'Real numbers'

'The Pythagoreans who ascertained the conception of disparate pieces , kept the breakthrough a recondite , because it is contrary to their deluxe article of belief of the unanimity of amounts in the world just ab come in us , it was hopeless to recognize their neat theory of integer basis of every subsistence, including geometric quantities . The Pythagoreans joined eternal flavor with eternal exploits of song, attributing this specific property offspring. world-wide , according to their doctrine , consisting of pure meter pool. This form of extreme high-mindedness manifested in the dedicated Trinity, the four Evangelists , the septet deadly sins and more. rise the incommensurability of the slanted of a squ ar by his side dealt a serious botch up to the entire Pythagorean school and contributed to its collapse.\nIt was short found that nesumirnist diagonal and side of a whole is no exception, that there atomic number 18 different set ​​for which it is impossible to gull the dimension of dickens ( integer ) numbers. Theodore of Cyrene ( Vst.do BC ) showed that side of the square , the ambit where lorivnyuyut 3, 5, 6, 7 , ..., 17, are repugnant with the side of the unit of measurement square. Rather than flesh out the concept of number , the Greeks came to the conclusion that it is infallible to separate the meditate of geometry, integers , set the charter limit betwixt arithmetic and geometry.\n all ludicrousity , which are solving quadratic equation equations , Euclid built strictly geometrically. Known hassle of doubling the mental block led the Greeks to the incoherentity of the highest order , they subscribe solved this line of work and also geometrically by constructing prove the existence of disparate segments of higher order.\nThe hypothesis of incommensurable value ​​provide of import back in the old age . Thus , a prominent antiquated Greek philosopher Aristotle ( 384- 3 22r.r.don.e .) Pointed out that it was surprising , handle any accredited scientific discovery.\nThe existence of disparate pieces of geometry is non impeded . The Greeks substantial the theory of the sexual relations of segments, which takes into account the accident of incommensurability , they were able to equivalence the largest much(prenominal) relation , to perform arithmetic operations on them ( in a purely geometrical form), in other words, using such ratios as numbers. To she-bop rid of absurd numbers , the Greeks utilize their approximation sufficiently accurate for unimaginative calculations. In this approximation Archimedes were scientific in nature . though Heron of Alexandria in the calculation of the area produces the square forerunner of the product of the numbers , and Diophantus of Alexandria says nothing of nonrational numbers , withal , the idea that the ratio of the lengths of incommensurable segments heap be regarded as a number, in Greek mat hs was not agnize until the end.\nSo: we tin can say that in solving problems in expanding the concept of the number of Greeks almost nothing done. How to Euclid, and , in fact, exist only for Diophantus integer. Indians and Arabs considered irrational numbers as numbers of recent species. They did not commemorate about whether legitimately add, multiply , carve up irrational numbers . For example, Bhaskar irratsionalnict destroys the denominator by multiplying numerator and denominator by the same irrational factor . The line irrational in the mathematical smack first employ in fourteen st.anhliyskyy mathematician Bradvardin (about 1290-1349 ). The concept of this precondition among the first associates (1544) German mathematician Shtifel . But it is when create verbally operations on irrational numbers can not, deal Euclid, to segments.'