Rose-Hulman Undergraduate Mathematics Journal Volume 17 Issue 1 Article 9 An Examination of Richard Dedekind’s “Continuity and Irrational Numbers” Chase Crosby University of Missouri — Kansas City Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation Crosby, Chase (2016) "An Examination of Richard Dedekind’s “Continuity and Irrational Numbers”," Rose- Hulman Undergraduate Mathematics Journal: Vol. 17 : Iss. 1 , Article 9. Available at: https://scholar.rose-hulman.edu/rhumj/vol17/iss1/9 ROSE- H ULMAN UNDERGRADUATE MATHEMATICS JOURNAL AN EXAMINATION OF RICHARD DEDEKIND’S “CONTINUITY AND IRRATIONAL NUMBERS” Chase Crosbya VOLUME 17, NO. 1, SPRING 2016 Sponsored by Rose-Hulman Institute of Technology Mathematics Department Terre Haute, IN 47803 Email: [email protected] http://www.rose-hulman.edu/mathjournal a University of Missouri – Kansas City ROSE-HULMAN UNDERGRADUATE MATHEMATICS JOURNAL VOLUME 17, NO. 1, SPRING 2016 AN EXAMINATION OF RICHARD DEDEKIND’S “CONTINUITY AND IRRATIONAL NUMBERS” Chase Crosby Abstract. This paper explicates each of the seven sections of mathematician Richard Dedekind’s 1858 essay “Continuity and Irrational Numbers”, which he eventually published in 1872. In this essay, he provides a simple, completely arithmetic proof of the continuity of the set of real numbers, a property on which the validity of many mathematical theorems, especially those in calculus, depend. The intent of this paper is to familiarize the reader with the details of Dedekind’s argument, which is exceptionally easy to follow and self-contained. Although the real numbers were often imagined as points lying on an infinite line, as a calculus instructor in Zürich, Switzerland, Dedekind became deeply troubled by the need to reference geometry when teaching his students concepts such as functions and limits. This inspired him to develop a rigorous arithmetic foundation for the set of real numbers, in which, through the use of what are now called “Dedekind cuts,” he cleverly defines both rational and irrational numbers, and demonstrates how they fit together to form the continuum of real numbers. Alternative viewpoints and criticisms of his work exist, and one is briefly discussed at the conclusion of the paper, though it is noted that Dedekind’s essay accomplishes the goal he set for himself in its preface. Acknowledgements: The author would like to thank Dr. Richard Delaware for the resources, guidance, and encouragement needed to complete this project. RHIT UNDERGRAD. MATH. J., VOL. 15, NO. 1 PAGE 135 1 Introduction In 1858, mathematician Richard Dedekind wrote a paper on the subject of the continuum entitled “Continuity and Irrational Numbers,” in which he provides an arithmetic proof of the continuity of the set of real numbers. Using a concept to which we today refer as “Dedekind cuts,” he defines rational and irrational numbers, and shows how they combine to form the continuum of real numbers. In Section 2, we are given biographical and background information relevant to Dedekind’s work. The next four sections pertain specifically to passages in “Continuity and Irrational Numbers.” In Section 3, we examine his explanation of the creation of rational numbers, and their properties. In Section 4, we are introduced to Dedekind cuts, irrational numbers, and their properties. In Section 5, we show how Dedekind proved the rational and irrational numbers defined by these cuts fit together to form a totally ordered continuum. In Section 6, we explicate Dedekind’s proofs of two theorems using his new, arithmetic proof of the continuity of the set of real numbers. Finally, in Section 7, we are given a brief conclusion mentioning an alternative viewpoint. 2 Background Information Julius Wilhelm Richard Dedekind, more well-known as simply Richard Dedekind, was born in 1831 in Braunschweig, Germany. As a youth, he was a student of science, but soon gained a preference for mathematics due to the precision it offered over other disciplines. In 1849, he began studying with Carl Friedrich Gauss – widely considered one of history’s greatest mathematicians – at the University of Göttingen, Germany, where he received his Doctorate in 1852 [1, p.38]. After studying number theory, definite integrals, and partial differential equations under the direction of Lejeune Dirichlet from 1855 to 1858, Dedekind was appointed to teach Calculus at the Polytechnikum in Zürich, Switzerland [1, p.38]. It is in preparation for his lectures there that he again became bothered by a lack of precision. At that point in time, references to geometry were necessary to introduce some of the basic concepts of functions and limits, but there existed no arithmetic definition of the concept of real numbers – a concept on which the idea of functions