# Dedekind, Julius Wilhelm Richard

### theory irrational logical arithmetic

[day duhkint] (1831–1916) German mathematician: made far-reaching contributions to number theory.

Dedekind studied at Brunswick and then formed a close association with at Göttingen, with each influencing the others. Dedekind learned about the method of least squares from Gauss, the theory of numbers, potential theory and partial differential equations from Dirichlet. After a short time he moved briefly to Zürich, and then returned to spend the rest of his long life as a professor at the Technical High School, Brunswick. He lived long enough for much of his influential work (eg on irrational numbers) to become familiar to a generation of students in his later years, and he became a legend. Some 12 years too soon, Teubner’s Calendar for Mathematicians recorded him as having died on 4 September 1899. Much amused, Dedekind wrote to the editor: ‘According to my own memorandum I passed this day in perfect health and enjoyed a very stimulating conversation … with my … friend Georg Cantor of Halle’.

Dedekind was one of the first to recognize the value of the work of Cantor (1845–1918) on infinite qualities. Dedekind himself made major steps towards modern standards of rigour and was ahead of his time in his approach to number theory.

In 1858 he produced an arithmetic definition of continuity and clarified the concept of an irrational number (that is, roughly, a number that cannot be represented as a fraction). In the first of three books he used Dedekind cuts (the categorization of irrational numbers by fractions) to rigorously examine the real number system.

Then in his second major work (1888) he established a logical foundation for arithmetic and described axioms that exactly represent the logical concept of whole numbers (these are now, incorrectly, called axioms). Finally, Dedekind described in 1897–1900 the factorization of real numbers using modern algebra.

## User Comments