Many of the theorems of real analysis, against the background of the ordered field axioms, are equivalent to Dedekind completeness, and hence can serve as completeness axioms for the reals. In the course of demonstrating this, the article offers a tour of some less-familiar ordered fields, provides some of the relevant history, and considers pedagogical implications.

Let [θ] denote the integer part and {θ} the fractional part of the real number θ. For θ > 1 and {θ^1/n}≠0, define M_θ(n)=[1/{θ^1/n}]. The arithmetic function M_θ(n) is eventually increasing, and the limit as n goes to infinity of M_θ(n)/n is 1/log θ. Moreover, M_θ(n) is "linearly periodic" if and only if log θ is rational. Other results and problems concerning the function M_θ(n) are discussed.

A ring satisfying x³ = x is necessarily commutative. We consider a variety of weaker forms of this condition and show that many, but not all of them, imply commutativity. We also present a variety of elementary proofs of the fact that x³ = x implies commutativity.

We show that two duelers with similar, lousy shooting skills (a.k.a. Galois duelers) will choose to take turns firing in accordance with the famous Thue-Morse sequence if they greedily demand their chances to fire as soon as the other's a priori probability of winning exceeds their own. This contrasts with a result from the approximation theory of complex functions, which says what more patient duelers would do, if they really cared about being as fair as possible. We note a consequent interpretation of the Thue-Morse sequence in terms of certain expansions in fractional bases close to, but greater than, 1.

If A is an n×n complex matrix and λ is an eigenvalue of A with geometric multiplicity k, then λ is in at least k of the n Geršgorin discs of A.

The Cauchy-Schwarz inequality is one of most widely used and most important inequalities in mathematics. The aim of this note is to show a new inequality that improves the Cauchy-Schwarz inequality.

We compute the limits of a class of continued radicals, extending the results of a previous note in which only periodic radicals of the class were considered.

In the following we discuss a well-known binomial identity. Many proofs by different methods are known for this identity. Here we present another proof, which uses linear ordinary differential equations of the first order.

We give an elementary proof, based on linear algebra and on a simple and well-known technique from the theory of dynamical systems, for the non-existence of supercyclic linear operators defined on a finite dimensional complex Banach space with dimension greater than or equal to two.

Symmetry: A Mathematical Exploration. By Kristopher Tapp. Reviewed by Kevin Woods.