History of the development of the number. Development of the concept of number

The development of ideas about numbers is an important part of our history. It is one of the basic mathematical concepts that allows you to express the results of a measurement or an account. The notion of number is the starting point for a lot of mathematical theories. It is also used in mechanics, physics, chemistry, astronomy and many other sciences. In addition, in everyday life we use numbers all the time.

The appearance of figures

The followers of the teachings of Pythagoras believed that the numbers contain the mystical essence of things. These mathematical abstractions guide the world, establishing order in it. The Pythagoreans assumed that all regularities existing in the world can be expressed with the help of numbers. It was with Pythagoras that the theory of number development began to interest many scientists. These symbols were considered the basis of the material world, and not just expressions of some regular order.

The history of the development of numbers and accounts began with the fact that a practical account of objects was created, as well as measurements of volumes, surfaces and lines.

Gradually, the notion of natural numbers was formed. This process was complicated by the fact that the primitive man was not able to separate the abstract from the concrete representation. The account as a result of this remained for a long time only material. We used notes, pebbles, fingers, etc. To memorize his results, nodules, nicks, etc. After the invention of writing, the history of the development of the number was marked by the fact that they began to use letters, as well as special icons used for the abbreviated image on writing large numbers . The principle of numbering, similar to that used in the language, was usually reproduced with such coding.

Later, the idea appeared to be counted in tens, not just in units. In 100 different Indo-European languages, the names of numbers from two to ten are similar, like the names of dozens. Consequently, the notion of an abstract number appeared long ago, even before these languages were separated.

The account on the fingers was originally widely distributed, and this explains the fact that in most nations, when forming numerals, a special position is occupied by the symbol denoting 10. The decimal number system is derived from here. Although there are exceptions. For example, 80 in translation from the French language - "four twenty", and 90 - "four twenty plus ten." The use of it goes back to counting on the fingers and toes. The numerals of Abkhazian, Ossetian and Danish are similarly arranged.

In the Georgian language, the score is even clearer in the twenties. The Aztecs and Sumerians were initially considered to be five. There are also more exotic options that mark the history of the development of the number. For example, in the scientific calculations the Babylonians used a sexagesimal system. In so-called "unary" systems, the number is formed by repeating the sign symbolizing the unit. Ancient people used this method approximately 10-11 thousand years BC. E.

There are also non-position systems in which the quantitative values of the symbols used for writing do not depend on their place in the number code. The addition of numbers is used.

Ancient Egyptian numbers

The knowledge of the mathematics of Ancient Egypt is based today on two papyri, which date back to around 1700 BC. E. The mathematical information presented in them goes back to the more ancient period, around 3500 BC. E. The Egyptians used this science to calculate the weight of various bodies, the volumes of granaries and the area of crops, the size of taxes, and the number of stones necessary for erecting structures. However, the main area of application of mathematics was astronomy, related to calendar calculations. The calendar was necessary to determine the dates of various religious holidays, as well as the predictions of the floods of the Nile.

The writing in Ancient Egypt was based on hieroglyphics. At that time the number system was inferior to the Babylonian. The Egyptians used the non-position decimal system, in which the number of vertical lines denoted numbers from 1 to 9. Individual symbols were entered for degrees of ten. The history of the development of the number in Ancient Egypt continued as follows. With the emergence of papyrus, a hieratic letter (ie cursive writing) was introduced. A special symbol was used in it to denote numbers from 1 to 9, as well as multiples of 10, 100, etc. The development of rational numbers at that time was slow. They were recorded as a sum of fractions with a numerator equal to one.

Numbers in Ancient Greece

Using the various letters of the alphabet, the Greek numeral system was based. The history of natural numbers in this country is marked by the fact that it was used from the 6th-3rd centuries BC. E. The attic system used the vertical line to designate the unit, and 5, 10, 100, etc. were written with the help of the initial letters of their names in the Greek language. In the ionic system, later, the working letters of the alphabet, and also 3 archaic ones, were used to denote the numbers. As the first 9 numbers (from 1 to 9) were designated as multiples of 1000 to 9000, but before the letter was placed a vertical line. "M" denoted tens of thousands (from the Greek word "mirio"). After it followed a number, which should be multiplied by 10,000.

In Greece in the 3rd century BC. E. There was a numerical system in which the own sign of the alphabet corresponded to each digit. The Greeks, beginning with the 6th century, began to use the first ten characters of their alphabet as figures. It was in this country that not only the history of natural numbers developed actively, but also mathematics was born in its modern sense. In other states of the time it was used either for everyday needs or for various magical rituals, through which the will of the gods was clarified (numerology, astrology, etc.).

Roman numbering

In ancient Rome, the numbering was used, which under the Roman name has survived to this day. We use it to denote anniversary dates, centuries, names of conferences and congresses, numbering stanzas of poems or chapters of the book. By repeating the digits 1, 5, 10, 50, 100, 500, 1000, denoted by them, respectively, as I, V, X, L, C, D, M all integers are written. If the big number is in front of the smaller one, they are added together, if the smaller one is before the larger one, then the last digit is subtracted from it. One and the same figure can not be put more than three times. For a long time, the countries of Western Europe were used as the main Roman numbering.

Position Systems

These are systems in which the quantitative values of the symbols depend on their place in the number code. Their main advantages are the simplicity of performing various arithmetic operations, as well as a small number of symbols necessary for writing numbers.

There are many such systems. For example, binary, octal, fivefold, decimal, twentieth, etc. Each has its own history.

