Ancient number systems. History of numbers and number systems – Knowledge Hypermarket

This is a way of representing numbers and the corresponding rules for operating with numbers.

The various number systems that existed in the past and that are used today can be divided into non-positional and positional.

Non-positional systems of antiquity

A study by archaeologists of “notes” from Paleolithic times on bone, stone, and wood showed that people sought to group marks of 3, 5, 7, and 10 pieces. This grouping made counting easier. People learned to count not only in units, but also in threes, fives, etc. Since the first computing tool people had fingers, so counting was most often done in groups of 5 or 10 items.

Later, ten tens (hundred), ten hundreds (thousand), etc. were given their name. For ease of recording, such key numbers began to be designated with special icons - numbers. If, when counting objects, there were 2 hundreds, 5 tens, and 4 more objects, then when recording this value, the hundreds sign was repeated twice, the tens sign five times, and the units sign four times.

In such number systems, the position of the sign in the number record does not depend on the value it denotes; therefore they are called non-positional number systems.

Non-positional systems were used by the ancient Egyptians, Greeks, Romans and some other peoples of antiquity.

Mayan numbers

You have been familiar with the positional decimal number system since early childhood, but perhaps you did not know that it was called that.

What the positional property of a number system means is easy to understand using the example of any multi-digit decimal number. For example, in the number 333, the first three means three hundreds, the second - three tens, the third - three ones. The same digit, depending on its position in the number notation, denotes different meanings.

333 = 3 100 + 3 10 + 3.

Another example:

32,478 = 3 10 000 + 2 1000 + 4 100 + 7 10 + 8 = 3 10 4 + 2 10 3 + 4 10 2 + 7 10 1 + 8 10 0.

This shows that any decimal number can be represented as the sum of the products of its constituent digits by the corresponding powers of ten. The same applies to decimals.

26.387 = 2 10 1 + 6 10 0 + 3 10 -1 + 8 10 -2 + 7 10 -3.

Obviously, the number “ten” is not the only possible basis for the positional system. The famous Russian mathematician N. N. Luzin put it this way: “The advantages of the decimal system are not mathematical, but zoological. If we had eight fingers instead of ten on our hands, then humanity would use the octal system.”

Any natural number greater than 1 can be taken as the base of the positional number system. The Babylonian system mentioned above had a base of 60. Traces of this system have survived to this day in the order of counting units of time (1 hour = 60 minutes, 1 minute = 60 seconds).

To write numbers in the base n positional system, you need to have an alphabet of n digits. Usually for this purpose when n < 10 uses the first n Arabic numerals, and when n > 10, letters are added to the ten Arabic numerals.

Here are examples of alphabets of several systems:

The base of the system to which a number belongs is usually indicated by a subscript to that number:

101101 2, 3671 8, ЗВ8F 16.

How is a series of natural numbers constructed in different positional number systems? This happens according to the same principle as in the decimal system. First there are single-digit numbers, then two-digit numbers, then three-digit numbers, etc. d. The largest single-digit number in the decimal system is 9. Then come two-digit numbers - 10, 11,12, ... The largest two-digit number is 99, then 100, 101, 102, etc. up to 999, then 1000, etc. d.

For example, consider the fivefold system. In it, the series of natural numbers looks like this:

1, 2, 3, 4, 10, 11, 12, 13, 14, 20, 21, 22, 23, 24, 30, 31, 32, 33, 34, 40, 41, 42, 43, 44, 100, 101, …, 444, 1000, ... .

It can be seen that here the number of digits “increases” faster than in the decimal system. The number of digits grows fastest in the binary number system. The following table compares the beginnings of the natural series of decimal and binary numbers:

Briefly about the main thing

A number system is a specific way of writing numbers and the corresponding rules for operating numbers.

Number systems can be positional or non-positional. An example of a non-positional system is the Roman number system.

In a positional number system, the quantitative value of each digit depends on the position of the digit in the number.

The alphabet of a number system is the set of numbers used in it. The base of the number system is equal to the power of the alphabet (the number of digits).

The smallest possible base of a positional number system is 2. Such a system is called binary.

The Arabic number system is decimal and positional.

Questions and tasks

1. What is a number system?
2. What is the main difference between positional and non-positional number systems?
3. What is the base of the number system?
4. Why is the Arabic number system called decimal positional?
5. What is the smallest base for a positional system?
6. What are the following numbers written in Roman numerals equal to in the decimal system:
XI; IX; LX; CLX; MDXLVIII?
7. Write in Roman numerals the numbers equal to the following decimals:
13; 99; 666; 444; 1692.
8. Write down a sequence of twenty numbers in the natural series, starting from one, for positional systems with bases 2, 3, 5, 8. Present the results in the form of a table:

9. Construct multiplication tables for single-digit numbers in binary and ternary number systems.

I. Semakin, L. Zalogova, S. Rusakov, L. Shestakova, Computer Science, 9th grade
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The history of numbers and the number system are closely interrelated, because the number system is a way of recording such an abstract concept as number. This topic does not relate 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 examined, many other aspects of the history of the civilizations that created them are briefly touched upon. Systems are generally divided into positional, non-positional and mixed. The entire history of numbers and number systems consists of their alternation. Positional systems are those in which the quantity denoted by a digit in the notation of a number depends on its position. In non-positional systems, accordingly, there is no such dependence. Humanity has also created mixed systems.

Studying number systems at school

Today the lesson “History of numbers and number systems” is taught in 9th grade as part of a computer science course. Its main practical significance is to teach how to convert numbers from one number system to another (primarily 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. It could also improve the interdisciplinary approach advocated today. As part of a general history course, in principle, not only the history of economic development, socio-political movements, governments and wars could be studied, but also, to a small extent, the history of numbers and number systems. In a 9th grade computer science course, in this case, in terms of converting numbers from one system to another, it would be possible to provide a significantly larger number of examples from previously covered material. And these examples are not without fascination, as will be shown below.

