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The Babylon mathematics had impact on the Greek mathematics. Traces of the Babylon six-denary numeral system kept in modern science at measurement of time and corners. Up to now division on the 60th hour min., for the 60th minutes with, circles on 360 degrees, degree for 60 min., minutes on 60s remained.

So there were first concrete fractions as certain parts of some certain measures. Only much later names of these concrete fractions started designating the same parts of other sizes, and then and abstract fractions.

The fact of existence of incommensurable pieces, nevertheless, did not slow down development of geometry in Ancient Greece. Greeks developed the theory of the relation of pieces which considered possibility of their incommensurability. They were able to compare such ratios in size, to carry out over them arithmetic actions in purely geometrical form, in other words, to use such ratios as numbers.

Written six-denary numbering of Babylonians was combined their two badges: a vertical wedge ▼, designating unit, and a conventional sign ◄, designating ten. The position numeral system for the first time occurs in the Babylon klinopisny texts. The vertical wedge designated not only 1, but also 60, 602, 603, etc. For zero in position six-denary system Babylonians had no sign in the beginning. Later the sign  replacing modern zero for office of categories among themselves was entered.

Indians considered irrational numbers as numbers of a new look, but the same arithmetic actions allowing over them, as well as over rational numbers. For example, the Indian mathematician Bkhaskara destroys irrationality in a denominator, multiplying numerator and a denominator by the same irrational multiplier. At it we meet expressions:

The natural numbers opposite to them (negative numbers and zero are called as integers. The whole and fractional numbers at the 2nd level of generalization received the general name - rational numbers. Them called also relative because any them can be presented them the relation of two integers. Each rational number can be presented as recurring periodic decimal decimal.

Over time practice of measurements and calculations showed that it is simpler and more convenient to use such measures which would have a constant relation of two next units of length and would equal to ten – the numbering basis. The metric system of measures meets these requirements.

Such beautiful theory the Cantor finished generalization of numbers at the 7th level. And so far is more abstract than it is not present: so far nothing absorbed transfinite numbers. However the truth and that transfinite numbers did not find still application outside the mathematics. The history with zero and complex numbers again repeats for transfinite numbers: what them it is possible to model? More eyelids does not know. Perhaps the Cantor generated the beautiful, but dead theory?

In Europe Leonardo Pizanscy rather close approached idea of negative quantity at the beginning of the XIII century, however in an explicit form negative numbers were applied for the first time at the end of the XV century by the French mathematician to Shyuka.