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Mathematics Experiments

Shangzhi Li, Falai Chen, Yaohua Wu, and Yunhua Zhang
World Scientific
Publication Date: 
Number of Pages: 
[Reviewed by
Bonnie Shulman
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I am always looking for ways to stimulate students to explore mathematical ideas on their own-to ask "what if" questions, to compute special cases, to create examples and generally get out a shovel and "dig mathematics." This book looked like a good tool to facilitate experimentation using the powerful software of Mathematica. I remained enthusiastic as I read through the table of contents which presented a varied array of topics including calculus, numerical analysis, real analysis, probability, number theory, mathematical physics and more. It has a laudatory forward by well respected Princeton mathematician Weinan E. A cursory glance at the chapters built up my expectations even further, as they were nonstandard, challenging, and open-ended, with a few historical references that caught my eye as I scanned the exercises (in particular an exploration of the "naturalness" of the base e using methods similar to Napier's in his invention of logarithms-sections 1.2 and 1.3).

Alas the promise that so excited me on my first pass was not fulfilled as I sat down to read carefully through the text. The most obvious problem is with the language. This is an English translation from the Chinese edition Experiments in Mathematics by Li Shangzhi et. al. But the book sorely needs a good English language editor, preferably one with mathematical expertise as well. For example, from p. 39:

The concept of limit is one of the most fundamental parts of calculus and mathematical analysis. As early as in the 3rd century B.C., Archimedes from the ancient Greece used the idea of the limit of sequences to calculate the area of curved triangles. This experiment is aiming at finding some rules of sequences and properties of the limit with the aid of computer.

After defining sequences and series ("An infinite sequence refers to a string of numbers arranged under a certain sequence"), the authors point out that the two are "tightly connected. The only sequence can be pined [sic] down by a given infinite series" (that is, the partial sums). They conclude (on p. 40): "Hence, an infinite sequence and an infinite series are interchangeable." Although a professional mathematician would understand this shorthand, my first thought for my students was: "Don't try this at home, kids!"

As I continued reading, I thought "well, this book would not work for students (there are too many cases of real sloppiness in mathematical terminology and they cannot make the necessary adjustments that someone who already knows the mathematics can), but perhaps it would be a good source for professors to get ideas they can use in class." But to add to my frustration, as I thought about actually using this as a teaching aid, I discovered there is no index. And as I read on, I found more and more passages that were incomprehensible, not just because of language, but completely devoid of meaning. In section 14.1, "Monoalphabetic Cipher" (p. 189):

As encryption is, you may think in an adequately peculiar encrypting way that one has kept himself alone indoors to invent a set of codes, what he encrypted can hardly be guessed by others. The following does agree with this thought.

There are also typographical errors that turn out to be disastrous. In the same chapter on cryptography, the authors repeatedly discuss the solution of a congruence relation (mod n) (where n=pq) when in fact it should be (mod b) (where b=(p-1)(q-1)). Indeed, there is evidence to suggest this may be more than a typo (solving the congruence equation is "very easy in case where n, h and (p-1)(q-1) are known" — p. 201). Be that as it may, these are not isolated instances-the book is riddled with such errors.

So, although the conception of this book is a good idea, and there is much good mathematics hidden within it, as it is written it is so flawed that I cannot recommend it.

Bonnie Shulman ( is associate professor of mathematics at Bates College in Lewiston, Maine. Her current research interests are in the history and philosophy of mathematics. One of her favorite pastimes is luring students who "always hated math" into her math classes and having them leave saying "I love math!" She also enjoys conversations about "what is mathematics, really" with people she meets on airplanes and at cocktail parties who are amazed at her confession of also not being able to balance her checkbook.