The emergence of 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 own counting systems, which means that the history of numbers and the number system originated in pre-civilization times. Stones and bones are not able to tell us what happened in the human mind, and written sources had not yet been created. Perhaps a person needed an account when dividing up spoils or much later, already during the Neolithic revolution, that is, during the transition to agriculture, for dividing plots of field. Any theories on this matter will be equally groundless. But some assumptions can still be made by studying the history of various languages.

Traces of the most ancient number system

The most logical initial counting system is the contrast between the concepts “one” and “many”. It is logical for us because in modern Russian there are only singular and plural numbers. But in many it was also used to designate two objects. It also 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 of “one,” “two,” and “many.” However, already in the most ancient civilizations known to us, more detailed number systems were developed.

Mesopotamian notation of numbers

We are used to the number system being decimal. This is understandable: there are 10 fingers on the hands. But nevertheless, the history of the emergence of numbers and number systems went through more complex phases. The Mesopotamian number system is sexagesimal. That’s why there are still 60 minutes in an hour and 60 seconds in a minute. Therefore, the year is divided by the number of months, which is a multiple of 60, and the day is divided by the same number of hours. Initially, these were sundials, that is, each of them was 1/12 of a day of light (in the territory of modern Iraq its duration did not vary much). Only much later did they begin to determine the duration of the hour not by the sun and also added 12 night hours.

It is interesting that the signs of this sexagesimal system were written down as if it were decimal - there were only two signs (to denote one and ten, not six or sixty, but ten), the numbers were obtained by combining these signs. It’s scary to even imagine how difficult it was to write down any large number in this way.

Ancient Egyptian number system

Both the history of numbers in the decimal number system and the use of numerous symbols to indicate numbers began with the ancient Egyptians. They combined the hieroglyphs that stood for one, one hundred, a thousand, ten thousand, one hundred thousand, a million and ten million, thus denoting the desired number. This system was much more convenient than the Mesopotamian one, which used only two signs. But at the same time, it had an obvious limitation: it was difficult to write down a number significantly 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 notation

The history of European philosophy, science, political thought and much more largely begins in Ancient Hellas (“Hellas” is a self-name, it is preferable to the “Greece” coined by the Romans). Mathematical knowledge was also developed in this civilization. The Hellenes wrote numbers in letters. A separate letter represented each number from 1 to 9, each ten from 10 to 90, and each hundred from 100 to 900. Only a thousand was represented by the same letter as one, but with a different sign next to the letter. The system allowed even large numbers to be denoted with relatively short inscriptions.

Slavic number system as a successor to the Hellenic

The history of numbers and number systems would not be complete without a few words about our ancestors. The Cyrillic alphabet, as you know, is based on the Hellenic alphabet, therefore the Slavic system of writing numbers was also based on the Hellenic. Here, too, each number from 1 to 9, each ten from 10 to 90, and each hundred from 100 to 900 were designated with separate letters. Only not Hellenic letters were used, but Cyrillic or Glagolitic. There was also an interesting feature: despite the fact that both Hellenic texts at that time, and Slavic ones from the very beginning of their history, were written from left to right, Slavic numbers were written as if 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 indirect evidence, 18 centuries before Copernicus developed the Heliocentric In all these complex calculations, Hellenic scientists were helped by their system of recording numbers.

But for ordinary people, such as merchants, the system often turned out to be too complex: in order to use it, it was necessary to memorize the numerical values ​​of 27 letters (instead of the numerical values ​​of 10 symbols that modern schoolchildren learn). Therefore, a simplified system appeared, called the Attic one (Attica is the region of Hellas, which at one time was the leader 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 the numbers one, five, ten, one hundred, one thousand and ten thousand began to be denoted by separate letters. It turns out that there are only six characters - they are much easier to remember, and the merchants still did not make too complex calculations.

Roman numerals

Both the number system and the history of numbers of the ancient Romans, and, in principle, the history of their science is a continuation of Hellenic history. The Attic system was taken as the basis, the Hellenic letters were simply replaced with Latin ones and a separate designation for fifty and five hundred was added. At the same time, scientists continued to make complex calculations in their treatises using the Hellenic notation system of 27 letters (and they usually wrote the treatises themselves in Hellenic).

The Roman system of recording numbers cannot be called particularly perfect. In particular, it is much more primitive than the Old Russian one. But historically it has developed so that it is still preserved on a par with Arabic (so-called) numerals. And you shouldn’t forget this alternative system and stop using it. In particular, today Arabic numerals are often used to denote ordinal numbers using Roman numerals.

Great ancient Indian invention

The numbers we use today originally appeared in India. It is not known exactly when the history of numbers and the number system made this significant turn, but most likely no later than the 5th century AD. It is often emphasized that it was the Indians who developed the concept of zero. This concept was known to mathematicians of other civilizations, but it was really only the Indian system that made it possible to fully include it in mathematical records, and therefore in calculations.

Spread of Indian Numeral System across the Earth

Presumably in the 9th century, Indian numerals were borrowed by the Arabs. While Europeans disdained the ancient heritage, and in some regions at one time even deliberately destroyed it as pagan, the Arabs carefully preserved the achievements of the ancient Greeks and Romans. From the very beginning of their conquests, translations of ancient authors into Arabic became a hot commodity. Mainly through the treatises of Arab scholars, medieval Europeans regained the heritage of ancient thinkers. Along with these treatises came Indian numerals, which in Europe began to be called Arabic. They were not immediately accepted, because for most people they turned out to be less understandable than the Roman ones. But gradually the convenience of mathematical calculations using these signs defeated ignorance. The leadership of European industrialized countries has led to the fact that the so-called Arabic numerals have spread throughout the world and are used almost everywhere today.

