The history of the development of number systems.
A modern person in everyday life is surrounded by a huge amount of the most diverse information, not a small proportion of which falls on numerical information. Indeed, we memorize phone numbers, calculate the cost of purchases, keep track of school lessons and their duration, etc. Historians have proven that even in ancient times people could write down numbers, performed various arithmetic operations on them, but the numbers were written in completely different ways. principles than we do today.
What is a number? Initially, the concept of number was "tied" to the objects that were counted. With the development of writing, the abstract concept of a natural number appears. The need to make measurements, i.e. comparison with another value of the same kind, chosen as a standard, led to the appearance of fractional numbers. The further development of the concept of number was directly related to the development of mathematics. Today, the number is a fundamental concept of mathematics and computer science, which is understood as its value, and not a symbolic notation. Conventional signs used to denote numbers are called digits.
The set of methods for naming and writing numbers is called calculus.
The number system is called the ways of writing numbers and the rules for operating on numbers.
The first mention of number systems can be attributed to the 10th - 11th millennium BC. During excavations of cultural layers belonging to this period, archaeologists found records in the form of a sequence of dashes - sticks. Scientists believe that numbers were written in this way and the number of sticks written in a line is equal to the value of the number. This numbering system was called single (stick) . Further development of the account led to the improvement and development of number systems. Throughout its history, mankind has used various number systems, and many evidence of this has survived to this day. For example, the fact that there are 60 minutes in an hour and 60 seconds in a minute indicates that people once used sexagesimal number system. Indeed, archaeologists have discovered, during excavations at the site of the ancient Babylonian civilization, traces of the use of such a number system. Twelve months in a year and twelve divisions on the watch dial indicate that it was most likely once used and duodecimal number system.
In ancient Rus', the so-called alphabetical number system, in which numbers were denoted by Cyrillic letters with a special sign, which was called title and served to distinguish numbers from letters.
The modern decimal number system originated in India around the 5th century BC. AD, the emergence of this system became possible after the number “0” was used to indicate the missing value.
Positional and non-positional number systems.
Number systems in which numbers are written as a sequence of digits can be divided into two classes: positional and non-positional. In non-positional systems, the meanings of the digits do not change when their position in the sequence changes. As an example of a non-positional system, let's take the well-known Roman number system. In the Roman numeral system, the symbol X is equal to 10 anywhere, but in the notation to the left of the older one (for example, XC), the symbol x is -10, and in combination before the younger one (for example, XV) is equal to +10. In non-positional number systems, operations on numbers are very difficult and have no rules. In these systems, negative and fractional numbers cannot be expressed, so non-positional systems are of limited use. They are mainly used to name dates, volumes, chapters, etc.
On the contrary, in positional number systems, the quantitative value of a digit in a number depends on its position.
We will give definitions of the main, most important, concepts of positional number systems, which include the base, alphabet and basis of number systems
Base number system shows how many times the quantitative value of a digit changes when moving to an adjacent position, and how many different characters (digits) are included in the so-called alphabet of the number system.
alphabet number system is a set of characters (digits) used in the positional number system to write numbers. So the alphabets of the number systems considered in the future are as follows:
Binary: 0.1.
Octal: 0,1,2,3,4,5,6,7.
Decimal: 0,1,2,3,4,5,6,7,8,9.
Hexadecimal: 0,1,2,3,4,5,6,7,8,9,A,B,C,D ,E ,F .
Basis A positional number system is a sequence of numbers, each of which sets the value of a digit by position. In other words, we can say that the basis of the number system is made up of numbers that are successive degrees of the base of the number system.
The base of the number system can be any natural number ≥ 2. One example of a positional number system is the decimal system, which is widely used in life. As decimal digits, Arabic numerals 0,1,2,3,4,5,6,7,8,9 are used - which are the alphabet of the decimal number system. The base of the number system is 10, which means that the values of the digits in adjacent positions differ by ten times, and that there are 10 digits in the alphabet. The base of the decimal number system is made up of numbers: 1, 10, 100, 1000, 10000 ... 10 n, this means that the digit in the zero position contributes - units, the digit in the first position contributes - tens, the digit in the second position contributes contribution - hundreds, etc..
