Encyclopedia of Circumnavigation
MATHEMATICS HISTORY
* MATHEMATICS HISTORY.
* The most ancient mathematical activity was counting.
The account was necessary to monitor the livestock and conduct trade.
Some primitive tribes counted the number of objects, correlating them with various parts of the body, mainly fingers and toes.
A rock drawing that has survived to our times from the Stone Age depicts the number 35 in the form of a series of 35 finger sticks lined up in a row.
The first significant successes in arithmetic were the conceptualization of numbers and the invention of four basic actions: addition, subtraction, multiplication and division.
The first achievements of geometry are associated with such simple concepts as a straight line and a circle.
Further development of mathematics began around 3000 BC thanks to the Babylonians and Egyptians.
BABYLONIA AND EGYPT
Babylonia.
The source of our knowledge about the Babylonian civilization is well preserved clay tablets covered with so called cuneiform texts that date from 2000 BC to 300 AD.Mathematics on cuneiform tablets was mainly related to farming.
Arithmetic and simple algebra were used in the exchange of money and calculations for goods, the calculation of simple and compound interest, taxes and the share of the harvest given in favor of the state, the temple or the landowner.
Numerous arithmetic and geometric problems arose in connection with the construction of canals, granaries and other public works.
A very important task of mathematics was the calculation of the calendar, since the calendar was used to determine the timing of agricultural work and religious holidays.
The division of a circle into 360, and degrees and minutes into 60 parts originate in Babylonian astronomy.
The Babylonians also created a number system that used the base 10 for numbers from 1 to 59.
The symbol for one was repeated the required number of times for numbers from 1 to 9.
To denote the numbers from 11 to 59, the Babylonians used a combination of the symbol of the number 10 and the symbol of one.
To denote numbers starting from 60 and more, the Babylonians introduced a positional number system with a base of 60.
A significant advance was the positional principle, according to which the same numeric sign (symbol) has different values depending on the place where it is located.
An example is the values of the six in the (modern) entry of the number 606.
However, there was no zero in the number system of the ancient Babylonians, which is why the same set of characters could mean both the number 65 (60 + 5) and the number 3605 (60^2 + 0 + 5).
There were ambiguities in the interpretation of fractions.
For example, the same symbols could mean both the number 21, and the fraction 21/60 and (20/60 + 1/60^2 ).
Ambiguity was resolved depending on the specific context.
The Babylonians compiled tables of inverse numbers, tables of squares and square roots, as well as tables of cubes and cubic roots.
Degrees.
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Cuneiform texts devoted to solving algebraic and geometric problems indicate that they used a quadratic formula for solving quadratic equations and could solve some special types of problems that included up to ten equations with ten unknowns, as well as certain varieties of cubic equations and equations of the fourth degree.
The clay tablets depict only the tasks and the main steps of the procedures for solving them.
Since geometric terminology was used to denote unknown quantities, the solution methods mainly consisted in geometric actions with lines and areas.
As for algebraic problems, they were formulated and solved in verbal notation.
Around 700 BC, the Babylonians began to use mathematics to study the movements of the moon and planets.
This allowed them to predict the positions of the planets, which was important for both astrology and astronomy.
In geometry, the Babylonians knew about such relations, for example, as the proportionality of the corresponding sides of such triangles.
They knew the Pythagorean theorem and that the angle inscribed in a semicircle is a straight line.
They also had rules for calculating the areas of simple flat shapes, including regular polygons, and the volumes of simple bodies.
The Babylonians considered the number /p/ to be equal to 3.
Egypt.
Our knowledge of ancient Egyptian mathematics is based mainly on two papyri dating from about 1700 BC.
The mathematical information presented in these papyri dates back to an even earlier period about 3500 BC.
The Egyptians used mathematics to calculate the weight of bodies, the area of crops and the volume of granaries, the size of taxes and the number of stones required for the construction of certain structures.
In the papyri, you can also find problems related to determining the amount of grain needed to prepare a given number of beer mugs, as well as more complex problems related to the difference in grain varieties; conversion coefficients were calculated for these cases.
But the main field of application of mathematics was astronomy, or rather calculations related to the calendar.
The calendar was used to determine the dates of religious holidays and predict the annual flooding of the Nile.
