Vol. 40, No. 2, pp. 78-155
Independent Axioms for the Field of Real Numbers
By Greg Oman
We give an irredundant axiomatization of the complete ordered field of real numbers. In particular, we show that all the field axioms for multiplication with the exception of the distributive property may be deduced as theorems in our system. We also provide a complete proof that the axioms we have chosen are independent.
CORDIC: How Hand Calculators Calculate
By Alan Sultan
We illustrate and discuss the method, called CORDIC, which many hand calculators use to calculate the trigonometric and other functions.
Topology Explains Why Automobile Sunshades Fold Oddly
By Curtis Feist and Ramin Naimi
Automobile sunshades always fold into an odd number of loops. The explanation why involves elementary topology (braid theory and linking number, both explained in detail here with definitions and examples), and an elementary fact from algebra about symmetric group.
Lobb's Generalization of Catalan's Parenthesization Problem
By Thomas Koshy
A. Lobb discovered an interesting generalization of Catalan's parenthesization problem, namely: Find the number L(n, m) of arrangements of n + m positive ones and n - m negative ones such that every partial sum is nonnegative, where 0 = m = n. This article uses Lobb's formula, L(n, m) = (2m + 1)/(n + m + 1) C(2n, n + m), where C is the usual binomial coefficient, to prove that L(n, m) is odd for all m if and only if either n = 0 or n is a Mersenne number. It follows that L(n, m) and the Catalan number Cn have the same parity. We also show that L(n, m) = C(2n, n - m) - C(2n, n - m - 1), so every Lobb number can be read from Pascal's triangle. In addition to other interesting combinatorial identities, we establish that every Catalan number C2n is the sum of n + 1 squares.
Eighty-eight Thousand, Four Hundred and Eighteen (More) Ways to Fill Space
By Anderson Norton
After Peano gave an arithmetic construction, Hilbert developed a geometric construction for space-filling curves. This paper describes the key idea of Hilbert's construction, here called ‘the nested-squares criterion,' implicit in Hilbert's writing but, once explicated, generalizes to a whole class of space-filling curves that correspond to a special class of rook's walks enumerated by Mobius 150 years ago!
A "Paperclip" Approach to Curvature, Torsion, and the Frenet-Serret Formulas
By Ulrich A. Hoensch
We explore how curvature and torsion determine the shape of a curve via the Frenet-Serret formulas. The connection is made explicit using the existence of solutions to ordinary differential equations. We use a paperclip as a concrete, visual example and generate its graph in 3-space using a CAS. We also show how certain physical deformations to the paperclip (e.g. bending) can easily be incorporated in the differential equations.
Student Research Project: Goursat's Other Theorem
By Joseph Petrillo
In an elementary undergraduate abstract algebra or group theory course, a student is introduced to a variety of methods for constructing and deconstructing groups. What seems to be missing from contemporary texts and syllabi is a theorem, first proved by Édouard Jean-Baptiste Goursat (1858-1936) in 1889, which completely describes the subgroups of a direct product and reveals beautiful connections among several elementary topics from group theory. We decompose the proof of Goursat's Theorem and its corollaries into a sequence of exercises which, when considered as a problem set for undergraduates, may be suitable for inclusion either in a mathematics capstone course, or even in the group theory section of a modern algebra final exam. On the other hand, the exercises may be strategically assigned to parallel the natural flow of the group theory presentation in abstract algebra so that by the end of the semester the theorem can be stated and proved without much difficulty.
PROBLEMS AND SOLUTIONS