История математики
| Разделы | Иностранные языки •Математика |
| Тип | |
| Формат |
 Microsoft Word
|
| Язык | Английский |
| Примечание | от автора: Доклад на тему истории математики на английском языке |
| Загрузить архив: | |
| Файл: ref-17317.zip (20kb [zip], Скачиваний: 95) скачать |
History of math. The most ancient mathematical activity was counting. The counting was necessary to keep up a livestock of cattle and to do business. Some primitive tribes counted up amount of subjects, comparing them various parts of a body, mainly fingers of hands and foots. Some pictures on the stone represents number 35 as a series of 35 sticks - fingers built in a line. The first essential success in arithmetic was the invention of four basic actions: additions, subtraction, multiplication and division. The first achievements of geometry are connected to such simple concepts, as a straight line and a circle. The further development of mathematics began approximately in 3000 up to AD due to Babylonians and Egyptians.
Babylonians have made tables of inverse
numbers (which were used at performance of division), tables of squares and
square roots, and also tables of cubes and cubic roots. They knew good
approximation of a number 
.
The texts devoted to the solving
algebraic and geometrical problems, testify that they used the square-law
formula for the solving quadratics and could solve some special types of the
problems, including up to ten equations with ten unknown persons, and also
separate versions of the cubic equations and the equations of the fourth
degree. On the clay tablets problems and the basic steps of procedures of their
decision are embodied only. About 700 AD babylonians
began to apply mathematics to research of, motions of the Moon and planets. It
has allowed them to predict positions of planets that were important both for astrology,
and for astronomy.
In geometry babylonians
knew about such parities, for example, as proportionality of the corresponding
parties of similar triangles, Pythagoras’ theorem and that a corner entered in half-circle-
was known for a straight line. They had also rules of calculation of the areas
of simple flat figures, including correct polygons, and volumes of simple
bodies. Number 
babylonians
equaled to 3.
But the main scope of mathematics was
astronomy, the calculations connected to a calendar are more exact. The
calendar was used find out dates of religious holidays and a prediction of
annual floods of
Ancient greek writing was based on hieroglyphs. They used their alphabet. I think it’s not efficient; It’s difficult to count using letters. Just think how they could multiply such numbers as 146534 to 19870503 using alphabet. May be they needn’t to count such numbers. Nevertheless they’ve built an incredible things – pyramids. They had to count the quantity of the stones that were used and these quantities sometimes reached to thousands of stones. I imagine their papyruses like a paper with numbers ABC, that equals, for example, to 3257.
The geometry at Egyptians was reduced to calculations of the areas of rectangular, triangles, trapezes, a circle, and also formulas of calculation of volumes of some bodies. It is necessary to say, that mathematics which Egyptians used at construction of pyramids, was simple and primitive. I suppose that simple and primitive geometry can not create buildings that can stand for thousands of years but the author thinks differently.
Problems and the solving resulted in
papyruses, are formulated without any explanations. Egyptians dealt only with
the elementary types of quadratics and arithmetic and geometrical progressions
that is why also those common rules which they could deduce, were also the most
elementary kind. Neither
THE GREEK MATHEMATICS
Classical
Insisting of Greeks on the deductive proof was extraordinary step. Any other civilization has not reached idea of reception of the conclusions extremely on the basis of the deductive reasoning which is starting with obviously formulated axioms. The reason is a greek society of the classical period. Mathematics and philosophers (quite often it there were same persons) belonged to the supreme layers of a society where any practical activities were considered as unworthy employment. Mathematics preferred abstract reasoning on numbers and spatial attitudes to the solving of practical problems. The mathematics consisted of a arithmetic - theoretical aspect and logistic - computing aspect. The lowest layers were engaged in logistic.
Deductive character of the Greek
mathematics was completely generated by Plato’s and Eratosthenes’ time. Other
great Greek, with whose name connect development of mathematics, was Pythagoras.
He could meet the
From simple geometrical configurations
there were some properties of integers. For example, Pythagoreans have found
out, that the sum of two consecutive triangular numbers is always equal to some
square number. They have opened, that if (in modern designations) n
- square number,
n+ 2n +1 = (n
+ 1). The number equal to the sum of all own dividers, except
for most this number, Pythagoreans named accomplished. As examples of the
perfect numbers such integers, as 6, 28 and 496 can serve. Two numbers
Pythagoreans named friendly, if each of numbers equally to the sum of dividers
of another; for example, 220 and 284 - friendly numbers (here again the number
is excluded from own dividers).
For Pythagoreans any number represented something the greater, than quantitative value. For example, number 2 according to their view meant distinction and consequently was identified with opinion. The 4 represented validity, as this first equal to product of two identical multipliers.
Pythagoreans also have opened, that the sum of some pairs of square numbers is again square number. For example, the sum 9 and 16 is equal 25, and the sum 25 and 144 is equal 169. Such three of numbers as 3, 4 and 5 or 5, 12 and 13, are called “Pythagorean” numbers. They have geometrical interpretation: if two numbers from three to equate to lengths of cathetuses of a rectangular triangle the third will be equal to length of its hypotenuse. Such interpretation, apparently, has led Pythagoreans tocomprehension more common fact known nowadays under the name of a pythagoras’ theorem, according to which the square of length of a hypotenuse is equal the sum of squares of lengths of cathetuses.
Considering a rectangular triangle with cathetuses equaled to 1, Pythagoreans have found out, that
the length of its hypotenuse is equal to ,
and it made them confusion because they tried to present number as
the division of two integers that was extremely important for their philosophy.