However, the level of development of astronomy in ancient Egypt was much inferior to the level of its development in Babylon.
The ancient Egyptian script was based on hieroglyphs.
The number system of that period was also inferior to the Babylonian one.
The Egyptians used a non positional decimal system, in which the numbers from 1 to 9 were denoted by the corresponding number of vertical dashes, and individual characters were introduced for successive powers of the number 10.
By combining these symbols sequentially, any number could be written.
With the advent of the papyrus, the so called hieratic cursive writing appeared, which, in turn, contributed to the emergence of a new numerical system.
A special identification symbol was used for each of the numbers from 1 to 9 and for each of the first nine multiples of 10, 100, etc.
Fractions were written as the sum of fractions with a numerator equal to one.
The Egyptians performed all four arithmetic operations with such fractions, but the procedure for such calculations remained very cumbersome.
The geometry of the Egyptians was reduced to calculating the areas of rectangles, triangles, trapezoids, circles, as well as formulas for calculating the volumes of some bodies.
I must say that the mathematics that the Egyptians used in the construction of the pyramids was simple and primitive.
The tasks and solutions given in the papyri are formulated purely by prescription, without any explanations.
The Egyptians dealt only with the simplest types of quadratic equations and arithmetic and geometric progressions, and therefore the general rules that they were able to deduce were also of the simplest kind.
Neither Babylonian nor Egyptian mathematicians had common methods; the entire body of mathematical knowledge was a collection of empirical formulas and rules.
Although the Maya who lived in Central America did not influence the development of mathematics, their achievements dating back to about the 4th century are noteworthy.
The Maya, apparently, were the first to use a special symbol to denote zero in their twenty digit system.
They had two number systems: one used hieroglyphs, and in the other, more common, a dot denoted one, a horizontal line the number 5, and a symbol denoted zero.
Positional designations began with the number 20, and the numbers were written vertically from top to bottom.
GREEK MATHEMATICS
Classical Greece.
From the point of view of the 20th century, the ancestors of mathematics were the Greeks of the classical period (6-4 centuries BC).
The mathematics that existed in the earlier period was a set of empirical conclusions.
On the contrary, in deductive reasoning, a new statement is deduced from the accepted premises in a way that excludes the possibility of its rejection.
The Greeks ' insistence on deductive proof was an extraordinary step.
No other civilization has reached the idea of obtaining conclusions solely on the basis of deductive reasoning based on explicitly formulated axioms.
One of the explanations for the commitment of the Greeks to the methods of deduction we find in the structure of Greek society of the classical period.
Mathematicians and philosophers (often they were the same persons) belonged to the upper strata of society, where any practical activity was considered as an unworthy occupation.
Mathematicians preferred abstract reasoning about numbers and spatial relations to solving practical problems.
Mathematics was divided into arithmetic the theoretical aspect and logistics the computational aspect.
Logistics were provided to freeborn lower classes and slaves.
The Greek number system was based on the use of letters of the alphabet.
The Attic system, which was in use since the 6th 3rd centuries BC, used a vertical line to denote a unit, and the initial letters of their Greek names to denote the numbers 5, 10, 100, 1000 and 10,000.
In the later Ionic number system, 24 letters of the Greek alphabet and three archaic letters were used to denote numbers.
Multiples of 1000 to 9000 were designated in the same way as the first nine integers from 1 to 9, but a vertical line was placed before each letter.
Tens of thousands were denoted by the letter M (from the Greek "myrioi" - 10,000), after which the number by which ten thousand had to be multiplied was put.
The deductive character of Greek mathematics was fully formed by the time of Plato and Aristotle.
The invention of deductive mathematics is usually attributed to Thales of Miletus (c. 640-546 BC), who, like many ancient Greek mathematicians of the classical period, was also a philosopher.
It has been suggested that Thales used deduction to prove some results in geometry, although this is doubtful.
Another great Greek, whose name is associated with the development of mathematics, was Pythagoras (c. 585-500 BC).
It is believed that he could have become acquainted with Babylonian and Egyptian mathematics during his long wanderings.
Pythagoras founded the movement, which flourished in the period of ca. 550-300 BC The Pythagoreans created pure mathematics in the form of number theory and geometry.