As an example, consider the number 5555, written in your usual number system with a base of 10.
5 3 5 2 5 1 5 0 = 5000+500+50+5
As you can see from the example, 5 standing in the 0th position contributes 5 units, 5 standing in the 1st position contributes 5 tens, 5 standing in the 2nd position contributes 5 hundreds, 5 standing in the 3rd position has a contribution of 5,000.
In any positional number system with a base greater than 1, a number is written as sequences of digits separated by a comma into two sequences
Positions , those located to the left of the decimal point are numbered from right to left with the numbers 0, 1, 2, ..., and to the right of the decimal point are numbered in a row from left to right -1, -2, -3, etc. The numbered positions are called discharges .
The sequence of digits located to the left of the comma is called the integer part of the number, and to the right of the comma is called the fractional part.
In modern computers, positional number systems with bases 2, 8, 16 and 10 are currently mainly used, although there have been attempts, though not entirely successful, to use other number systems (for example, ternary).
An important feature of the base of the number system should be noted - in any positional number system, the base is written as 10, but it has a different quantitative value. For example, in binary 10 is two, in ternary 10 it is three, and in decimal 10 it is ten.
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MINISTRY OF EDUCATION AND SCIENCE OF THE RUSSIAN FEDERATION
FEDERAL STATE BUDGET EDUCATIONAL INSTITUTION OF SECONDARY VOCATIONAL EDUCATION
"TYUMEN STATE UNIVERSITY"
SURGUT INSTITUTE OF ECONOMICS, MANAGEMENT AND LAW (BRANCH) Tyumen State University
Topic: "History of number systems"
Performed:
1st year student BD-154-O
Kutova A. A.
Checked:
Volkova T.G
Surgut 2015
1. History of number systems
2. Decimal number system
Literature
1. History of number systems
Notation is a set of techniques and rules for designating and naming numbers.
A modern person in everyday life is constantly faced with numbers: we memorize bus and telephone numbers, calculate the cost of purchases in a store, keep our family budget in rubles and kopecks (hundredths of a ruble), etc. Numbers, figures... they are with us everywhere. And what did a person know about numbers several thousand years ago? The question is not easy, but very interesting. Historians have proven that even five thousand years ago people could write down numbers and perform arithmetic operations on them. Of course, the principles of recording were not at all the same as they are now. But in any case, the number was depicted using one or more characters.
These symbols involved in writing a number, in mathematics and computer science, are called numbers.
But what do people understand then by the word "number"?
Initially, the concept of an abstract number was absent, the number was "tied" to those specific objects that were counted. The abstract concept of a natural number appears along with the development of writing. Fractional numbers were invented when it became necessary to make measurements. Measurement, as you know, is a comparison with another value of the same kind, chosen as a standard.
The standard is also called a unit of measurement. It is clear that the unit of measurement did not always fit an integer number of times in the measured value. Hence the practical need to introduce "smaller" numbers than natural ones. The further development of the concept of number was already due to the development of mathematics.
The concept of number is a fundamental concept of both mathematics and computer science. In the future, when presenting the material, by number we will understand its value, and not its symbolic notation.
Today, at the very end of the 20th century, mankind mainly uses the decimal number system to write numbers. What is a number system?
Notation is a way of writing (imaging) numbers.
The various number systems that existed before and are currently in use are divided into two groups: positional and non-positional.
The most perfect are positional number systems, i.e. systems of writing numbers, in which the contribution of each digit to the value of the number depends on its position (position) in the sequence of digits representing the number. For example, our usual decimal system is positional: in the number 34, the number 3 indicates the number of tens and "contributes" to the value of the number 30, and in the number 304 the same number 3 indicates the number of hundreds and "contributes" to the value of the number 300.
Number systems in which each digit corresponds to a value that does not depend on its place in the notation of the number are called non-positional.
Positional number systems are the result of a long historical development of non-positional number systems.
