Decimal number system: basis, examples and translation into other numeral systems

Since the moment when man first realized himself as an autonomous object in the world, he looked around, interrupting the vicious circle of thoughtless survival, he began to study. I watched, compared, counted, made conclusions. It is on these seemingly elementary actions, which are now under the force and the child, began to build modern science.

With what will we work?

To begin with, it is necessary to determine what is generally a number system. This is a conditional principle of writing numbers, their visual representation, which simplifies the process of cognition. In themselves, the numbers do not exist (let Pythagoras forgive us, who considered the number the basis of the universe). This is just an abstract object, which has a physical justification only in the calculations, a kind of measure. Numbers are objects from which the number is made up.

Start

The first conscious account was of the most primitive nature. Now it is called the non-position 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 |||. Whatever one may say, three dashes are the same three dashes. If we take more similar examples, the ancient Novgorodians used the Slavic alphabet when counting. If it is necessary to select the number above the letter, simply put a ~. Also, the alphabetic number system was in favor with the ancient Romans, where numbers are again letters, but belonging to the Latin alphabet.

Because of the separateness of the ancient powers, each of them developed science on its own, who is in that much. Noteworthy is the fact that the alternative decimal number system was derived by the Egyptians. However, the "relative" of the concept we are familiar with can not be considered, since the principle of counting was different: the inhabitants of Egypt used the number ten as the base, operating with degrees.

With the development and complication of the process of cognition of the world, there was a need to allocate discharges. Imagine that you need to somehow record the strength of the army of the state, which is measured in thousands (at best). What now, endlessly write out the wand? Because of this Sumerian scientists of those years singled out the number system, in which the location of the symbol was due to its rank. Again, an example: the numbers 789 and 987 have the same "composition", but, due to the change in the arrangement of the digits, the second is much larger.

What is a decimal number system? Justification

Of course, the positiveness and regularity were not uniform for all counting methods. For example, in Babylon the number was 60, in Greece - alphabetical system (the number was letters). It is noteworthy that the method of counting the inhabitants of Babylon is alive to this day - it has found its place in astronomy.

However, the one with which the base of the number system is ten has settled down and spread, as there is a direct parallel with the fingers of human hands. Judge for yourself - alternately bending your fingers, you can count to almost infinite set.

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 written in a word and as "eighteen", and as "without two twenty." It was also Indian scientists who brought out such a notion as "zero", officially its appearance was fixed in the IX century. It was this step that became fundamental in the formation of classical positional number systems, because zero, despite what symbolizes emptiness, nothing, is capable of maintaining 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 denote emptiness, absence, and the second number is just one. Logically, they should be equal, but in practice this is far from the case. Zeros in 100,000 denote the presence of those categories, which are not in the second number. Here to you and "nothing".

Modernity

The decimal number system consists of numbers from zero to nine. The numbers compiled in its framework are built on the following principle:

The rightmost digit denotes units, shift one step to the left - get dozens, one more step to the left - hundreds and so on. Complicated? Nothing like this! In fact, the decimal system examples can provide very visual, take at least the number 666. It consists of three digits 6, each of which indicates its rank. And this form of recording is collapsed. If you want to emphasize what kind of number it is, you can deploy it, giving it a written form to what your inner voice "speaks" each time you see a number - "six hundred and sixty-six." The writing itself includes all the same units, tens and hundreds, that is, each position digit is multiplied by a certain power of 10. The unfolded form is the following expression:

666 ₁₀ = 6х10 ² + 6 * 10 ¹ + 6 * 10 ⁰ = 600 + 60 + 6.

Topical Alternatives

The second most popular after the decimal number system is a fairly young version - binary (binary). She appeared because of the omnipresent Leibniz, who believed that in particularly difficult cases, in the study of the theory of numbers, binarity would be more convenient than decimal. Its widespread distribution, it has received with the development of digital technologies, since it has a number 2 in the base, and the elements in it are made up of digits 1 and 2. Coding information occurs in this system, since 1 - the presence of a signal, 0 - its absence. On the basis of this principle, several illustrative examples demonstrating the conversion to the decimal number system can be shown.

With the passage of time, the processes associated with programming became more complex, and therefore introduced ways of writing numbers that have 8 and 16 at the bottom. Why exactly? Firstly, the number of characters is greater, which means that the number itself will be shorter, and secondly, the basis of this is the power of two. The octal system consists of the digits 0-7, and the hexadecimal is the same digits as the decimal, plus the letters A through F.

Principles and methods of translation of the number

Translate into the decimal number system simply, 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 digit degree.

The basic formula for computing:

_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 fix, 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 problem, because the system includes the translation of fractional numbers, for this we consider separately the integer and separately fractional part - 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, the examples of which we considered above, are still "on horseback", and it is not worth writing off it. It is she who becomes the mathematical basis in the school, on her example the laws of mathematical logic are learned, the ability to build reconciled relationships is derived. But what really there - almost the whole world uses this system, not embarrassed by its irrelevance. The reason for this is one: it is convenient. In principle, you can take out the basis of the account, it will be an apple, if necessary, but why complicate it? Perfectly verified number of digits can be counted if necessary and on fingers.