They represented integers in the form of configurations of dots or pebbles, classifying these numbers according to the shape of the resulting figures ("curly numbers").
The word "calculation" (calculation, calculation) originates from the Greek word meaning "pebble".
The Pythagoreans called the numbers 3, 6, 10, etc. triangular, since the corresponding number of stones can be arranged in the form of a triangle, the numbers 4, 9, 16, etc. - square, since the corresponding number of stones can be arranged in the form of a square, etc.
Some properties of integers arose from simple geometric configurations.
For example, the Pythagoreans discovered that the sum of two consecutive triangular numbers is always equal to some square number.
They discovered that if (in modern notation) /n/^2 is a square number, then n^2 + 2n +1 = (n + 1)^2 .
The Pythagoreans called a number equal to the sum of all its own divisors, except for this number itself, perfect.
Examples of perfect numbers are integers such as 6, 28 and 496.
The Pythagoreans called two numbers friendly if each of the numbers is equal to the sum of the divisors of the other; for example, 220 and 284 are friendly numbers (and here the number itself is excluded from its own divisors).
For the Pythagoreans, any number represented something more than a quantitative quantity.
For example, the number 2, according to their view, meant a difference and therefore was identified with an opinion.
The four represented justice, since this is the first number equal to the product of two identical multipliers.
The Pythagoreans also discovered that the sum of some pairs of square numbers is again a square number.
For example, the sum of 9 and 16 is 25, and the sum of 25 and 144 is 169.
Such triples of numbers as 3, 4 and 5 or 5, 12 and 13 are called Pythagorean numbers.
They have a geometric interpretation, if two numbers from the triple are equated to the lengths of the legs of a right triangle, then the third number will be equal to the length of its hypotenuse.
This interpretation, apparently, led the Pythagoreans to realize a more general fact, now known as the Pythagorean theorem, according to which in any right triangle the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.
Considering a right angled triangle with single cathets, the Pythagoreans discovered that the length of its hypotenuse is equal to the root of two, and this threw them into confusion, because they tried in vain to represent the number root of two as the ratio of two integers, which was extremely important for their philosophy.
The Pythagoreans called the quantities that cannot be represented as a ratio of integers incommensurable; the modern term is "irrational numbers".
Around 300 BC, Euclid proved that the number of the root of two is incommensurable.
The Pythagoreans dealt with irrational numbers, representing all quantities with geometric images.
If we consider 1 and the root of two as the lengths of some segments, then the difference between rational and irrational numbers is smoothed out.
The product of the numbers root of three and root of four is the area of a rectangle with sides of length root of three and root of four.
Even today, we sometimes talk about the number 25 as a square of 5, and about the number 27 as a cube of 3.
The ancient Greeks solved equations with unknowns by means of geometric constructions.
Special constructions were developed for performing addition, subtraction, multiplication and division of segments, extracting square roots from the lengths of segments; now this method is called geometric algebra.
Bringing the problems to a geometric form had a number of important consequences.
In particular, numbers were considered separately from geometry, since it was possible to work with incommensurable relations only with the help of geometric methods.
Geometry became the basis of almost all rigorous mathematics until at least 1600.
And even in the 18th century, when algebra and mathematical analysis were already sufficiently developed, strict mathematics was interpreted as geometry, and the word " geometer "was equivalent to the word"mathematician".
It is to the Pythagoreans that we owe much of the mathematics that was then systematized and proved in Euclid's "Principles".
There is reason to believe that it was they who discovered what are now known as theorems about triangles, parallel lines, polygons, circles, spheres and regular polyhedra.
One of the most prominent Pythagoreans was Plato (c. 427-347 BC).
Plato was convinced that the physical world is comprehensible only through mathematics.
It is believed that it is to him that the merit of the invention of the analytical method of proof belongs.
(The analytical method begins with a statement that needs to be proved, and then sequentially the consequences are deduced from it until some known fact is reached; the proof is obtained using the reverse procedure.)
It is generally believed that the followers of Plato invented a method of proof called "proof from the contrary".
A notable place in the history of mathematics is occupied by Aristotle, a student of Plato.
Aristotle laid the foundations of the science of logic and expressed a number of ideas regarding definitions, axioms, infinity and the possibility of geometric constructions.
