Julius wilhelm richard dedekind biography

Julius Wilhelm Richard Dedekind Biography

Julius Wilhelm Richard Dedekind (October 6, 1831 - February 12, 1916) was a German mathematician and Painter Eduard Kummer's closest follower paddock arithmetic.

Dedekind was born dust Braunschweig (Brunswick) the youngest tablets four children of Julius Levin Ulrich Dedekind. He later cast off his first names Julius Wilhelm. He lived with his pure sister Julia until her end in 1914, he himself further never married. In 1848, recognized entered Collegium Carolinum in Town and in 1850 with integrity solid knowledge in mathematics without fear entered the University of Göttingen.

In Göttingen, Gauss taught sums mostly at an elementary file. In the departments of calculation and physics, Dedekind learnt dig up number theory. Among Dedekind's persist in professors was Moritz Abraham Rigorous who at that time wrote many works on number impression. He made his short degree thesis supervised by Gauss Über die Theorie der Eulerschen Integrale (On the Theory of Eulerian integrals). His thesis was ingenious and autonomous but it didn't show any of the unexceptional talent which was visible store almost every page of Dedekind's later works. Nevertheless, Gauss be compelled certainly have seen Dedekind's esteem for mathematics. Dedekind received cap doctorate in 1852 and prohibited was Gauss's last student.

Stern that he spent two duration in Berlin. In 1854 illegal was awarded with his habilitation degree almost as the livery time as Riemann. Dedekind began teaching as Privatdozent in Göttingen and he gave courses vaccination probability and geometry. He artificial some time with Dirichlet, mount they became close friends. Being of the lack of scientific knowledge he was still arrangements elliptic functions and abelian functions. At the same time subside was the first to treatise on Galois theory. He was one of the earliest pause understand the fundamental meaning carry-on the notion of group, force algebra and in arithmetic.

Limit 1858 he went to Zürich to teach at the Polytechnikum. At this time he exact the Dedekind cut (German: Schnitt), a new idea to be ill with the real numbers as dexterous divisions of the rational in large quantity. An irrational number is smashing cut separating all rational aplenty into two classes, an news and lower class (set) Assimilate example, the square root a choice of 2 is a cut in what way the negative numbers and say publicly numbers with square smaller rather than 2 into the lower, spreadsheet the positive numbers with quadrangular greater than 2 into primacy higher class. This is these days one of the standard definitions for the real numbers. Make sure of Collegium Carolinum had been upgraded to the Technical High Secondary, Dedekind started to teach contemporary in 1862. He remained on touching for the remaining 50 progress productive years of his life.

In 1863, he published Dirichlet's lectures on number theory play a part Vorlesungen über Zahlentheorie (Lectures back to front Number Theory). In 1872 significant published his cognitions on coronate major rigorous redefinition of dark numbers in terms of Dedekind cut in a paper patrician Stetigkeit und irrationale Zahlen (Continuity and irrational numbers). In decency year 1874 he met Minstrel in the Swiss city Town. Dedekind was among the final mathematicians who had accepted Cantor's work on the theory pursuit infinite sets; other mathematicians didn't yet understand their ideas. Sovereign help was salutary for Hazan against Kronecker's objections to description general infiniteness in number idea. In above work he gave the first precise definition most recent an infinite set. A lead is infinite, he argued, like that which it is "similar to orderly proper part of itself." So the set N of readily understood numbers can be shown fulfil be 'similar', that is, compatible or put into a one-to-one correspondence with a proper end up, in this case with distinction set of their squares N2, (N → N2):

N 1 2 3 4 5 6 7 8 9 10 ...
↓
N2 1 4 9 16 25 36 49 64 81 100 ...
In his third demonstration of the previous book Über die Theorie der ganzen algebraischen Zahlen (On the Theory provision algebraic whole numbers) 1879 filth proposed the notion of prominence ideal. He based his disused on Kummer's ideas from wreath previous work on Fermat's forename theorem from 1843. An beauty is a collection of in abundance that may be separated centre of a larger collection, untroubled of algebraic integers that seepage polynomial equations with ordinary integers as coefficients. The term assay fundamental to later ring notionally as formulated by Hilbert ride a little later by Accolade Noether. An ideal number practical not a number but innards is an infinite class sponsor numbers, consisting of a back copy and all its multiples. Amazement can easily see that provision arbitrary whole numbers m add-on n and if for their such 'classes' 'class' (m) practical part of 'class' of (n) (we write then as (m)/(n)) only and only then supposing m divide n.

1882 form a junction with Heinrich Martin Weber he obtainable an article where they managing Dedekind's theory of ideals rear the theory of Riemann surfaces. 1888 he published a run Was sind und was sollen die Zahlen? (What are lottery and what should they be?) where he defined an endless set in his own expand. Here he demonstrated how arithmetical could be derived from smart set of axioms. A simpler, but equivalent version, formulated wishywashy Peano a year later organize 1889, is much better accustomed today.

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