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American Mathematical Monthly: December, 1997

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December 1997

Ramanujan's Association with Radicals in India
by Bruce C. Berndt, Heng Huat Chan, and Liang-Cheng Zhang,,

Before he came to Cambridge, Ramanujan submitted several problems involving radicals to the Journal of the Indian Mathematical Society. The pinnacle of Ramanujan's work on radicals, however, is contained in his massive amount of work on calculating class invariants, which are certain very beautiful radicals. After giving a few examples, the authors provide a lengthy discussion on the attempts of Ramanujan, H. Weber, G. N. Watson, and the authors to calculate class invariants. Weber's calculations of class invariants were motivated by the constructions of Hilbert class fields of imaginary quadratic fields. Ramanujan, on the other hand, was interested in their applications in approximations to ¹ and in explicit evaluations of the Rogers-Ramanujan continued fraction and theta functions.

Ramanujan, Taxicabs, Birthdates, ZIP Codes, and Twists
by Ken Ono

In 1916 S. Ramanujan published a beautiful article on quadratic forms where he proclaimed that the odd numbers not of the form x^2+y^2+10z^2 "do not seem to obey a simple law." Here we examine the odd integers that are not of the form x^2+y^2+10z^2, and show that they possess some striking properties in the theory of elliptic curves.

Simplicity and Surprise in Ramanujan's "Lost" Notebook
by George Andrews

The Indian genius, Ramanujan, is famous for his wild, complex and amazing formulas. In this article, we examine some of his less complicated infinite series formulas. Our object is to reveal Ramanujan's uncanny knack for picking out results filled with surprises.

The Catalan Numbers, the Lebesgue Integral, and 4^n-2
by Louis W. Shapiro, Douglas Rogers, and Wen-Jin Woan,

This article presents a short and motivated proof of a lovely result in counting. The number of path pairs of a given length is counted by the Catalan numbers, as was shown by Polya in 1969. The total area inside all the path pairs of a given length is a power of four and this surprising and elegant result is the focus of this paper. Lebesgue enters the picture through his parable about the two ways a shopkeeper might total the receipts at the end of the day. We develop a generating function that is analogous to counting the number of m dollar bills.

Good Matrices: Matrices that Preserve Ideals
by R. Bruce Richter and William P. Wardlaw,

Let A be a matrix with entries in the commutative ring R with unity. Several statements about the extendibility of A to an invertible matrix are considered. One example, new to this article, is the concept of "goodness": a matrix A is "left good" if, for every vector x, the ideal generated by the entries in x is equal to the ideal generated by the entries in xA. Among other results, it is shown that if A extends (by adding rows) to an invertible matrix, then A is left good. If R is a principal ideal ring, then the converse holds.

A Converse of the Mean Value Theorem
by Peter A. Braza and Jingcheng Tong,

The mean value theorem is one of the most important theorems in calculus. But is a converse ever discussed? In this paper we give a converse and show that it is not true in general. As a matter of fact, we give an example of a function where the converse fails at a countable number of points. Are there functions that fail this converse at a dense set of point? An uncountable set of points?

Primes at a (Somewhat Lengthy) Glance
by Andrew Granville, Takashi Agoh, and Paul Erdos,

Every prime n ³ 11 may be proved to be prime by expressing it in the form n = N_1 + N_2 + . . . + N_k, where p_1 = 2 The Mathematics Education Reform: Why You Should be Concerned and What You Can Do
by H. Wu

The purpose of this article is to bring greater awareness to the mathematical community of the current mathematics education reform. Beyond the citation of specific examples of mathematical anomalies common in the reform, it discusses potentially far-reaching consequences of this reform were it to go unchecked. It also argues why a greater involvement by all members of the mathematical community in the education enterprise is essential for the well being of mathematics. The article concludes with recommendations of possible actions for different segments of the community. A fuller discussion of the same topics is given in "The mathematics education reform: what is it and why should you care?", available at

Confronting Reform
by Jeremy Kilpatrick

Reform of school mathematics in the United States has been attempted three times in this century: at the beginning, when an effort was made to unify the secondary curriculum; just after mid-century, when the new math reformers attempted to use abstract structures of mathematics as organizing themes; and during the last decade, as a standards-based curriculum has become the goal. Each time, opposition has arisen within the mathematical community. The article briefly examines these reform efforts and reactions to them, giving particular attention to the recent critique by H. Wu. It concludes with thoughts on the value to the community of an informed dialogue in facing the challenges of changing school mathematics.


A Shorter Proof of the Ramanujan Congruence Modulo 5
by John L. Drost

An Amusing Representation of x/(sin x)
by Scott Ahlgren, Lars English, and Ron Winters,,


Monthly Unsolved Problems, 1969-1997
by Richard J. Nowakowski and Richard K. Guy,



Conceptual Mathematics: A First Introduction to Categories. By F. William Lawvere and Steven Schanuel

Reviewed by Saunders Mac Lane

Early Astronomy. By Hugh Thurston

Reviewed by Ezra Brown