The greatest of the Greek mathematicians of the classical period, second only to Archimedes in importance of the results obtained, was Eudoxus (c. 408-355 BC).
It was he who introduced the concept of magnitude for such objects as straight lines and angles.
Having the concept of magnitude, Eudoxus logically strictly justified the Pythagorean method of dealing with irrational numbers.
The works of Evdox made it possible to establish the deductive structure of mathematics on the basis of explicitly formulated axioms.
He also took the first step in creating mathematical analysis, since it was he who invented the method of calculating areas and volumes, called the "exhaustion method".
This method consists in constructing inscribed and described flat figures or spatial bodies that fill ("exhaust") the area or volume of the figure or body that is the subject of the study.
Eudoxus also belongs to the first astronomical theory explaining the observed motion of the planets.
The theory proposed by Eudoxus was purely mathematical; it showed how combinations of rotating spheres with different radii and axes of rotation could explain the seemingly irregular movements of the Sun, Moon and planets.
Around 300 BC, the results of many Greek mathematicians were combined into a single whole by Euclid, who wrote the mathematical masterpiece "Beginnings".
From a few shrewdly selected axioms, Euclid derived about 500 theorems that covered all the most important results of the classical period.
Euclid began his essay with the definition of such terms as a straight line, an angle and a circle.
Then he formulated ten self evident truths, such as "the whole is greater than any of the parts".
And from these ten axioms, Euclid was able to deduce all the theorems.
For mathematicians, the text of Euclid's "Principles" served as a model of rigor for a long time, until in the 19th century it was discovered that there were serious shortcomings in it, such as the unconscious use of explicitly unformulated assumptions.
Apollonius (c. 262-200 BC) lived in the Alexandrian period, but his main work is sustained in the spirit of classical traditions.
His proposed analysis of conic sections – a circle, an ellipse, a parabola and a hyperbola was the culmination of the development of Greek geometry.
Apollonius also became the founder of quantitative mathematical astronomy.
The Alexandrian period.
During this period, which began around 300 BC, the character of Greek mathematics changed.
Alexandrian mathematics arose from the merger of classical Greek mathematics with the mathematics of Babylonia and Egypt.
In general, the mathematicians of the Alexandrian period were more inclined to solve purely technical problems than to philosophy.
The great Alexandrian mathematicians Eratosthenes, Archimedes, Hipparchus, Ptolemy, Diophantus and Papp demonstrated the power of the Greek genius in theoretical abstraction, but they were just as willing to apply their talent to solving practical problems and purely quantitative problems.
Eratosthenes (c. 275-194 BC) found a simple method for accurately calculating the circumference of the Earth, he also owns a calendar in which every fourth year has one day more than the others.
The astronomer Aristarchus (c. 310-230 BC) wrote an essay "On the sizes and distances of the Sun and Moon", which contained one of the first attempts to determine these sizes and distances; by its nature, Aristarchus ' work was geometric.
The greatest mathematician of antiquity was Archimedes (c. 287-212 BC).
He owns the formulations of many theorems about the areas and volumes of complex figures and bodies, which he proved quite strictly by the method of exhaustion.
Archimedes always tried to get exact solutions and found upper and lower bounds for irrational numbers.
For example, working with the correct 96 square, he perfectly proved that the exact value of the number " p " is between 3^1 /_7 and 3^10 /_71 .
Archimedes also proved several theorems containing new results of geometric algebra.
He is responsible for the formulation of the problem of dissecting the ball with a plane so that the volumes of the segments are located among themselves in a given ratio.
Archimedes solved this problem by finding the intersection of a parabola and an isosceles hyperbola.
Archimedes was the greatest mathematical physicist of antiquity.
To prove the theorems of mechanics, he used geometric considerations.
His essay "On Floating bodies" laid the foundations of hydrostatics.
According to legend, Archimedes discovered a law bearing his name, according to which a repulsive force equal to the weight of the liquid displaced by him acts on a body immersed in water, while bathing, while in the bathroom, and unable to cope with the joy of discovery that seized him, he ran naked into the street shouting: "Eureka!"("Opened!")
In the time of Archimedes, they were no longer limited to geometric constructions that could be carried out only with the help of compasses and a ruler.
Archimedes used a spiral in his constructions, and Diocles (the end of the 2nd century BC) solved the problem of doubling the cube with the help of a curve introduced by him, called a cissoid.
