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Prime Obsession
from the claire dept.
| 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.
This is long overdue (Score:5, Funny)
Re:This is long overdue (Score:2, Funny)
lay person? (Score:4, Insightful)
Ummm...what would its peers be? Just how many "classic" math books does the lay-person have now?
Could it be that the lay-person wouldn't be interested in any book about math, no matter how well written?
I dunnnoooo...almost sounds completely probable.
Examples of Math books for lay people (Score:5, Informative)
Parent
Re:Examples of Math books for lay people (Score:3, Insightful)
I second this - Mathematics for the Million is truly a classic that belongs in this category that the story author referred to. It should be handed out to every child at a reasonable age, so that they can use it as an extra-curricular guide as they learn their way through the horrors of school-taught math.
Re:lay person? (Score:5, Insightful)
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.
Parent
Re:lay person? (Score:2)
Re:lay person? (Score:3, Funny)
I for one welcome our new, math-knowing, Finnish overloards!
And our math-knowing Korean overloards.
And our math-knowing... eh, forget it. At least we beat Portugal [go.com].
Re:lay person? (Score:5, Insightful)
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?
Parent
Re:lay person? (Score:3, Funny)
Achilles says, 'I think you mean "couldn't be bothered to" rather than "could not." In my experience most people are guilty of lack or interest rather than actual stupidity.'
Re:lay person? (Score:2, Funny)
It should become a classic, alongside with this book: Everyday Math For Dummies http://www.dummies.com/WileyCDA/DummiesTitle/produ ctCd-1568842481.html [dummies.com]
Re:lay person? (Score:3, Insightful)
I am always looking for "laypersons" math books, because after reading Richard Feynmans (non-math) works, I want to understand his Physics Lectures.
As a helpful AC http://slashdot.org/comments.pl?s [slashdot.org]
Re:lay person? (Score:4, Interesting)
Consider yourself "shown".
I am not speaking of general math - rather I am speaking of the esoteric stuff such as "new math" stuff that has no "purpose" other than to be a neat trick.
I was deeply impressed by Richard Feynmans chapter on his reviewing high school math books. He was livid that a number of things being taught were useless. He wanted the books to teach the students not only what they were learning, but why. One example has him in an uproar because there was a question about taking the average tempurature of a number of stars. This made Feynman angry because there is no reason to get an average star tempurature for a number of stars, it is just not something that you do. Feynman called it "a trick to get the students to add".
Furthermore, he was furious at a physics problem in one book, that had wrong answers, and in fact, Feynman actually performed the experiment listed in the book, and found out the "observed" results were wrong. The author did not even take the time to DO the experiment listed.
Again, this made him furious because he felt that teaching students math in a deceptive manner would never give them a feeling as to where the math can take you in fields in the sciences. I agree.
So, I don't want to learn fluff. I was at a disadvantage because I was just told "learn this" and in answer to the question of "why?" I was only given "so you can pass the exams."
In high school I deeply wanted the answers to some questions in Physics, that were available with mathematics, but I was not shown these, and I developed an unfortunate disgust with mathematics because of this.
So many people here on slashdot can take me to task for being bad at math - and I know I am. I don't know if you would have been so interested in it either if it was drilled into you in a dull manner, and a feeling that it lacked a purpose.
Am I learning math now? Yes, but then I understand much more about the why, the how, and the history now than I did then. I don't know about the rest of you but I detest rote learning. So take me to task on my math skills if you wish (or my typing
Parent
Re:lay person? (Score:3, Insightful)
Re:lay person? (Score:2)
Not to mention, of course, his Code Book (assuming you consider cryptography as a branch of mathematics, which I think most people do).
Douglas Adams (Score:5, Funny)
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.
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.
Re:Douglas Adams (Score:2)
That's all there is.
Re:Douglas Adams (Score:2)
Re:Douglas Adams (Score:5, Informative)
Sorry to burst the bubble.
Parent
Re:Douglas Adams (Score:2)
Offtopic...rant... (Score:5, Insightful)
Re:Offtopic...rant... (Score:5, Insightful)
I get the same thing all the time. Last year, my mother-in-law got me a put-it-together kinetic flashlight kit for kids. (I'm an Electrical Engineer.) She tried.
This book might be an interesting read. That's probably what they thought.
Parent
MOD PARENT UP! (Score:5, Insightful)
People get those gifts because they try. They don't understand math at all, but they know that you do "something mathy".
Exactly right. They are trying to get you something that they think you might like even though they don't know very much about math. Instead of the grandparent getting all hot under the collar that his family and friends dare insult his grant intellect by purchasing him a "Math for Dummies" book (as he seems to think this historical work is), he should feel gratified to know that he family cares enough about him to actually put forth some effort to getting something that attempts to match his interests. There are lots of people who simply buy generic gifts for family like socks or shit like that. Isn't this book a lot better than a gift like that?
