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 count. The starting point for many mathematical theories is the concept of number. It is also used in mechanics, physics, chemistry, astronomy, and many other sciences. In addition, in everyday life we ​​constantly use numbers.
The appearance of numbers
Followers of the teachings of Pythagoras believed that numbers contain the mystical essence of things. These mathematical abstractions guide the world, establishing order in it. The Pythagoreans assumed that all the laws existing in the world can be expressed using numbers. It was from Pythagoras that the theory of the development of numbers 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 counts began with the creation of a practical count of objects, as well as measurements of volumes, surfaces and lines.
Gradually the concept of natural numbers was formed. This process was complicated by the fact that primitive man did not know how to separate the abstract from a concrete representation. The score as a result of this remained for a long time only material. Notes, pebbles, fingers, etc. were used. Nodules, nicks, etc. were used to memorize its results. After the invention of writing, the history of the development of the number was noted by the fact that letters and special signs were used to use abbreviated images in large numbers . The numbering principle, similar to that used in the language, was usually reproduced with such coding.
Later, the idea came to count in dozens, and not just units. In 100 different Indo-European languages, the names of numbers from two to ten are similar, as are the names of dozens. Consequently, the concept of abstract number appeared long ago, even before these languages ​​were divided.
Counting on the fingers was initially widespread, and this explains the fact that the majority symbolizing the formation of numerals is occupied by the symbol denoting 10. The decimal number system comes from here. Although there are exceptions. For example, 80 in translation from French is “four twenty”, and 90 is “four twenty plus ten”. The use of this goes back to counting on the toes and hands. The numerals of the Abkhazian, Ossetian and Danish languages ​​are similarly arranged.
In Georgian, the score of twenty is even clearer. The Aztecs and Sumerians were initially considered fives. There are also more exotic options that mark the history of the development of numbers. For example, the Babylonians used a six-decimal system in scientific calculations. In the so-called "unary" systems, a number is formed by repeating the sign symbolizing the unit. Ancient people used this method around 10-11 thousand years BC. e.
There are also non-positional systems in which the quantitative values ​​of the characters used for writing are independent of their place in the number code. The addition of numbers is used.
Ancient egyptian numbers
The knowledge of mathematics in Ancient Egypt is based today on two papyrus dating back to around 1700 BC. e. The mathematical information presented in them dates back to a more ancient period, about 3500 BC. e. The Egyptians used this science in order to calculate the weight of various bodies, the volume of granaries and the area of ​​crops, the size of taxes, as well as the number of stones necessary for the construction of structures. However, the main field of application of mathematics was astronomy, calendar related calculations. A calendar was needed to determine the dates of various religious holidays, as well as predict the floods of the Nile.
Writing in Ancient Egypt was based on hieroglyphs. At that time, the number system was inferior to the Babylonian. The Egyptians used a non-positional decimal system, in which the number of vertical bars denotes numbers from 1 to 9. Individual characters were entered for powers of ten. The history of the development of numbers in ancient Egypt continued as follows. With the advent of the papyrus, an hieratic letter was introduced (i.e. cursive). 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 the sum of fractions with an equal unit numerator.
Numbers in Ancient Greece
The Greek number system was based on the use of various letters of the alphabet. The history of natural numbers in this country is marked by the fact that it was used from 6-3 centuries BC. e. The Attic system used a vertical line to designate a unit, and 5, 10, 100, etc., were written using the initial letters of their names in Greek. In the ionic system, later, 24 active letters of the alphabet, as well as 3 archaic ones, were used to designate the numbers 24. As the first 9 numbers (from 1 to 9) multiples of 1000 to 9000 were designated, however, a vertical line was placed before the letter . "M" denoted tens of thousands (from the Greek word "myrioi"). After it followed a number by which 10,000 should be multiplied.
In Greece, in the 3rd century BC. e. a numerical system arose in which the proper sign of the alphabet corresponded to each digit. The Greeks, starting from the 6th century, began to use the first ten characters of their alphabet as numbers. It was in this country that not only the history of natural numbers was actively developing, but also mathematics was born in its modern sense. In other countries of that time, it was used either for everyday needs, or for various magical rituals, with the help of which they clarified the will of the gods (numerology, astrology, etc.).
Roman numbering
In ancient Rome, numbering was used, which, under the name of Roman, has been preserved to this day. We use it to designate anniversaries, centuries, naming conferences and congresses, numbering stanzas of a poem or chapters of a book. By repeating the numbers 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 larger digit is in front of the smaller one, they are summed, but if the larger one is in front of the smaller one, the last one is subtracted from it. The same number must not be set more than three times. For a long time, the countries of Western Europe used as the main Roman numbering.
Position systems
These are such systems in which the quantitative values ​​of characters depend on their place in the code of the number. Their main advantages are the simplicity of performing various arithmetic operations, as well as the small number of characters needed to write numbers.
