for relative numbers:
“a negative multiplied by a negative gives a positive, while a negative multiplied by a positive gives a negative, and the distinguishing sign for a negative is an inverted and shortened (letter) ψ.”
“After I explained multiplication to you, the division of the proposed terms also becomes clear; Now it will be good to start practicing the addition, subtraction and multiplication of such terms. Add positive and negative terms with different coefficients to other terms that are either positive or equally positive and negative, and from positive and other negative terms subtract other positive and equally positive and negative terms.”
Note that although Diophantus seeks only rational positive solutions, in intermediate calculations he willingly uses negative numbers.
We can thus note that Diophantus expanded the number field into a field of rational numbers in which all four operations of arithmetic can be performed without hindrance.
3. Diophantine equation
Definition - algebraic equations or systems of algebraic equations with integer coefficients, having a number of unknowns exceeding the number of equations, and for which integer or rational solutions are sought.
ax + by = 1
Where A And b- coprime integers
Coprime numbers several integers such that the common divisors for all these numbers are only + 1 and - 1. The smallest multiple of a pair of prime numbers is equal to their product.
has infinitely many solutions:
If x0 And y0- one solution, then the numbers
X = x0 + bn
at = y0 -an
(n- any integer) will also be solutions.
Another example of D. u.
x2 + y2 = z2
Positive integer solutions to this equation represent the lengths of the legs X , at and hypotenuse z right triangles with integer side lengths are called Pythagorean numbers.
triples of natural numbers such that a triangle whose side lengths are proportional (or equal) to these numbers is rectangular.
All triplets of coprime Pythagorean numbers can be obtained using the formulas
X = m2 - n2
at = 2mn
z = m2 + n2
Where m And n- whole numbers ( m > n > 0).
This equation defines on the plane R 2 algebraic curveΓ. We will call the rational solution (2) rational point curve Γ. In what follows, we will often resort to the language of geometry, although Diophantus himself does not use it anywhere. However, geometric language has now become such an integral part of mathematical thinking that many facts will be easier to understand and explain with its help.
First of all, it is necessary to give some classification of equations (2) or, which is the same, algebraic curves. The most natural and earliest to arise is their classification by order.
Let us remind you that in order curve (2) is the maximum order of the terms of the polynomial f (x , y), where the order of a term is understood as the sum of powers at x And y. The geometric meaning of this concept is that a straight line intersects a curve of order n exactly at n points. When counting points, one must, of course, take into account the multiplicity of intersection points, as well as complex and “infinitely distant” points. So, for example, a circle x 2 + y 2 = 1 and straight x + y= 2 intersect at two complex points, and the hyperbola x 2 – y 2 = 1 and straight y =x- at two points at infinity, the same hyperbola with a straight line x=1 has one common point of multiplicity 2.
However, for the purposes diophantine analysis(this name was given to the field of mathematics that grew out of the problems of solving indeterminate equations; however, now it is more often called Diophantine geometry) classification by order turned out to be too rough.
Rice. 1.
Let us explain this with an example. Let a circle be given C : x 2 + y 2 = 1 and any straight line with rational coefficients, for example, L : y=0. Let us show that the rational points of this circle and the line can be put into a one-to-one correspondence. This can be done, for example, like this: fix the point A(0,–1) circles and assign each rational point B straight L point B" circle C, lying at the intersection C and straight AB(Fig. 1). That the coordinates of the point B" will be rational, we will let the reader prove it for himself or read a similar proof from Diophantus (it will be presented in the next paragraph). Obviously, the same correspondence can be established between the rational points of any conic section, if at least one rational point lies on it, and a rational line. We see that from the point of view of Diophantine analysis the circle C and straight L are indistinguishable: their sets of rational solutions are equivalent. And this despite the fact that the orders of both curves are different.
More subtle is the classification of algebraic curves by genus, which was introduced only in the 19th century by Abel and Riemann. This classification takes into account the number of singular points of the curve Γ.
We assume that in equation (2) of the curve Γ the polynomial f (x , y) is irreducible over the field of rational numbers, i.e. it does not expand into a product of polynomials with rational coefficients. As is known, the equation of the tangent to the curve Γ at the point P (x 0 , y 0) will
y – y 0 = k (x – x 0),
| k = – |
fx" (x 0 , y 0) fy" (x 0 , y 0) |
If at the point P derivative fx" or fy" is different from zero, then the slope k tangent has a very definite meaning (if fy" (x 0 , y 0) = 0, a fx" (x 0 , y 0) ≠ 0, then k=∞ and tangent at P will be vertical).
If at the point P both partial derivatives vanish,
fx" (x 0 , y 0) = 0 and fy" (x 0 , y 0) = 0,
then point P called special .
For example, at the curve y 2 = x 2 + x 3 point (0, 0) will be special, since in it fx" = –2x – 3x 2 and fy" = 2y go to zero.
Rice. 2.
The simplest singular points are double ones, at which at least one of the derivatives f xx "" , f xy "" And f yy "" is different from zero. In Fig. Figure 2 shows a double point where the curve has two different tangents. Other more complex singular points are shown in Fig. 3.
