Calculus systems. Table of systems of calculation. Calculation systems: informatics

People did not immediately learn to count. Primitive society was guided by a small number of objects - one or two. All that was more, by default was named "a lot". This is considered the beginning of the modern system of calculus.

A Brief Historical Reference

In the process of the development of civilization, people began to appear the need to separate small collections of objects, united by common signs. Began to arise the corresponding concepts: "three", "four" and so on to "seven." However, it was a closed, limited series, the latter concept in which continued to carry the semantic load of the earlier "much." A vivid example of this is folk folklore, which reached us in its original form (for example, the proverb "Seven times measure - cut once").

The emergence of complex account methods

Over time, life and all processes of people's activities have become more complicated. This, in turn, led to the emergence of a more complex system of calculus. At the same time, people used the simplest calculation tools for clarity of expression. They found them around themselves: they drew sticks on the walls of the cave with improvised means, did notches, spread out the numbers of stones and stones they were interested in, just a small list of the variety that existed then. Later, modern scientists assigned this name to the unique name "unary calculus". Its essence is to write a number using a single kind of sign. Today, this is the most convenient system, allowing you to visually compare the number of objects and signs. The greatest distribution was in primary school classes (counting sticks). The legacy of "pebble account" can safely be considered modern devices in their various modifications. Interesting and the emergence of the modern word "calculation", whose roots come from the Latin calculus, which translates only "pebble."

Counting on the fingers

In the conditions of the extremely poor vocabulary of primitive man, gestures quite often served as an important addition to the information transmitted. The advantage of the fingers was in their versatility and in constant standing with the object that wanted to convey the information. However, here there are significant drawbacks: significant limitations and short-term transmission. Therefore, the entire account of people who used the "finger method", 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 on the arms and legs. Due to the relatively slow development of the numerical reserve, this system existed for quite a long time period.

First improvements

With the development of the system of calculus and the expansion of the capabilities and needs of mankind, the maximum number used in the cultures of many peoples was 40. It was also understood to mean an indefinite (not measurable) quantity. In Russia, the expression "forty magpies" became widespread. Its meaning was reduced to the number of items that can not be counted. The next stage of development is the appearance of the number 100. Then the division into dozens began. Subsequently, the numbers 1000, 10 000 and so on, each of which carried a semantic load, similar to the 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 systems of calculation for the least number of objects take a unit. In most cases, it is an indivisible quantity. However, with more accurate measurements, it also undergoes crushing. It is with this that the concept of a fractional number appeared at a certain stage of development. For example, the Babylon system of money (weights) was 60 minutes, which was 1 talan. In turn, 1 mine was equated to 60 shekels. It is on this basis that Babylonian mathematics widely used the sexagesimal crushing. Widely used in Russia fractions came to us from the ancient Greeks and Indians. The records themselves are identical to the Indian ones. A slight difference is the absence of a fractional line in the last. The Greeks were prescribed by the numerator from above, and the denominator from below. The Indian variant of writing fractions was widely developed in Asia and Europe thanks to two scientists: Mohammed Khorezmsky and Leonardo Fibonacci. The Roman system of calculus equated 12 units, called ounces, to the whole (1 acc), respectively, at the base of all the calculations were twelve fractions. Together with the generally accepted, quite often, special divisions were used. So, for example, by the 17th century astronomers used the so-called sixty-decimal fractions, which were later superseded by decimals (Simon Stevin, an engineer scientist, introduced it). As a result of the further progress of mankind, there was a need for an even more significant expansion of the numerical series. So there were negative, irrational and complex numbers. A familiar zero has appeared relatively recently. It began to be applied when introducing into the modern systems of calculus of negative numbers.

Use of the non-position alphabet

What is such an alphabet? For a given calculus system, it is characteristic that the value of the digits does not change from their arrangement. The non-position alphabet is characterized by the presence of an unlimited number of elements. At the core of systems built on the basis of this type of alphabet is the principle of additivity. In other words, the total value of a number consists of the sum of all the digits that the record includes. The emergence of non-position systems occurred earlier than positional systems. Depending on the method of calculation, the total value of the number is determined as the difference or sum of all the digits that make up the number.

There are disadvantages of such systems. Among the main should be allocated:

• The introduction of new figures in the formation of a large number;
• Impossibility to reflect negative and fractional numbers;
• Complexity of performing arithmetic operations.

