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Prime Obsession
Bernhard Riemann came to the University of Goettingen in 1846 at the age of 19, originally to study theology. The University, however, was home to Carl Friedrich Gauss, "the greatest mathematician of his age and possibly of any age," and the impressionable young Riemann, succumbing to the privilege of Gauss's presence and following his already blossoming interest in mathematics, refocused his studies on the area in which he would soon attain distinct immortality. As early as 1851 he was impressing even Gauss with the results of his doctoral dissertation and in 1859 was appointed a corresponding member of the Berlin Academy. To this honor, Riemann responded with his most famous paper, entitled "On the number of prime numbers less than a given quantity," containing therein what became known as the Riemann Hypothesis.
At the heart of the RH is the Zeta function which, in its basic form, looks like this: Z(s)=1 + 1/2^s + 1/3^s + 1/4^s + ... and which, through some simple algebraic manipulation as demonstrated by the mathematically gifted journalist Derbyshire, can be given in the form (1 - 2^-s)^-1 * (1 - 3^-s)^-1 * (1 - 5^-s)^-1 * (1 - 7^-s)^-1 * ... And it is in this second form which Derbyshire calls "The Golden Key" where the non-mathematician gets the first glimpse of the Zeta function's relationship with prime numbers.
But where this Golden Key appears as this "novel's" turning point--its central conflict-- it is not until Prime Obsession's climax when the Key is at last turned and the Zeta function's true relationship to the prime counting function pi(x)--the number of primes less than a given x--is at last made clear. Along the way, from the introduction of the Zeta function to the final explanation of its relevance to prime numbers (the turning of the Key), Derbyshire enlightens us with clear, mostly English language descriptions of the mathematics involved, as well as plentiful anecdotes that give readers a sense of the life and work of the major figures in the history surrounding the RH from Euler, Gauss and Dedekind in the late 18th century through Riemann's 1859 paper, and from 1859 onward to recent advancements in the '80s and '90s.
The Riemann Hypothesis states that "all nontrivial zeros of the Zeta function have real part one-half." Understanding the statement of the hypothesis is Derbyshire's first mission for the reader. In short, most functions with a dependent variable, say f(x)=x^2-2x+1, have a value for which if you replace x with this value, the function returns zero. In the example given, it is at the value x=1 where f(x)=0. The Zeta function has an infinite number of these zeroes and an infinite number of these is "non-trivial." The non-trivial zeroes come from complex number values. Riemann's guess, his hypothesis, is that the real part of each of these non-trivial zeroes is equal to one-half. The imaginary part can be anything.
Derbyshire explains all of the mathematics in very readable language. It's unlikely that anyone who did well in high school mathematics will not be able to follow Derbyshire's mathematics (and it's unlikely that those who didn't do well will pick up a 400-page book on this topic). The Zeta function is explored from a number of angles--numerically, graphically, algebraically, statistically, and there's even a link between the non-trivial zeroes of the Zeta function and quantum physics! By a larger margin, however, Prime Obsession's intrigue lies in Derbyshire's expositions on Riemann, Hilbert, Turing, Gauss, et al, as well as those modern mathematicians he's interviewed personally. The line between the mathematical half of the book and the historical is clearly defined; the odd-numbered chapters are devoted to the former, the even to the latter.
Those fans and foes of Derbyshire's most public line of work as a journalist/editorial writer for National Review will be comforted to know all political polemics have been set aside. John Derbyshire gives a virtuoso performance as an informed journalist and maintains his stance as a personable and careful guide through a sometimes difficult terrain. Anyone with some interest in the topic will find it hard to put down Derbyshire's book once begun. If we are lucky (hint, hint, JD) perhaps Derbyshire's next book will cover the newly-proven Poincare Conjecture ...
You can purchase Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics from bn.com. Slashdot welcomes readers' book reviews -- to see your own review here, carefully read the book review guidelines, then visit the submission page.
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Superior mathematician.
The answer? 42.
The question? What is 6 times 9.
The part he didn't tell you is that the question/answer machine was devised by a group of aliens that had 13 fingers. They wouldn't count in base 10, they would count in base 13, naturally.
6 x 9 does in fact equal 42. In base 13.
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Why is it, that if you have studied math that people get you these books for Christmas, etc. People say, "Wow, he's into math, I'm sure he'd like that", when books like this are written for the lay person, as a fun introduction to the subject. People don't get Literature majors "Shakespeare for Dummies".
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The author must be sad.
Try Mathematics for the Million by Hogben - it's fantastic, and the most coherent Calculus explanation I've ever encountered.
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What a sad, sad assumption: That lay-people have no interest in math.
Martin Gardner's series of Mathematical Games books certainly qualifies as classic.
I would put some of Douglas Hofstadter's books in there too. Certainly _Godel, Escher, Bach_ is highly (though not entirely) mathematical.
Richard Smullyan also has a number of very good math/puzzle books.
There are others, too, but you get the idea. I don't think you need to be professional mathematician to enjoy any of these.
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Mathematics And Sex (2004)
Pi: A Biography of the World's Most Mysterious Number (2004)
Chance: A Guide to Gambling, Love, the Stock Market and Just About Everything Else (2004)
Entanglement: The Unlikely Story of How Scientists, Mathematicians, and Philosphers Proved Einstein's Spookiest Theory (2003)
The Mathematical Century : The 30 Greatest Problems of the Last 100 Years (2003)
The Golden Ratio : The Story of PHI, the World's Most Astonishing Number (2003)
When Least Is Best : How Mathematicians Discovered Many Clever Ways to Make Things as Small (or as Large) as Possible (2003)
The Honors Class: Hilbert's Problems and Their Solvers (2001)
An Imaginary Tale (1998)
e: The Story of a Number (1998)
Just to pick some recent examples (i.e. not including the masterpieces of Martin Gardner and other recreational mathematicians in the 1960s and 70s, and apologies if I left off your favorite). I would agree, however, that good pop-math books are a great deal more rare.
I just checked and the first five pages of GEB are about Bach. Nothing complex.
One of the great things about GEB is that it completely obliterates the standard humanities vs. sciences distinction.
I find the anti-science attitude particularly irritating because I've heard it a lot from otherwise intelligent "Liberal Arts" people.
Tell me, do you think the average lay person could understand the first five pages of _Moby Dick_, or a Shakespeare play, or an Emily Dickinson poem? I think not -- yet I don't hear anyone dissuading them from trying.
What's the practical benefit of, say, reading Jane Austen in high school? None. Yet the main complaint you hear is that "I don't want to learn algebra because I'll never use it in my real life".
Why are "average" people discourgaged from learning about science and math?
The most rabid believers in American Exceptionalism are the exact same people whose policies are destroying it.