Using The Infinite Descent Method To Find Convenient Rational And Non-Rational Numbers Using Dedekind Cuts

It is well-known that irrational numbers play a relevant role in mathematics and basic sciences e.g., the number introduced by the Babylonians and Egyptians of ancient times, Euler´s number e to explain exponentially-varying processes, and the l of Conway´s cosmological theory. Therefore, a strong understanding of real numbers is important. Many mathematicians such as R. Dedekind and W. Rudin, when introducing the real numbers via the rational and irrational numbers and the concept of Dedekind cuts, make use of “convenient numbers†such as Rudin's h which seem to be “taken out of a hat.†From a pedagogical point of view, the use of these numbers has proven to be a sticky issue to both students and professors because there has been little, if any, justification for their ""convenience"". In this paper the authors, using Dedekind cuts explain the introduction of those “convenient†numbers using the infinite descent method. The Extended Euclidean Convergent Algorithm is used to create convergent fractions to approximate irrational numbers with a desired approximation via the computer.