In the history of mankind, various systems of calculus were used. The most famous are: Greek, Roman, alphabetic, unary, ancient Egyptian, Babylonian.

One of the most common ways to account

The Roman numbering, which has survived to this day practically unchanged, is one of the most famous. With the help of it, various dates are marked, including jubilees. It also found wide application in the literature, science and other areas of life. In the Roman system of calculation 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.

Occurrence

The very origin of Roman numerals is incomprehensible, the history has not preserved the exact data of their appearance. At the same time, there is an unquestionable fact: a significant influence on the Roman numbering was rendered by the fivefold system of calculating numbers. However, in Latin there is no mention of it. On this basis, a hypothesis arose about the borrowing by the ancient Romans of their system from another people (presumably in the Etruscans).

Features

The record of all integers (up to 5000) is made by repeating the figures described above. The key feature is the arrangement of signs:

• Addition occurs under the condition that the greater one stands before the smaller one (XI = 11);
• Subtraction occurs if the smaller figure stands before the larger (IX = 9);
• The same sign can not stand more than three times in a row (for example, 90 is written XC instead of LXXXX).

Its disadvantage is the inconvenience of performing arithmetic operations. At the same time, it existed for a rather long time and ceased to be used in Europe as the main calculus system relatively recently - in the 16th century.

The Roman system of calculation is not considered absolutely nonpositional. This is due to the fact that in a number of cases there is a subtraction of a smaller number from a larger one (for example, IX = 9).

Method of Account in Ancient Egypt

The third millennium BC is considered the moment of origin of the calculus system in Ancient Egypt. Its essence consisted in the recording of special digits 1, 10, 102, 104, 105, 106, 107. All other numbers were recorded as a combination of the data of the original signs. At the same time, there was a restriction - each figure should be repeated no more than nine times. At the heart of this method of counting, which modern scientists call the "non-position decimal system of calculus," lies a simple principle. The meaning of it is that the number written was equal to the sum of all the digits of which it consisted.

Unary account method

The system of calculus, in which, when writing numbers, one sign is used - I - is called unary. Each successive number is obtained as a result of adding a new I to the previous one. In this case, the number of such I is equal to the value of the number written with their help.

Octal system of calculation

This is a positional way of counting, at the base of which the number 8 is. Numbers from 0 to 7 are used for displaying numbers. The system has been widely used in the production and use of digital devices. Its main advantage is the easy translation of numbers. They can be converted to a binary system and vice versa. These manipulations are carried out by replacing numbers. From the octal system they are converted to binary triplets (for example, 28 = 0102, 68 = 1102). This method of account was distributed in the field of computer production and programming.

Hexadecimal system of calculation

Recently, in the computer sphere, this method of account is used quite actively. At the root of this system is the base - 16. The calculus based on it assumes the use of numbers from 0 to 9 and a series 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 Has already been noted, is used in the production of software and documentation related to computers and their components. This is based on the properties of a modern computer, the basic unit of which is 8-bit memory. It is convenient to convert and write using two hexadecimal digits. The founder 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.

Ways to write

Mathematical design of the account method is based on indicating it in the lower index in the decimal system. Example, the number 1444 is written in the form 144410. Programming languages for writing hexadecimal systems have different syntaxes:

• In C and Java languages use the prefix "0x";
• In Ada and VHDL the following standard is applied - "1516 # 5A3 #";
• Assemblers suggest the use of the letter "h", which is put after the number ("6A2h") or the prefix "$", which is typical for AT & T, Motorola, Pascal ("$ 6B2");
• Also there are records like "# 6A2", the combination "& h", which is placed before the number ("& h5A3") and others.

Conclusion

How are calculus systems studied? Informatics is the main discipline in the framework of which data is accumulated, the process of their registration in a form convenient for consumption. With the use of special tools, all available information is written and translated into a programming language. It is further used in the creation of software and computer documentation. Studying various systems of calculus, informatics involves the use, as already mentioned above, of different instruments. Many of them contribute to the implementation of the rapid translation of numbers. One such "tool" is the table of calculus systems. It is quite convenient to use it. With the help of these tables it is possible, for example, to quickly convert the number from a hexadecimal system to a binary one without having special scientific knowledge. Today, the possibility to carry out digital transformations is almost everyone interested in this person, since the necessary tools are offered to users on open resources. In addition, there are also online translation programs. This greatly simplifies the task of converting numbers and shortens the time of operations.