Rice. 3.
4. Solutions
Rule 1. If c is not divisible by d, then the equation ax + vy = c has no solutions in integers. N.O.D.(a,b) = d.
Rule 2. To find a solution to the equation ax + vy = c with coprime a and b, you must first find a solution (X o; y o) to the equation ax + y = 1; the numbers CX o, Su o form a solution to the equation ax + vy = c.
Solve the equation in integers (x,y)
5x - 8y = 19 ... (1)
First way. Finding a particular solution using the selection method and recording the general solution.
We know that if N.O.D.(a;b) =1, i.e. a and b are coprime numbers, then equation (1)
has a solution in integers x and y. N.O.D.(5;8) =1. Using the selection method we find a particular solution: X o = 7; y o =2.
So, the pair of numbers (7;2) is a particular solution to equation (1).
This means that the equality holds: 5 x 7 – 8 x 2 = 19 ... (2)
Question: How, given one solution, write down all the other solutions?
Let us subtract equality (2) from equation (1) and get: 5(x -7) – 8(y - 2) =0.
Hence x – 7 = . From the resulting equality it is clear that the number (x – 7) will be an integer if and only if (y – 2) is divisible by 5, i.e. y – 2 = 5n, where n is some integer. So, y = 2 + 5n, x = 7 + 8n, where n Z.
Thus, all integer solutions of the original equation can be written in the following form:
Second way . Solving an equation for one unknown.
We solve this equation with respect to the unknown that has the smallest (modulo) coefficient. 5x - 8y = 19 
x = .
Remainders when divided by 5: 0,1,2,3,4. Let's substitute these numbers for y.
If y = 0, then x = =.
If y = 1, then x = =.
If y = 2, then x = = = 7 Z.
If y = 3, then x = =.
If y = 4 then x = =.) Conclusion
Meanwhile, most historians of science, as opposed to mathematicians, have so far underestimated the works of Diophantus. Many of them believed that Diophantus was limited to finding only one solution and used artificial techniques for this, different for different problems. But in fact, in most Diophantine equations we observe similar solution algorithms.
Today, as we see, there are several different solutions, the algorithms of which are easy to remember. As mentioned earlier, this equation is usually found in task C6 on the Unified State Exam. Studying algorithms for solving Diophantine equations can help in solving this task, which is worth a significant number of points.
Bibliography
1. Diophantus of Alexandria. Arithmetic and a book about polygonal numbers (translation from ancient Greek by I. N. Veselovsky; editing and comments by I. G. Bashmakova). M., “Science”, 1974.
2. B. L. Van der Waerden, Awakening Science (translation by I. N. Veselovsky). M., Fizmatgiz, 1959.
3. G. G. Tseyten, History of mathematics in antiquity and the Middle Ages (translation by P. Yushkevich). M.–L., Gostekhizdat, 1932
4. A. V. Vasiliev, Integer. Petersburg, 1919
5. I. V. Yashchenko, S. A. Shestakov, P. I. Zakharov, Mathematics, Unified State Examination, MTsNMO, 2010
The date of celebration was approved by the Decree of the Presidium of the USSR Armed Forces in 1965. In 2020 prof. The holiday turns 55 years old. Since the date is an anniversary, it should be celebrated on a special scale.
Unlike the Russian date, World Fishing Day celebrated on June 27, in 2020 - on Saturday June 27, 2020.
Today, fishing is one of the most common entertainments in the world.
In Russia, the Day of the Baptism of Rus' was included in the number of official memorial dates from June 13, 2010.
The fireworks will begin at 22:30 and will last 10 minutes.
30 volleys of artillery guns and more than 2,000 fireworks will be launched into the evening sky of St. Petersburg.
Where is the best place to watch the fireworks display on Navy Day, July 28, 2019:
To hold the festive salute and fireworks on July 28, 2019, 2 sites will be organized. The first will be located on the Big Beach of the Peter and Paul Fortress, and the second - in Kronstadt.Fireworks on Navy Day 2019 will be visible from different parts of the city. However, it is best to watch the fireworks from a safe distance as close as possible to the launch sites. To view it, it is better to take a place in advance on the Palace Embankment, the spit of Vasilyevsky Island, on one of the bridges (Dvortsovy, Liteiny, Birzhevoy, Troitsky).
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Over the course of 10 minutes, 30 salvos will be fired by a battery of 12 D-44 guns, and two thousand fireworks will be launched using 12 fireworks installations based on KamAZ.
Military sports festivals in honor of Navy Day will be held in 7 cities of Russia: Astrakhan, Vladivostok, Baltiysk, Severomorsk, Sevastopol, Novorossiysk and, of course, in St. Petersburg.
Also in the Northern capital of Russia a naval parade will be held, in which more than 40 ships, boats and submarines, as well as 41 aircraft will take part.
Parade start time Navy ships in St. Petersburg on July 28, 2019 - 11:00 (local/Moscow time).
