I.
The history of the development of mathematics.
1. Introduction.
In short, mathematics can be described as the science of numbers and figures.
Its name comes from the Greek "mathema" - science.
Until the beginning of the XVIII century, mathematics was mainly the science of numbers, scalar quantities and relatively simple geometric figures, the quantities studied by it lengths, areas, volumes are considered as constants.
The emergence of arithmetic, geometry, and later — algebra and trigonometry belongs to this period.
The field of application of mathematics was accounting, trade, surveying, architecture, astronomy.
In the XVII and XVIII centuries, the needs of the rapidly developing natural science and technology (navigation, astronomy, ballistics, hydraulics) led to the introduction of the ideas of motion and change into mathematics, primarily in the form of variables and functional dependencies.
This led to the creation of analytical geometry, differential and integral calculus.
In the XVIII century.
the theory of differential equations and differential geometry arise and develop.
In the XIX XX centuries.
mathematics rises to a new level of abstraction.
Ordinary quantities and numbers turn out to be only special cases of objects studied by modern algebra, geometry turns to the study of spaces.
New disciplines are being developed: the theory of functions of a complex variable, group theory, non Euclidean geometry, set theory, mathematical logic, functional analysis.
Practical mastering of the results of theoretical mathematical research requires obtaining an answer to the problem in numerical form.
Therefore, in the twentieth century, numerical methods of mathematics are separated into an independent branch — computational mathematics.
The desire to simplify and speed up the solutions of a number of time consuming computational tasks led to the appearance of computers.
The needs of mathematics itself, the "mathematization" of various fields of science, the penetration of mathematical methods into many spheres of human activity, the rapid progress of computer technology have led to the emergence of a number of new mathematical disciplines: game theory, information theory, graph theory, discrete mathematics, optimal control theory.
Starting the study of this section, you can recall the words with which the famous French mathematician of the XVIII century.
Joseph Louis Lagrange (who already had a degree in mathematics at the age of 19) appealed to young mathematicians:
"Read it, understanding will come later."
2. The state of science in different historical periods.
1.1. The origin of mathematics
It is impossible to accurately date the emergence of the most important concepts —an integer, a quantity, a figure—.
When writing appeared, the idea of them was already formed.
By this time, various systems of written numbering of integers were also developed.
/ 2000-1700 BC — / the first mathematical texts that have come down to us: two Egyptian papyri and numerous clay tablets from ancient Babylon containing formulations and solutions of problems.
The Egyptians used decimal non positional numbering and fractions with the numerator 1 ("basic" fractions).
The Babylonians had a sexagesimal positional number system without zero and systematic sexagesimal fractions.
Later, in the middle of the first millennium BC, the Babylonians introduced a sign to indicate a missed sexagesimal digit.
Geometry in Babylon and Egypt was primarily computational.
So, the rules for calculating the areas of a triangle by side and height, a circle by its radius were known (the Babylonians took the number 3 as /p/, and the Egyptians took the number 3.16), as well as the volumes of a pyramid and a truncated pyramid with a square base.
The Babylonians knew that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs, as well as the inverse sentence.
Apparently, both of these proposals were discovered by them using examples and they did not yet know how to prove them in a general form.
The most remarkable achievement of this period is the creation of elements of algebra in ancient Babylon and the discovery of the rule for solving quadratic equations.
The Babylonians were also able to find approximate values of square roots from non square numbers.
They knew the rules for summing an arithmetic progression and a series of squares of natural numbers.
Mathematical knowledge was presented in this era in the form of recipes, the correctness of which was not proved; usually the same type of numerical examples and their solutions were given.
There was no mathematics as a science yet.
1.2. The emergence of mathematics as a science.
Construction of the first mathematical theories (mathematics of Ancient Greece)
/ VI century BC - / systematic introduction of logical proofs, which was a turning point in the development of mathematics.
In the Pythagorean scientific school, the construction of geometry as an abstract science was begun, the truths of which are deduced from a few initial axioms with the help of proofs.
The first mathematical theories go back to the Pythagoreans: planimetry of rectilinear figures (including a strict proof of the famous Pythagorean theorem) and elements of number theory (introduction of the concepts of prime numbers, mutually prime numbers, the study of divisibility, the construction of perfect numbers).
