History of numbers and the number system, position systems (briefly)

The history of numbers and the number system are closely interrelated, because the number system is a way of writing such an abstract concept as a number. This topic does not apply specifically to the field of mathematics, because all this is an important part of the culture of the people as a whole. Therefore, when the history of numbers and number systems is being analyzed, many other aspects of the history of the civilizations that created them are briefly touched upon. Systems in general are divided into positional, non-position and mixed. They consist of the entire history of numbers and number systems. Positional systems are those in which the quantity denoted by a digit in a number entry depends on its position. In non-position systems, accordingly, there is no such dependence. Mankind has created mixed systems.

The study of number systems in school

Today, the lesson "History of numbers and number systems" is held in the 9th grade within the framework of the course in computer science. The main practical value of it is to teach you to translate numbers from one number system to another (first of all, from decimal to binary). However, the history of numbers and number systems is an organic part of history as a whole and could well complement this subject of the school curriculum as well. It could also improve the interdisciplinary approach advocated today. In the framework of the general course of history, in principle, not only the history of economic development, socio-political movements, boards and wars could be studied, but also to a small extent the history of numbers and number systems. 9 class in the course of informatics in this case it would be possible in the part of the transfer of numbers from one system to another to provide a significantly greater number of examples from the previously traversed material. And these examples are not without fascination, which will be shown below.

The origin of the number systems

It is difficult to say when, and most importantly, how a person learned to count (just as it is impossible to find out for certain when, and most importantly, how language arose). It is only known that all ancient civilizations already had their systems of account, which means that the history of numbers and the number system originated in pre-civilization time. Stones and bones are not able to tell us what happened in the human mind, and written sources were not yet created. Perhaps, the account was needed for a person in the division of production or much later, already during the Neolithic revolution, that is, in the transition to agriculture, for the division of field sections. Any theories on this subject will be equally unfounded. But some assumptions can still be made by studying the history of different languages.

Traces of the oldest number system

The most logical initial system of the account is the opposition of the concepts "one" - "many". It is logical for us because in modern Russian there is only a single and plural number. But in many ancient languages there was also a dual number for the designation of two objects. It existed in the first Indo-European languages, including Old Russian. Thus, the history of numbers and the number system began with the separation of the concepts "one", "two", "many". However, even in the most ancient civilizations known to us, more detailed numbering systems were developed.

Mesopotamian number writing

We are used to the fact that the number system is decimal. This is understandable: on the hands of 10 fingers. But nevertheless the history of the appearance of numbers and number systems passed through more complicated phases. The Mesopotamian number system is sixty. Because still in the hour 60 minutes, and in a minute - 60 seconds. Therefore, the year is divided by the number of months, a multiple of 60, and the day is divided by the same number of hours. Initially, it was a sundial, that is, each of them was 1/12 light days (in the territory of present-day Iraq its duration did not vary much). Only a long time later the hours began to be determined not by the sun and added 12 hours of night.

It is interesting that the signs of this sixties system were written, as if it were a decimal one - there were only two signs (for the designation of a unit and a dozen, not six and not sixty, namely a dozen), the numbers were obtained by combining these signs. It's scary even to imagine how difficult it was to write down any large number in this way.

Ancient Egyptian number system

And the history of numbers in the decimal number system, and the use of numerous icons to designate numbers began with the ancient Egyptians. They combined hieroglyphs that denoted one, one hundred, one thousand, ten thousand, one hundred thousand, one million and ten million, thus denoting the required number. Such a system was much more convenient than the Mesopotamian system, which used only two signs. But it had an obvious limitation: it was difficult to write down a number much larger than ten million. True, the ancient Egyptian civilization, like most civilizations of the Ancient World, did not encounter such numbers.

Hellenic letters in mathematical records

The history of European philosophy, science, political thought and many other things begins in many respects in Ancient Hellas ("Hellas" is a self-name, it is preferable than the "Greece" invented by the Romans). Developed in this civilization were also mathematical knowledge. The numbers of the Hellenes were written down in letters. Individual letters denoted every number from 1 to 9, every ten from 10 to 90 and every hundred from 100 to 900. Only one thousand were denoted by the same letter as the one, but with another sign next to the letter. The system allowed even large numbers to be designated with relatively short inscriptions.