Single system
The need to record numbers appeared in very ancient times, as soon as people began to count. The number of objects, such as sheep, was depicted by drawing lines or serifs on some solid surface: stone, clay, wood (before the invention of paper, it was still very, very far away). Each sheep in such a record corresponded to one line. Archaeologists have found such "records" during excavations of cultural layers belonging to the Paleolithic period (10 - 11 thousand years BC).
Scientists called this way of writing numbers the unit ("stick") number system. In it, only one type of sign was used to write numbers - the "stick". Each number in such a number system was designated using a string made up of sticks, the number of which was equal to the designated number.
The inconveniences of such a system of writing numbers and the limitations of its application are obvious: the larger the number to be written, the longer the string of sticks. Yes, and when writing a large number, it is easy to make a mistake by inflicting an extra number of sticks or, conversely, without adding them.
It can be suggested that in order to facilitate counting, people began to group objects into 3, 5, 10 pieces. And when recording, they used signs corresponding to a group of several objects. Naturally, the fingers were used in the counting, so the first signs appeared to indicate a group of objects of 5 and 10 pieces (units). Thus, more convenient systems for notating numbers arose.
Ancient Egyptian decimal non-positional system
In the ancient Egyptian number system, which arose in the second half of the third millennium BC, special numbers were used to denote the numbers 1, 10, 102, 103, 104, 105, 106, 107. Numbers in the Egyptian numeral system were written as combinations of these digits, in which each of them was repeated no more than nine times.
Example. The ancient Egyptians wrote the number 345 like this:
Units Tens Hundreds
Both the stick and ancient Egyptian numeral systems were based on the simple principle of addition, according to which the value of a number is equal to the sum of the values of the digits involved in its recording. Scientists attribute the ancient Egyptian number system to decimal non-positional.
Babylonian sexagesimal system
Also far from our days, two thousand years BC, in another great civilization - the Babylonian - people wrote the numbers in a different way.
The numbers in this number system were composed of signs of two types: a straight wedge served to denote units, and a recumbent wedge to denote tens.
To determine the value of a number, it was necessary to divide the image of the number into digits from right to left. A new discharge began with the appearance of a straight wedge after a recumbent one, if we consider the number from right to left.
For example: The number 32 was written like this:
The signs of a straight wedge and a lying wedge served as numbers in this system. The number 60 was again denoted by the same straight wedge as 1, the numbers 3600=60 2 , 216000=60 3 and all other powers of 60 were also denoted by the same sign. Therefore, the Babylonian number system was called sexagesimal.
The value of the number was determined by the values of its constituent digits, but taking into account the fact that the digits in each subsequent digit meant 60 times more than the same digits in the previous digit.
Example. The number 92=60+32 was written like this:
and the number 444 in this notation system had the form
because 444=7*60+24.
Solely for clarity, it is separated by a space (which the Babylonians did not have) the senior digit (left) and the junior.
The Babylonians wrote all numbers from 1 to 59 in a decimal non-positional system, and the number as a whole - in a positional system with base 60.
The record of the number among the Babylonians was ambiguous, because. there was no number to represent zero. The record of the number 92 given above could mean not only 92=60+32, but also, for example, 3632=3600+32. Additional information was needed to determine the absolute value of the number. Subsequently, the Babylonians introduced a special character to indicate the missing sexagesimal digit.
which corresponds to the appearance of the digit 0 in the decimal notation.
Example. The number 3632 now had to be written like this:
But at the end of the number, this symbol was usually not put, i.e. this symbol was still not the number "zero" in our understanding, and again, additional information was required in order to distinguish 1 from 60, from 3600, etc.
The Babylonians never memorized the multiplication table, because it was practically impossible. When calculating, ready-made multiplication tables were used.
Sexagesimal Babylonian system - the first number system known to us, based in part on the positional principle.
The Babylonian system played an important role in the development of mathematics and astronomy, traces of which have survived to this day. So, we still divide an hour into 60 minutes, and a minute into 60 seconds. Following the example of the Babylonians, we also divide the circle into 360 parts (degrees).
Roman system
familiar to us Roman the system is not too fundamentally different from the Egyptian. In it to denote numbers 1, 5, 10, 50, 100, and 1000 capital latin letters are used I, V, X, C, D and M respectively, which are the digits of this number system.
