The origin of a six-denary numeral system at Babylonians is connected as scientists believe, that the Babylon monetary and weight units of measure were subdivided owing to historical conditions into 60 equal parts:
In Ancient Greece there were two systems of written numbering: attic and Ionic or alphabetic. They were so called on Ancient Greek areas - Attica and Ionia. In the attic system called also gerodianovy the majority of numerical signs are the first letters of the Greek corresponding numerals, for example, of GENTE (Ghent or cent – five, Δ (Dec – ten, etc. This system was applied in Attica till the I century AD, but in other areas of Ancient Greece it was replaced with more convenient alphabetic numbering which quickly extended across all Greece even earlier.
Some size (length, volume, weight, etc.) is always the cornerstone of any measurement. The need for more exact measurements led to that initial units of measure started splitting up for 2, 3 and more parts. To smaller unit of measure which was received as a result of smashing, gave the individual name, and sizes measured already by this smaller unit.
To imaginary numbers there was no place on a coordinate axis. However scientists noticed that if to take a real number of b on positive part of a coordinate axis and to increase it on, we will receive imaginary number b, it is not known where located. But if once again to increase this number on, we will receive - b, that is initial number, but already on negative part of a coordinate axis. So, two umnozheniye on we threw number b with positive in negative, and exactly on the middle of this throw the number was imaginary. So found a place to imaginary numbers in points on an imaginary coordinate axis, perpendicular to the middle of the valid coordinate axis. Plane points between imaginary and valid axes represent the numbers found Cardano which in a general view of a + b · i contain real numbers and and imaginary b · i in one complex (structure therefore are called as complex numbers.
Gradually the technology of operations over imaginary numbers developed. At a turn of XVII and XVII centuries the general theory of roots of n-nykh of degrees at first from negative, and then was constructed of any complex numbers, based on the following formula of the English mathematician A. Moivre:
Only after emergence of geometry of Descartes (1637 application irrational as however, and negative numbers began. Descartes's ideas led to generalization of concept about number. Between points of a straight line and numbers univocity was defined. The variable was entered into mathematics.
Algebraic call numbers which are roots of algebraic polynomials with the whole coefficients, for example, 4. All others (nonalgebraic numbers belong to the transcendental. As each rational number of p/q is a root of the corresponding polynomial of the first degree with the whole coefficients of qx – p, all transcendental numbers are irrational.
It was talked of search and research of size which we designate now. Opening of the fact that between two pieces – the party and a diagonal of a square – does not exist the general measure, led to the real crisis of fundamentals, at least, of Ancient Greek mathematics.
In the V-VI centuries negative numbers appear and very widely extend in the Indian mathematics. In India negative numbers systematically used generally as we do it now.
Inconveniently to manage only natural numbers. For example, them it is impossible to subtract bigger of the smaller. Negative numbers were for such a case entered: Chinese – in the X century BC, Indians – in the VII century, Europeans – only in the XIII century.
The sixtieth shares were habitual in life of Babylonians. That is why they used the six-denary fractions having a denominator always number 60 or its degrees: 602 = 3600, 603 = 216000, etc. In this regard six-denary fractions can be compared to our decimal fractions.
Let's mark out characteristics considered (natural, rational, valid) numbers: they model only one property – quantity; they are odnomerna and all are represented by points on one direct, called coordinate axis.