52 of 52 people found the following review helpful
Prime Obsession, is a wonderful book based on the history and insight of the brilliant mathematician, Bernhard Riemann. As the title suggests, the main aim of the book is to give the reader a clear and understandable definition of what the Riemann Hypothesis actually is. To do this, Derbyshire has structured the book so the reader is given a chapter of mathematical tools, followed by a chapter of the history of Riemann and other great mathematicians, such as Gauss, Euler, Hardy, followed by a math chapter etc... However, don't let the math sections put you off this book, as Derbyshire explains, he uses minimal calculus to get the reader through the book. He takes the reader though basic analysis, then onto prime numbers, domain streching, followed by what he calls the Golden Key which uses the Euler product. Then he introduces basic complex number theory, and finally he pulls them all together to start to explain the RH (Riemann Hypothesis). Riemanns ideas and visualizations of complex functions are difficult to comprehend for even the most accomplished mathematician, but Derbyshire employs a method that any lay person can understand perfectly, using his "Argument Ant". Any person interested in mathematics, should read this book, as it serves as a wonderful insight into one of the greatest mathematicians, and problems that has ever existed. And for those who are just interested in the RH but were never quite sure where the zeros come from, then the chapter on domain streching and subsequent chapters will make it all clear. This is the best popular science book I have read since Feynmans "QED: The strange theory of light and matter".
45 of 45 people found the following review helpful
I have read this book and two of the other three popularisations about the Riemann hypothesis. Instead of interviewing mathematicians who may be near to solving it or writing around the subject, this book actually works through the mathematics of Riemann's 1859 paper.
"Prime obsession" emphasises the centrality of the other parts of Riemann's paper apart from the famous Hypothesis. By doing this it helps to explain why some 30 years later that mathematicians were able to prove the Prime Number Theorem, independently of the truth or otherwise of the famous Hypothesis. The Prime Number Theorem states, roughly that: as numbers get larger the number of primes less than that number tends to about the number divided by its logarithm (base e). The reason the Prime Number Theorem could be proved, irrespective of Riemann's Hypothesis' truth, is because of the techniques that Riemann invented in his 1859 paper.
Riemann's starting point was to generalise Euler's formula which relates the sum of the reciprocals of natural numbers:
1+1/2+1/3+1/4+...
to the product of the inverses of the prime numbers
(1/2)*(1/3)*(1/5)*(1/7)*(1/11)*.....
Derbyshire's explanation is far clearer and much easier to follow than those in the other popularisations.
This book is precise and clear: one really feels that one has some insight into an astonishing piece of creative mathematical work by the time one has read the book. That alone in my opinion should qualify it as one of the greatest pieces of popular science writing of this or any other decade.
This book needs to be more actively marketed: whatever its faults, the author has made a genuine attempt to really explain a great piece of science technically to a non -technical audience, rather than just waffling around the subject and making us all feel these things are so far above our heads we will never understand them in any way. This courage on the author's part needs to be more widely feted.
I cannot do more than endorse the other reviewers' praise for this classic-to-be. for those interested in pursuing this fascinating subject further, I found Gamma: Exploring Euler's Constant (Princeton Science Library) by Havil to be a wonderful book.
22 of 23 people found the following review helpful
I am a bit of a junkie for books on maths, revisiting my degree of 15-20 years ago. The quality varies a lot though and I am very often disappointed. This I supose is not surprising: I want not to be patronised but I also want accessibility, context (historical, personal), and some insight into the underlying beauty of the mathematics in question. But this book pushes all the right buttons.
The Riemann Hypothesis is really quite advanced - you wouldn't find much in-depth study of it in any compulsory modules of undergraduate courses. But Derbyshire brings it to life. The book is challenging but accessible, and ultimately a very fulfilling read.
I think the key to his success is the interleaving of chapters on the lives of the protagonists with those on the maths leading up to and surrounding the Hypothesis. Because an understanding of the relevant mathematics helps understand the importance of a given mathematician's life, and an understanding of historical context helps bring the maths to life, these chapters are mutually reinforcing. As such the whole is greater than the sum of the parts (I think I might just have found that 1+1>2). And because so many of the great mathematicians contributed to the foundations of number theory and analysis, and many subsequently worked on the Riemann Hypothesis itself, this book kind of doubles as a selective history of modern (from Newton) mathematics.
