Julius Wilhelm Richard Dedekind (1831 - 1916) was one of four children of a Lutheran family from Braunschweig, Germany. He entered Gottingen at nineteen and at twenty-two earned his doctorate with a thesis on calculus, praised even by Gauss. He was a student of Dirichlet and devoted himself to secondary education in Brunswick until the last years of his life.
Concerned about the nature of functions and numbers, he focused on the problem of irrational numbers since 1858 when he taught calculus, publishing his most celebrated book, "Continuity and the Irrational Numbers." One of his great doubts was about what is in the continuous geometric line that distinguishes it from rational numbers, since Galileo and Leibniz had concluded that between any two points there is always a third, and thus the rational numbers form a dense set but not. continuous.
Rereading, Dedekind noted that the essence of the continuity of the line is not linked to the density but to the nature of dividing the line into two parts, which he called classes through a single point on the line. This division of the line was called "schnitt" or "cut", which would become the support of Analysis, for with this observation "the secret of continuity would be revealed." Dedekind also saw that the points of a line can be matched in one-to-one correspondence with the real numbers, which he did by broadening the set of rationals. This conclusion is known to us as the Cantor-Dedekind Axiom.
Another of his observations was about the fundamental limit theorem, thinking that to obtain a rigorous demonstration of this concept it was necessary to develop it only through arithmetic, without interference of geometric methods although these were responsible for its brilliant results. In 1879 he was the first to give an explicit definition of numerical body as a collection of numbers that form an abelian (commutative) group with respect to addition and multiplication, in which multiplication is distributive with respect to addition. This concept, which was fundamental to the development of Algebra, is also responsible for the algebraic integer theorem, as well as introducing in Arithmetic the concept of "ideal".
Dedekind lived so many years after his famous introduction of the "cuts" that the famous publisher Tebner gave as his date of death, September 4, 1899. This amused Dedekind who lived twelve more years and wrote to the editor who had passed the date in question. stimulating conversation with his friend Georg Cantor.
Source: Fundamentals of Elementary Mathematics, Gelson Iezzi - Current Publisher