Binary number system of modern computers

With the advent of computers, many areas of knowledge gradually took a significant turn. The history of numbers and number systems was no exception. The photo of the first computer bears little resemblance to the modern device on the monitor of which you are reading this article, but the work of both of them is based on notation, a code consisting only of zeros and ones. For ordinary consciousness, it still remains surprising that with the help of a combination of just two symbols (actually a signal or its absence), you can perform the most complex calculations and automatically (if you have the appropriate program) convert numbers in the decimal number system into numbers in binary, hexadecimal, sixty-sexadecimal and any other system. And with the help of such a binary code, this article is displayed on the monitor, which reflects the history of numbers and the number system of different civilizations in history.

After studying this topic, you will learn and repeat:

What number systems exist;
- how numbers are converted from one number system to another;
- what number systems the computer works with;
- how different numbers are represented in computer memory.

Since ancient times, people have been faced with the problem of designating (coding) numerical information.

Little children show their age on their fingers. A pilot shot down a plane, he gets an asterisk for it, Robinson Crusoe counted the days with notches.

The number denoted some real objects whose properties were the same. When we count or recount something, we seem to depersonalize the objects, i.e. we imply that their properties are the same. But the most important property of a number is the presence of an object, i.e. unit and its absence, i.e. zero.

What is a number?

This is the alphabet of numbers, a set of symbols with which we encode numbers. Numbers are the numerical alphabet.

Numbers and numbers are two different things! Let's consider two numbers 5 2 and 2 5. The numbers are the same - 5 and 2.

How are these numbers different?

In order of numbers? - Yes! But it’s better to say - the position of the digit in the number.

Let's think about what a number system is?

Is this writing numbers? Yes! But we cannot write as we please - other people must understand us. Therefore, it is also necessary to use certain rules for recording them.

The concept of a number system

Numbers are used to record information about the number of objects. Numbers are written using special sign systems called number systems. The alphabet of number systems consists of symbols called digits. For example, in the decimal number system, numbers are written using ten well-known digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

A number system is a signed system in which numbers are written according to certain rules using symbols of a certain alphabet called digits.

All number systems are divided into two large groups: positional and non-positional number systems. In positional number systems, the value of a digit depends on its position in the number, but in non-positional number systems it does not depend.

Non-positional number systems arose earlier than positional ones, so we will first consider various non-positional number systems.

Non-positional number systems

A non-positional number system is a number system in which the quantitative equivalent (“weight”) of a digit does not depend on its location in the number record.

Non-positional systems include: the Roman number system, alphabetic number systems and others.

At first, people simply distinguished between ONE object in front of them or not. If there was more than one item, they said “MANY.”

The first concepts of mathematics were “less”, “more”, “same”.

If one tribe exchanged caught fish for stone knives made by people of another tribe, there was no need to count how many fish and how many knives they brought. It was enough to place a knife next to each fish for the exchange between the tribes to take place.

The account appeared when a person needed to inform his fellow tribesmen about the number of objects he found.

And, since many peoples in ancient times did not communicate with each other, different peoples developed different number systems and representations of numbers and numbers.

Numerals in many languages ​​indicate that primitive man's counting tools were primarily fingers.

Fingers turned out to be an excellent computing machine. With their help, one could count up to 5, and if you take two hands, then up to 10. In ancient times, people walked barefoot. Therefore, they could use their fingers and toes to count. There are still tribes in Polynesia that use the 20th number system.

However, peoples are known whose units of counting were not fingers, but their joints.

The duodecimal number system was quite widespread. Its origin is connected with counting on fingers. They counted the phalanges of the other four fingers with the thumb: there are 12 in total.

Elements of the duodecimal number system were preserved in England in the system of measures (1 foot = 12 inches) and in the monetary system (1 shilling = 12 pence). Often in everyday life we ​​come across the duodecimal number system: tea and table sets for 12 people, a set of handkerchiefs - 12 pieces.

Numbers in English from one to twelve have their own name, subsequent numbers are composite:

For numbers from 13 to 19, the ending of the words is teen. For example, 15 -- fiveteen.

Finger counting has been preserved in some places to this day. For example, at the world's largest grain exchange in Chicago, offers and requests, as well as prices, are announced by brokers on their fingers without a single word.

It was difficult to memorize large numbers, so various devices were added to the “counting machine” of the arms and legs. There was a need to write down numbers.

The number of objects was depicted by drawing dashes or serifs on any hard surface: stone, clay...

Unit (“stick”) number system

The need to write numbers appeared in very ancient times, as soon as people began to count. The number of objects was depicted by drawing lines or serifs on any hard surface: stone, clay, wood (the invention of paper was still very, very far away). Each object in such a record corresponded to one line. Archaeologists have found such “records” during excavations of cultural layers dating back to the Paleolithic period (10 - 11 thousand years BC).

Scientists called this method of writing numbers the unit (“stick”) number system. In it, only one type of sign was used to record numbers - “stick”. Each number in such a number system was designated using a line made up of sticks, the number of which was equal to the designated number. The Peruvians used multi-colored cords with knots tied on them to remember numbers. An interesting way to write numbers was used by Indian civilizations around the 8th century BC. They used “knot writing” - threads tied together. The symbols on these threads were knots, often with stones or shells woven into them. The knotted recording of numbers allowed the Incas to transmit information about the number of warriors, indicate the number of deaths or births in a particular province, and so on.

