Prime Obsession 325
Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics | |
author | John Derbyshire |
pages | 422 |
publisher | Plume |
rating | 9/10 |
reviewer | jkauzlar |
ISBN | 0452285259 |
summary | History of the attempt to prove the Riemann Hypothesis |
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.
Evens and odds in GEB (Score:3, Informative)
It's been a long time since I read Douglas Hofstadter's "Godel, Escher, Bach", but didn't it use the same kind of formula, alternating between dialogs and discussion chapters? I really loved that book. I've heard a lot of criticism of it from mathematicians and musicians, but that noise always sounded like so much professional nitpicking to me.
Examples of Math books for lay people (Score:5, Informative)
Re:Douglas Adams (Score:3, Informative)
"What do you get when you multiply six by nine?"
People have argued that since Arthur Dent got this by picking letters out of his homemade scrabble set at random, that this is impossbile, as there are not 4 Y's in a standard Scrabble set.
Motivation (Score:2, Informative)
Don't come along often? (Score:5, Informative)
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.
Re:Douglas Adams (Score:5, Informative)
Sorry to burst the bubble.
John Derbyshire (Score:2, Informative)
Re:What is bad about him? (Score:4, Informative)
Re:Examples of Math books for lay people (Score:2, Informative)
In his own words (Score:2, Informative)
Why not cite his own explaination [olimu.com] of his homophobia?
The odds are against such a pair existing (Score:1, Informative)
Consider a set of size n (with n fairly large) and a random function f from that set back to itself. How many pairs (x,y) are there with f(x)=y and f(y)=x?
Well the number of pairs in the set is n*(n-1)/2 which is approximately n^2/2. Each pair has odds 1/n^2 of satisfying the condition. It turns out that pairs are not perfectly independent in satisfying/not satisfying (if (x,y) satisfies the condition then (x,z) cannot for any other z), but in a detailed analysis it is close enough to independent for what I'm going to say next to still be true.
The situation that we have here is that we are observing a large number of unlikely possible events, with a ratio between the number and the probability indicating that we expect to see a fixed amount on average (in our case 0.5 since the number of pairs is n^2/2 and the probability of a given pair satisfying our condition is 1/n^2). How many do we deserve? Assuming that the number is large, probability says that the distribution that we see is well-approximated by the Poisson Distribution [nist.gov], which says that the odds of seeing k occurances is exp(-lambda)*lambda^k/k! with lambda being the expected number of occurances.
Therefore the odds of our seeing 0 occurances are exp(-0.5)*0.5^0/0! = 0.60653065971263... - hence my claim that they are 60% against.
FREE! (Score:3, Informative)
I've started it and it is very good so far. Haven't had time to get past the first few chapters unfortunately.