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American Mathematical Monthly -January 2009

January 2009

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Why Did Lagrange "Prove" the Parallel Postulate?
By: Judith V. Grabiner
In 1806, Joseph-Louis Lagrange read a memoir "proving" Euclid's parallel postulate to the Institut de France in Paris. The memoir still exists in manuscript, and we’ll look at what it says. We ask why he tried to prove the postulate, and why he attacked the problem in the way that he did. We also look at how the ideas in this manuscript are related to such things as Lagrange’s philosophy of mathematics, artists’ ideas about space, Newtonian mechanics, and Leibniz's Principle of Sufficient Reason. Finally, we reflect on how this episode changes our views about eighteenth-century attitudes toward geometry, space, and the nature of science.


By: Mike Paterson and Uri Zwick,
How far off the edge of a table can we reach by stacking n identical, homogeneous, frictionless blocks? A classical solution achieving a logarithmic overhang was widely believed to be optimal. We show, however, that this classical solution is exponentially far from optimality.


A Probabilistic Proof of the Lindeberg-Feller Central Limit Theorem
By: Larry Goldstein
The Central Limit Theorem, or CLT, is one of the most striking and useful results in probability and statistics, and explains why the normal distribution appears in areas as diverse as gambling, measurement error, sampling, and statistical mechanics. Equipped with a probabilistic "zero-bias" explanation for the appearance of the Lindeberg condition, a sufficient and nearly necessary condition for the CLT, a simplified proof of the Lindeberg-Feller Theorem, and its partial converse, is given.


Does Lipschitz with Respect to x Imply Uniqueness for the Differential equation y' = f(x,y)?
By: José Ángel Cid and Rodrigo López Pouso,
Most uniqueness tests for differential equations need assumptions about how the nonlinear part depends on the unknown. A paradigm for this is the classical Lipschitz Theorem. We show that the theorem on differentiation of inverse functions yields, in a significant situation, an elementary technique for turning such uniqueness results into alternative versions of them with assumptions transferred from the dependent to the independent variable.



A Simple Complex Analysis and an Advanced Calculus Proof of the Fundamental Theorem of Algebra
By: Anton R. Schep
In this note we present two proofs of the Fundamental Theorem of Algebra which do not seem to have been observed before and which we think are worth recording. The first one uses Cauchy's Integral Theorem and is, in the author's opinion, as simple as the most popular complex analysis proof based on Liouville's Theorem. The second one considers the integral obtained by parameterizing the contour integral from the first proof and uses only results from advanced calculus.

Yet Another Proof of the Irrationality of
By: Natalia Casás Ferreño
This note presents a new proof of the irrationality of the square root of two combining an argument about a one-dimensional discrete dynamical system with the binomial theorem.

A General Lagrange Theorem
By: Giovanni Panti
A real number has an eventually periodic continued fractions expansion if and only if it is a quadratic irrational. This fact was proved by Lagrange in 1769; variants of his algebraic proof are still commonly presented in textbooks. In terms of dynamical systems, Lagrange's result amounts to a characterization of the set of points which are preperiodic under a certain piecewise-fractional map, the so-called Gauss map. We give a computation-free, simple geometric proof of a generalization of Lagrange's Theorem to a large class of piecewise-fractional maps, including the maps corresponding to the even and the odd continued fractions expansions, the Farey expansion, the nearest integer expansion, and other more esoteric continued fractions algorithms.

A Note on the Congruence
By: Romeo Meštrovic
In this note, we prove a partial converse assertion of a consequence of Lucas’s Theorem. Further, various congruences modulo prime powers related to our result are discussed.


A Simple Solution to a Multiple Player Gambler's Ruin Problem
By: Sheldon M. Ross
We consider a multiple player gambler's ruin problem. In each stage two of the players are chosen to play a fair game, with the winner of the game receiving one unit from the loser. A player whose fortune hits zero is eliminated, and the problem continues until a single surviving player remains. We present elementary arguments for finding the probability that each player is the survivor as well as the mean number of games played by any given pair of players, and show that neither depends on the method for choosing the players in each stage.


A Garden of Integrals
By: Frank E. Burk
Reviewed by: Erik Talvila