Monday, October 6, 2014

Richard Dedekind and the Real Numbers

Richard Dedekind
(1831 – 1916)
On October 6, 1831, German mathematician Julius Wilhelm Richard Dedekind was born. He is known for making important contributions to abstract algebra (particularly ring theory), algebraic number theory and the foundations of the real numbers.

Richard Dedekind was the son of an administrator at Collegium Carolinum in Braunschweig. His family was known as pretty influential and the young Richard enrolled in 1848 at the Collegium in order to study mathematics as well. Two years later, he continued his studies in Göttingen and was taught by the famous Carl Friedrich Gauß who motivated him to write his dissertation in 1852 'On the Theory of Eulerian Integrals'.

However, Dedekind preferred to continue his studies at the University of Berlin, which was known to have the best reputation in the field of mathematics. There, he was awarded the habilitation and returned to Göttingen shortly after, where he was appointed Privatdozent of geometry and probability. It is assumed that Dedekind was the first to teach Galois' theory and to understand the notion of groups in algebra and arithmetic. After a short period of teaching in Zurich, Switzerland, Dedekind returned to his native Braunschweig where he spent the rest of his working career, because the Collegium Carolinum was upgraded to an Institute of Technology. [1]

One of Dedekind's best known contributions to mathematics is the 'Dedekind cut'. The idea behind a cut is that an irrational number divides the rational numbers into two classes, with all the members of one class being strictly greater than all the members of the other class. His thought on irrational numbers and Dedekind cuts was published in his pamphlet "Continuity and irrational numbers", in modern terminology better known as completeness. In the later 1870s, Dedekind began his first works fundamental to ring theory, even though the term 'ring' never appeared in his writings. Dedekind defined an ideal as a subset of a set of numbers, composed of algebraic integers that satisfy polynomial equations with integer coefficients. The concept underwent further development in the hands of Hilbert and, especially, of Emmy Noether. [2]

Dedekind was elected to the Academies of Berlin and Rome, and to the French Academy of Sciences. He received honorary doctorates from the universities of Oslo, Zurich, and Braunschweig.

At yovisto, you may be interested in a lecture on Gauss' Law.

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