A number in the Roman numeral system is denoted by a set of consecutive digits. The value of the number is:
1. the sum of the values \u200b\u200bof several consecutive identical digits (let's call them a group of the first type);
2. the difference between the values of two digits, if there is a smaller one to the left of the larger digit. In this case, the value of the smaller digit is subtracted from the value of the larger digit. Together they form a group of the second kind. Note that the left digit can be less than the right digit by a maximum of one order: for example, before L (50) and C (100) of the "younger" ones, only X (10) can stand, before D (500) and M (1000) - only C(100), before V(5) - only I(1);
3. the sum of the values of groups and numbers that are not included in the groups of the first or second type.
Example 1. The number 32 in the Roman numeral system has the form XXXII=(X+X+X)+(I+I)=30+2 (two groups of the first type).
Example 2. The number 444, which has 3 identical digits in its decimal notation, in the Roman numeral system will be written as CDXLIV=(D-C)+(L-X)+(V-I)=400+40+4 (three groups of the second type).
Example 3. The number 1974 in the Roman numeral system will look like MCMLXXIV=M+(M-C)+L+(X+X)+(V-I)=1000+900+50+20+4 (along with groups of both types, individual "numbers").
2. Decimal number system
Decimal sysnumber theme- this is a familiar and well-known positional number system for all of us, but we will start studying from it and consider it from positions that will help us understand other number systems that are unusual for us.
So, the base of the system is the number ten (10), which means that ten digits are used to represent numbers (0,1,2,3,4,5,6,7,8,9).
Let's just count in this system, we will count and write numbers from the numbers at our disposal:
Zero - 0 ;
One - 1 ;
Eight - 8 ;
Nine - 9 ;
What to do next? All numbers are gone. How to represent the number ten? To get out of the situation, we introduce a new concept - "ten" and say that ten is one ten and zero units. And this can already be written down - "10".
So, Ten - 10 (one ten, zero ones)
Eleven - 11 (one ten, one unit)
Twenty - 20 (two tens, zero ones)
Ninety nine - 99 (nine tens, nine ones)
One hundred - 100 (one hundred, zero tens, zero units)
And so always, when we no longer have enough digits to display the next number, we enlarge the units of the account (i.e., we count in tens, hundreds, etc.) and write the number with an extension by one digit.
Consider the number 4329 written in decimal notation. It can be said that it contains: four thousand, three hundred, two tens and nine units. And you can get its value through the numbers included in it as follows.
4329 = 4 *1000+3 *100+2 *10+9 *1, hereinafter the * (asterisk) sign means multiplication.
But the series of numbers 1000, 100, 10, 1 is nothing but the integer powers of the number 10 (the base of the number system) and therefore we can write:
4329 = 4 *10 3 +3 *10 2 +2 *10 1 +9 *10 0
Similarly for a fractional number (decimal) for example: 0.235 (zero point two hundred and thirty-five thousandths), we can say about it that it contains: two tenths, three hundredths and five thousandths. And its value can be calculated as follows:
0.235 = 2 *0.1 + 3 *0.01 + 5 *0.001
And here the series of numbers 0.1 0.01 0.001 1 is nothing but the integer powers of the number 10 and we can also write:
0.235 = 2 *10 -1 + 3 *10 -2 + 5 *10 -3
For the mixed number 752.159, we can similarly write:
752.369 = 7 *10 2 +5 *10 1 +2 *10 0 +3 *10 -1 +6 *10 -2 +9 *10 -3
Now, if we number the digits of the integer part of any number, from right to left, as 0,1,2 ... n (numbering starts from zero!). And the digits of the fractional part, from left to right, as -1, -2, -3 ... -m, then the value of any arbitrary decimal number can be calculated by the formula:
N= d n10 n +d n-110 n-1 +…+d 1 10 1 +d 0 10 0 +d -1 10 -1 +d -2 10 -2 +…+d -(m-1)10 -(m-1) +d -m10 -m
Where: n- the number of digits in the integer part of the number minus one;
m- the number of digits in the fractional part of the number
d i- number in i-th category
This formula is called the formula for the bitwise expansion of a decimal number, i.e. number written in decimal notation. But if in this formula the number ten is replaced by some natural number q, then we get the expansion formula for the number expressed in the number system with the base q:
N= d nqn +d n-1qn-1 +…+d 1 q 1 +d 0 q 0 +d -1 q -1 +d -2 q -2 +…+d -(m-1)q-(m-1) +d -mq-m
With the help of the last formula, we can always get the value of the number written in any positional number system.