Reading the grandparents rant, I was reminded of an article in The Onion awhile back about some film snob getting all upset because his family -- damn their incompetance! -- dared buy him the widescreen edition of one of the Matrix sequels when he actually wanted the letterboxed edition (opportunity for karma whoring here if someone can link to it). For chrissake, your family and friends are trying their goddamned best and you get your panties in a bunch over details? That's so incredibly childlike, I can hardly believe this above "rant". Christmas isn't really about getting exactly what you want -- at least once you're an adult it's not. Christmas is just an opportunity to get together with loved ones and exchange gifts as a token of affection. It doesn't have to be the "perfect gift"; as long as it's somewhere in the ballpark you should feel happy that your family is at least aware of your interests.
GMD
Parent
Re:Offtopic...rant... (Score:5, Insightful)
The problem is that a lack mathematical understanding verging on innumeracy is socially acceptable, cool even. Imagine boasting that you couldn't "do reading", and found books aimed at ten-year-olds too much of a challenge. If that was true, then you wouldn't admit it - but go out to eat with half a dozen friends or workmates, and when it comes to the bill people will cheerfully admit that they're rubbish at Maths and can't divide the total by six. I had one colleague who was impressed that I could divide £45 between seven people...
Now, if you've ever shown any ability to do any Maths, however basic, from their point of view you're forever "good at Maths". They don't know this book from Landau & Lifshitz, but you're "good at Maths" so you'll like it. Won't you?
Parent
Re:Offtopic...rant... (Score:3, Insightful)
PRESSURE PRESSURE PRESSURE! BUY BUY BUY! STUFF STUFF STUFF!
Jesus who?
Majored in math, away from it for a year (Score:2, Insightful)
I would probably need to do a few laps before I could go playing around near the high-dive again or anything. I don't think this speaks to my grasp of the subject or my intelligence, but to my complete abandonment of study for a long period.
This wouldn't be a book to get someone that works in a heavily mathematical field, but its a great choice for the coder in your live that likes math
Re:Majored in math, away from it for a year (Score:3, Interesting)
This wouldn't be a book to get someone that works in a heavily mathematical field...
Why not? I have a Ph.D. in math from one of the top scientific institutions in the world and I think the book sounds interesting. Quoting from the above review:
Propaganda (Score:3, Insightful)
Why must we use such slanted terms to describe the views of people we disagree with?
perhaps I just answered my own question.
Re:Propaganda (Score:2, Insightful)
Trolls are conservative? (Score:3, Insightful)
I'm in charge in Moria!!
Re:Propaganda (Score:3, Funny)
Kinda sad actually. If some frothing deaniac at the NYT or somewhere had written a totally non-political book the editors wouldn't have even found it worth mentioning the 'day' job of the author.
But then what do I know... I read Derbyshire's NRO columns so I'm irredeemably wicked in the eyes of the
If you are interested in solving math puzzles (Score:4, Interesting)
Not quite related to primes, but close and can certainly create an obsession. Also, look behind the scenes for something simpler to solve.
Re:If you are interested in solving math puzzles (Score:3)
They're both right. It might be possible to solve, but will certainly require some developments in cryptography that have yet to be made.
Failing that, a brute force search of all 128 bit inputs would do the job. I wouldn't expect it to finish quickly though.
All in all, I'd wager that your money's safe for at least the next ten years.
Re:If you are interested in solving math puzzles (Score:3, Interesting)
Actually, it is possible that the problem is unsolvable. You have 2^160 (not 128) inputs, 2^160 outputs. If there is a strict one-to-one mapping, it might not hold true for any pair. Personally, I think I'd be willing to give $50 for a proof that a solution exists at all. A lot more for a proof that a solution didn't exist.
To demonstrate, I'll use a silly 2 bit "hash" function.
00->
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.
ISBN not prime (Score:5, Funny)
The author must be sad.
Motivation (Score:2, Informative)
Re:Motivation (Score:3, Funny)
Plus chicks will dig you. Yessiree, I'm telling you. If you prove the Riemann Hyphothesis you'll be getting into some prime poontang. If you know what I mean.
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:Don't come along often? (Score:2, Insightful)
Not to forget N. Bourbaki, "Elements de Mathematique".
Re:Don't come along often? (Score:2)
Review: Review: Prime Obsession (Score:5, Insightful)
It's called white space. Look into it. Humans parse on it much faster then they parse on operators.
My favorite. (Score:4, Interesting)
It's very readable, and has chapters on interesting stuff like knot theory, cellular automata and primes.
I highly recommend it. It isn't going to turn anyone into a math professor, but it is very interesting reading.
Why I believe this book to be of interest. (Score:5, Interesting)
I agree with the reviewer's sentiment that the book is well written, and it is very enjoyable. The author writes in a very audience-centric fashion, even going as far to discuss the "scaffolding" of the book itself (all of the "hard math" stuff is found in odd chapters, the author had debated putting this information in only the "prime" chapters, but then said "there is such a thing as being too cute.")
Anywho, if you have a math friend you need to buy a gift for, definitely consider this book.
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.
Re:What is bad about him? (Score:2, Funny)
It's entirely people's prerogative to mix politics and pleasure, but my God, what a silly prerogative to exercise.
Re:What is bad about him? (Score:4, Informative)
Parent
Re:uh... (Score:2)
Re:uh... (Score:2)