At the same school, three of the five regular bodies were discovered: a cube, a tetrahedron and a dodecahedron.
/ V century BC - /In the Pythagorean school, the greatest discovery was made about the incommensurability of the side of a square and its diagonal.
It showed that rational numbers (i.e. integers and fractions) are not enough to measure geometric quantities and substantiate the doctrine of similarity.
Thanks to this discovery, it became necessary to create a theory of relations of both commensurate and incommensurable quantities.
/ V century BC/(the second half) - the so called geometric algebra was created, which made it possible in a general way to solve problems that are reduced to a quadratic equation or a sequence of such equations, purely geometrically, using a compass and ruler.
Geometric algebra played the role of our letter algebra in ancient mathematics, but its apparatus was much less convenient.
At the same time, three famous problems of antiquity were formulated:
1) doubling the cube (build a cube with a volume twice as large as this one),
2) 2) trisection of the angle (divide an arbitrary angle into three equal parts) and
3) 3) quadrature of the circle (build a square equal to this circle).
All these constructions, as it was proved in the XIX century, are impossible with the help of a compass and a ruler.
The ancients used new curves to solve them: conic sections (ellipse, hyperbola and parabola) and quadraris (the first transcendental curve).
In search of the quadrature of the circle, Hippocrates of Chios discovered quadratable lunules (called Hippocratic ones), t, E. figures bounded by arcs of circles for which it is possible to construct squares equal to them.
At the end of the V century.
Hippocrates compiled the first "Beginnings" — a systematic presentation of the foundations of mathematics of his time.
This work has not reached us.
/ IV century BC/(first half) - the Athenian mathematician Theaetetus undertook a study of algebraic irrationalities and began to classify them.
Defined the simplest classes of quadratic irrationalities,
such as:, _ ±_, __,// which were later described in Euclid's "Principles".
He also showed that _ is irrational if it is not a cube.
He also owns the discovery of the octahedron and icosahedron.
/ IV century BC/(middle) - the mathematician and astronomer Eudoxus of Cnidus created a general theory of relations for any homogeneous quantities (both commensurate and incommensurable).
This theory coincides, in essence, with the theory of real numbers proposed at the end of the XIX century by R. Dedekind.
To determine the areas and volumes, Evdox developed the so called "exhaustion method".
Both theories were based on the general doctrine of quantities, and for the first time the most important axiom, now known as the axioms of Archimedes, was formulated:
if a>b, then you can repeat b as many times as/nb>a.
/
With the help of new methods, Eudoxus proved for the first time that the cone is equal to the cylinder having the same base and height, and the pyramid is equal to the corresponding prism.
He also proved that the areas of two circles are related as the squares of their diameters.
300 BC - / Euclid created the "Beginnings", in which he summed up all the previous development of ancient mathematics.
The deductive method of presenting the "Principles" has become a model for building a mathematical theory.
The Principles systematically describe geometry, elements of number theory, algebra, the theory of relations and the method of exhaustion.
Here we formulate Euclid's algorithm for finding the greatest common divisor of two numbers, we prove that the product of the numbers/ab/is divisible by a prime number /p/ if and only if one of the factors is divisible by /p,/ and also that there are infinitely many primes.
For the first time, a strict derivation of the formula for the sum of a finite number of terms of a geometric progression is found in the" Beginnings " and it is shown that there are only five regular polyhedra: a cube, a tetrahedron, an octahedron, a dodecahedron and an icosahedron.
/III century BC//// e//.
— / Archimedes developed methods for finding areas and volumes, as well as methods for determining tangents and the largest and smallest values of quantities, which he applied to solve problems of statics, hydrostatics and the theory of equilibrium of floating bodies.
The methods of Archimedes formed the basis of differential and integral calculus, created in the XVII century.
Archimedes found all semi regular polyhedra.
With the help of conic sections, he solved cubic equations of the form
/x//^2 //(a±x)=//b/
and I conducted a complete study of them.
/ Ill// - II centuries BC - / Apollonius systematically and comprehensively studied conic sections.
His books on conic sections served as the basis for the creation of analytical geometry by R. Descartes and P. Fermat (XVII century), projective geometry by V. Pascal and J. Desargues (XVII century), and also were a mathematical apparatus for the research on mechanics and astronomy of I. Kepler, G. Galileo and I. Newton.