Conclusion
Today we are accustomed to using the decimal number system in everyday life. Decimal digits express time, house and telephone numbers, prices, budget, and the metric system of measures is based on them.
Arithmetic operations on decimal numbers are performed using fairly simple operations, which are based on multiplication and addition tables known to every student. Learned at a very early age, these rules are assimilated so firmly as a result of daily practice that we already operate with them subconsciously. For this reason, many people today are not even aware of the existence of other number systems.
Literature
1. http://sch69.narod.ru/mod/1/6506/system.html
2. https://en.wikipedia.org/wiki/%D0%94%D0%B5%D1%81%D1%8F%D1%82%D0%B8%D1%87%D0%BD%D0%B0 %D1%8F_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0_%D1%81%D1%87%D0%B8%D1%81%D0 %BB%D0%B5%D0%BD%D0%B8%D1%8F
3. http://comp-science.narod.ru/Demenev/files/history.htm
4. Bosova L.L. Informatics and ICT: Textbook for grade 6. - M.: BINOM. Knowledge Lab, 2012
5. http://www.reshinfo.com/desytichnaja_systema.php
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The history of recording numbers and number systems has been going on since the appearance of counting among people. People depicted the number of different objects using serifs or dashes. They were applied to surfaces that served as “paper” at that time: clay tablets, tree bark or stones. Archaeologists attribute the first information about such records to the Paleolithic period, that is, to 10-11 millennium BC.
This way of writing is called the unit number system. All numbers were indicated by a line of dashes (or any other signs, for example, dots): the more characters in the line, the larger the number. This counting system was not convenient, because, with large numbers, it was easy to make a mistake in the number of sticks. Each time they had to be recalculated.
To simplify the counting, objects began to be combined into small groups of 3, 5 and 10 units. At the same time, each group had its own sign-designation on the letter. Since the most convenient account has always been the account on the fingers, the first to be designated were combinations of objects from 10 and 5 units. This is what laid the foundation for a convenient number system.
The system used by the ancient Greeks was called Attic. The first four numbers were written in dashes. For the number five, there was a sign - "pi", as for the number ten - the first letter of the word "deca". Hundred, thousand and ten thousand in writing were designated as H, X, M.
This system was replaced by the Ionian system in the third century BC. The numbers from one to nine in it were denoted by the letters of the Greek alphabet: from the first to the ninth. The letters from ten to eighteen were tens - from ten to ninety. And the last nine were recorded hundreds - from one hundred to nine hundred.
With the help of the alphabet, the Eastern and Southern Slavs also wrote down the numbers. Some of them used the Slavic alphabet, endowing each letter with a numerical value. The other - only those letters that occur in the Greek alphabet. To distinguish letters from numbers allowed a special icon, which was placed above the number - "titlo". This numbering was used in Russia until the 18th century.
The beginning of the reign of Peter I brought Arabic numeration to the country, which is still used today. However, liturgical books still use the Slavic system of writing.
Each of us is at least somewhat familiar with the "Roman system", which denotes centuries, anniversaries, conference titles, verse stanzas and book chapters. It was she who was once used by the ancient Romans. Researchers believe that it was borrowed by the inhabitants of Rome from the Etruscans. All integers in this system up to 5000 are written using the numbers I, V, X. If there is a large number in front and a smaller one after it, they add up. If on the contrary - the smaller one before the larger one - are subtracted. The same number is placed in a row no more than three times. Any arithmetic operation in such a record of numbers becomes a difficult task. However, until the 13th century in Italy and until the 16th century in the countries of Western Europe, they used it.
