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Supplementary Reading (page 2 of 3)

Sections: Study helps and math reports, More math report options, Math biography, studying ahead, and math in literature


Another book of this genre is "An Imaginary Tale: The Story of i", by Paul Nahin (where "i" is the square root of 1). While very complete historically, I found the text to be a bit confusing, both in organization and explanation. I would definitely wait until after calculus to delve into this history of the "imaginary". However, for the recreational or precocious student, or for someone with a background in practical science (such as engineering), the history and science contained in this paperback could be invigorating.
Recommendation: homeschool or college students, or engineers or similar scientific types

 

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Robert Kaplan has written "The Nothing that Is: A Natural History of Zero", a highly literary book on the development and eventual acceptance of the zero concept. He spends a lot of time in wonderings and ponderings; if you can "trip the light fantastic" with words, Mr. Kaplan does. This book was a bit flowery for my taste, but if you lean toward the liberal arts, this book could be your cup of tea.
Recommendation: homeschool or college students

On the other hand, "Zero: The Biography of a Dangerous Idea", by Charles Seife, was just my speed. This book spends more time on the known history than in wondering what happened during the gaps in our knowledge of the past. He humanizes the subject with lots of detail, and his writing style is very enjoyable. (It's hard to go wrong with a book that contains a "proof" that Winston Churchill is a carrot, and which ponders the implications of humans and gods having infinite amounts of sex.) The author makes the mathematics very approachable; you don't need to be familiar with complex numbers, calculus, physics, etc., in order to follow his reasoning. Starting with chapter 7, the book turns from the history of zero to the implications of zero within modern physics, so you might want to restrict your book report to the first six chapters.
Recommendation: high school, homeschool, or college students

In addition to his "History of Zero" (above), Robert Kaplan, together with his wife Ellen Kaplan, has also written "The Art of the Infinite: The Pleasures of Mathematics". This books considers questions whose answers require a consideration of some aspect of infinity. I enjoyed this book more than his book on zero, but would recommend this text only for the gifted or mathematically-inclined student. It is not that the material is too difficult, but it is sort of "out there", and you'd need to be really "into" math to want to wade through this. Any student could probably benefit from the earlier chapters covering sequences, series, and proofs by induction, and some of the geometry is quite accessible. But the second half of the book is more for devotees of mathematics, such as the chapter on such topics as pencils of points and duality in the projective plane. (If your eyes just glazed over when you read that last sentence, then maybe this book isn't for you.) This text covers mathematical thinking, and refers to biographical aspects of mathematicians' lives, as well as literature and history. It's a good read, if you're willing to put in the effort.
Recommendation: gifted students or math majors

Barry Mazur has written "Imagining Numbers: Particularly the Square Root of Minus Fifteen". In this book, Mr. Mazur attempts to lead the reader through the invention (discovery?) of imaginary numbers. Along the way, he compares the act of "doing mathematics" with other acts of creative imagination, such as painting or writing a poem. The author assumes the reader has a literary background, making references to historical facts, novelists, and philosophers, and occasionally quoting French sayings (in French). I found the first two-thirds or so of the book to be fairly good, though it seemed to trail off a bit in the last third. Still, the exposure to the actual work of mathematicians, with all the sweat and tears, the messiness, and the bickering, will be quite illuminating to many. If you think that math is really as sterile as many books present it as being, this text could be an eye-opener.
Recommendation: homeschool or college students

A book that your teacher will probably like is John Paulos' "Innumeracy: Mathematical Illiteracy and Its Consequences". Mr. Paulos expounds on why it can be harmful to be unable to deal intelligently with numbers (mostly statistics and probability). While his examples are often dated (for instance, Margaret Thatcher has not been the prime minister in England for quite a few years now), and his politics tends toward the "correct" end of the spectrum, his point is good: you can get in trouble if you don't know enough about numbers to keep yourself from being fooled by scam-artists. The book is widely available, easy to read, and relatively short. The only annoyance is when he gets cute and asks a question and then doesn't answer it, as though he's giving you a homework problem. But this doesn't come up much, and the discussion of real-life statistics and probability is worth the trouble.
Recommendation: high school, homeschool, or college students

If you would like to investigate the practical use of statistics, then try "Damned Lies and Statistics" and "More Damned Lies and Statistics", by Joel Best. Neither of these books requires much math, as the discussion is more aimed at the creation, use, and misuse of the numbers, rather than their calculation. The author says, "[w]e sometimes talk about statistics as though they are facts that simply exist, like rocks, completely independent of people, and that people gather stitistics much as rock collectors pick up stones. This is wrong...All statistics are social products, the results of people's efforts." The author then discusses the poor use of statistics, illustrating possible problems with examples that span the political spectrum (in order to combat the "weak assumption that our side's numbers are better than the other side's numbers, simply because they're ours"). He encourages the reader not to blindly accept or reject profferred numbers, but to examine them critically. "[F]ailing to adopt a Critical mind-set makes us powerless to evaluate what others tell us. When we fail to think cricially, the statistics we hear might just as well be magical." If you want to learn about the power of mathematics to enable true critical thinking (as opposed to innumerate and mindless criticism), these books are an excellent source. The second book ends with a listing of further resources, some of which are quite a lot of fun.
Recommendation: high school, homeschool, college, or recreational

To learn something of the history of the use of numbers (mostly in the form of statistics) in modern life, and how surprisingly recent this use of numbers is, consider "The Triumph of Numbers: How Counting Shaped Modern Life", by I.B. Cohen. You'll learn how modern statistical techniques were initially developed in an effort to increase the odds of winning when gambling, and how Florence Nightingale was famous in her own day not so much for nursing as for introducing statistics into medical considerations, thereby saving thousands of lives.
Recommendation: high school, homeschool, or college

If you're looking for a book on logic, there are various options. "Conned Again, Watson! Cautionary Tales of Logic, Math, and Probability", by Colin Bruce, presents original Sherlock Holmes stories, illustrating basic concepts of logic and probability in both common everyday contexts (where the errors may be hidden by their familiarity) and in simplified contexts (where the error is more easily extracted and refuted). This is an easy read; not only can you get a good book report out of this, but you might learn something useful, too. For a consideration of logic separate from mathematics, I highly recommend "Crimes Against Logic", by Jamie Whyte. Not only is this book practical and even-handed (slaying sacred cows on both ends of the spectrum rather than, as is usually the case, on the right-of-centre "wrong" end), but the writing is deft and the examples practical and easily understood. This slim volume is a delightful read.
Recommendation: high school, homeschool, college, or recreational

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