People did not immediately learn to count. Primitive society focused on a small number of objects - one or two. Anything more than that was named "many" by default. This is what is considered the beginning of the modern number system.
Brief historical background
In the process of development of civilization, people began to have the need to separate small collections of objects, united by common features. Corresponding concepts began to appear: "three", "four" and so on up to "seven". However, it was a closed, limited series, the last concept in which continued to carry the semantic load of the earlier "many". A vivid example of this is the folklore that has come down to us in its original form (for example, the proverb "Measure seven times - cut once").
The emergence of complex methods of counting
Over time, life and all the processes of people's activities became more complicated. This, in turn, led to the emergence of a more complex systemcalculus. At the same time, people used the simplest counting tools for clarity of expression. They found them around themselves: they drew sticks on the walls of the cave with improvised means, made notches, laid out the numbers they were interested in from sticks and stones - this is just a small list of the variety that existed then. In the future, modern scientists gave this species a unique name "unary calculus". Its essence is to write a number using a single type of sign. Today it is the most convenient system that allows you to visually compare the number of objects and signs. She received the greatest distribution in the primary grades of schools (counting sticks). The heritage of the "pebble account" can be safely considered modern devices in their various modifications. The emergence of the modern word "calculation" is also interesting, the roots of which come from the Latin calculus, which translates only as "pebble".
Counting on fingers
In the conditions of the extremely poor vocabulary of primitive man, gestures quite often served as an important addition to the transmitted information. The advantage of the fingers was their versatility and constant presence with the object that wanted to convey information. However, there are also significant drawbacks: a significant limitation and short duration of transmission. Therefore, the entire count of people who used the "finger method" was limited to numbers that are multiples of the number of fingers: 5 - corresponds to the number of fingers on one hand; 10 - on both hands; 20 - the total number ofhands and feet. Due to the relatively slow development of the numerical reserve, this system has existed for quite a long time period.
First improvements
With the development of the number system and the expansion of the possibilities and needs of mankind, the maximum used number in the cultures of many nations was 40. It also meant an indefinite (incalculable) amount. In Russia, the expression "forty forties" was widely used. Its meaning was reduced to the number of objects that cannot be counted. The next stage of development is the appearance of the number 100. Then the division into tens began. Subsequently, the numbers 1000, 10,000 and so on began to appear, each of which carried a semantic load similar to seven and forty. In the modern world, the boundaries of the final account are not defined. To date, the universal concept of "infinity" has been introduced.
Integer and fractional numbers
Modern calculus systems take one for the smallest number of items. In most cases, it is an indivisible value. However, with more accurate measurements, it also undergoes crushing. It is with this that the concept of a fractional number that appeared at a certain stage of development is connected. For example, the Babylonian system of money (weights) was 60 min, which was equal to 1 Talan. In turn, 1 mina was equal to 60 shekels. It was on the basis of this that Babylonian mathematics widely used sexagesimal division. Fractions widely used in Russia came to usfrom the ancient Greeks and Indians. At the same time, the records themselves are identical to the Indian ones. A slight difference is the absence of a fractional line in the latter. The Greeks wrote the numerator on top and the denominator on the bottom. The Indian version of writing fractions was widely developed in Asia and Europe thanks to two scientists: Muhammad of Khorezm and Leonardo Fibonacci. The Roman system of calculus equated 12 units, called ounces, to a whole (1 ass), respectively, duodecimal fractions were the basis of all calculations. Along with the generally accepted ones, special divisions were also often used. For example, until the 17th century, astronomers used the so-called sexagesimal fractions, which were later replaced by decimal ones (introduced by Simon Stevin, a scientist-engineer). As a result of the further progress of mankind, a need arose for an even more significant expansion of the number series. This is how negative, irrational and complex numbers appeared. The familiar zero appeared relatively recently. It began to be used when negative numbers were introduced into modern calculus systems.
Using a non-positional alphabet
What is this alphabet? For this system of calculation, it is characteristic that the meaning of the numbers does not change from their arrangement. A non-positional alphabet is characterized by the presence of an unlimited number of elements. The systems built on the basis of this type of alphabet are based on the principle of additivity. In other words, the total value of a number consists of the sum of all the digits that the entry includes. The emergence of non-positional systems occurred earlier than positional ones. Depending on the counting method, the total value of a number is defined as the difference or sum of all the digits that make up the number.
There are drawbacks to such systems. Among the main ones should be highlighted:
- introducing new numbers when forming a large number;
- inability to reflect negative and fractional numbers;
- complexity of performing arithmetic operations.