and the theory of limits were dependent [3, p. 764]. In 1872, he wrote a preface to his then unpublished 1858 essay “Stetigkeit und irrationale Zahlen,” or “Continuity and Irrational Numbers,” in which he explains: For myself this feeling of dissatisfaction was so overpowering that I made the fixed resolve to keep meditating on the question till I should find a purely arithmetic and perfectly rigorous foundation for the principles of infinitesimal analysis. The statement is so frequently made that the differential calculus deals with continuous magnitude, and yet an explanation of this continuity is nowhere given; even the most rigorous expositions of the differential calculus do not base their proofs upon continuity but, with more or less consciousness of the fact, they either appeal to geometric notions or those suggested by geometry, or depend upon theorems which are never established in a purely arithmetic manner. PAGE 136 RHIT UNDERGRAD. MATH. J., VOL. 17, NO. 1 The seven sections in the essay which follow attempt to provide such a “rigorous foundation.” It should be noted that when quoting sections of this work, any words appearing in square brackets are my own. 3 Definition and Properties of Rational Numbers In Section I of “Continuity and Irrational Numbers,” entitled “Properties of Rational Numbers,” Dedekind gives a brief explanation of the set of rational numbers, pointing out how they can be created from the set of natural numbers {1, 2, 3, …}. The natural numbers, he explains, are a natural consequence of the simple act of counting, where each object (number) is defined by the one preceding it. The acts of addition, multiplication, subtraction, and division in combination with human intuition led to the creation of the number zero, negative and fractional numbers, which together compose the set of rational numbers. Dedekind denotes this “system” by “R,” though today Q is more commonly used, and goes on to briefly explain some commonly known, intuitive properties of the set. He specifically points out the property of the rational numbers most important to his purposes in the essay, that they form “a well-arranged [totally ordered] domain of one dimension extending to infinity on two opposite sides,” and notes that any two rational numbers a, b differ when the expression a – b has either a positive (when a > b) or negative (when b > a) value. Three laws are then established in regard to these ways in which numbers can differ. First law for rationals: If a > b, b > c, then a > c, and b lies between the numbers a and c. Second law for rationals: Infinitely many numbers lie between any two different numbers a, and c. Third law for rationals (original form): If a is any definite [rational] number, then all numbers of the system R [Q] fall into two classes, A1 and A2, each of which contains infinitely many individuals; the first class A1 comprises all numbers a1 that are < a, the second class A2 comprises all numbers a2 that are > a; the number a itself may be assigned at pleasure to the first or second class, being respectively the greatest number of the first class or the least of the second. In every case, the separation of the system into the two classes A1, A2 is such that every number of the first class A1 is less than every number of the second class A2. In Section II of his essay, entitled “Comparison of the Rational Numbers with the Points of a Straight Line,” Dedekind shows how the properties and laws just established for the “arithmetic” rational numbers relate to the position of “geometric” points on a straight line L distinguished as having the directions “right” and “left.” This is easily accomplished, but is valuable preparation for the ideas to be presented later on in the work. Given two different points p and q on L, p either lies to the right or left of q. The three laws offered above for rational numbers stand in a similar fashion for L. RHIT UNDERGRAD. MATH. J., VOL. 15, NO. 1 PAGE 137 First law for points on L: If p lies to the right of q, and q to the right of point r, then q lies between p and r. Second law for points on L: There lie infinitely many points between any two different points p and r on L. Third law for points on L: The line L is divided into two classes P1 and P2 by any definite point p on L such that every point p1 in P1 lies to the left of every point p2 in P2, and the point p can be assigned to either class. Every rational number a corresponds to a point on L. The position of this point is found by laying a line of length |a| on L from an origin point (corresponding to the rational number zero) to the end-point p (to the right when a > 0, to the left when a < 0) which then corresponds to a. This correspondence is very intuitive, and easy to imagine.