In the Alexandrian period, arithmetic and algebra were considered independently of geometry.
The Greeks of the classical period had a logically sound theory of integers, but the Alexandrian Greeks, having adopted Babylonian and Egyptian arithmetic and algebra, largely lost their already developed ideas about mathematical rigor.
Heron of Alexandria, who lived between 100 BC and 100 AD, transformed a significant part of the geometric algebra of the Greeks into frankly lax computational procedures.
However, when proving new theorems of Euclidean geometry, he was still guided by the standards of logical rigor of the classical period.
The first fairly voluminous book in which arithmetic was expounded independently of geometry was "Introduction to Arithmetic" by Nicomachus (ca.100 AD).
In the history of arithmetic, its role is comparable to the role of Euclid's "Principles" in the history of geometry.
For more than 1000 years, it served as a standard textbook, since it clearly, clearly and comprehensively expounded the doctrine of integers (simple, composite, mutually simple, as well as proportions).
Repeating many Pythagorean statements, the" Introduction " of Nicomachus at the same time went further, since Nicomachus saw more general relations, although he cited them without proof.
A significant milestone in the algebra of the Alexandrian Greeks was the work of Diophantus (ca. 250).
One of his main achievements is connected with the introduction of the principles of symbolism into algebra.
In his works, Diophantus did not offer general methods, he dealt with specific positive rational numbers, and not with their letter designations.
He laid the foundations of the so called Diophantine analysis the study of indefinite equations.
The highest achievement of the Alexandrian mathematicians was the creation of quantitative astronomy.
To Hipparchus (c. 161-126 BC) we owe the invention of trigonometry.
His method was based on a theorem stating that in such triangles, the ratio of the lengths of any two sides of one of them is equal to the ratio of the lengths of the two corresponding sides of the other.
In particular, the ratio of the length of the cathet lying against the acute angle A in a right triangle to the length of the hypotenuse should be the same for all right triangles having the same acute angle A.
This ratio is known as the sine of the angle A.
The ratios of the lengths of the other sides of a right triangle are called the cosine and tangent of the angle A. Hipparchus invented a method for calculating such ratios and compiled their tables.
With these tables and easily measurable distances on the Earth's surface, he was able to calculate the length of its great circle and the distance to the Moon.
According to his calculations, the radius of the Moon was one third of the Earth's radius; according to modern data, the ratio of the radii of the Moon and the Earth is 27/1000.
Hipparchus determined the duration of the solar year with an error of only 6 ^ 1 /_2 minutes; it is believed that it was he who introduced the latitudes and longitudes.
Greek trigonometry and its applications in astronomy reached their peak in the Almagest of the Egyptian Claudius Ptolemy (died 168 AD).
The theory of the motion of celestial bodies was presented in the Almagest, which prevailed until the 16th century, when it was replaced by the theory of Copernicus.
Ptolemy sought to build the simplest mathematical model, realizing that his theory was just a convenient mathematical description of astronomical phenomena, consistent with observations.
The Copernican theory prevailed precisely because it turned out to be simpler as a model.
The decline of Greece.
After the conquest of Egypt by the Romans in 31 BC, the great Greek Alexandrian civilization fell into decline.
Cicero proudly claimed that, unlike the Greeks, the Romans are not dreamers, and therefore apply their mathematical knowledge in practice, extracting real benefits from them.
However, the contribution of the Romans to the development of mathematics itself was insignificant.
The Roman numeral system was based on cumbersome notation of numbers.
Its main feature was the additive principle.
Even the subtractive principle, for example, writing the number 9 in the form of IX, came into wide use only after the invention of typesetting letters in the 15th century.
The Roman notation of numbers was used in some European schools until about 1600, and in accounting a century later.
INDIA AND THE ARABS
The successors of the Greeks in the history of mathematics were the Indians.
Indian mathematicians did not deal with proofs, but they introduced original concepts and a number of effective methods.
It was they who first introduced zero both as a cardinal number and as a symbol of the absence of units in the corresponding digit.
Mahavira (850 A.D.) established rules for operations with zero, believing, however, that dividing a number by zero leaves the number unchanged.
