# Richard Dedekind

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caption =Richard Dedekind, c. 1850

birth_date =October 6 ,1831

birth_place = Braunschweig

death_date =February 12 ,1916

death_place = Braunschweig

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nationality = German

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field = mathematician

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known_for =abstract algebra algebraic number theory real number s

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footnotes =**Julius Wilhelm Richard Dedekind**(October 6 ,1831 –February 12 ,1916 ) was a German mathematician who did important work inabstract algebra ,algebraic number theory and the foundations of thereal number s.**Life**Dedekind was the youngest of four children of Julius Levin Ulrich Dedekind. As an adult, he never employed the names Julius Wilhelm. He was born, lived most of his life, and died in Braunschweig (often called "Brunswick" in English).

In 1848, he entered the Collegium Carolinum in Braunschweig, where his father was an administrator, obtaining a solid grounding in mathematics. In 1850, he entered the

University of Göttingen . Dedekind studiednumber theory underMoritz Stern . Gauss was still teaching, although mostly at an elementary level, and Dedekind became his last student. Dedekind received his doctorate in 1852, for a thesis titled "Über die Theorie der Eulerschen Integrale" ("On the Theory ofEulerian integral s"). This thesis did not reveal the talent evident on almost every page Dedekind later wrote.At that time, the University of Berlin, not Göttingen, was the leading center for mathematical research in Germany. Thus Dedekind went to Berlin for two years of study, where he and Riemann were contemporaries; they were both awarded the

habilitation in 1854. Dedekind returned to Göttingen to teach as a "Privatdozent", giving courses onprobability andgeometry . He studied for a while with Dirichlet, and they became close friends. Because of lingering weaknesses in his mathematical knowledge, he studied elliptic and abelian functions. Yet he was also the first at Göttingen to lecture onGalois theory . Around this time, he became one of the first to understand the fundamental importance of the notion of groups foralgebra andarithmetic .In 1858, he began teaching at the Polytechnic in

Zürich . When the Collegium Carolinum was upgraded to a "Technische Hochschule " (Institute of Technology) in 1862, Dedekind returned to his native Braunschweig, where he spent the rest of his life, teaching at the Institute. He retired in 1894, but did occasional teaching and continued to publish. He never married, instead living with his unmarried sister Julia.Dedekind was elected to the Academies of Berlin (1880) and Rome, and to the Paris Académie des Sciences (1900). He received honorary doctorates from the universities of Oslo, Zurich, and Braunschweig.

**Work**While teaching calculus for the first time at the Polytechnic, Dedekind came up with the notion now called a

Dedekind cut (in German: "Schnitt"), now a standard definition of the real numbers. The idea behind a cut is that anirrational number divides therational number s into two classes (sets), with all the members of one class (upper) being strictly greater than all the members of the other (lower) class. For example, the square root of 2 puts all the negative numbers and the numbers whose squares are less than 2 into the lower class, and the positive numbers whose squares are greater than 2 into the upper class. Every location on the number line continuum contains either a rational or an irrational number. Thus there are no empty locations, gaps, or discontinuities. Dedekind published his thought on irrational numbers andDedekind cut s in his paper " [*http://www.ru.nl/w-en-s/gmfw/bronnen/dedekind2.html Stetigkeit und irrationale Zahlen*] " ("Continuity and irrational numbers." Ewald 1996: 766. Note that Dedekind's terminology is old-fashioned: in the present context, one now says "Vollständigkeit" instead of "Stetigkeit", so a modern translation would have "continuity" replaced with "completeness").In 1874, while on holiday in

Interlaken , Dedekind met Cantor. Thus began an enduring relationship of mutual respect, and Dedekind became one of the very first mathematicians to admire Cantor's work on infinite sets, proving a valued ally in Cantor's battles with Kronecker, who was philosophically opposed to Cantor'stransfinite numbers .If there existed a

one-to-one correspondence between two sets, Dedekind said that the two sets were "similar." He invoked similarity to give the first precise definition of an infinite set: a set is infinite when it is "similar to a proper part of itself," in modern terminology, isequinumerous to one of its proper subsets. (This is known asDedekind's theorem .) Thus the set**N**ofnatural number s can be shown to be similar to the subset of**N**whose members are the squares of every member of**N**, (**N**→**N**^{2}):**N**1 2 3 4 5 6 7 8 9 10 ... ↓**N**^{2}1 4 9 16 25 36 49 64 81 100 ...Dedekind edited the collected works of