The first local or positional numbering was "created" in Babylon in 4000 BC. Its essence is that one digit can mean different numbers, depending on the place where it stands. A striking example is the modern decimal system. Depending on the position in the number, the number can mean ten, one, and one hundred.
The Babylonian system was sexagesimal, since initially they took not 10, but 60 as the basis. All numbers less were written with two signs - tens and units. The numbers themselves were written on clay tablets with triangular sticks, so they looked like a wedge. The signs were repeated depending on the number.
The sexagesimal system did not spread beyond Ancient Babylon, but sexagesimal fractions were used in the countries of Central Asia, Western Europe, the Middle East and North Africa. Before the advent of decimal fractions, they played an important role in astronomy and other sciences. Today, we are reminded of this system by dividing a minute into 60 seconds, and an hour into 60 minutes, an angle into 360 degrees.
All number systems can be conditionally divided into positional and non-positional. Those signs that we use in them to write numbers are called numbers.
The position of a digit in a written number in non-positional systems does not affect the value that it denotes. These are, for example, systems that use letters to write numbers - Slavic and Roman.
The position of a digit in positional systems determines the value of the value that is written to it. In this case, the position is the place that this digit occupies in the number. And the number of digits that are used to write is called the base of the system. Examples of such a system are Babylonian sexagesimal and modern decimal.
Positional systems use a small number of characters, which makes it easy to write large numbers. That is why it is more common today in the world. In addition, it provides convenience and simplicity when performing arithmetic operations on numbers.
In our time, the Indo-Arabic decimal system has received the greatest distribution. It first appeared zero when writing numbers. It bears this name because it uses ten digits.
The easiest way to understand the differences between a positional system and a non-positional system is to compare two numbers written in one and the other. The first compares the numbers in the same place, from left to right. The larger the number, the larger the value itself. For example, the number 245 will be greater than the number 123, because the 2 in this position is greater than 1. For a non-positional system, this law does not apply. If we compare Roman IX and VI, then the first will be greater than the second, although I in the same position is less than V.
The base 2 binary number system represents the positive positional number system with integers. It allows you to write all numerical values using two characters. The most commonly used numbers are 0 and 1.
The basis for the octal positive positional system is 8. Any number in it can be written using the numbers from 0 to 7. This system is used by digital and computer devices. It was she who was used at the dawn of the computer era, but now has given way to a more advanced - hexadecimal.
The most recognizable in the world, the decimal system is a positional system with base 10. It uses Arabic numerals from 0 to 9 to indicate numbers.
One of the most popular systems of antiquity - duodecimal - is still used in some areas of science. It is also the main one among some peoples of Tibet and Nigeria, but reminds of itself in other cultures. For example, in our language the word "dozen" has been preserved, and in English "dozen", which refer us to the number twelve. Its base is 12. The letters A and B and the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 0 are used as signs.
Hexadecimal number system - represents a positional positive system with a base of 16 digits. As its numbers, the letters of the Latin alphabet A, B, C, D, E, F are used to denote numbers from ten to fifteen and the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 0.
The hexadecimal number system is used in modern computer programs to encode fonts. Hexadecimal numbers in many modern computer graphics programs encode colors. Also, web designers encrypt the color with a hexadecimal code. For example, the code #00ff00 represents green. The two letters f in the middle of this code correspond to the number 256 in decimal notation.
When working with computers, binary, octal and hexadecimal number systems are most often used. Both humans and computers are great at working with these systems. But some cases force us to turn to less popular number systems. Such systems are septenary, ternary and the number system with base 32. All arithmetic operations in them do not differ from the usual ones.
Primitive man almost did not have to count. "One", "two" and "many" are all his numbers. Modern people have to deal with numbers literally at every turn. You need to be able to correctly name and write down any number, no matter how large it may be. If each number were called by a special name and denoted in a letter by a special sign, then it would be impossible for anyone to remember all these words and signs. How to cope with this task? A good notation helps us out.
The collection of a few names and signs that allow you to write down any number and give it a name is called the number system or numbering.
Almost all over the globe, the alphabet in the language of numbers is 10 digits, from 0 to 9. Nine of them are used to denote the first nine natural numbers, and the tenth - zero - does not denote any number, it is the so-called "positional plug". This language is called the decimal number system.