On which channel to watch the live broadcast of the Navy parade in St. Petersburg:
Naval parade July 28, 2019 Channel One will show live. To prepare a colorful broadcast of this grandiose spectacle, about 100 television cameras will be used, which are located in the water, on ships, on land, in the sky (on airplanes), and even under water.That is, the 2019 Navy Parade in St. Petersburg:
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Biography
Latin translation Arithmetic (1621)
Almost nothing is known about the details of his life. On the one hand, Diophantus quotes Hypsicles (2nd century BC); on the other hand, Theon of Alexandria (about 350 AD) writes about Diophantus, from which we can conclude that his life took place within the boundaries of this period. A possible clarification of the life time of Diophantus is based on the fact that he Arithmetic dedicated to “the most venerable Dionysius.” It is believed that this Dionysius is none other than Bishop Dionysius of Alexandria, who lived in the middle of the 3rd century. n. e.
Arithmetic Diophanta
The main work of Diophantus - Arithmetic in 13 books. Unfortunately, only the first 6 books out of 13 have survived.
The first book is preceded by an extensive introduction, which describes the notation used by Diophantus. Diophantus calls the unknown “number” ( ἀριθμός ) and is denoted by the letter ς , square unknown - symbol (short for δύναμις - "degree"). Special signs are provided for the following degrees of the unknown, up to the sixth, called the cube-cube, and for the degrees opposite to them. Diophantus does not have an addition sign: he simply writes positive terms next to each other, and in each term the degree of the unknown is first written, and then the numerical coefficient. The subtracted terms are also written side by side, and a special sign in the form of an inverted letter Ψ is placed in front of their entire group. The equal sign is represented by two letters ἴσ (short for ἴσος - “equal”). A rule for bringing similar terms and a rule for adding or subtracting the same number or expression to both sides of an equation were formulated: what al-Khorezmi later began to call “algebra and almukabala.” A sign rule has been introduced: minus times minus gives plus; This rule is used when multiplying two expressions with subtracted terms. All this is formulated in general terms, without reference to geometric interpretations.
Most of the work is a collection of problems with solutions (there are a total of 189 in the six surviving books), skillfully selected to illustrate general methods. Main issues Arithmetic- finding positive rational solutions to uncertain equations. Rational numbers are treated by Diophantus in the same way as natural numbers, which is not typical for ancient mathematicians.
First, Diophantus examines systems of 2nd order equations in 2 unknowns; it specifies a method for finding other solutions if one is already known. Then he applies similar methods to equations of higher degrees.
In the 10th century Arithmetic was translated into Arabic, after which mathematicians from Islamic countries (Abu Kamil and others) continued some of Diophantus’s research. In Europe, interest in Arithmetic increased after Raphael Bombelli discovered this work in the Vatican Library and published 143 problems from it in his Algebra(). In 1621, a classic, thoroughly commented Latin translation appeared Arithmetic, executed by Bachet de Meziriac. Diophantus' methods greatly influenced François Viète and Pierre Fermat; however, in modern times, indefinite equations are usually solved in integers, and not in rational numbers, as Diophantus did.
In the 20th century, under the name of Diophantus, the Arabic text of 4 more books was discovered Arithmetic. I. G. Bashmakova and E. I. Slavutin, having analyzed this text, put forward a hypothesis that their author was not Diophantus, but a commentator well versed in Diophantus’ methods, most likely Hypatia.
Other works of Diophantus
Treatise of Diophantus About polygonal numbers (Περὶ πολυγώνων ἀριθμῶν ) not completely preserved; in the preserved part, a number of auxiliary theorems are derived using geometric algebra methods.
From the works of Diophantus About measuring surfaces (ἐπιπεδομετρικά ) And About multiplication (Περὶ πολλαπλασιασμοῦ ) also only fragments have survived.
Book of Diophantus Porisms known only from a few theorems used in Arithmetic.
Literature
Categories:
- Ancient Greek mathematicians
- Mathematicians of Ancient Rome
- Personalities in alphabetical order
- Mathematicians by alphabet
- 3rd century mathematicians
- Mathematicians in number theory
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See what “Diophantus of Alexandria” is in other dictionaries:
- (c. 3rd century) ancient Greek mathematician. In the main work Arithmetic (6 books out of 13 have survived) he gave solutions to problems leading to the so-called. Diophantine equations, and for the first time introduced letter symbols into algebra... Big Encyclopedic Dictionary
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- (born 325, died 409 AD) famous Alexandrian mathematician. There is almost no information about his life; even the dates of his birth and death are not entirely reliable. D. lived 84 years, as can be seen from the epitaph, composed as follows... ... Encyclopedic Dictionary F.A. Brockhaus and I.A. Ephron
Diophantus- DIOPHANTUS of Alexandria (c. 3rd century), other Greek. mathematician. In the main tr. Arithmetic (6 books out of 13 have been preserved) gave solutions to problems leading to the so-called. Diophantine equations, and for the first time introduced letter symbols into algebra... Biographical Dictionary