Around 1100 AD e. English King Henry I invented one of the most unusual monetary systems in history, called the “measuring rod” system. This monetary system lasted 726 years and was abolished in 1826.

A polished wooden strip with notches indicating the denomination was split along its entire length so as to preserve the notches.

The inconveniences of such a system for writing numbers and the limitations of its application are obvious: the larger the number that needs to be written, the longer the string of sticks. And when writing down a large number, it’s easy to make a mistake by adding an extra number of sticks or, conversely, not writing them down.

Ancient Egyptian decimal number system (2.5 thousand BC)

Around the third millennium BC, the ancient Egyptians came up with their own numerical system, in which the key numbers were 1, 10, 100, etc. special icons were used - hieroglyphs.

All other numbers were composed from these key numbers using the operation of addition. The number system of Ancient Egypt is decimal, but non-positional and additive.

The digits of the number were recorded starting with the largest values ​​and ending with the smaller ones. If there were no tens, units, or some other digit, then we moved on to the next digit.

Try adding these two numbers, knowing that you cannot use more than 9 identical hieroglyphs, and you will immediately understand that a special person is needed to work with this system. An ordinary person cannot do this.

Roman decimal number system (2 thousand years BC to the present day)

The most common of the non-positional number systems is the Roman system.

The main problem with Roman numerals is that multiplication and division are difficult. Another disadvantage of the Roman system is: Writing large numbers requires introducing new symbols. Fractional numbers can only be written as a ratio of two numbers. However, they were basic until the end of the Middle Ages. But in our time they are still used.

Remember where?

The meaning of a digit does not depend on its position in the number.

For example, in the number XXX (30), the number X appears three times and in each case denotes the same value - the number 10, three numbers of 10 add up to 30.

The size of a number in the Roman numeral system is defined as the sum or difference of the digits in the number. If the smaller number is to the left of the larger one, then it is subtracted, if to the right, it is added.

Remember: 5, 50, 500 are not repeated!

Which ones can be repeated?

If there is a minor digit to the left of the major digit, it is subtracted. If the lowest digit is to the right of the highest one, then it is added - I, X, C, M can be repeated up to 3 times.

For example:

1) MMIV = 1000+1000+5-1 = 2004

2) 149 = (One hundred is C, forty is XL, and nine is IX) = CXLIX

For example, writing the decimal number 1998 in the Roman numeral system would look like this: MSMХСVIII = 1000 + (1000 - 100) + (100 - 10) + 5 + 1 + 1 + 1.

Alphabetic number systems
Slavic Cyrillic decimal alphabetic

This numbering was created together with the Slavic alphabetic system to translate the sacred biblical books for the Slavs by the Greek monks brothers Cyril and Methodius in the 9th century. This form of writing numbers became widespread due to the fact that it was completely similar to the Greek notation of numbers. Until the 17th century, this form of recording numbers was official in the territory of modern Russia, Belarus, Ukraine, Bulgaria, Hungary, Serbia and Croatia. Until now, Orthodox church books use this numbering.

Numbers were written from digits in the same way from left to right, from large to small. Numbers from 11 to 19 were written in two digits, with the unit coming before the ten:

We read literally “fourteen” - “four and ten”. As we hear, we write: not 10+4, but 4+10, - four and ten. Numbers from 21 and above were written in reverse, with the full tens sign written first.

The number notation used by the Slavs is additive, that is, it only uses addition:

= 800+60+3

In order not to confuse letters and numbers, titles were used - horizontal lines above the numbers, which we see in the figure.

To indicate numbers greater than 900, special icons were used that were added to the letter. This is how the numbers were formed:

Slavic numbering existed until the end of the 17th century, until the positional decimal number system came to Russia from Europe with the reforms of Peter I.

Ancient Indian number systems

The Kharoshti number system was in use in India between the 6th century BC and the 3rd century AD. This was a non-positional additive number system. Little is known about her, since few written documents from that era have survived. The Kharoshti system is interesting in that the number four is chosen as an intermediate step between one and ten. Numbers were written from right to left.

Along with this system, there was another Brahmi number system in India.

Brahmi numbers were written from left to right. However, both systems had quite a bit in common. In particular, the first three digits are very similar. The common thing was that up to a hundred the additive method was used, and after that the multiplicative method was used. An important difference between Brahmi numbers was that numbers from 4 to 90 were represented by only one sign. This feature of Brahmi numerals was later used to create a positional decimal system in India.

Ancient India also had a verbal number system. It was multiplicative and positional. The zero sign was pronounced as "empty", or "sky", or "hole". The unit is like “moon” or “earth”. Two is like “twins”, or “eyes”, or “nostrils”, or “lips”. Four as “oceans”, “cardinal directions”. For example, the number 2441 was pronounced like this: the eyes of the oceans are the cardinal directions of the moon.

Disadvantages of non-positional number systems:

1. There is a constant need to introduce new symbols for recording large numbers.

2. It is impossible to represent fractional and negative numbers.

3. It is difficult to perform arithmetic operations, since there are no algorithms for performing them. In particular, all nations, along with number systems, had methods of finger counting, and the Greeks had an abacus counting board - something like our abacus.

Until the end of the Middle Ages, there was no universal system for recording numbers. Only with the development of mathematics, physics, technology, trade, and the financial system did the need for a single universal number system arise, although even now many tribes, nations and nationalities use other number systems.

But we still use elements of the non-positional number system in everyday speech, in particular, we say one hundred, not ten tens, a thousand, a million, a billion, a trillion.

Positional number systems

A positional number system is a number system in which the quantitative equivalent (“weight”) of a digit depends on its location in the notation of the number.

Any positional number system is characterized by its base.