I can't recommend this book enough. Even for those with no background in maths, but with an enquiring spirit, there is enough here (crucially, without turgidity) to dimly comprehend the profound beauty and true mystery of maths. It makes you believe somehow in the Platonic Ideals and that those blessed with true insight get closer to them than the rest of us. I have always felt that advanced pure mathematics is as worthy an art as painting or sculpture, and the great mathematicians as worthy artists as Van Gogh etc. But because of the inaccessibility of the subject matter to the layman this great art couldn't be widely-enough shared. With more books like Prime Obsession this wrong will be righted.
5 of 5 people found the following review helpful
If you've come from Simon Singh's "Fermat's Last Theorem" or Marcus du Sautoy's "The Music of the Primes" then prepare yourself for some much heavier maths. To be honest it took me two weeks to get through, and the last few chapters utterly defeated me. This is not a failing of the book though. The maths is explained to the lay person, me, with great clarity and lashings of insight. I haven't touched integration since leaving college at 18, and I think it is testament to the author's skill that I now understand its application and manipulation better than ever. I followed all the reasoning and methods as we went, and there were even small epiphanies along the way. But somewhere around half way I found it had been too much for me. I couldn't recall all of the steps as we went higher and higher. Riemann's Hypothesis draws on so much maths that it requires a huge, concerted effort to follow. What kept me going were the larger themes of how different theories were so deeply related to each other, and the astonishing ways in which those zeros kept cropping up. Truly amazing stuff.
While I half expected to find the maths too difficult, I was pleasantly surprised to thoroughly enjoy the historical chapters too. These chapters detailed the characters and the often difficult environment in which much of the maths was developed. Really interesting stuff and it helped me to get over the maths as well as to appreciate it better.
So while I failed this time round, I was happy to be stretched beyond my limits and I will be re-reading this great book; I don't think I could take it right now though! I simply wanted to encourage anyone hesitating, like I did, to take on Riemann's Hypothesis. It is impossible to express the depth of Riemann's insight without something on the scale of Prime Obsession.
4 of 4 people found the following review helpful
Georg Friedrich Bernhard Riemann is one of the most significant and influential mathematicians of all time. He has singlehandedly created whole new branches of mathematics (such as differential geometry) and has made seminal contributions to several others. Unfortunately, Riemann's life was cut short at the age of thirty-nine, and we can only speculate on how much more he could have given us had he lived longer.
One of the areas of mathematics that Riemann made a major contribution in is the analytic number theory. Analytic number theory is one of the most fascinating chimeras in all of mathematics. It combines two at the first glance completely unrelated fields: analysis, which deals with continuity, and number theory, which works with the discrete countable numbers. Prime numbers in particular are the most discrete numbers of them all, and from the very early on people have been fascinated with their properties and distribution amongst other numbers. In one of his many papers Riemann made a conjecture about the exact form of this distribution, and this conjecture has become known as the Riemann Hypothesis. As of this writing this hypothesis remains unproved, and attempts to shed some light on it are among the most important areas of the contemporary mathematical research.
Writing about mathematics for the general audience is a major challenge. In most sciences it is possible to appreciate the objects that are studied even though one does not understand any of the research tools or conceptual frameworks that are being used. Black holes and genetic engineering have become parts of the general culture. To paraphrase a statement attributed to Euclid, there really isn't any royal road to mathematics. Having that in mind, one ought to applaud John Derbyshire and his valiant attempt to bring the Riemann Hypothesis to as wide of an audience as possible. Roughly speaking, the book alternates between chapters that deal with mathematics and those that deal with mathematicians. Mathematics is ultimately a human invention and learning about remarkable individuals who have built it to what we have today is a fascinating story in its own right. The math chapters of this book can prove a bit challenging. It is probably impossible to explain Riemann Hypothesis in anywhere nearly accurate way without employing calculus and complex numbers. However, if you are comfortable with these two concepts then John Derbyshire's explanation will be the gentlest yet the most comprehensive one that you will come across. Short of opening a math textbook you will not find a batter explanation of what Riemann Hypothesis is all about.
This is an eminently well written book. John Derbyshire combines good, smooth-flowing prose with historical insights that help the reader navigate through the maze of modern mathematics. He exhibits a deep appreciation for his subject, and this book conveys much of this enthusiasm. "Prime Obsession" is easily one of my favorite popular mathematics books.