/I// - II**centuries AD — / wide development of computational algebraic methods in ancient mathematics.
/I V./(end) - Menelaus created a systematic course of spherical geometry, modeled on the" Principles " of Euclid, and developed spherical trigonometry.
/II c.
- /Ptolemy, in his astronomical works, expounded planar and spherical trigonometry; he derived a formula equivalent to the formula
sin(a± b) = = sin a.
cos b± cos a.
sin b, and compiled detailed tables of chords (instead of the sine line, the ancients considered the entire chord).
In the tables, Ptolemy used the symbol to indicate the missing sexagesimal digit.
It is possible that this symbol was the prototype of zero.
/ Ill/ / century AD — / Diophantus of Alexandria wrote "Arithmetic", in which he expanded the numerical domain to the field of rational numbers, formulated the rule of multiplication of relative numbers, introduced algebraic symbolism: signs for the first six positive and negative degrees of the unknown, for subtraction and equality.
There are also rules for transferring terms from one part of the equation to another and bringing similar ones.
In "Arithmetic" the problems of solving indefinite equations in rational numbers are considered and methods for finding rational solutions of indefinite equations of the second and third degree are given.
1.3. Mathematics of the countries of the Far, Middle and Middle East
/ II century BC - / creation of the oldest extant Chinese mathematical treatise "Mathematics in nine books", which contained information on arithmetic and geometry.
When solving problems in the treatise, the Pythagorean theorem was used.
The most remarkable in it is a uniform method for solving a system of linear equations.
At the same time, negative numbers appear, for which the rules of addition and subtraction are formulated.
The treatise also describes the calculation algorithm .square and cubic roots, similar to the modern one.
This algorithm was transferred in the VII XIII centuries to the case of calculating the roots of the general equation of the third and fourth degrees.
It coincides mainly with the so called Gorner scheme, obtained in Europe in the XIX century.
/ Ill/ / century AD - /**there are named decimals in Sun Tzu's treatise.
/V VI centuries.
- /creation of a decimal positional number system in India and the introduction of zero as a special digit into it.
/ 499 - /in the astronomical treatise of Ariabhata, he solved an indefinite equation in integers
/ah+bw=c.
///
/ Around 628 - / Brahmagupta, operating with negative numbers, gave a single rule for solving any quadratic equation, formulated rules for actions with zero, which thanks to this became a number equal to other numbers.
Brahmagupta used algebraic symbols: special signs for unknowns and their degrees, signs for the square root, for addition and subtraction operations.
/ IX c.
- /Muhammad al Khorezmi explained the rules of operations with numbers written in the decimal positional system, and studied quadratic equations.
The words "algebra" and" algorithm " first appeared in the translation of his treatises.
The first of them meant the operation of transferring terms from one part of the equation to another, and the second the distorted name of the author (al Khorezmi Algorithmi), it was originally used only to denote the rules of calculation according to the decimal positional system.
/ XI century.
- /the mathematician and poet Omar Khayyam in a treatise on algebra solved geometrically cubic equations (according to the method of Archimedes).
Commenting on Euclid's "Beginnings", he brought the concepts of relation and number closer together.
By the time of Khayyam, the formula for raising a binomial to any positive integer was known .the degree and method of extracting the root of any degree.
/XII century.
- /Bhaskara akarya formulated all the rules for actions with negative numbers.
Bhaskara knew that due to the two digit nature of the square root, a square equation can have two solutions.
/XIII century.
- /Nasiraddin Tuey wrote a treatise on spherical geometry and trigonometry, which contained the doctrine of solving triangles.
The treatise played a crucial role for the development of trigonometry in Europe.
/XV century.
- /Jamshid al Kashi, who worked at the Ulugbek Observatory near Samarkand, introduced and used decimal fractions: the decimal positional system was extended to record any real numbers.
He calculated the number pc with an accuracy of 17 decimal places.
1.4. Mathematics of the European Middle Ages and the Renaissance
/XII XIII centuries.
- /Arabic and Greek works on mathematics have been translated into Latin.
Gradually, the decimal positional system spread in Europe.
/XIII century.
- /Leonardo of Pisa (Fibonacci) outlined a new positional numbering, gave information on algebra and arithmetic, considered various numerical series.