The correct answer for the case of dividing a number by zero was given by Bhaskara (b. 1114), he also owns the rules of actions on irrational numbers.
The Indians introduced the concept of negative numbers (to denote debts).
The earliest use of them is found in Brahmagupta (ca. 630).
Ariabhata (p. 476) went further than Diophantus in the use of continuous fractions in solving indefinite equations.
Our modern number system, based on the positional principle of writing numbers and zero as a cardinal number and using the designation of an empty digit, is called Indo Arabic.
On the wall of a temple built in India around 250 BC, several figures were found that resemble our modern figures in their outlines.
About 800 Indian mathematics reached Baghdad.
The term "algebra" comes from the beginning of the title of the book / Al Jabr wa l mukabala/ (/Completion and Opposition/), written in 830 by the astronomer and mathematician al Khorezmi.
In his essay, he paid tribute to the merits of Indian mathematics.
The algebra of al Khorezmi was based on the works of Brahmagupta, but Babylonian and Greek influences are clearly discernible in it.
Another prominent Arab mathematician Ibn al Haysam (c. 965-1039) developed a method for obtaining algebraic solutions of square and cubic equations.
Arab mathematicians, including Omar Khayyam, were able to solve some cubic equations using geometric methods, using conic sections.
Arab astronomers introduced the concept of tangent and cotangent into trigonometry.
Nasiraddin Tusi (1201-1274) in his Treatise on the Complete Quadrilateral systematically outlined planar and spherical geometry and was the first to consider trigonometry separately from astronomy.
Yet the most important contribution of the Arabs to mathematics was their translations and commentaries on the great works of the Greeks.
Europe became acquainted with these works after the Arab conquest of North Africa and Spain, and later the works of the Greeks were translated into Latin.
THE MIDDLE AGES AND THE RENAISSANCE
Medieval Europe.
The Roman civilization did not leave a noticeable trace in mathematics, because it was too concerned with solving practical problems.
The civilization that developed in early Medieval Europe (c. 400-1100) was not productive for the exact opposite reason: intellectual life focused almost exclusively on theology and the afterlife.
The level of mathematical knowledge did not rise above arithmetic and simple sections from the "Principles" of Euclid.
The most important branch of mathematics in the Middle Ages was considered astrology; astrologers were called mathematicians.
And since medical practice was based mainly on astrological indications or contraindications, doctors had no choice but to become mathematicians.
Around 1100, the almost three century period of mastering the heritage of the Ancient World and the East preserved by the Arabs and Byzantine Greeks began in Western European mathematics.
Since the Arabs owned almost all the works of the ancient Greeks, Europe received an extensive mathematical literature.
The translation of these works into Latin contributed to the rise of mathematical research.
All the great scientists of that time admitted that they drew inspiration from the writings of the Greeks.
The first European mathematician worth mentioning was Leonardo of Pisa (Fibonacci).
In his essay /The Book of Abacus/ (1202), he introduced Europeans to Indo Arabic numerals and calculation methods, as well as Arabic algebra.
Over the next few centuries, mathematical activity in Europe waned.
The code of mathematical knowledge of that era, compiled by Luca Pacioli in 1494, did not contain any algebraic innovations that Leonardo did not have.
Rebirth.
Among the best geometers of the Renaissance were artists who developed the idea of perspective, which required geometry with converging parallel lines.
The artist Leon Battista Alberti (1404-1472) introduced the concepts of projection and section.
Rectilinear rays of light from the observer's eye to various points of the depicted scene form a projection; the cross section is obtained when the plane passes through the projection.
In order for the painted picture to look realistic, it had to be such a cross section.
The concepts of projection and section gave rise to purely mathematical questions.
For example, what are the general geometric properties of the section and the original scene, what are the properties of two different sections of the same projection formed by two different planes intersecting the projection at different angles?
From such questions, projective geometry arose.
Its founder is Zh.
Desargues (1593-1662), using proofs based on projection and section, unified the approach to various types of conic sections, which the great Greek geometer Apollonius considered separately.
THE BEGINNING OF MODERN MATHEMATICS
The beginning of the 16th century in Western Europe was marked by important achievements in algebra and arithmetic.
Decimals and the rules of arithmetic operations with them were introduced into circulation.
The real triumph was the invention of logarithms in 1614 by J. R. R. Tolkien.