The base of the positional number system - the number of different digits used to represent numbers in a given number system.

Any natural number can be taken as a base - two, three, four, ..., forming a new positional system: binary, ternary, quaternary, etc.

Babylonian decimal/sexagesimal

In ancient Babylon around the 2nd millennium BC there was such a number system - numbers less than 60 were indicated using two signs: for one and for ten. They had a wedge-shaped appearance, since the Babylonians wrote on clay tablets with triangular sticks. These signs were repeated the required number of times, for example

It is believed that the Sumerians had a decimal system, and after they were conquered by the Semites, their system was adapted to the sexagesimal system of the Semites.

The sexagesimal notation of integers was not widely used outside the Assyro-Babylonian kingdom, but sexagesimal fractions are still used in measuring time. For example, one minute = 60 seconds, one hour = 60 minutes.

Ancient Chinese decimal

This system is one of the oldest and most progressive, since it contains the same principles as the modern “Arab” one that we use. This system arose about 4,000 thousand years ago in China.

Numbers in this system, just like ours, were written from left to right, from largest to smallest. If there were no tens, units, or some other digit, then at first they did not put anything and moved on to the next digit. (During the Ming Dynasty, a sign for an empty digit was introduced - a circle - an analogue of our zero). In order not to confuse the digits, several service hieroglyphs were used, written after the main hieroglyph, and showing what value the hieroglyph-digit takes in a given digit.

This is multiplicative notation because it uses multiplication. It is decimal, it has a zero sign, and besides this it is positional. Those. it almost corresponds to the “Arabic” number system.

The Mayan base numeral system or long counting

This system is very interesting because its development was not influenced by any of the civilizations of Europe and Asia. This system was used for the calendar and astronomical observations. Its characteristic feature was the presence of a zero (an image of a shell). The base of this system was the number 20, although traces of the fivefold system are strongly visible. The first 19 numbers were obtained by combining dots (one) and dashes (five).

The number 20 was depicted with two digits, zero and one at the top, and was called uinalu. The numbers were written down in a column, with the smallest digits at the bottom and the largest at the top, resulting in a “bookcase” with shelves. If the number zero appeared without a unit at the top, this meant that there were no units for this digit. But, if at least one unit was in this digit, then the zero sign disappeared, for example, the number 21, this will be . Also in our number system: 10 – with zero, 11 – without it. Here are some example numbers:

There is an exception to the ancient Maya's base-20 counting system: if you add only one first-order unit to the number 359, this exception immediately takes effect. Its essence boils down to the following: 360 is a starting number of the third order and its place is no longer on the second, but on the third shelf.

But then it turns out that the initial number of the third order is not twenty times greater than the initial number of the second (20x20 = 400, not 360!), but only eighteen! This means that the principle of twenty-fold has been violated! That's right. This is the exception.

The fact is that among the Mayan Indians, 20 kin days formed a month or uinal. 18 months-uinals formed a year or tuna (360 days a year) and so on:

K"in = 1 day. Vinal = 20 k"in = 20 days. Tun = 18 Vinal = 360 days = about 1 year. K"atun = 20 tun = 7200 days = about 20 years. Bak"tun = 20 k"atun = 144,000 days = about 400 years. Pictun = 20 bak"tun = 2,880,000 days = about 8,000 years. Kalabtun = 20 pictuns = 57,600,000 days = about 160,000 years. K"inchiltun = 20 kalabtun = 1152000000 days = about 3200000 years. Alavtun = 20 k"inchiltun = 23040000000 days = about 64000000 years.

This is a rather complex number system, mainly used by priests for astronomical observations; another Mayan system was additive, similar to the Egyptian one, and was used in everyday life.

History of "Arabic" numbers.

The history of our familiar “Arabic” numbers is very confusing. It is impossible to say exactly and reliably how they happened. Here is one version of this origin story. One thing is certain: it is thanks to the ancient astronomers, namely their precise calculations, that we have our numbers.

As we already know, in the Babylonian number system there is a sign to indicate missing digits. Around the 2nd century BC. Greek astronomers (for example, Claudius Ptolemy) became acquainted with the astronomical observations of the Babylonians. They adopted their positional number system, but they wrote down whole numbers not using wedges, but in their own alphabetical numbering, and fractions in the Babylonian sexagesimal number system. But to indicate the zero value of the digit, Greek astronomers began to use the symbol “0” (the first letter of the Greek word Ouden - nothing).

Between the 2nd and 6th centuries AD. Indian astronomers became acquainted with Greek astronomy. They adopted the sexagesimal system and the round Greek zero. The Indians combined the principles of Greek numbering with the decimal multiplicative system taken from China. They also began to denote numbers with one sign, as was customary in the ancient Indian Brahmi numbering. This was the final step in creating the positional decimal number system.

The brilliant work of Indian mathematicians was adopted by Arab mathematicians and Al-Khwarizmi in the 9th century wrote the book “The Indian Art of Counting”, in which he describes the decimal positional number system. Simple and convenient rules for adding and subtracting arbitrarily large numbers written in the positional system made it especially popular among European merchants.

In the 12th century. Juan of Seville translated the book “The Indian Art of Counting” into Latin, and the Indian counting system spread widely throughout Europe. And since Al-Khorezmi’s work was written in Arabic, the Indian numbering in Europe received the wrong name - “Arabic”. But the Arabs themselves call numbers Indian, and arithmetic based on the decimal system - Indian counting.

The form of "Arabic" numerals has changed greatly over time. The form in which we write them was established in the 16th century.

Even Pushkin proposed his own version of the form of Arabic numbers. He decided that all ten Arabic numerals, including zero, fit in a magic square.

