The positionality principle is the basic theme of our reasonings and researches. It appeared with the greatest force in the system of the notation (numbering), based on the positionalality principle or place value of figures.

The purpose of any numbering is the image of any natural number with the aid of the small group of individual marks. The most known method of notation of numbers is that, on which is based our decimal system of numeration. In this numeration all numbers from 1 up to 9 are designated by individual symbols 1, 2..., 9, to which is joined sign 0 for zero. Well-known French mathematician and physicist P.S.Laplace (1749 - 1827) thus expressed his admiration by this principle [1]:

"Idea to express all numbers by 9 marks, betraying to them, apart from value under the form, another value on the place too, it is so simple, what namely because of this simplicity is difficult to understanding as it is surprising. As not easy it was to become to this method, we see based on the example of the greatest geniuses of Greek learning Archimedes and Apollonius, from whom this idea remained latent."

The great mathematician of Russia M.V.Ostrogradsky (1801-1862) has even greater admiration expressed one [2]:

"It seems to us that after the invention of writing the largest discovery was the use by humanity of the so-called decimal notation.

We want to say that the agreement, with the aid of which we can express all useful figures by twelve words and by their endings is one of the most remarkable creations of human genius...

But immediately whether did give results this remarkable discovery, made at conceiving of society? No. On the hours of history were required fifty centuries, in order to arrive at such remarkable method, which we posses for the writing of the figures.

Only about nine centuries passed since we have learned to write the numbers with the aid of the figures, each of which has their value and the value, which depends on position. This principle of the relative value of figures is very simple and, however, only it is possible to say, randomly it became conventional, besides it is very slow, in Europe and in the remaining peace.

It seems to us that importance of this deep discovery emphasize insufficiently and attention to it give too little.

Actually, what calculations were possible before this discovery?

Everything was impeded because of the absence of this simple notation of the numbers.

All achievements of mathematical sciences, astronomy, mechanics, even chemistry depended on fulfillment in the mind of extremely difficult actions.

Now ten-year child without effort can execute calculations, which could not even visualize great Archimedes, Pythagoras or Hipparchus.

Thus, arithmetics, modest arithmetics, is relatively recent discovery. Now it is amazingly simple, if it is not complicated for the amusement by pedantic contrivances."

In these statements of two great creators of science the positionality principle is not mentioned, but it is discussed actually it, when each of them speaks about the value of the figure, which depends on the position (Ostrogradsky) or the place (Laplace). Very important is the fact that each of them specifies that, in spite of apparent simplicity of this writing system, it is the product of prolonged historical development, and in its creation is participated entire peoples, it is possible to say even that the creation of this system is the matter of entire mankind, although the realization of the very fact of the invention began considerable later (I would add that realization it is partial, but there is no complete, untill now, that it will be explained below).

Though each of them speaks about the decimal system, the decimal system itself in reality does not posses any special advantages, which release it from the positional systems with another base. In facty, positional representation with the base (or on the basis) b is determined by the rule:

where each symbol a_i obtains the value, determined: 1) by its protraction, 2) by its position in the writing of the number. Our traditional decimal notation - this, it goes without saying, that special case, when b is equal to ten, and when values a_i are selected from decimal figures 0, 1, 2, 3, 4, 5, 6, 7, 8, 9; in this case the index b in (1) can be omitted. In the general case as b is taken any integer, is larger 1 and the numbers a_i are integers of the interval: 0<a_i< b. So receive standard binary (b = 2), threefold (b = 3), fourfold (b = 4)... numeration. The choice of the basis is essentially any. The understanding of this fact means comprehension of positionality principle. There are all bases to consider that this merit belongs to B.Pascal (1623 - 1662). In 1654 he for the first time has disassembled essence of numbering with the any basis in the composition "About the divisibility of numbers deduced with the help of one addition of their figures", published in 1665.

Having acquired with the childhood an positional method of numbering we are intent to underestimate this remarkable cultural achievement of mankind. However the history of mathematics speaks [1, 3] that such highly cultured peoples of an antiquity as Egyptians and even Greeks with their amazingly thin and deep mathematical culture have not created positional numbering. Difficulties of a way to this great discovery are visible from a history of development of systems of a designation of numbers at different peoples [4, 5]. These systems (behind absolutely small exception) were not positional. Now only one of them was kept and used (in part) - Roman, in which central numbers are: I - unity, V - five, X - ten, L - fifty, C - hundred, D - five hundred, M - one thousand. Zero is not present [1, 3, 5].