Values, not representable as the division of
integers, Pythagoreans have named incommensurable; the modern term - «
irrational numbers ». About 300 AD is
incommensurable. Pythagoreans dealt with irrational numbers, representing all
sizes in the geometrical images. If 1 and to
count lengths of some pieces distinction between rational and irrational
numbers smoothes out. Product of numbers 
also
is
the area of a rectangular with the sides in length and
.Today
sometimes we speak about number 25 as about a square of 5, and about number 27
- as about a cube of 3.
Ancient Greeks solved the equations with unknown values by means of geometrical constructions. Special constructions for performance of addition, subtraction, multiplication and division of pieces, extraction of square roots from lengths of pieces have been developed; nowadays this method is called as geometrical algebra.
Reduction of problems to a geometrical
kind had a number of the important consequences. In particular, numbers began
to be considered separately from geometry because to work with incommensurable divisions
it was possible only with the help of geometrical methods. The geometry became
a basis almost all strict mathematics at least to 1600 AD. And even in 18
century when the
algebra and the mathematical analysis have already been advanced enough, the strict
mathematics was treated as geometry, and the word "geometer" was
equivalent to a word "mathematician".
One of the most outstanding Pythagoreans was Plato. Plato has been convinced, that the physical world is conceivable only by means of mathematics. It is considered, that exactly to him belongs a merit of the invention of an analytical method of the proof. (the Analytical method begins with the statement which it is required to prove, and then from it consequences, which are consistently deduced until any known fact will be achieved; the proof turns out with the help of return procedure.) It is considered to be, that Plato’s followers have invented the method of the proof which have received the name "rule of contraries". The appreciable place in a history of mathematics is occupied by Aristotle; he was the Plato’s learner. Aristotle has put in pawn bases of a science of logic and has stated a number of ideas concerning definitions, axioms, infinity and opportunities of geometrical constructions.
About 300 AD results of many Greek
mathematicians have been shown in the one work by Euclid, who had written a
mathematical masterpiece “the Beginning”. From few selected axioms
Apollonius lived during the
The
Eratosthenes has found a simple method of exact calculation of length of a circle of the Earth, he possesses a calendar in which each fourth year has for one day more, than others. The astronomer the Aristarch has written the composition “About the sizes and distances of the Sun and the Moon”, containing one of the first attempts of definition of these sizes and distances; the character of the Aristarch’s job was geometrical.
The greatest mathematician of an antiquity
was Archimedes. He possesses formulations of many theorems of the areas and
volumes of complex figures and the bodies. Archimedes always aspired to receive
exact decisions and found the top and bottom estimations for irrational
numbers. For example, working with a correct 96-square, he has irreproachably
proved, that exact value of number is between 3
and 3
Архимед
has proved also some theorems, containing new results of geometrical algebra.
Archimedes also was the greatest mathematical physicist of an antiquity. For the proof of theorems of mechanics he used geometrical reasons. His composition “About floating bodies” has put in pawn bases of a hydrostatics.
Decline of
Successors of Greeks in a history of
mathematics were Indians. Indian mathematics were not engaged in proofs, but
they have entered original concepts and a number of effective methods. They
have entered zero as cardinal number and as a symbol of absence of units in the
corresponding category.
Our modern notation based on an item
principle of record of numbers and zero as cardinal number and use of a
designation of the empty category, is called Indo-Arabian. On a wall of the
temple constructed in
About 800 Indian mathematics has achieved
And still the most important contribution
of arabs to mathematics of steel their translations
and comments to great creations of Greeks.
MIDDLE AGES AND REVIVAL
Medieval
About 1100 in the West-European
mathematics began almost three-century period of development saved by arabs and the Byzantian Greeks of
a heritage of the Ancient world and the East.
The first European mathematician deserving
a mention became Leonardo Byzantian (Fibonacci). In
the composition “the Book Abaca” (1202) he has acquainted Europeans with
the Indо-Arabian figures and methods of calculations and
also with the Arabian algebra. Within the next several centuries mathematical
activity in
Revival. Among the best geometers of Renaissance there were the artists developed idea of prospect which demanded geometry with converging parallel straight lines. The artist Leon Batista Alberty (1404-1472) has entered concepts of a projection and section. Rectilinear rays of light from an eye of the observer to various points of a represented stage form a projection; the section turns out at passage of a plane through a projection. That the drawn picture looked realistic, it should be such section. Concepts of a projection and section generated only mathematical questions. For example, what general geometrical properties the section and an initial stage, what properties of two various sections of the same projection, formed possess two various planes crossing a projection under various corners? From such questions also there was a projective geometry. Its founder - Z. Dezarg (1593-1662 AD) with the help of the proofs based on a projection and section, unified the approach to various types of conic sections which great Greek geometer Apollonius considered separately.
I think that mathematics developed by attempts and mistakes. There is no perfect science today. Also math has own mistakes, but it aspires to be more accurate. A development of math goes thru a development of the society. Starting from counting on fingers, finishing on solving difficult problems, mathematics prolong it way of development. I suppose that it’s no people who can say what will be in 100-200 or 500 years. But everybody knows that math will get new level, higher one. It will be new high-tech level and new methods of solving today’s problems. May in the future some man will find mistakes in our thinking, but I think it’s good, it’s good that math will not stop.
Bibliography:
Ван-дер-Варден Б.Л. «Пробуждающаяся наука». Математика древнего Египта, Вавилона и Греции. МОСКВА, 1959
Юшкевич A.П. История математики в средние века. МОСКВА, 1961
Даан-Дальмедико А., Пейффер Ж. Пути и лабиринтыю Очерки по истории математики МОСКВА, 1986
Клейн Ф. Лекции о развитии математики в XIX столетии. МОСКВА, 1989