The history of mathematics
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 by comparing 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.
The 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 (602 + 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 + 1/602).
Ambiguity was resolved depending on the specific context.
The Babylonians compiled tables of inverse numbers (which were used when performing division), tables of squares and square roots, as well as tables of cubes and cubic roots.
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 an angle inscribed in a semicircle would only be 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 "pi" 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 tasks related to determining the amount of grain needed to prepare a given number of beer mugs, as well as more complex tasks related to the difference in grain varieties: conversion coefficients were calculated for these cases.
But the main field of application of mathematics was astronomy, more precisely, 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 have an impact on the development of mathematics, their achievements dating back to about IV b deserve attention.
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 notation 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 XX century.
the ancestors of mathematics were the Greeks of the classical period (VI IV centuries BC).
Mathematics, which existed in an 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 insisted on deductive proof, and this 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 is found 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 VI III 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 mirioi - 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 Pythagorians 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 Pythagorians 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 Pythagorians discovered that the sum of two consecutive triangular numbers is always equal to some square number.
They discovered that if (in modern notation) n2 is a square number, then n2 + 2n +1 = (n + 1)2.
The Pythagorians 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 Pythagorians 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 Pythagorians, 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 Pythagorians 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 Pythagorians 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.