It is impossible to agree with that, "ionium"-system of numbering within the limits of numbers, with which the Greek mathematics had to operate, quite met the requirements of practice, therefore there was no necessity of search of different system, more perfect. For this purpose it is enough to pay attention to necessity of introduction "octad's" (it is 10⁸) in Archimedes (287 - 212) and similar to them "tetrad's" (it is 10⁴ ) in Apollonius (260 - 170). The basic purpose of work by Archimedes "Psammit" ("Calculation grain of sand") in which were used "octad's" (see [6], consist in creation of regular reception of construction and a verbal designation of as much as big numbers (it in a modern terminology - potential practicability).

The attentive analysis of the developed numbering by these two geniuses of an antiquity shows that they rather close came to idea about positionality, but nevertheless this ingenious idea has escaped them, has escaped and accompanying idea on introduction zero, about which Van der Varden (born in 1903) has told [3]: "the most important figure is zero. It was the ingenious idea - to make something from anything, appropriate to name for this something and to invent for it a symbol".

I think that the reasons [1] treating in a counterbalance [2] are not solvent that this discovery could not be casual, because there was an occurring at different times and independent occurrence of positional system at least at three peoples: 1) more than for two thousand years B.C. in a valley of the Tiger and Euphrates at Babylonian, 2) in the beginning of our era at tribes Maya, former inhabitants of peninsula Yucatan in Central America, and 3) in VIII - IX centuries of our era in India.

All these three discovery's were (as in them the positionality principle is looked through actually) and all these three discovery's were not (or, more precisely, all of them were casual empirical receptions) because the understanding made was not, otherwise each of these systems would receive the improvement basing on positionality principle. The understanding (again incomplete, about more full will be even lower) made was fixed only by B.Pascal in the mentioned above work because the understanding of positionality principle does not mean to have a 60-notation (not having, by the way speaking, absolute character) as at Babylonian, 20-notation (with attributes 5-th) as at Maya, decimal as at Indian, and - any.

Thus, the age of positionality principle (as in part realized discovery) does not exceed 350 years. I understand that to me can specify [1] that the first precisely dated inscription in which there is a mark of zero concerns to 876 year (in it number 270 is written down with use of zero) or to cite from [3]: "... Eventually the victory was awarded the Indian figures. As well as in all different areas of culture, here on the first place there was an Italy. In 1202 the fine book on arithmetics - "Liber Abaci" by Leonardo of Pisa, nicknamed Phibonacci - has appeared.

I shall not speak that these reasons are not convincing because as the new system was torn away: in 1299 the decree of authorities of city of Florence has appeared, forbidding to use Indian figures, and in 1494 the Frankfurt burgomaster admonished sellers to know when to stop with these figures, that is not too frequently to use them. I understand that it is not so convincing. Therefore I answer on it so: occurrence of zero or the book with propagation of the Indian figures yet does not mean that propagandists understood all advantages of that they propagandized. I think that they intuitively felt that this system has advantages, but it is not enough of it, when it is not known what. And that before occurrence of the mentioned above work, B.Pascal has designed summing machine (1642), speaks about deep understanding of positionality principle: consistently carried out positional method of a designation has huge advantages to engineering of the account. It is enough to compare even multiplication in modern figures to calculation, for example, in the Roman figures:

If for the first calculation we address with tens, hundreds ... completely the same as if they were units and only we move them on one, two... places to the left, but for calculation in the Roman figures it will be perfect another. Really, for Roman there is nothing general in multiplications: CCC * LXX and III * VII.

Basic distinction between literal (anyone not positional) and positional notations not only in mthod of the account, but also in escaping attention of researchers the fact: if in positional system at writing of sizes the length of the text grows linearly, but in literal - with the final alphabet for writing of the same sizes the length of the text will grow to exponent.

The last means that if we have refused a positionality principle in representation of numbers, that the modern calculus mathematics could not exist and presence of modern computers (their presence is problematic) a little that would change.

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