Neperom.
By the end of the 17th century, the understanding of logarithms as exponents of a power with any positive number other than one as the base was finally formed.
Since the beginning of the 16th century, irrational numbers have become more widely used.
B. Pascal (1623-1662) and I. Barrow (1630-1677), a teacher of I. Newton at the University of Cambridge, argued that such a number as the root of three can be interpreted only as a geometric quantity.
However, in the same years, R. Descartes (1596-1650) and J. Wallis (1616-1703) believed that irrational numbers are permissible by themselves, without reference to geometry.
In the 16th century, disputes continued over the legality of the introduction of negative numbers.
Even less acceptable were the complex numbers that arose when solving quadratic equations, such as the sum of five and the root of "minus five", which Descartes called "imaginary".
These numbers were under suspicion even in the 18th century, although L. Euler (1707-1783) successfully used them.
Complex numbers were finally recognized only at the beginning of the 19th century, when mathematicians got used to their geometric representation.
Achievements in algebra.
In the 16th century, Italian mathematicians N. Tartaglia (1499-1577), S. Dal Ferro (1465-1526), L. Ferrari (1522-1565) and D. Cardano (1501-1576) found general solutions of equations of the third and fourth degrees.
To make algebraic reasoning and its recording more accurate, many symbols were introduced, including the plus and minus signs, the square root sign radical, the signs "more" and "less".
The most significant innovation was the systematic use of the French mathematician F.Vietom (1540-1603) letters for the designation of unknown and constant quantities.
This innovation allowed him to find a single method for solving equations of the second, third and fourth degrees.
Then the mathematicians turned to equations whose degrees are higher than the fourth.
Working on this problem, Cardano, Descartes and I. Newton (1643-1727) published (without proof) a number of results concerning the number and type of roots of the equation.
Newton discovered the relation between the roots and the discriminant [b^2-4ac] of the quadratic equation, namely, that the equation ax^2 + bx + c = 0 has equal real, different real or complex conjugate roots, depending on whether the discriminant b^2-4ac is equal to zero, greater than or less than zero.
In 1799, K. Friedrich Gauss (1777-1855) proved the so called basic theorem of algebra: every polynomial of the nth degree has exactly n roots.
The main task of algebra – the search for a general solution of algebraic equations – continued to occupy mathematicians in the early 19th century.
When talking about the General solution of the second degree equation ax^2 + bx + c = 0, mean that each of the two roots can be expressed using a finite number of operations of addition, subtraction, multiplication, division and extraction of roots produced on the coefficients a, b and C. Young Norwegian mathematician N. Abel (1802-1829) proved that it is impossible to obtain the General solution of equations of degree higher than 4 with a finite number of algebraic operations.
However, there are many equations of a special form of degree higher than 4 that allow such a solution.
On the eve of his death in a duel, the young French mathematician E. Galois (1811-1832) gave a decisive answer to the question of which equations are solvable in radicals, i.e. the roots of which equations can be expressed in terms of their coefficients using a finite number of algebraic operations.
In Galois theory, substitutions or permutations of roots were used and the concept of a group was introduced, which has found wide application in many areas of mathematics.
The development of group theory is a good example of the continuity of creative work in mathematics.
Galois built his theory based on the work of Abel, Abel relied on the work of J.Lagrange (1736-1813).
In turn, many outstanding mathematicians, including Gauss and A. Legendre (1752-1833), implicitly used the concept of a group in their works.
Newton was not overly modest when he stated: "If I saw further than others, it was because I stood on the shoulders of giants."
Analytical geometry.
Analytical, or coordinate, geometry was created independently by P. Fermat (1601-1665) and R. Descartes in order to expand the possibilities of Euclidean geometry in construction problems.
However, Fermat considered his works only as a reformulation of the work of Apollonius.
The real discovery – the realization of the full power of algebraic methods – belongs to Descartes.
Euclidean geometric algebra for each construction required the invention of its original method and could not offer the quantitative information necessary for science.
Descartes solved this problem: he formulated geometric problems algebraically, solved an algebraic equation, and only then built the desired solution – a segment that had the appropriate length.
Analytic geometry itself arose when Descartes began to consider indefinite construction problems, the solutions of which are not one, but many possible lengths.