3 of 3 people found the following review helpful
Prime Obsession, is a wonderful book based on the history and insight of the brilliant mathematician, Bernhard Riemann. As the title suggests, the main aim of the book is to give the reader a clear and understandable definition of what the Riemann Hypothesis actually is. To do this, Derbyshire has structured the book so the reader is given a chapter of mathematical tools, followed by a chapter of the history of Riemann and other great mathematicians, such as Gauss, Euler, Hardy, followed by a math chapter etc... However, don't let the math sections put you off this book, as Derbyshire explains, he uses minimal calculus to get the reader through the book. He takes the reader though basic analysis, then onto prime numbers, domain streching, followed by what he calls the Golden Key which uses the Euler product. Then he introduces basic complex number theory, and finally he pulls them all together to start to explain the RH (Riemann Hypothesis). Riemanns ideas and visualizations of complex functions are difficult to comprehend for even the most accomplished mathematician, but Derbyshire employs a method that any lay person can understand perfectly, using his "Argument Ant". Any person interested in mathematics, should read this book, as it serves as a wonderful insight into one of the greatest mathematicians, and problems that has ever existed. And for those who are just interested in the RH but were never quite sure where the zeros come from, then the chapter on domain streching and subsequent chapters will make it all clear. This is the best popular science book I have read since Feynmans "QED: The strange theory of light and matter".
38 of 42 people found the following review helpful
Having read Marcus de Sautoy's book on prime numbers my appetite was sufficiently wetted to go out and by Edwards book on the Zeta function. Unfirtunately one look at this told me I wasn't going to be able to get through it. I picked this book up by accident and it was fascinating in that the author goes through the whole of Riemanns 1859 paper and explains the whole theorem, which is quite breathtaking in its brilliance. He loses it a bit at the end, but he can be forgiven for that as it does become very complicated. That combined with the way he weaves the history of prime numbers in alternative chapters makes this a thoroughly enjoyable book. If you like maths go and buy it!
2 of 2 people found the following review helpful
"...so it was with Riemann. Outwardly he was pitiable; inwardly, he burned brighter than the sun."
So says the author of this work on Bernhard Riemann and his famous hypothesis.
This is an excellent book on the subject. But the lay reader will find it difficult. Knowledge of Maths is required to navigate it.
Riemann was an incredible genius. Another area of Maths that his name has been lent to is Riemannian Geometry. Einstein later availed of Riemann's creation to build the mathematical formalism of General Relativity. On the shoulders of giants.
The Riemann Hypothesis is currently unsolved. We don't know if the hypothesis is true or false. (It has nothing to do with Riemannian geometry.)
Why is the Riemann Hypothesis so important? One reason is there is a connection between it and the distribution of the prime numbers. Is there a pattern hidden in the primes? Nobody knows.
Tantalising hints of some hidden order remain. See Chapter 18 - "Number Theory Meets Quantum Mechanics." (Hugh Montgomery meets Freeman Dyson ).
Then there is the strange story recounted in Chapter 22, the last chapter of the book. Two mathematicians go to Gottingen to see Riemann's notes from 1859. One wants to look at some pure number theory stuff. The other, the applied mathematician with no interest in all that, wants to look at some work Riemann had done on perturbation theory. Both put in their separate requests to the librarian.
She comes back with just one set of notes. It turned out that Riemann had been working on both of these problems at the SAME TIME.
3 of 3 people found the following review helpful
If you are interested in maths (and number theory in particular) this is one of the better "popular" books. Whereas other books in the genre often dumb things down a tad too much and leave you wanting for more (what does that formula look like? what does a graph of that function look like? how was that theorem proved? etc.) this book is not afraid to show formulas and graphs and at least the gist of complex issues. It also breathes life into the people who did the work (as well as their historical, political and mathematical context) and whom we otherwise mostly know because they have a theorem named after them.
Minor gripes: does the Lindelöf hypothesis really not deserve more than a footnote to an appendix? Why weren't the often informative footnotes put at the bottom of the page they refer to?
16 of 18 people found the following review helpful
This book by John Derbyshire is absolutely fantastic. Giving a thorough insight into the history of Riemann, and mathematics for that matter, provides the reader with a fuller knowledge before the author tries to smash through the hypothesis bit by bit. Breaking down the ideas mathematicians have developed over the past 140 years in trying to solve this greatest 'unsolved' problem in mathematics, Derbyshire gives the reader the feeling that Riemann truly was a fantastic mathematician and his innovative ideas are truly unique. The proof in which is that this hypothesis is still today unsolved. If you want a book about this complex hypothesis, I reccomend this. Easily illustrated and not too difficult to understand, Derbyshire makes this hypothesis seem so trivial in complexity and worthiness to the lay mathematician, yet to those with a keener knowledge this book relays the hidden answers beautifully.