People did not immediately learn to count. Primitive society focused on a small number of objects - one or two. Everything that was bigger was called "a lot" by default. That is what is considered the beginning of the modern calculus system.
Brief historical background
In the process of development of civilization, people began to need to share small collections of objects, united by common signs. The corresponding concepts began to arise: "three", "four" and so on until "seven". However, it was a closed, limited series, the last concept in which continued to carry the semantic load of the earlier “many”. A striking example of this is folklore that has come down to us in its original form (for example, the proverb "Measure seven times - cut once").
The emergence of complex counting methods
Over time, life and all processes of human activity became more complicated. This, in turn, led to the emergence of a more complex calculus. At the same time, people used the simplest tools of counting for clarity of expression. They found them around them: they drew sticks on the walls of the cave with improvised means, made nicks, laid out numbers of interest to them from sticks and stones - this is just a small list of the diversity that existed then. Later, modern scientists assigned this species the unique name "unary calculus". Its essence is to record numbers using a single type of character. Today it is the most convenient system that allows you to visually compare the number of objects and signs. It received the greatest distribution in elementary grades of schools (counting sticks). The legacy of the "pebble bill" can be safely considered modern devices in their various modifications. The emergence of the modern word “costing”, whose roots come from the Latin calculus, is also interesting, which translates as “pebble”.
Finger count
In the conditions of an extremely scarce 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 the constant presence of an object that wanted to convey information. However, there are significant drawbacks: a significant limitation and short duration of transmission. Therefore, the whole account of people using 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 on the arms and legs. Due to the relatively slow development of the numerical reserve, this system existed for a rather long time period.
First enhancements
With the development of the calculus system and the expansion of the possibilities and needs of mankind, the maximum number used in the cultures of many peoples was 40. By it, we also understood an indefinite (non-countable) quantity. In Russia, the expression "forty magpies." 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. Next, the division into dozens has begun. 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 the unit for the smallest number of objects. In most cases, it is an indivisible quantity. However, with more accurate measurements, it is also crushed. The concept of fractional number that appeared at a certain stage of development is connected with this. For example, the Babylonian system of money (weights) was 60 minutes, which amounted to 1 talent. In turn, 1 mine was equated to 60 shekels. It was on the basis of this that Babylonian mathematics widely applied six-decimal fragmentation. Fractions widely used in Russia came to us from the ancient Greeks and Indians. At the same time, the records themselves are identical to the Indian ones. An insignificant difference is the absence of fractional features in the latter. The Greeks prescribed the numerator above, and the denominator below. The Indian version of the spelling was widely developed in Asia and Europe thanks to two scholars: Muhammad Khorezm and Leonardo Fibonacci. The Roman system of calculation equated 12 units, called ounces, to the whole (1 ass), respectively, all calculations were based on decimal fractions. Along with the generally accepted, quite often special divisions were also applied. So, for example, astronomers until the XVII century used the so-called six-decimal fractions, which were subsequently replaced by decimals (introduced into use by Simon Stevin - a scientific engineer). As a result of further progress of mankind, the need arose for an even more significant expansion of the number series. Thus appeared negative, irrational, and complex numbers. The familiar zero appeared relatively recently. It began to be applied when introducing negative numbers into modern systems.
Using the non-positional alphabet
What is such an alphabet? For this system of calculus it is characteristic that the meaning of the numbers does not change from their arrangement. The non-positional alphabet is characterized by the presence of an unlimited number of elements. The systems based on 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 before positional. Depending on the method of calculation, the total value of the number is defined as the difference or the sum of all the digits that make up the number.
There are drawbacks to such systems. Among the main should be allocated:
- the introduction of new numbers in the formation of a large number;
- inability to reflect negative and fractional numbers;
- the complexity of arithmetic.
In the history of mankind, various calculus systems have been used. The most famous are: Greek, Roman, alphabetic, unary, ancient Egyptian, Babylonian.
One of the most common ways to count
The Roman numbering, preserved to this day almost unchanged, is one of the most famous. With the help of it, various dates, including anniversaries, are indicated. She also found wide application in literature, science and other areas of life. In the Roman system of 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.
Occurrence
The very origin of the Roman numerals is incomprehensible, the history has not preserved the exact data of their appearance. At the same time, the fact is obvious: a five-fold system of calculating numbers had a significant influence on Roman numbering. However, in Latin there is no mention of it. On this basis, a hypothesis arose that the ancient Romans borrowed their system from another people (presumably from the Etruscans).
Features
All integers (up to 5000) are recorded by repeating the numbers described above. A key feature is the location of the characters:
- addition occurs under the condition that the greater is before the smaller (XI = 11);
- subtraction occurs if a smaller digit is in front of a larger one (IX = 9);
- the same character cannot stand in a row more than three times (for example, 90 is written in XC instead of LXXXX).
Its disadvantage is the inconvenience of arithmetic. 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 numeral system is not considered absolutely non-positional. This is due to the fact that in some cases, a smaller figure is subtracted from a larger one (for example, IX = 9).
Counting Method in Ancient Egypt
The third millennium BC is considered the moment of the origin of the calculus system in Ancient Egypt. Its essence consisted in writing the digits 1, 10, 102, 104, 105, 106, 107 with special characters. All other numbers were recorded as a combination of the data of the original characters. At the same time, there was a limitation - each figure had to be repeated no more than nine times. This method of counting, which modern scientists call the "non-positional decimal system of calculus," is based on a simple principle. Its meaning is that the written number was equal to the sum of all the digits of which it consisted.
Unary way of counting
A calculus system in which one character is used when writing numbers - I - 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 recorded using them.
Octal number system
This is a positional method of counting, based on the number 8. To display the numbers, a digital series from 0 to 7 is used. This 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 binary and vice versa. These manipulations are carried out thanks to the replacement of numbers. From the octal system, they are translated into binary triplets (for example, 28 = 0102, 68 = 1102). This method of counting was common in the field of computer production and programming.
Hexadecimal number system
Recently, in the computer sector, this method of counting is used quite actively. At the root of this system is the foundation - 16. The calculating system based on it involves the use of numbers from 0 to 9 and a number of letters of the Latin alphabet (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. This is based on the properties of a modern computer, the main unit of which is 8-bit memory. It is convenient to convert and record using two hexadecimal digits. The founder of this process was the IBM / 360 system. The documentation for her was first translated in this way. The Unicode standard provides for writing any character in hexadecimal using at least 4 digits.
Recording Methods
The mathematical design of the calculation method is based on indicating it in the lower index in the decimal system. For example, the number 1444 is written as 144410. The 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 applies - "1516 # 5A3 #";
- 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 records like "# 6A2", the combination "& h", which is preceded by a number ("& h5A3") and others.
Conclusion
How are calculus systems studied ? Informatics is the main discipline in which data is accumulated, the process of their design in a form convenient for consumption. With the use of special tools, all information is compiled and translated into a programming language. It is further used in the creation of software and computer documentation. Studying various calculus systems, computer science involves the use, as mentioned above, of various tools. Many of them contribute to the rapid translation of numbers. One of these "tools" is a table of calculus systems. Using it is quite convenient. 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 is able 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.