From the moment that a person first realized himself as an autonomous object in the world, looked around, breaking the vicious circle of thoughtless survival, he began to study. He looked, compared, believed, drew conclusions. It is on these seemingly elementary actions that are now under the power of the child, modern science began to be based.
What will we work with?
To begin with, it is necessary to determine what the number system is in general. This is a conditional principle of recording numbers, their visual representation, which simplifies the process of cognition. The numbers themselves do not exist (forgive us Pythagoras, who considered the number the basis of the universe). This is just an abstract object that has physical justification only in calculations, a kind of measure. Numbers are the objects from which the number is composed.
Start
The first conscious account was of the most primitive nature. Now it is customary to call it a non-positional number system. In practice, it is a number in which the position of its constituent elements is unimportant. Take, for example, ordinary dashes, each of which corresponds to a specific object: three people are equivalent to |||. Like it or not, three dashes are the same three dashes. If we take closer examples, then the ancient Novgorodians used the Slavic alphabet when counting. If it was necessary to select exactly the numbers above the letter, they simply put the ~ sign. Also, the alphabetic number system was held in high esteem by the ancient Romans, where numbers are again letters, but already belonging to the Latin alphabet.
Due to the isolation of the ancient powers, each of them developed science on their own, who is what much.
It is noteworthy that an alternative decimal number system was derived by the Egyptians. However, it cannot be considered a “relative” of the usual notion, since the principle of counting was different: the inhabitants of Egypt used the number ten as a basis, operating in degrees.
With the development and complication of the process of cognition of the world, the need arose to distinguish categories. Imagine that you need to somehow fix the size of the state’s army, which is measured in thousands (at best). Well now, endlessly write out sticks? Because of this, the Sumerian scientists of those years identified a number system in which the location of the symbol was due to its discharge. Again, an example: the numbers 789 and 987 have the same “composition”, but, due to a change in the arrangement of numbers, the second is much larger.
What is a decimal number system? Justification
Of course, positioning and regularity were not uniform for all methods of counting. For example, in Babylon, the base was the number 60, in Greece - the alphabetical system (the number were letters). It is noteworthy that the method of counting the inhabitants of Babylon is still alive today - it has found its place in astronomy.
However, one has taken root and spread, in which the base of the number system is a dozen, since an open parallel can be traced with the fingers of human hands. Judge for yourself - alternately bending your fingers, you can count almost to an infinite number.
The beginning of this system was laid in India, and it appeared immediately on the basis of "10". The formation of the names of numbers was twofold - for example, 18 could be spelled with the word both as “eighteen” and as “without two twenty”. Also, it was Indian scientists who derived such a concept as “zero”, officially its appearance was recorded in the 9th century. It was this step that became fundamental in the formation of classical positional number systems, because zero, despite the fact that it symbolizes emptiness, nothing is able to maintain the digit capacity of the number so that it does not lose its meaning. For example: 100000 and 1. The first number includes 6 digits, the first of which is a unit, and the last five indicate emptiness, absence, and the second number is simply a unit. Logically, they should be equal, but in practice this is far from the case. Zeros of 100,000 indicate the presence of those digits that are not in the second number. So much for nothing.
Modernity
The decimal number system consists of numbers from zero to nine. The numbers compiled within its framework are constructed according to the following principle:
the rightmost digit denotes units, shift one step to the left - get tens, another step to the left - hundreds and so on. Difficult? Nothing like this! In fact, the decimal system can provide very illustrative examples, take at least the number 666. It consists of three digits 6, each of which designates its own category. Moreover, this form of recording is minimized. If you want to emphasize what kind of number it is, you can expand it by giving a written form to what your inner voice “says” every time you see the number - “six hundred sixty six”. The spelling itself includes all the same units, tens and hundreds, that is, each digit of the position is multiplied by a certain power of 10. The expanded form is the following expression:
666 10 = 6x10 2 + 6 * 10 1 + 6 * 10 0 = 600 + 60 + 6.
Actual alternatives
The second most popular after the decimal number system is a fairly young variety - binary (binary). It appeared thanks to the ubiquitous Leibniz, who believed that in particularly difficult cases in the study of number theory, binary would be more convenient than decimal. It received its widespread use with the development of digital technologies, as it has the number 2 in the base, and the elements in it are made up of the numbers 1 and 2.
Information coding takes place in this system, since 1 is the presence of a signal, 0 is its absence. Based on this principle, several illustrative examples can be shown that demonstrate the conversion to the decimal number system.
Over time, the processes associated with programming became more complicated, so they introduced ways to write numbers with 8 and 16 at the base. Why are they? Firstly, the number of characters is greater, which means that the number itself will be shorter, and secondly - they are based on the power of two. The octal system consists of the numbers 0-7, and the hexadecimal system consists of the same digits as the decimal, plus the letters A to F.
Principles and methods of number translation
To translate into a decimal number system is simple, it is enough to adhere to the following principle: the original number is written as a polynomial, which consists of the sums of the products of each number on the basis of "2", raised to the corresponding bit depth.
The basic formula for calculating:
x2 = y k 2 k-1 + y k-1 2 k-2 + y k-2 2 k-3 + ... + y 2 2 1 + y 1 2 0 .
Translation Examples
To consolidate, consider several expressions:
101111 2 = (1x2 5 ) + (0x2 4 ) + (1x2 3 ) + (1x2 2 ) + (1x2 1 ) + (1x2 0 ) = 32 + 8 + 4 + 2 + 1 = 47 10 .
We complicate the task, because the system includes the translation of fractional numbers, for this we consider separately the whole and separately fractional parts - 111110.11 2. So:
111110.11 2 = (1x2 5 ) + (1x2 4 ) + (1x2 3 ) + (1x2 2 ) + (1x2 1 ) + (0x2 0 ) = 32 + 16 + 8 + 4 + 2 = 62 10 ;
11 2 = 2 -1 x1 + 2 -2 x1 = 1/2 + 1/4 = 0.75 10.
As a result, we get that 111110.11 2 = 62.75 10 .
Conclusion
Despite all the “antiquity”, the decimal number system, examples of which we examined above, is still “on the horse”, and you should not write it off. It is she who becomes the mathematical basis in school, the laws of mathematical logic are learned on her example, the ability to build verified relationships is derived. But what is it - almost the whole world uses this system, not embarrassed by its irrelevance. There is only one reason for this: it is comfortable. In principle, you can deduce the basis of the account any, it will even become an apple if necessary, but why complicate it? An ideal number of digits can be counted if necessary and on the fingers.