Quite a lot of such systems exist. For example, binary, octal, quaternary, decimal, decimal, etc. Each has its own history.
Inca system
Kipu is an ancient counting and mnemonic system that existed among the Incas, as well as their predecessors in the Andes. She is quite peculiar. These are complex knots and rope plexuses made of wool of llamas and alpacas, or of cotton. It can be in a pile from several hanging threads to two thousand. It was used by messengers to send messages on imperial roads, as well as in various aspects of society (as a topographic system, calendar, for fixing laws and taxes, etc.). Read and write a pile of interpreters, specially trained. They felt the nodules with their fingers, picking up a pile. Most of the information in it is numbers represented in the decimal system.
Babylonian numbers
The Babylonians wrote cuneiform signs on clay tablets. They have survived 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 Babylonian culture were inherited to a large extent from the Sumerians - the counting technique, cuneiform writing, etc.
The Babylonian system of counting was much more perfect than the Egyptian one. The Babylonians and Sumerians used a 60-hex positional, which is now immortalized by 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 carried out in ancient China. In this country, numbers were indicated using special characters that appeared about 2 thousand years BC. e. However, their final mark was established only by the 3rd century BC. e. And today these hieroglyphs are applied. At first, the recording method was multiplicative. The number 1946, for example, can be represented using Roman numerals instead of hieroglyphs, like 1M9S4X6. But the calculations in practice were made on the counting board, where there was a different notation of numbers - positional, as in India, and not decimal, like the Babylonians. Empty space was designated zero. Only about the 12th century AD e. a special character appeared for him.
Numeral 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 the decimal positional system, familiar to us, was invented. The Indians proposed symbols for writing 10 digits, with some changes that are used nowadays everywhere. It was in this country that the foundations of decimal arithmetic were also laid.
Modern figures came from Indian icons, the inscription of which was used as early as the 1st century AD. e. Initially, Indian numbering was exquisite. Tools for writing numbers up to ten to the fiftieth degree were used in Sanskrit. At first, the so-called "Syro-Phoenician" system was used for numbers, and from the 6th century BC. e. “Brahmi,” with separate signs for them. These icons, somewhat modified, have become modern numbers, called today Arabic.
Unknown Indian mathematician around 500 AD e. invented a new recording system - decimal positional. Performing various arithmetic operations in it was immeasurably easier than in others. The Indians later used counting boards that were adapted to positional recording. They developed algorithms for arithmetic operations, including obtaining cubic and square roots. The Indian mathematician Brahmagupta, who lived in the 7th century, introduced negative numbers. Far advanced Indians in algebra. Their symbolism is richer than that of Diophantus, although somewhat littered with words.
The historical development of numbers in Russia
Numbering is the main prerequisite for mathematical knowledge. She had a different look from various peoples of antiquity. The emergence and development of numbers at an early stage coincided in different parts of the world. At first, all nations marked them with notches on sticks called tags. This method of recording taxes or debt was used by the illiterate population of the whole world. They made rifles on a stick that corresponded to the amount of tax or debt. Then it was split in half, leaving one half with the payer or debtor. Another was kept at the treasury or from the lender. Both halves during payment were checked by folding.
The numbers appeared with the advent of writing. At first they resembled nicks on sticks. Then special icons appeared for some of them, such as 5 and 10. At that time, all the numbers were not positional, but reminiscent of Roman ones. In Ancient Russia, while in the states of Western Europe they used Roman numbering, they used an alphabetical one similar to Greek, since our country, like other Slavic ones, 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 depicted by the letters of the Slavic alphabet (Cyrillic alphabet introduced in the ninth century).
Some exceptions were from this rule. So, 2 was not designated “beeches”, the second in the alphabet, but “lead” (third), since the letter Z was transmitted in the old Russian way with the sound “in”. The “phyta” at the end of the alphabet stood for 9, the “worm” for 90. Individual letters were not used. To indicate that this sign is a number, not a letter, a sign called “titlo”, “~” was written above it. "Darkness" was called tens of thousands. Designated them, circling the signs of units. Hundreds of thousands were called "legions." They were depicted by circling dots around the unit signs. Millions are leodra. These signs were depicted as circled in commas or rays.
Further development of the natural number occurred at the beginning of the seventeenth century, when Indian numbers became known in Russia. Until 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 due to the fact that the formula for calculating the roots of the cubic equation was singled out. Tartaglia, an Italian mathematician, received in the first half of the sixteenth century a calculation expression for the root of the equation in terms of some parameters, for finding which it was necessary to compose a system. However, it was found that such a system did not have a solution for all cubic equations in real numbers. This phenomenon was explained by Rafael Bombelli in 1572, which was essentially the introduction of complex numbers. However, the results obtained were long 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 advent of the works of K. F. Gauss.