The
Purplemath Forums 
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.
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 mathematicallyinclined
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 twothirds 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 eyeopener. 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 scamartists. 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 reallife statistics and probability is worth the trouble. 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 mindset 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 evenhanded (slaying
sacred cows on both ends of the spectrum rather than, as is usually the
case, on the rightofcentre "wrong" end), but the writing is
deft and the examples practical and easily understood. This slim volume
is a delightful read. 


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