February 2011 Contents
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Historical Reflections on Teaching the Fundamental Theorem of Integral Calculus
David M. Bressoud
Abstract: This article explores the history of the Fundamental Theorem of Integral Calculus, from its origins in the 17th century through its formalization in the 19th century to its presentation in 20th century textbooks, and draws conclusions about what this historical development tells us about how to teach this fundamental insight of calculus.
The Mathematics of Doodling
Abstract: Wondering about a childhood doodle compels us to ask a series of questions, which will lead us on a tour of increasingly sophisticated ideas from many parts of mathematics.
Ford Circles, Continued Fractions, and Rational Approximation
Abstract: We give an elementary geometric proof using Ford circles of a well-known theorem of Lagrange, which says that the convergents of the continued fraction expansion of a real number α coincide with the rationals that are best approximations of α.
How to Add a Noninteger Number of Terms: From Axioms to New Identities
Markus Müller; Dierk Schleicher
Abstract: Starting from a small number of well-motivated axioms, we derive a unique definition of sums with a noninteger number of addends. These “fractional sums” have properties that generalize well-known classical sum identities in a natural way. We illustrate how fractional sums can be used to derive infinite sum and special functions identities; the corresponding proofs turn out to be particularly simple and intuitive.
When Does Appending the Same Digit Repeatedly on the Right of a Positive Integer Generate a Sequence of Composite Integers?
Abstract: Let k be a positive integer, and suppose that k •a1 a2 . . . at, where ai is the ith digit of k (reading from left to right). Let d •• 0, 1, . . . , 9•. For n • 1, define
In this article, we examine when sn is composite for all n.
On Dickson's Theorem Concerning Odd Perfect Numbers
Abstract: A 1913 theorem of Dickson asserts that for each fixed natural number k, there are only finitely many odd perfect numbers N with at most k distinct prime factors. We show that the number of such N is bounded by 4k2.
Every Simple Arrangement of n Lines Contains an Inducing Simple n-gon
Eyal Ackerman; Rom Pinchasi; Ludmila Scharf; Marc Scherfenberg
Abstract: We show that for any arrangement A of n ≥ 3 lines in general position in the plane there exists a simple closed polygon with n edges having the property that every edge of the polygon lies on a distinct line of A.
A Characterization of Continuity Revisited
José L. Gámez-Merino; Gustavo A. Muñoz-Fernández; Juan B. Seoane-Sepúlveda
Abstract: It is well known that a function f : R → R is continuous if and only if the image of every compact set under f is compact and the image of every connected set is connected. We show that there exist two 2c-dimensional linear spaces of nowhere continuous functions that (except for the zero function) transform compact sets into compact sets and connected sets into connected sets respectively.
A Short Proof and Generalization of Lagrange's Theorem on Continued Fractions
Abstract: We present a short new proof that the continued fraction of a quadratic irrational eventually repeats. The proof easily generalizes; we construct a large class of functions which, when iterated, must eventually repeat when starting with a quadratic irrational.
Generalization and Probabilistic Proof of a Combinatorial Identity
Guisong Chang; Chen Xu
Abstract: In this note, we give a probabilistic proof of a combinatorial identity which involves binomial coefficients. Our results consist of showing that the identity essentially computes the moment of order n of the chi-square random variable with m degrees of freedom.