Higher Trigonometry, Hyperreal Numbers, and Euler’s Analysis of Infinities
Mark McKinzie and Curtis D. Tuckey
In a textbook published in 1748, without the barest mention of the derivative, Euler explained the series for the exponential and logarithmic functions, proved the identities relating the exponential function, sine, and cosine, and computed series for the trigonometric functions, among many other facts. Â“All this follows from ordinary algebra,Â” he claimed, and all this in a textbook explicitly intended to educate beginners: Introductio in Analysin Infinitorum (Introduction to Analysis of Infinities). Euler's Â“ordinary algebraÂ” included the arithmetic of infinite and infinitesimal quantities, and because of this, he is often portrayed in popular accounts as a reckless symbol-manipulator who worked in a number system fraught with nonsense and contradiction but who through sheer intuitive brilliance somehow came to correct conclusions. In contrast we take Euler's calculations involving infinite and infinitesimal quantities seriously, and provide an alternative interpretation based on the hyperreals, a rigorous, modern number system that includes infinite and infinitesimal numbers. In this system, we prove all of the results mentioned above, without using limits or derivatives, instead using only the arithmetic of the hyperreals, together with an Eulerian notion of the determinacy of an infinite series.
Smullyan’s Vizier Problem
Michael Khoury, Jr.
In his book, Satan, Cantor, and Infinity, renowned logician and puzzler Raymond Smullyan poses the following problem in the familiar genre of knights (who always tell the truth) and knaves (who never tell the truth). It is easy to distinguish a knight from a knave by asking a question, like Â“Is two prime?Â”, whose answer is already known. Is it, however, possible to distinguish a knight from a knave without ever asking a question whose truthful answer is known in advance? Smullyan does not solve the problem himself except by evading the issue (albeit masterfully). In this article the problem is stated and treated more precisely than in Smullyan to avoid Â“trickÂ” solutions like Smullyan's. I also present my own two solutions. Simple generalizations are also considered.
The Wallet Paradox Revisited
Maureen T. Carroll, Michael A. Jones, and Elyn K. Rykken
In Martin Gardner’s Â“Wallet GameÂ” two players agree to wager the contents of their wallets. The player carrying the lesser amount of money wins the other player’s amount. Assuming infinitely repeated trials, Merryfield, Viet, and Watson view this game probabilistically and ask if an optimal strategy exists when the distribution of the players’ amounts are required to have the same mean. In this paper we show that no such strategy exists in both the discrete and continuous cases. We also consider the analogous restriction on the median.
Extriangles and Excevians
Suppose squares are constructed outward on the three sides of a triangle. Suppose further that the remote vertices of the squares are joined to form three triangles with each consisting of a vertex of the original triangle and the two adjoining sides of the squares. Are similarly formed cevians, one from each Â“extriangle,Â” concurrent? If so, do three of these form an Â“Euler line?Â”
Boxlike Domains in the Complex Plane
John A. Feroe, Benjamin A. Lotto, and Charles I. Steinhorn
The interplay between the geometry of a domain in the complex plane and the analytic properties of holomorphic functions defined on that domain is central in complex analysis. In this paper, we introduce a new class of domains called "boxlike domains" that arise naturally from a generalization of one proof of Cauchy's Theorem. We give a geometric characterization of boxlike domains and derive from this characterization a heuristic for identifying boxlike domains just by "looking" at them.
Integrals of Periodic Functions
Sean F. Ellermeyer and David G. Robinson
As calculus students become adept at integration, they learn to recognize the general forms of antiderivatives of several classes of elementary functions including, for example, polynomials and rational functions. However, it seems that a common pattern that is present in integrals of periodic functions is less likely to be noticed. In this note, we provide a result on the general form of integrals of periodic functions and we illustrate our result using examples from calculus and differential equations.
An Elementary Proof of a Particular Case of Dirichlet’s Theorem on Primes in an Arithmetic Progression
The paper presents simple and elementary proof of the fact that every arithmetic progression with the first term equal 1 contains infinitely many primes. Although previous elementary proofs are known, the proof presented in this note is probably the simplest and the shortest.
A Nowhere Differentiable Continuous Function Constructed by Cantor’s Series
The examples of continuous nowhere differentiable functions given in most analysis texts involve the uniform convergence of a series of functions. In the last twenty years this subject received again a good deal of attention. The purpose of this note is to construct a new elementary example by using the Cantor series, which needs only the basic notion of limit and is very accessible and appropriate for a first calculus course.
PROOFS WITHOUT WORDS
The Weierstrass Substitution
Sidney H. Kung
edited by Elgin H. Johnston
edited by Paul J. Campbell
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