/XTV//—XV centuries.
- /the algebraic symbolism has been improved, notations for the degree, for the radical and for the degrees of the unknown have been introduced.
/XVI century.
— / the first major success of European mathematics: S. Ferro, N. Tartaglia and J. Cardano solved the equation of the third degree in radicals and Cardano's student, L. Ferrari, solved the equation of the fourth degree.
/ 1572 - / in" Algebra " R. Bombelli for the first time considered imaginary numbers /a+b_/and formulated rules for actions with them.
He interpreted these numbers themselves as symbols convenient for obtaining results relative to real numbers.
/1585 - / p. Stevin entered decimals.
/ XVI century.
/ (the second half) - F.
Viet introduced letter designations for unknown and constant quantities and created a mathematical formula.
1.5. The period of mathematics of variables (XVII XVIII centuries)
In the/ XVII century.
/ the mechanics of terrestrial and celestial bodies made great progress, and in this connection there were problems of studying the dependencies of some quantities on others, problems of determining velocities, accelerations, areas of curved figures, centers of gravity, etc.
There was no ready made analytical apparatus for solving these problems in mathematics.
Scientists began to look for ways to study variables in mathematics, using the creations of ancient mathematicians.
As a result, the function (the term of G. Leibniz) has become the same basic object of mathematics as a number and a quantity.
/1614 - / J.
Neper introduced logarithms and published the first logarithmic tables.
A little later, the tables of logarithms were published by I. Burgi.
/1636-1637 - / p. Descartes and P. Fermat introduced the coordinate method into mathematics, which made it possible to reduce geometric problems to algebraic ones.
Independently of each other, Descartes and Fermat constructed analytical geometry using a new method.
Descartes gave algebraic symbolism a modern look.
/ 1608-1660 - / development of the analysis of infinitesimals (methods for determining volumes, areas, centers of gravity, tangents, extremes, velocities, accelerations) in the works of I. Kepler, B. Cavalieri, E. Torricelli, P. Fermat, B. Pascal, J. Wallis, I. Barrow, et al.
/40 —50s of the XVII century .
- / P.
Fermat formulated the famous problems of number theory, which for 200 years remained central in this science.
/1665 - / B.
Pascal in the" Treatise on the Arithmetic Triangle", deducing the properties of binomial coefficients and the relations between them, formulated and applied the principle of complete mathematical induction.
/ 60-80 ies of the XVII century .
— / I.
Newton (since 1665) and G. Leibniz (since 1673) independently created differential and integral calculus and introduced into mathematics the most important analytical apparatus for the representation and study of functions — power series.
Newton extended the formula for raising a binomial to a power to the case when the exponent is any rational number.
/ 1687 - / the book of I. Newton "Mathematical principles of Natural Philosophy" was published, in which the mathematical construction of the foundations of classical mechanics of terrestrial and celestial bodies was given.
/17//13//d.
— / a work by J. Bernoulli containing the simplest form of the law of large numbers, one of the basic laws of probability theory, was published (posthumously).
/40s of the XVIII century — L. / Euler developed the doctrine of the functions of both real and complex variables.
He studied in detail the elementary functions */x/**/^n /**/, /**/a^x, log x, sin x, cos x/*/, / found expressions for* * them in the form of infinite series and determined the logarithms of negative and imaginary numbers.
/ 30-60s of the XVIII century — - / L.
Euler proved the basic theorems of elementary number theory, studied quadratic deductions and discovered the quadratic law of reciprocity — one of the main ones in higher arithmetic.
Euler proved a special case of Fermat's great theorem, namely that the equation
*x**^n +**y**^n =**z^n*
for n=3 and n=4, it has no solutions in integers.
At the same time, he considered expressions of the form */t+n_/* as new integers, which was the first generalization of the concept of an integer.
/1770/—/1771 G. - / J.
Lagrange analyzed all the methods for solving equations of the first four degrees in radicals and showed why all these methods are not suitable for solving equations of the fifth degree.
He discovered that the solvability of equations in radicals depends on the properties of the group of permutations of the roots of this equation, and thus drew attention to the importance of studying groups.
/1796 - / k.
