История математики
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.
BABYLONIA AND EGYPT
Babylonia. The source of our knowledge about the Babylon civilization are
well saved clay tablets covered with texts which are dated from 2000 AD and
up to 300 AD . The mathematics on tablets basically has been connected to
housekeeping. Arithmetic and simple algebra were used at an exchange of
money and calculations for the goods, calculation of simple and complex
percent, taxes and the share of a crop which are handed over for the
benefit of the state, a temple or the land owner. Numerous arithmetic and
geometrical problems arose in connection with construction of channels,
granaries and other public jobs. Very important problem of mathematics was
calculation of a calendar. A calendar was used to know the terms of
agricultural jobs and religious holidays. Division of a circle on 360 and
degree and minutes on 60 parts originates in the Babylon astronomy.
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
[pic]. 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 [pic] babylonians equaled to 3.
Egypt. Our knowledge about ancient greek mathematics is based mainly on two
papyruses dated approximately 1700 AD. Mathematical data stated in these
papyruses go back to earlier period - around 3500 AD. Egyptians used
mathematics to calculate weight of bodies, the areas of crops and volumes
of granaries, the amount of taxes and the quantity of stones required to
build those or other constructions. In papyruses it is possible to find
also the problems connected to solving of amount of a grain, to set number
necessary to produce a beer, and also more the challenges connected to
distinction in grades of a grain; for these cases translation factors were
calculated.
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 Nile. However the
level of development of astronomy in Ancient Egypt was much weaker than
development in Babylon.
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
Babylon, nor Egyptian mathematics had no the common methods; the arch of
mathematical knowledge represented a congestion of empirical formulas and
rules.
THE GREEK MATHEMATICS
Classical Greece. From the point of view of 20 century ancestors of
mathematics were Greeks of the classical period (6-4 centuries AD). The
mathematics existing during earlier period, was a set of the empirical
conclusions. On the contrary, in a deductive reasoning the new statement is
deduced from the accepted parcels by the way excluding an opportunity of
its aversion.
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 Babylon and
Egyptian mathematics during the long wanderings. Pythagoras has based
movement which blossoming falls at the period around 550-300 AD.
Pythagoreans have created pure mathematics in the form of the theory of
numbers and geometry. They represented integers as configurations from
points or a little stones, classifying these numbers according to the form
of arising figures (« figured numbers »). The word "accounting" (counting,
calculation) originates from the Greek word meaning "a little stone".
Numbers 3, 6, 10, etc. Pythagoreans named triangular as the corresponding
number of the stones can be arranged as a triangle, numbers 4, 9, 16, etc.
- square as the corresponding number of the stones can be arranged as a
square, etc.
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[pic] - square number,
n[pic] + 2n +1 = (n + 1)[pic]. 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 to
comprehension 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
[pic], and it made them confusion because they tried to present number
[pic]as the division of two integers that was extremely important for their
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