Decimal positional number system

Indian scientists made one of the most important discoveries in mathematics - they invented the positional number system, which is now used by the whole world. Al-Khwarizmi described Indian arithmetic in detail in his book.

Muhammad bin Musa al-Khorezm

Around 850 AD. he wrote a book about the general rules for solving arithmetic problems using equations. It was called "Kitab al-Jabr". This book gave its name to the science of algebra.

Three hundred years later (in 1120) this book was translated into Latin, and it became the first textbook of “Indian” arithmetic for all European cities.

History of zero.

Zero can be different. First, zero is a digit that is used to indicate an empty place; secondly, zero is an unusual number, since you cannot divide by zero and when multiplied by zero, any number becomes zero; thirdly, zero is needed for subtraction and addition, otherwise, how much will it be if you subtract 5 from 5?

Zero first appeared in the ancient Babylonian number system; it was used to indicate missing digits in numbers, but numbers such as 1 and 60 were written the same way, since they did not put a zero at the end of the number. In their system, the zero served as a space in the text.

The great Greek astronomer Ptolemy can be considered the inventor of the form of zero, since in his texts in place of the space sign there is the Greek letter omicron, very reminiscent of the modern zero sign. But Ptolemy uses zero in the same sense as the Babylonians. On a wall inscription in India in the 9th century AD. The first time the zero symbol occurs is at the end of a number. This is the first generally accepted designation for the modern zero sign. It was Indian mathematicians who invented zero in all its three senses. For example, the Indian mathematician Brahmagupta back in the 7th century AD. actively began to use negative numbers and operations with zero. But he argued that a number divided by zero is zero, which is of course an error, but a real mathematical audacity that led to another remarkable discovery by Indian mathematicians. And in the 12th century, another Indian mathematician Bhaskara makes another attempt to understand what will happen when divided by zero. He writes: “a quantity divided by zero becomes a fraction whose denominator is zero. This fraction is called infinity.”

Leonardo Fibonacci, in his work “Liber abaci” (1202), calls the sign 0 in Arabic zephirum. The word zephirum is the Arabic word as-sifr, which comes from the Indian word sunya, i.e. empty, which served as the name for zero. From the word zephirum comes the French word zero (zero) and the Italian word zero. On the other hand, the Russian word digit comes from the Arabic word as-sifr. Until the mid-17th century, this word was used specifically to refer to zero. The Latin word nullus (nothing) came into use to mean zero in the 16th century.

Zero is a unique sign. Zero is a purely abstract concept, one of man's greatest achievements. It is not found in the nature around us. You can easily do without zero in mental calculations, but it is impossible to do without accurately recording numbers. In addition, zero is in contrast to all other numbers, and symbolizes the infinite world. And if “everything is number,” then nothing is everything!

Bases used today:

10 - the usual decimal number system (ten fingers on the hands). Alphabet: 1, 2, 3, 4, 5, 6, 7, 8, 9, 0

60 - invented in Ancient Babylon: dividing an hour into 60 minutes, minutes into 60 seconds, and an angle into 360 degrees.

12 - spread by the Anglo-Saxons: there are 12 months in a year, two periods of 12 hours in a day, 12 inches in a foot

7 - used to count days of the week

The most perfect principle for representing numbers is positional(local) principle, according to which the same numerical sign (digit) has different meanings depending on the place where it is located.

This number system is based on the fact that a certain number N of units (base SS) are combined into one unit of the second digit, N units of the second digit are combined into one unit of the third digit, etc.

The base of number systems can be any number greater than one. Such systems include the modern decimal number system (with base N = 10). In it, the numbers 0,...,9 are used to indicate the first ten numbers.

Despite the apparent naturalness of such a system, it was the result of a long historical development.

Emergence decimal number system associated with counting on fingers. There were number systems with other bases: 5, 6, 12 (counting in dozens), 20 (traces of such a system are preserved in French, for example quatre - vingts, i.e. literally four - twenty, means 80), 40, 60 and etc.

When performing calculations on a PC, a number system with base 2 is used. Representation of information in the binary system has been used by humans since ancient times. Thus, the inhabitants of the Polynesian islands transmitted the necessary information with the help of drums: alternating ringing and dull beats. The sound above the surface of the water spread over a fairly large distance, which is how the Polynesian telegraph “worked.” In the telegraph in the 19th-20th centuries. information was transmitted using Morse code- in the form of a sequence of dots and dashes. Often we agree to open the front door only on a “conventional signal” - a combination of short and long bells. The binary system is used to solve puzzles and construct winning strategies in some games.

Modern decimal positional The number system arose on the basis of numbering, which originated no later than the 5th century. V India. Before this, India had number systems that used not only the principle of addition, but also the principle of multiplication (the unit of some digit is multiplied by the number on the left).

At that time, there were many different numbering systems in different areas of India, one of which spread throughout the world and is now generally accepted. In it, the numbers looked like the initial letters of the corresponding numerals in the ancient Indian language - Sanskrit(Devangari alphabet).

Initially, these signs represented the numbers 1, 2, 3 ... 9, 10, 20, 30 ... 90, 100, 1000; with their help other numbers were described. Subsequently, a special sign (bold dot, circle) was introduced to indicate an empty digit; signs for numbers greater than 9 fell out of use, and the Devangari numbering system turned into a decimal place system. How and when this transition took place is still unknown. By the middle of the 8th century, the positional numbering system was widely used in India.