Gauss showed that if n is a prime number, then a regular n square can be constructed using a compass and a ruler when /n/ has the form *_ + 1*.
/1799 - / k.
Wessel gave a geometric interpretation of complex numbers.
However, his works remained unknown.
In 1806, a similar geometric interpretation was proposed by J. Argan.
Complex numbers received universal recognition in mathematics only after the works of K. Gauss in 1832.
/1801 - / k.
Gauss created the foundations of number theory.
He first developed the theory of comparisons, proved the main theorems of this theory, studied the theory of quadratic residues to the end, and outlined the theory of the equations of division of the circle.
/1821-1823 - / Fr. Cauchy developed the theory of limits and based on it built the doctrine of functions, defined the concept of the sum of a series, the continuity of a function, and later put the doctrine of limits as the basis of all mathematical analysis.
Cauchy also developed the foundations of the theories of functions of a complex variable (1825).
/1824-1826 - / N.
Abel proved that algebraic equations of degree n ?
5 are generally undecidable in radicals.
/1827 - / k.
Gauss developed the so called internal geometry of surfaces, in which each surface acts as a carrier of properties of a special geometry.
/1829-1830— / N. I. Lobachevsky published his first works on non Euclidean geometry, which opened a new era in the history of geometry.
Independently of N. I. Lobachevsky, the system of non Euclidean geometry was built by Ya.
Bolyai (1831).
/ 1830-1832 — / E. Galois found a criterion for whether this equation is solved with numerical coefficients in radicals.
At the same time, he developed methods of group and field theory, which have become of great importance in mathematics and its applications.
1832 in connection with his research on the theory of numbers, K. Gauss extended the concept of an integer to the complex numbers a+ bi, where a and b are integers.
He transferred the algorithm for finding the greatest common divisor to new numbers and developed all the arithmetic of complex integers.
/1840-1851 - / U.
Hamilton generalized the concept of a complex number by constructing quaternions numbers of the form */a+/ / bi+/ /cj+/ / dk/*/, / where /i/ / ^2 / / =j^2 =k//^2 //=//-1//, a, //b//, c,//d— / are real numbers.
It turned out that not all the laws of ordinary arithmetic are fulfilled for these numbers.
Thus, the multiplication of quaternions does not have the property of displacement/.
/  /1847 E//.
/Kummer using the introduction so for ideal numbers, I built the arithmetic of integers of the fields of division of the circle.
This gave him the opportunity to prove Fermat's great theorem for all n<100).
/ 1849 — / P. L. Chebyshev obtained the first accurate results after Euclid on the distribution of prime numbers in a natural series.
/1864 - / B. Riemann, generalizing the ideas of Gauss on the internal geometry of surfaces, gave a way to construct all possible metric non Euclidean geometries.
Riemannian geometries later became the main mathematical apparatus of the general theory of relativity.
A special case of Riemannian geometries is the Euclidean geometry and the Lobachevsky geometry.
/1871-1889 - / p. Dedekind, E. I. Zolotarev and L. Kronecker independently and by different methods constructed the arithmetic of integers of any field of algebraic numbers.
At the same time, Dedekind introduced with the help of axiom systems such basic concepts of modern algebra as a ring, a module and an ideal.
/1881-1882 - / p. Dedekind, G. Kantor and K. Weierstrass constructed the theory of real numbers in three different ways.
Soon, in the works of Dedekind and especially Cantor, a new important field of modern mathematics appeared — set theory.
/1899 - / d.
Hilbert, in the Foundations of Geometry, constructed a complete axiomatics of Euclid's geometry and analyzed the relations between various groups of axioms.
Since that time, the axiomatic method has received great development in mathematics.
3. Modern questions of mathematics.
In the XX century, new mathematical theories were created, such as topology, mathematical logic, and the old ones were radically transformed, the language of mathematics itself changed, so that the mathematics of the XIX century.
to read modern books, you would have to relearn again.
The concepts, methods and constructions of modern mathematics are very general in nature.
Accordingly, the field of application of mathematical methods has expanded enormously.
Mathematical methods have penetrated almost all departments of physics, chemistry, and in recent decades — biology, medicine, linguistics, economics.
Mathematics itself has expanded enormously quantitatively and has undergone profound qualitative changes.
In general, it has risen to a higher level of abstraction.
