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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 [the square root of minus one]",
by Paul Nahin. 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
could be invigorating.
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.
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.
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.
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.
A book that your teacher will probably like is John
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
If you would like to investigate the practical use
of statistics, then try "Damned Lies and Statistics"
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.
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.
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