However, not at all times and not everywhere people used the decimal system. From a purely mathematical point of view, it has no special advantages over other number systems, and this system owes its ubiquitous distribution not at all to the general laws of mathematics, but to reasons of a completely different nature.
Recently, the binary and, to some extent, ternary systems, which modern computers "prefer" to use, have seriously competed with the decimal system.
How people counted and how they called numbers before the invention of writing, no one knows for sure. This can only be guessed at. Undoubtedly, one thing: humanity mastered the account very slowly. However, by the time writing was invented, people already knew how to count well.
Four thousand years ago, the most developed peoples (Egyptians, Chaldeans) were able to write and use not only integers, but also the simplest fractional numbers. Moreover, at that time there were already schools in which they taught the art of counting.
There were no letters in the primitive writing. Every thing, every action was depicted by a picture. Gradually, the pictures became simpler. Along with the image of objects and actions, special figures appeared, denoting various properties of things, as well as icons for words corresponding to our prepositions and conjunctions.
Thus arose the writing called hieroglyphs; in hieroglyphic writing, each icon does not correspond to a sound, like ours, but to a whole word.
There were no special characters (numbers) for writing numbers then. But the words "one", "two", ... "seventeen" and so on corresponded to certain hieroglyphs. There were not so many of them, since people did not know large numbers then.
In some countries (for example, China and Japan), hieroglyphic writing has survived to this day. Here, for example (see Fig. 2), a few hieroglyphs:
Rice. 2
Among the Slavs, the order of numbers when writing a number was the same as in its oral name. They say, for example, "fifteen" (in Slavic - "five by ten"), calling forward the number of units, then ten. The Slavs wrote like that, that is, they wrote a five in front, and a dozen behind it. On the contrary, in the number "twenty-three" first they call tens, then ones, among the Slavs, first three then twenty, this was displayed in the letter.
To distinguish numbers from letters, a special icon was placed above them - a title. It was placed only above one of the numbers. The place of the digit, its position in the notation of the number did not matter.
With the help of these signs, large numbers were easily written. The titlo sign stood for thousands. By repeating this sign, it was possible to write very large numbers
Numbers up to a thousand in Ancient Rus' were called almost the same as now. There was a slight difference in pronunciation (for example, "one" was called "one" and the like). Ten thousand was called "darkness", and this number was considered so huge that the same word denoted any multitude that could not be counted.
At a later time (XVI - XVII centuries), a peculiar system of naming numbers appeared, the so-called "great Slavic number", in this system, numbers up to 999999 were called almost the same as now. The word "darkness" already means a million. In addition, the following names appear: "the darkness of topics", or "legion" (that is, a million million, or a trillion, equals 10); "legion of legions", or "modr" (septillion, 1024); finally, "modr modr", or "raven" (that is, 1048).
Positional numbering appeared, apparently, in ancient Babylon (about four thousand years ago). She will be discussed a little later. In India, it has taken the form of positional decimal numbering using zero. From the Hindus, this system of numbers was borrowed by the Arabs, who became in the VIII - IX centuries. one of the most cultured nations in the world. Europeans adopted it from the Arabs (hence the name - "Arabic numerals").
Of particular interest is Babylonian mathematics. Babylonian numbering existed for one and a half thousand years (from the 18th to the 3rd century BC) and was widely used throughout the Middle East. She influenced Chinese, Indian and Greek mathematics.
The Babylonians wrote with sticks on plates of soft clay and then burned their "manuscripts". Strong brick "documents" were obtained, partially surviving to our time, they are often found during excavations in Mesopotamia (now Iraq). Therefore, the study of Babylonian history and mathematics in particular was quite good.
At the turn of the XIX - XVIII centuries. BC, there was a merger of two peoples: the Sumerians and the Akkadians. Each of these peoples had a fairly developed trade, weight and monetary units, but none of these peoples had a developed numbering.
Among the Akkadians, the basic unit - "mekel" - was about 60 times less than the unit among the Sumerians - "mines" (about half a kilogram). A mina of silver served as the monetary unit.