The system that existed in the Incas

Kipu is an ancient counting and mnemonic system that existed among the Incas, as well as their predecessors in the Andes. It is quite peculiar. These are complex knots and rope plexuses, made of wool of lamas and alpac, or from cotton. Maybe in a bale from a few hanging threads to two thousand. It was used by messengers to convey messages on imperial roads, as well as in various aspects of society (like a topographic system, a calendar, for fixing laws and taxes, etc.). Readers and writers of the pile were interpreted, specially trained. They felt the nodules with their fingers, picking up a pile. Most of the information in it is the numbers represented in the decimal system.

The Babylonian Numbers

On the clay tablets cuneiform inscriptions were written by the Babylonians. They have come down to our days in considerable numbers (more than 500 thousand, about 400 of which are related to mathematics). It should be noted that the roots of the culture of the Babylonians were inherited to a large extent from the Sumerians - counting methodology, cuneiform writing, etc.

The Babylonian account system was much more perfect than the Egyptian one. The Babylonians and Sumerians used the 60-positional positional, which today is immortalized in dividing the circle by 360 degrees, as well as hours and minutes by 60 minutes and seconds, respectively.

Account in Ancient China

The development of the concept of number was also carried out in ancient China. In this country, the figures were indicated with the help of special hieroglyphs, which appeared about 2 thousand years BC. E. However, the final mark of them was established only to the 3rd century BC. E. And today these hieroglyphs are used. First, the method of recording was multiplicative. The number 1946, for example, can be represented using Roman numerals instead of hieroglyphs, like 1M9C4X6. But the calculations were carried out in practice on a counting board, where there was another record of numbers - positional, as in India, and not decimal, as in the Babylonians. An empty place was designated zero. Only about the 12th century BC. E. A special hieroglyph appeared for him.

Numerical history in India

The achievements of mathematics in India are diverse and wide. This country has made a great contribution to the development of the concept of number. It was here that a decimal position system was invented, familiar to us. The Indians offered symbols for writing 10 digits, with some changes being used nowadays everywhere. It was in this country that the foundations of decimal arithmetic were also laid.

Modern figures originated from Indian badges, the inscription of which was used as far back as the 1st century AD. E. Initially, the Indian numbering was exquisite. Means for writing numbers to ten at the fiftieth degree were applied in Sanskrit. First, for the figures, the so-called "Syro-Phoenician" system was used, and from the 6th century BC. E. - "Brahmi", with separate signs for them. These icons, somewhat modified, became modern figures, called today's Arabian.

Unknown Indian mathematician around 500 AD. E. Invented a new recording system - a decimal positional. Performing various arithmetic operations in it was immeasurably simpler than in others. Indians subsequently used counting boards that were adapted to positional recordings. They developed algorithms for arithmetic operations, including the obtaining of cubic and square roots. The Indian mathematician Brahmagupta, who lived in the 7th century, introduced negative numbers. The Indians advanced in algebra. Their symbolism is richer than that of Diophantus, although somewhat clogged with words.

Historical development of numbers in Russia

Numbering is the main prerequisite of mathematical knowledge. It had a different appearance among different peoples of antiquity. The emergence and development of the number at an early stage coincided in different parts of the world. At first, all nations designated them with notches on sticks, called tags. This way of recording taxes or debt obligations was used by a low literate population throughout the world. They made rifles on a stick, which corresponded to the amount of tax or debt. Then it was split in half, leaving one half of the payer or the debtor. The other was kept in the treasury or at the lender. Both halves were checked by folding during the payment.

The figures appeared with the appearance of writing. They resembled first the notches on sticks. Then there were special icons for some of them, such as 5 and 10. All the numberings at the time were not positional, but reminiscent of Roman. In ancient Russia, while in the states of Western Europe Roman numerals were used, they used an alphabetic one, similar to the Greek one, as our country, like other Slavic languages, was known to be in cultural communication with Byzantium.

Numbers from 1 to 9, and then tens and hundreds in the Old Russian numbering were represented by the letters of the Slavic alphabet (Cyrillic alphabet, introduced in the ninth century).

Some exceptions were from this rule. So, 2 was designated not "beeches", the second in the account in the alphabet, and "drive" (third), as the letter "З" in Old Russian was transmitted by the sound "в". The "phytus" at the end of the alphabet was 9, the "worm" was 90. Individual letters were not used. To indicate that this sign is a digit, not a letter, above it a sign was written, called "titlo", "~". "Darkness" was called tens of thousands. They marked them, circling the signs of units. Hundreds of thousands were called "legions." They were depicted by circling the signs of units from the points. Millions are leoders. These signs were depicted as circled in circles of commas or rays.

Further development of the natural number occurred at the beginning of the seventeenth century, when Indian figures became known in Russia. Up to the eighteenth century Slavic numbering was used in Russia. After that, it was replaced by a modern one.

History of complex numbers

These numbers were introduced for the first time in connection with the fact that the formula for calculating the roots of the cubic equation was singled out. Tartalay, an Italian mathematician, received in the first half of the sixteenth century an expression for calculating the root of the equation through some parameters, for which it was necessary to compile a system. However, it was found that such a system had a solution not for all cubic equations in real numbers. This phenomenon was explained by Rafael Bombelli in 1572, which was in fact the introduction of complex numbers. However, the results obtained for a long time were considered doubtful by many scientists, and only in the nineteenth century the history of complex numbers was marked by an important event - their existence was recognized after the appearance of the works of KF Gauss.