Slavic number system as the heiress of Hellenic

The history of numbers and number systems would not be complete without a few words about our ancestors. Cyrillic alphabet is known to be based on the Hellenic alphabet, therefore the Slavic system of recording numbers was also based on the Hellenic alphabet. Here, too, in individual letters, each number from 1 to 9, every ten from 10 to 90, and every hundred from 100 to 900. Only not Hellenic letters were used, but Cyrillic or Glagolitic. There was also an interesting feature: in spite of the fact that Hellenic texts at the time, and Slavic texts from the very beginning of their history were recorded from left to right, the Slavic figures were written from right to left, that is, the letters denoting tens were placed to the right of the letters denoting units, letters , Denoting hundreds to the right of letters denoting tens, etc.

Attic simplification

Hellenic scientists reached enormous heights. The Roman conquest did not interrupt their research. For example, judging by circumstantial evidence, Aristarchus of Samos for 18 centuries before Copernicus developed the Heliocentric system of the world. In all these complex calculations Hellenic scientists were helped by their number writing system.

But for ordinary people, for example, traders, the system was often too complicated: to use it, it was required to memorize the numerical values of 27 letters (instead of the numerical values of 10 symbols taught by modern schoolchildren). Therefore, a simplified system, called Attica (Attica - the region of Hellas, at one time was leading in the region as a whole and especially in the maritime trade of the region, since the capital of Attica was the famous Athens). In this system, only numbers one, five, ten, one hundred, one thousand and ten thousand became known as separate letters. It turns out only six signs - they are much easier to remember, and too complex calculations, merchants still did not produce.

Roman numerals

And the number system, and the history of the numbers of the ancient Romans, and in principle the history of their science is a continuation of Hellenic history. The Attic system was used as a basis, just Hellenic letters were replaced with Latin letters and added a separate designation of fifty and five hundred. In this case, scientists continued to make complex calculations in their treatises by the Hellenic system of writing in 27 letters (and they themselves usually wrote the treatises in Greek).

The Roman system of recording numbers can not be called a particularly perfect one. In particular, it is much more primitive than the Old Russian one. But historically it turned out that it is still preserved on a par with the Arab (so-called) figures. And forget this alternative system, stop using it is not worth it. In particular, today, numerically numerals are often denoted by Arabic numerals, and Roman ones are ordinal numerals .

Great ancient Indian invention

The figures that we use today, appeared originally in India. It is not exactly known when the history of numbers and the number system made this significant turn, but, most likely, not later than the 5th century from the Nativity of Christ. It is often emphasized that it was the Indians who developed the concept of zero. This concept was known to mathematicians and other civilizations, but really only the system of Indians allowed to fully include it in mathematical records, and hence in calculations.

Distribution of the Indian system of radixing on the Earth

Presumably in the IX century, the Indian figures borrowed the Arabs. While the Europeans disregarded the ancient heritage, and in some regions even at one time even purposely destroyed it as pagan, the Arabs carefully preserved the achievements of the ancient Greeks and Romans. From the very beginning of their conquests, the translations of ancient authors into Arabic became the market commodity. Basically, through the treatises of Arab scholars, medieval Europeans regained the legacy of ancient thinkers. Along with these treatises came Indian figures, which in Europe began to be called Arabic. They were not immediately accepted, because for most people they were less intelligible than Roman ones. But gradually the convenience of mathematical calculations with the help of these signs was won by ignorance. Leadership of the European industrialized countries has led to the fact that the so-called Arabian figures have spread around the world and are now used almost everywhere.

The binary system of modern computers

With the advent of computers, many fields of knowledge have gradually made a significant turn. The history of numbers and number systems has not become an exception. The photo of the first computer does not look like a modern device, on the monitor of which you are reading this article, but the work of both of them is based on the binary system , the code consisting only of zeros and ones. For ordinary consciousness it is still surprising that using a combination of only two symbols (in fact a signal or its absence) it is possible to perform the most complicated calculations and automatically (if there is an appropriate program) to translate numbers in the decimal system of the calculus into numbers in binary, hexadecimal, hexadecimal And any other system. And with the help of such a binary code on the monitor, this article is depicted, which reflects the history of numbers and the number system of different civilizations in history.