Around the same time, it penetrated into other countries ( Indochina, China, Tibet, to the territory of our Central Asian republics, V Iran and etc.). A manual compiled at the beginning of the 9th century played a decisive role in the spread of Indian numbering in Arab countries. Muhammad of Khorezm(now Khorezm region of Uzbekistan). It was translated into Latin in Western Europe in the 12th century. In the 13th century. Indian numbering takes precedence in Italy. In other countries Western Europe it is established in the 16th century. The Europeans, who borrowed Indian numbering from the Arabs, called it Arabic(the historically incorrect name is still used today).

Borrowed from the Arabic language and layer " number" (in Arabic "syfr"), literally meaning "empty place" (from the Sanskrit word "sunya", which has the same meaning). This word was originally used to name the sign of an empty digit, and retained this meaning back in the 18th century, although already in the 15th century. the Latin term " zero" The form of Indian numerals has undergone various changes. The form in which we write them was established in the 16th century.

In the 9th century Manuscripts appeared in Arabic, which set out this number system, in the 10th century. decimal positional numbering goes up to Spain, at the beginning of the 12th century. it also appears in other European countries. The new number system is called Arabic, because in Europe they first became acquainted with it through Latin translations from Arabic. Only in the 16th century. the new numbering became widespread in science and everyday life. In Russia it begins to spread in the 17th century. and at the very beginning of the 18th century. supersedes alphabetical numbering. With the introduction of decimal fractions, the decimal system became a universal means of recording all real numbers. It makes it possible in principle to write arbitrarily large numbers. Writing numbers in it is compact and convenient for performing arithmetic operations. Therefore, this system begins to quickly spread from India to the West and East.

The language of numbers has its own alphabet. In that language of numbers, the alphabet is ten digits from 0 to 9. This is the decimal number system.

Number system is a way of representing a number by symbols of some alphabet, which are called digits. The ancient image of decimal digits is not accidental: each digit represents a number by the number of angles in it. For example, 0 - no corners, 1 - one corner, 2 - two corners, etc. The writing of decimal numbers has undergone significant changes. The form we use was established in the 16th century.

They were built similarly old chinese number system and some others.

According to the famous African explorer Stanley, a number of African tribes had fivefold SS. For a long time they used the five-fold number system and China. The connection between this number system and the structure of the human hand is obvious. Thus, a person has five fingers on his hand, which are convenient to use for visual counting.

The Aztecs and Mayans, peoples who inhabited vast areas of the American continent for many centuries and created the highest culture there, including mathematics, adopted twentieth SS. This number system was also adopted by the Celts, who inhabited Western Europe starting from the 2nd millennium BC. The basis for counting is fingers and toes. Some traces of this system in the French monetary system: the basic monetary unit, the franc, is divided by 20

(1 franc = 20 sous).

Was widespread duodecimal notation. Its origin is also connected with counting on fingers. They counted the thumb and phalanges of the other four fingers: there are 12 of them in total. Elements of the duodecimal number system were preserved in the system of measures (1 foot = 12 inches) and in the monetary system

(1 shilling = 12 pence). We often encounter duodecimal SS in everyday life: tea and table sets for 12 people, a set of handkerchiefs - 12 pieces.

The southern and eastern Slavic peoples used alphabetical numbering to record numbers. Among some Slavic peoples, the numerical values ​​of letters were established in the order of the Slavic alphabet, while for others (including Russians), not all letters played the role of numbers, but only those that are in the Greek alphabet. At the same time, the numerical values ​​of the letters increased in the same order as the letters in the Greek alphabet (the order of the letters of the Slavic alphabet was slightly different)

Slavic numbers up to 18th century were the main digital designation in Russia. Slavic numbering was preserved in Russia until the end of the 17th century. Under Peter I, the so-called Arabic numbering prevailed. Slavic numbering was preserved only in liturgical books. The Armenians used the alphabetical numbering principle. But the ancient Armenian and ancient Georgian alphabets had many more letters than the ancient Greek. This made it possible to introduce special notations for the numbers 1000, 2000, 3000, 4000, 5000, 6000, 7000, 8000, 9000. The numerical values ​​followed the order of letters in the Armenian and Georgian alphabets.

History of the development of number systems . 2

Binary number systems 6

Binary arithmetic 10

Forms of representation of numbers with fixed and floating point. 13

Adding fixed point numbers. 16

Addition of floating point numbers. 16

Multiplying fixed-point numbers. 17

Multiplying floating point numbers. 18

9. Direct, reverse and additional codes. Modified code. 20

History of the development of number systems.

Calculus, numbering, is a set of techniques for representing natural numbers. In any number system, some symbols (words or signs) are used to designate certain numbers, called node numbers, the remaining numbers (algorithmic) are obtained as a result of some operations from node numbers. Number systems differ in the choice of key numbers and methods of generating algorithmic ones, and with the advent of written notations for numerical symbols, number systems began to differ in the nature of numerical signs and the principles of their recording.

The most perfect principle for representing numbers is the positional principle, according to which the same numerical sign (digit) has different meanings depending on the place where it is located. Such a number system is based on the fact that a certain number n units (the base of the number system) are combined into one unit of the second digit, n units of the second digit are combined into one unit of the third digit, etc. The base of the number system can be any number greater than one. Such systems include the modern decimal number system (with base n=10). In it, the numbers 0,1,...,9 are used to indicate the first ten numbers.

Despite the apparent naturalness of such a system, it was the result of a long historical development. The emergence of the decimal number system is associated with counting on fingers. There were number systems with another base: 5.12 (counting in dozens), 20 (traces of such a system are preserved in the French language, for example quatre - vingts, i.e. literally four - twenty, means 80), 40, 60, etc. When calculating Computers often use a base 2 number system.