Dirichlet , Gauss, andRiemann . Dedekind's study of Dirichlet's work was what led him to his later study ofalgebraic number field s and ideals. In 1863, he published Dirichlet's lectures onnumber theory as "Vorlesungen über Zahlentheorie " ("Lectures on Number Theory") about which it has been written that:"Although the book is assuredly based on Dirichlet's lectures, and although Dedekind himself referred to the book throughout his life as Dirichlet's, the book itself was entirely written by Dedekind, for the most part after Dirichlet's death." (Edwards 1983)

The 1879 and 1894 editions of the "Vorlesungen" included supplements introducing the notion of an ideal, fundamental to ring theory. (The word "Ring", introduced later by

Hilbert , does not appear in Dedekind's work.) Dedekind defined an ideal as a subset of a set of numbers, composed ofalgebraic integer s that satisfy polynomial equations withinteger coefficients. The concept underwent further development in the hands ofHilbert and, especially, ofEmmy Noether . Ideals generalizeErnst Eduard Kummer 'sideal number s, devised as part of Kummer's 1843 attempt to proveFermat's Last Theorem . (Thus Dedekind can be said to have been Kummer's most important disciple.) In an 1882 article, Dedekind andHeinrich Martin Weber applied ideals toRiemann surface s, giving an algebraic proof of theRiemann-Roch theorem .Dedekind made other contributions to algebra. For instance, around 1900, he wrote the first papers on

modular lattice s.In 1888, he published a short monograph titled "Was sind und was sollen die Zahlen?" ("What are numbers and what should they be?" Ewald 1996: 790), which included his definition of an infinite set. He also proposed an

axiom atic foundation for thenatural number s, whose primitive notions were one and thesuccessor function . The following year, Peano, citing Dedekind, formulated an equivalent but simpler set of axioms, now the standard ones.**ee also***

Dedekind domain

*Dedekind eta function

*Dedekind-infinite set

*Dedekind sum

*Dedekind zeta function

*Ideal (ring theory)

*Ideal number

*Vorlesungen über Zahlentheorie **Bibliography**Primary literature in English:

*1890. "Letter to Keferstein" inJean van Heijenoort , 1967. "A Source Book in Mathematical Logic, 1879-1931". Harvard Univ. Press: 98-103.

* 1963 (1901). "Essays on the Theory of Numbers". Beman, W. W., ed. and trans. Dover. Contains English translations of " [*http://www.ru.nl/w-en-s/gmfw/bronnen/dedekind2.html Stetigkeit und irrationale Zahlen*] " and "Was sind und was sollen die Zahlen?"

* 1996. "Theory of Algebraic Integers". Stillwell, John, ed. and trans. Cambridge Uni. Press. A translation of "Über die Theorie der ganzen algebraischen Zahlen".

* Ewald, William B., ed., 1996. "From Kant to Hilbert: A Source Book in the Foundations of Mathematics", 2 vols. Oxford Uni. Press.

**1854. "On the introduction of new functions in mathematics," 754-61.

**1872. "Continuity and irrational numbers," 765-78. (translation of "Stetigkeit...")

**1888. "What are numbers and what should they be?", 787-832. (translation of "Was sind und...")

**1872-82, 1899. Correspondence with Cantor, 843-77, 930-40.Secondary:

*Edwards, H. M., 1983, "Dedekind's invention of ideals," "Bull. London Math. Soc. 15": 8-17.

*cite book

author =William Everdell

year = 1998

title = _es. "The First Moderns"

publisher =University of Chicago Press

location = Chicago

id = ISBN 0-226-22480-5

*Gillies, Douglas A., 1982. "Frege, Dedekind, and Peano on the foundations of arithmetic". Assen, Netherlands: Van Gorcum.

