Richard Dedekind

Richard Dedekind (born October 6, 1831, Braunschweig, duchy of Braunschweig [Germany]—died February 12, 1916, Braunschweig) was a German mathematician who developed a major redefinition of irrational numbers in terms of arithmetic concepts. Although not fully recognized in his lifetime, his treatment of the ideas of the infinite and of what constitutes a real number continues to influence modern mathematics.

Dedekind was the son of a lawyer. While attending the Gymnasium Martino-Catharineum in 1838–47 in Braunschweig, he was at first interested primarily in chemistry and physics. At the Caroline College in 1848–50, however, he turned to calculus, algebra, and analytic geometry, which helped qualify him to study advanced mathematics at the University of Göttingen under the mathematician Carl Friedrich Gauss.

After two years of independent study of algebra, geometry, and elliptic functions, Dedekind served as Privatdozent (“unsalaried lecturer”) in 1854–58 at the University of Göttingen, where, in his lectures, he introduced, probably for the first time, the Galois theory of equations and attended the lectures of the mathematician Peter Gustav Lejeune Dirichlet. These experiences led Dedekind to see the need for a redefinition of irrational numbers in terms of arithmetic properties. The geometric approach had led Eudoxus in the 4th century bce to define them as approximations by rational numbers (e.g., a series of nonrepeating decimals, as Square root of√2 = 1.414213 . . . ).

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Numbers and Mathematics

In 1858 Dedekind joined the faculty of the Zürich Polytechnic, where he remained for five years. In 1862 he accepted a position in the Technical High School in Braunschweig, where he remained in comparative isolation for the rest of his life.

While teaching there, Dedekind developed the idea that both rational and irrational numbers could form a continuum (with no gaps) of real numbers, provided that the real numbers have a one-to-one relationship with points on a line. He said that an irrational number would then be that boundary value that separates two especially constructed collections of rational numbers.

Dedekind perceived that the character of the continuum need not depend on the quantity of points on a line segment (or continuum) but rather on how the line submits to being divided. His method, now called the Dedekind cut, consisted in separating all the real numbers in a series into two parts such that each real number in one part is less than every real number in the other. Such a cut, which corresponds to a given value, defines an irrational number if no largest or no smallest is present in either part; whereas a rational is defined as a cut in which one part contains a smallest or a largest. Dedekind would therefore define the square root of 2 as the unique number dividing the continuum into two collections of numbers such that all the members of one collection are greater than those of the other, or that cut, or division, separating a series of numbers into two parts such that one collection contains all the numbers whose squares are larger than 2 and the other contains all the numbers whose squares are less than 2.

Dedekind developed his arithmetical rendering of irrational numbers in 1872 in his Stetigkeit und Irrationale Zahlen (Eng. trans., “Continuity and Irrational Numbers,” published in Essays on the Theory of Numbers). He also proposed, as did the German mathematician Georg Cantor, two years later, that a set—a collection of objects or components—is infinite if its components may be arranged in a one-to-one relationship with the components of one of its subsets. By supplementing the geometric method in analysis, Dedekind contributed substantially to the modern treatment of the infinitely large and the infinitely small.

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While vacationing in Interlaken, Switzerland, in 1874, Dedekind met Cantor. Dedekind gave a sympathetic hearing to an exposition of the revolutionary idea of sets that Cantor had just published, which later became prominent in the teaching of modern mathematics. Because both mathematicians were developing highly original concepts, such as in number theory and analysis, which were not readily accepted by their contemporaries, and because both lacked adequate professional recognition, a lasting friendship developed.

Continuing his investigations into the properties and relationships of integers—that is, the idea of number—Dedekind published Über die Theorie der ganzen algebraischen Zahlen (1879; “On the Theory of Algebraic Whole Numbers”). There he proposed the “ideal” as a collection of numbers that may be separated out of a larger collection, composed of algebraic integers that satisfy polynomial equations with ordinary integers as coefficients. The ideal is a collection of all algebraic integer multiples of a given algebraic integer. For example, the notation (2) represents such a particular collection, as . . . -8, -6, -4, -2, 0, 2, 4, 6, 8 . . . . The sum of two ideals is an ideal that is composed of all the sums of all their individual members. The product of two ideals is similarly defined. Ideals, considered as integers, can then be added, multiplied, and hence factored. By means of this theory of ideals, he allowed the process of unique factorization—that is, expressing a number as the product of only one set of primes, or 1 and itself—to be applied to many algebraic structures that hitherto had eluded analysis.