In the history of mankind, various systems of calculation were used. The most famous are: Greek, Roman, alphabetic, unary, ancient Egyptian, Babylonian.
One of the most common counting methods
Roman numeration, which has survived to this day almost unchanged, is one of the most famous. With the help of it, various dates are indicated, including anniversaries. It has also found wide application in literature, science and other areas of life. In the Roman calculus, only seven letters of the Latin alphabet are used, each of which corresponds to a certain number: I=1; V=5; x=10; L=50; C=100; D=500; M=1000.
Rise
The very origin of Roman numerals is not clear, history has not preserved the exact data of their appearance. At the same time, the fact is undoubted: the quinary numbering system had a significant impact on the Roman numbering. However, there is no mention of it in Latin. On this basis, a hypothesis arose about the borrowing by the ancient Romans of theirsystems from another people (presumably the Etruscans).
Features
Write all integers (up to 5000) by repeating the numbers described above. The key feature is the location of the signs:
- addition occurs under the condition that the larger one comes before the smaller one (XI=11);
- subtraction occurs if the smaller digit comes before the larger one (IX=9);
- the same character cannot be used more than three times in a row (for example, 90 is written XC instead of LXXXX).
The disadvantage of it is the inconvenience of performing arithmetic operations. At the same time, it existed for quite a long time and ceased to be used in Europe as the main system of calculation relatively recently - in the 16th century.
The Roman numeral system is not considered absolutely non-positional. This is due to the fact that in some cases the smaller number is subtracted from the larger one (for example, IX=9).
Method of counting in ancient Egypt
The third millennium BC is considered the moment of the emergence of the number system in ancient Egypt. Its essence was to write the numbers 1, 10, 102, 104, 105, 106, 107 with special characters. All other numbers were written as a combination of these original characters. At the same time, there was a restriction - each digit had to be repeated no more than nine times. This method of counting, which modern scientists call "non-positional decimal calculus", is based on a simple principle. Its meaning is that the written numberwas equal to the sum of all the digits of which it consisted.
Unary counting method
The number system in which one sign - I - is used when writing numbers is called unary. Each subsequent number is obtained by adding a new I to the previous one. Moreover, the number of such I is equal to the value of the number written with them.
Octal number system
This is a positional counting method based on the number 8. Numbers are displayed from 0 to 7. This system is widely used in the production and use of digital devices. Its main advantage is the easy translation of numbers. They can be converted to binary and vice versa. These manipulations are carried out due to the replacement of numbers. From the octal system, they are converted to binary triplets (for example, 28=0102, 68=1102). This counting method was widespread in the field of computer production and programming.
Hexadecimal number system
Recently, in the computer field, this method of counting is used quite actively. The root of this system is the base - 16. The calculus based on it involves the use of numbers from 0 to 9 and a number of letters of the Latin alphabet (from A to F), which are used to indicate the interval from 1010 to 1510. This method of counting, as It has already been noted that it is used in the production of software and documentation related to computers and their components. It is based on the propertiesmodern computer, the basic unit of which is 8-bit memory. It is convenient to convert and write it using two hexadecimal digits. The pioneer of this process was the IBM/360 system. The documentation for it was first translated in this way. The Unicode standard provides for writing any character in hexadecimal form using at least 4 digits.
Writing methods
The mathematical design of the counting method is based on specifying it in a subscript in the decimal system. For example, the number 1444 is written as 144410. Programming languages for writing hexadecimal systems have different syntaxes:
- in C and Java languages use "0x" prefix;
- in Ada and VHDL the following standard applies - "15165A3";
- assemblers assume the use of the letter "h", which is placed after the number ("6A2h") or the prefix "$", which is typical for AT&T, Motorola, Pascal ("$6B2");
- there are also entries like "6A2", combinations "&h", which is placed before the number ("&h5A3") and others.
Conclusion
How are calculus systems studied? Informatics is the main discipline within which the accumulation of data is carried out, the process of their registration in a form convenient for consumption. With the use of special tools, all available information is designed and translated into a programming language. It is later used forcreation of software and computer documentation. Studying various systems of calculus, computer science involves the use, as mentioned above, of different tools. Many of them contribute to the implementation of a quick translation of numbers. One of these "tools" is the table of calculus systems. It is quite convenient to use it. Using these tables, you can, for example, quickly convert a number from a hexadecimal system to binary without having special scientific knowledge. Today, almost every person interested in this has the opportunity to carry out digital transformations, since the necessary tools are offered to users on open resources. In addition, there are online translation programs. This greatly simplifies the task of converting numbers and reduces the time of operations.
