College Mathematics Journal Contents—May 2018

May 2018 College Mathematics Journal Cover

The May issue features two articles that draw from historic Chinese mathematics. Jeff Chen highlights the work of Mei Wending by comparing two treatments of a linear algebra problem separated by 80 years and the first Chinese translation of Euclid's Elements. Bud Brown and Matt Crawford look at the stand-out "well ropes problem'' from the ancient Nine Chapters (whose solution uses linear algebra) from a new perspective, prompted by a 265 in the solution to look for combinatorial connections. Burkard Polster and Marty Ross invoke precision marching with merging squares to illustrate the irrationality of the square root of two and extend their engaging visual mathematics to address related "near misses.'' The geometric art of Portugal's Almada Negreiros, economic's Gini index applied to visual contrast, finance's famed Black–Scholes equation, and much more await your spring and summer mathematical reading.

Vol. 49, No. 3, pp. 161-240

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ARTICLES

Five Families Around a Well: A New Look at an Old Problem

p. 162.

Ezra Brown & Matthew Crawford

The Nine Chapter on the Mathematical Arts from ancient China includes a problem concerning five families who share a well. We generalize the problem and offer a combinatorial perspective that appears to be new.

DOI: 10.1080/07468342.2018.1447203

A Systematic Treatment of “Linear Algebra” in 17th-Century China

p. 169.

Jiang-Ping Jeff Chen

To examine mathematical reasoning in 17th century China, we focus on the textual practices of explaining in a treatise on linear algebra problems and the Gaussian-elimination-like solutions collectively known as Fangcheng. In this work, the mode of reasoning is not proof, rather explication of the rules of operations and narratives of steps. In contrast to earlier works that give examples of algorithms without explanation, this treatise presents a logically consistent, self-contained system of treating Fangcheng problems.

DOI: 10.1080/07468342.2018.1448670

Proof Without Words: Sums of Squares in a Thin Rectangle

p. 180.

Stephan Berendonk

We fit six copies of the finite sum of initial squares into a thin rectangle.

DOI: 10.1080/07468342.2018.1424424

Marching in Squares

p. 180.

Burkard Polster & Marty Ross

We begin with a new take (in the context of precision marching choreography) on a beautiful proof by contradiction. Then we turn around the destructive part of this proof to construct some very pretty near-misses.

DOI: 10.1080/07468342.2018.1440054

Dividing the Circle

p. 187.

Pedro J. Freitas & Hugo Tavares

Some but not all regular polygons can be constructed using only straightedge and compass; the Gauss–Wantzel theorem states precisely which. The known constructions differ from one polygon to the other. There are, however, general processes for determining the side of an arbitrary n-gon approximately, but sometimes with great precision. We describe two such methods, named for Bion and Tempier, analyze their errors, and explain why these approximate constructions work. We also highlight some related geometric artwork of Almada Negreiros.

DOI: 10.1080/07468342.2018.1440871

A New Angle on the Fermat–Torricelli Point

p. 195.

David Benko & Dan Coroian

Around 1640, Torricelli found a geometrical solution to a problem allegedly formulated in the early 1600s by Fermat: Given three points in a plane, find a fourth point such that the sum of its distances to the three given points is as small as possible. Later, Simpson and Weber looked at the more general case of minimizing the sum of the transportation costs toward the three points. Here, we investigate an interesting special case using undergraduate mathematics.

DOI: 10.1080/07468342.2018.1440865

Proof Without Words: An Alternating Geometric Series

p. 200.

Ángel Plaza

We give a visual proof of the geometric series of the constant negative one-half.

DOI: 10.1080/07468342.2018.1437302

Fakin’ Flips

p. 201.

C. Ray Rosentrater

A random sequence of many coin flips should have some long runs where the flips have the same value. How long? We develop a recurrence relation to compute the probability distribution of the length of the longest run.

DOI: 10.1080/07468342.2017.1446655

The Gini Index and Grayscale Images

p. 205.

Roberta La Haye & Petr Zizler

We provide a new definition of global contrast in digital imaging that is linked to the Gini index from economics. We explore properties of this global contrast, including examples and exercises.

DOI: 10.1080/07468342.2018.1426330

Derivation of the Black–Scholes Equation from Basic Principles

p. 212.

Granville Sewell

The Black–Scholes partial differential equation, used for pricing options, is derived here from basic principles, in a way which we believe is unique in that it does not require any background in financial mathematics or stochastic calculus.

DOI: 10.1080/07468342.2018.1439634

Classroom Capsules

Normal Limit of the Binomial via the Discrete Derivative

p. 216.

Ajoy Thamattoor

We derive the normal limit to the binomial distribution by looking at the derivative of the distribution function as the change for a unit step as the parameters are blown up in scale. This gives us the differential equation that identifies the Gaussian, avoiding the more difficult routes through Stirling’s formula or the central limit theorem.

DOI: 10.1080/07468342.2018.1440872

An Unusual Proof of the Triangle Inequality

p. 218.

Mehtaab Sawhney

A standard proof of triangle inequality requires using Cauchy–Schwarz inequality. The proof here bypasses such tools by instead relying on expectations.

DOI: 10.1080/07468342.2017.1397993