After the merger of these peoples, both systems of units "circulated": mines and mekels were used in the same way as kilograms and grams (rubles and kopecks) are now used, with the only difference that the larger unit was not 100, but 60 small units. Over time, a larger unit appeared - "talent": 1 talent = 60 min, 1 min = 60 mekels.
How did the Babylonians write down numbers? They wrote with sticks, pressing them into the clay, so their main graphic elements were wedges. The first denoted units, the second - tens, see fig. 3.
Rice. 3
These signs are very clear, the number of wedges is striking, so there is no need to count them. But cuneiform writing is very inconvenient for estimating the size of the gaps between numbers, and the need to rewrite everything by hand led to frequent typos. The sign of separation was needed, and it appeared. Starting from some time, the icon ^ appears on the Babylonian bricks, corresponding to our zero.
However, having introduced a "positional stopper" in the middle of the numbers, the Babylonians did not think of putting it at the end. And until the very fall of the Babylonian culture, the numbers 1, 60, 3000 were written in the same way.
Only the Hindus, who borrowed positional numbering from them, learned to use the zero sign correctly, and, having introduced base 10 instead of 60, gave the number its modern form.
Three thousand years ago, the Hindus already used modern numbering, although numbers greater than 100,000 are not mentioned in the monuments of that time. Much larger numbers are found in later sources - up to one hundred quadrillions (1017). One of the relatively young legends about Buddha says that he knew the names of numbers before 1054. However, the Hindus, apparently, did not imagine the infinity of the natural series, they believed that there was some largest number known only to the gods.
The proof of the infinity of the number series is the merit of ancient Greek scientists.
ancient number systems
are very diverse, since the way we are used to writing numbers using ten characters did not appear immediately.
First of all, it should be noted that there were two main number systems - five and the usual decimal. In addition to them, there was also a 12-decimal one, which generally dominated in England until the 19th century. From Ancient Babylon, the 60-number system came to us, which is still used in measuring angular values - a circle consisting of 360 degrees is divided without a trace into many convenient numbers. It is worth noting that in ancient number systems a number of peoples trace the remains of a more ancient five-fold system - among the ancient Romans and the Maya, for example.
The variety in is actually small - mostly decimal or quinary-decimal. But, when it came to writing on paper or stone, then, as they say, everyone was their own head. There were no academies of sciences then, there were no ministries either, no one had heard of the standards of school education, the Chinese knew little about the achievements of the Greeks, to put it mildly, and vice versa. Therefore, everyone invented their own way of writing.
Perhaps the most ancient designation of the number can be considered a vertical stick. In almost all ancient peoples, it naturally depicted a unit. Next came two, three, less often four sticks, respectively. Further, new signs were mainly introduced upon reaching a certain number, at which it was simply inconvenient to write down a large number of sticks.
The Incas in South America came up with a generally unique numbering system - type - the numbers were indicated by knots on the shoelaces! The shape of the knots, the color of the laces, their location on the lace differed. The system was quite complicated, it required special training, but it completely satisfied the Incas, even allowing you to keep a double account in the accounting department!
In ancient Egypt, there was a decimal number system and there were several systems for notating numbers. A hieroglyphic form of writing, when all degrees of ten, including one, had their own sign. Like other number systems, any number could be denoted by adding the numerical values of these signs. This is a "ceremonial", rather cumbersome form of notation, therefore there was a priestly (hieratic) number system in which for units, tens, etc. there were separate marks. It was also necessary to fold in such a record, but the inscription was noticeably shorter. Later, an even simpler demotic letter arose. So far, the Egyptian number systems have not been made in mine, due to difficulties with encodings and fonts for ancient Egyptian inscriptions.
The real revolution was the discovery of a full-fledged concept of zero by Indian mathematicians. Thanks to this, the decimal POSITIONAL number system, familiar to us, appeared, which makes little sense to talk about. Many countries have their own designations for numbers, but in fact - they all differ from each other only in the appearance of signs (numbers) and nothing more.
I tried not only to collect all these number systems of the ancient world and different peoples together, but also make it convenient to use. The result is a program "Titlo" - translator of numbers .