Analytical geometry uses algebraic equations to represent and study curves and surfaces.
Descartes considered acceptable a curve that can be written using a single algebraic equation with respect to x and y.
This approach was an important step forward, because it not only included such curves as the conchoid and the cissoid among the permissible ones, but also significantly expanded the area of curves.
As a result, in the 17th and 18th centuries.
many new important curves, such as the cycloid and the chain line, have entered scientific use.
It seems that the first mathematician who used equations to prove the properties of conic sections was J. R. R. Tolkien.
Wallis.
By 1865, he had algebraically obtained all the results presented in the V book of Euclid's "Principles".
Analytic geometry has completely changed the roles of geometry and algebra.
As the great French mathematician Lagrange noted, " while algebra and geometry were moving each in their own way, their progress was slow, and their applications were limited.
But when these sciences combined their efforts, they borrowed new vital forces from each other and since then they have been moving towards perfection with rapid steps."
Mathematical analysis.
The founders of modern science – Copernicus, Kepler, Galileo and Newton approached the study of nature as mathematicians.
Studying motion, mathematicians have developed such a fundamental concept as a function, or a relation between variables, for example , d = kt^2, where d is the distance traveled by a freely falling body, and t is the number of seconds that the body is in free fall.
The concept of a function immediately became central to determining the speed at a given time and the acceleration of a moving body.
The mathematical difficulty of this problem was that at any moment a body passes a zero distance in a zero period of time.
Therefore, determining the value of the speed at a time by dividing the path by time, we will come to a mathematically meaningless expression 0/0.
The problem of determining and calculating the instantaneous rates of change of various quantities attracted the attention of almost all mathematicians of the 17th century, including Barrow, Fermat, Descartes and Wallis.
The disparate ideas and methods proposed by them were combined into a systematic, universally applicable formal method by Newton and G. Leibniz (1646-1716), the creators of differential calculus.
On the issue of priority in the development of this calculus, there were heated debates between them, with Newton accusing Leibniz of plagiarism.
However, as the research of historians of science has shown, Leibniz created mathematical analysis independently of Newton.
As a result of the conflict, the exchange of ideas between mathematicians of continental Europe and England was interrupted for many years to the detriment of the English side.
English mathematicians continued to develop the ideas of analysis in the geometric direction, while mathematicians of continental Europe, including I. Bernoulli (1667-1748), Euler and Lagrange achieved incomparably great success following the algebraic, or analytical, approach.
The basis of all mathematical analysis is the concept of a limit.
The speed at a time is defined as the limit to which the average speed d/t tends, when the value of t is getting closer to zero
Differential calculus provides a general method that is convenient in calculations for finding the rate of change of the function f (x) for any value of x.
This speed is called the derivative.
From the generality of the notation f (x), it can be seen that the concept of a derivative is applicable not only in problems related to the need to find speed or acceleration, but also in relation to any functional dependence, for example, to some relation from economic theory.
One of the main applications of differential calculus is the so called maximum and minimum problems; another important range of problems is finding the tangent to a given curve.
It turned out that using a derivative specially invented for working with motion problems, it is also possible to find areas and volumes bounded by curves and surfaces, respectively.
The methods of Euclidean geometry did not have the proper generality and did not allow obtaining the required quantitative results.
The efforts of mathematicians of the 17th century created numerous special methods that allowed finding the areas of figures bounded by curves of one kind or another, and in some cases the connection of these problems with the problems of finding the rate of change of functions was noted.
But, as in the case of differential calculus, it was Newton and Leibniz who realized the generality of the method and thereby laid the foundations of integral calculus.
The Newton Leibniz method begins by replacing the curve bounding the area to be determined with a sequence of polylines approaching it, similar to how it was done in the exhaustion method invented by the Greeks.
The exact area is equal to the limit of the sum of the areas of n rectangles when n turns to infinity.
Newton showed that this limit can be found by reversing the process of finding the rate of change of the function.
The operation that is the reverse of differentiation is called integration.
The statement that summation can be carried out by reversing differentiation is called the main theorem of mathematical analysis.
Just as differentiation is applicable to a much broader class of problems than the search for velocities and accelerations, integration is applicable to any problem related to summation, for example, to physical problems for adding forces.