Primitive peoples did not have a developed number system. Back in the 19th century, many tribes in Australia and Polynesia had only two numerals: one and two; their combinations formed the numbers: 3 - two - one, 4 - two - two, 5 - two - two - one and 6 - two - two - two. All numbers greater than 6 were talked about “a lot” without individualizing them. With the development of social and economic life, the need arose to create number systems that would make it possible to designate increasingly large collections of objects. One of the most ancient number systems is the Egyptian hieroglyphic numbering, which arose as early as 2500 - 3000 BC. e. It was a decimal non-positional number system, in which only the principle of addition was used to record numbers (numbers expressed by adjacent digits add up). There were special signs for the unit ▯ , ten ⋓, one hundred and other decimal places up to . The number 343 was written like this:

Similar number systems were Greek Herodian, Roman, Syriac, etc.

Roman numerals are the traditional name for a sign system for designating numbers, based on the use of special symbols for decimal places:

1 5 10 50 100 500 1000

Originated around 500 BC. e. among the Etruscans and was used in Ancient Rome; sometimes still used today. In this number system, natural numbers are written by repeating these digits. Moreover, if a larger number is in front of a smaller one, then they are added (the principle of addition), but if a smaller one is in front of a larger one, then the smaller one is subtracted from the larger one (the principle of subtraction). The last rule applies only to avoid repeating the same number four times. For example, I, X, C are placed respectively before X, C, M to indicate 9, 90, 900 or before V, L, D to indicate 4, 40, 400.

For example, VI=5+1=6, IV=5-1=4 (instead of IIII), XIX=10+10-1=19 (instead of XVIIII), XL=50-10=40 (instead of XXXX), XXXIII= 10+10+10+1+1+1=33, etc. Performing arithmetic operations on multi-digit numbers in this system is very inconvenient.

More advanced number systems are alphabetic: Ionian, Slavic, Hebrew, Arabic, as well as Georgian and Armenian. The first alphabetic number system was apparently Ionian, which arose in the Greek colonies in Asia Minor in the middle of the 5th century BC. e. In alphabetic number systems, numbers from 1 to 9, as well as all tens and hundreds, are usually designated by successive letters of the alphabet (over which dashes are placed to distinguish the entries of numbers from words). The number 343 in the Ionian system was written as follows:
(Here - 300, - 40, - 3).

Digital meaning of Slavic alphabet. So for Cyrillic:

To indicate numbers above the letters, a special sign is the title (sometimes above each letter, sometimes only above the first or above the entire number). When writing numbers greater than 10, the numbers were written from left to right in descending order of decimal places (however, sometimes for numbers from 11 to 19 units were written earlier than ten). To designate thousands, a special sign was placed in front of their number (bottom left). For example:

To designate and name higher decimal places (more
) there were two systems: “small number” and “great number”; the latter system included numbers up to
or even
(“the human mind cannot comprehend more than this”):

Slavic numbers were the main digital designation in Russia until the 18th century.

In alphabetic number systems, numbers are written much shorter than in previous ones; in addition, it is much easier to perform arithmetic operations on numbers written in alphabetical numeration. However, in alphabetic number systems you cannot write arbitrarily large numbers. The Greeks expanded the Ionian numbering: they denoted the numbers 1000, 2000,...,9000 with the same letters as 1,2,...,9, but put a stroke at the bottom left: so,
stood for 1000, - 2000, etc. A new sign was introduced for 10,000. Nevertheless, the Ionian number system turned out to be unsuitable for astronomical calculations of the Hellenistic era, and Greek astronomers of that time began to combine the alphabetic system with the Babylonian sexagesimal - the first number system known to us based on the positional principle. In the number system of the ancient Babylonians, which arose approximately 2000 BC. e. all numbers were written using two signs: (for one) and (for ten). Numbers up to 60 were written as combinations of these two signs using the principle of addition. The number 60 was again designated by a sign, being a unit of the highest category. To record numbers from 60 to 3600, the principle of addition was again used, and the number 36,000 was denoted by the same sign as one, etc. The number 343 = 5*60+4*10+3 in this system was written like this:

However, due to the absence of a sign for zero, which could be used to mark the missing digits, the recording of numbers in this number system was not unambiguous. The peculiarity of the Babylonian number system was that the absolute value of the numbers remained uncertain.

Another number system based on the positional principle arose among the Mayan Indians, inhabitants of the Yucatan Peninsula (Central America) in the middle of the 1st millennium AD. e. The Mayans had two number systems: one, reminiscent of the Egyptian one, was used in everyday life, the other - positional, with a base of 20 and a special sign for zero, was used in calendar calculations. Recording in this system, as in our modern one, was absolute.

The modern decimal positional number system arose on the basis of numbering, which originated no later than the 5th century. in India. Before this, India had number systems that used not only the principle of addition, but also the principle of multiplication (the unit of some digit is multiplied by the number on the left). The Old Chinese number system and some others were constructed in a similar way. If, for example, we conventionally designate the number 3 as the symbol III, and the number 10 as the symbol X, then the number 30 will be written as IIIX (three tens). Such number systems could serve as an approach to creating decimal positional numbering.

The decimal positional system makes it possible in principle to write arbitrarily large numbers. Writing numbers in it is compact and convenient for performing arithmetic operations. Therefore, soon after its inception, the decimal positional number system begins to spread from India to the West and East. In the 9th century, manuscripts appeared in Arabic, which set out this number system; in the 10th century, decimal positional numbering reached Spain; at the beginning of the 12th century, it appeared in other European countries. The new number system was called Arabic because in Europe it was first introduced to it through Latin translations from Arabic. Only in the 16th century did the new numbering become widespread in science and everyday life. In Russia it begins to spread in the 17th century and at the very beginning of the 18th century. displaces the alphabetic one. With the introduction of decimal fractions, the decimal positional number system became a universal means for writing all real numbers.