*Ivor Grattan-Guinness , 2000. "The Search for Mathematical Roots 1870-1940". Princeton Uni. Press.There is an [

*http://www-groups.dcs.st-and.ac.uk/~history/References/Dedekind.html online bibliography*] of the secondary literature on Dedekind. Also consult Stillwell's "Introduction" to Dedekind (1996).**External links***

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* [*http://www.archive.org/details/essaysintheoryof00dedeuoft Dedekind, Richard, "Essays on the Theory of Numbers." Open Court Publishing Company, Chicago, 1901.*] at theInternet Archive .Persondata

NAME=Dedekind, Julius Wilhelm Richard

ALTERNATIVE NAMES=

SHORT DESCRIPTION=Mathematician specializing inabstract algebra andreal number s

DATE OF BIRTH=1831-10-06

PLACE OF BIRTH=Braunschweig ,Germany

DATE OF DEATH=1916-02-12

PLACE OF DEATH=Braunschweig ,Germany

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**Richard Dedekind**— Naissance 6 octobre 1831 Brunswick … Wikipédia en Français**Richard Dedekind**— Porträt (1850) Julius Wilhelm Richard Dedekind (* 6. Oktober 1831 in Braunschweig; † 12. Februar 1916 ebenda) war ein deutscher Mathematiker. Inhaltsverzeichnis … Deutsch Wikipedia**Julius Wilhelm Richard Dedekind**— Richard Dedekind fundamentó la teoría de la recta real y creó la teoría de los ideales Julius Wilhelm Richard Dedekind (6 de octubre de 1831 12 de febrero de 1916), matemático alemán. Dedekind nació en Brunswick (Braunschweig en alemán), el más… … Wikipedia Español**Julius Wilhelm Richard Dedekind**— (6 de octubre de 1831 12 de febrero de 1916), matemático alemán. Dedekind nació en Brunswick (Braunschweig en alemán), el más joven de los cuatro hijos de Julius Levin Ulrich Dedekind. Vivió con Julia, su hermana soltera, hasta que falleció en… … Enciclopedia Universal**Dedekind cut**— Dedekind used his cut to construct the irrational, real numbers. In mathematics, a Dedekind cut, named after Richard Dedekind, is a partition of the rationals into two non empty parts A and B, such that all elements of A are less than all… … Wikipedia**Dedekind-Unendlichkeit**— ist ein Begriff aus der Mathematik, der eine scheinbar paradoxe Eigenschaft unendlicher Mengen einfängt. Eine endliche Menge M, etwa mit n Elementen, ist niemals zu einer echten Teilmenge gleichmächtig, d.h., es kann keine bijektive Abbildung von … Deutsch Wikipedia**Dedekind-endlich**— Dedekind Unendlichkeit ist ein Begriff aus der Mathematik, der eine scheinbar paradoxe Eigenschaft unendlicher Mengen einfängt. Eine endliche Menge M, etwa mit n Elementen, ist niemals zu einer echten Teilmenge gleichmächtig, d.h., es kann keine… … Deutsch Wikipedia**Dedekind-unendlich**— Dedekind Unendlichkeit ist ein Begriff aus der Mathematik, der eine scheinbar paradoxe Eigenschaft unendlicher Mengen einfängt. Eine endliche Menge M, etwa mit n Elementen, ist niemals zu einer echten Teilmenge gleichmächtig, d.h., es kann keine… … Deutsch Wikipedia**Dedekind**— is the name of: People Brendon Dedekind (born 1976), South African swimmer Friedrich Dedekind (1524 1598), German humanist, theologian, and bookseller Richard Dedekind (1831 1916), German mathematician Other 19293 Dedekind, asteroid named after… … Wikipedia**DEDEKIND (R.)**— Le mathématicien allemand Richard Dedekind est un des fondateurs de l’algèbre moderne. Sa théorie des idéaux, systématisation et rationalisation des « nombres idéaux» de Kummer, est en effet devenue l’outil essentiel pour étudier la divisibilité… … Encyclopédie Universelle