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Yearning for the Impossible: The Surprising Truths of Mathematics | 
 enlarge | Author: John Stillwell
Publisher: AK Peters, Ltd.
Category: Book
List Price: $34.95
Buy New: $21.85
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Rating:  9 reviews
Sales Rank: 183853
Media: Hardcover
Pages: 244
Number Of Items: 1
Shipping Weight (lbs): 1.1
Dimensions (in): 9.1 x 6.1 x 0.9
ISBN: 156881254X
Dewey Decimal Number: 510
EAN: 9781568812540
Publication Date: May 22, 2006
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Product Description
This book explores the history of mathematics from the perspective of the creative tension between common sense and the "impossible" as the author follows the discovery or invention of new concepts that have marked mathematical progress: - Irrational and Imaginary Numbers - The Fourth Dimension - Curved Space - Infinity and others The author puts these creations into a broader context involving related "impossibilities" from art, literature, philosophy, and physics. By imbedding mathematics into a broader cultural context and through his clever and enthusiastic explication of mathematical ideas the author broadens the horizon of students beyond the narrow confines of rote memorization and engages those who are curious about the place of mathematics in our intellectual landscape.
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| Customer Reviews: Read 4 more reviews...
Beautiful, substantial, unusual topics October 15, 2006
Viktor Blasjo
38 out of 49 found this review helpful
The chapters on geometry---projective geometry and hyperbolic geometry in particular---are extremely beautiful. We study many picturesque ideas, wonderful in themselves, that arise as "impossibly" neat solutions to interesting problems (perspective drawing, axiomatisation of geometry, shape of the universe, etc.), and then pay off by supplying unexpected insights elsewhere (e.g., the connection between projectively generated arithmetic and hypercomplex number systems that deserves to be better known). The chapters on complex numbers and quaternions are also very interesting. There are "unrecognised appearances" of complex numbers in already in Diophantus's number theory, namely the equivalent of complex multiplication in the context of sums of two squares. Thousands of years later, when the geometry of complex numbers was established, the search for a three dimensional analog failed and one had to settle for the analog in four dimensions. The historical circle closed beautifully when Graves noted with surprise and satisfaction that this state of affairs is precisely mirrored in classical number theory where Diophantus's theorem on sums of two squares generalises to four squares but not three. Stemming from the same roots in classical number theory, there is also an excellent chapter on algebraic number theory. Just as in his proof of the non-existence of three dimensional hypercomplex numbers in the quaternion chapter, Stillwell here takes on some very serious mathematics that many mathematicians would tell you require plenty of abstract algebra. But Stillwell knows better, cutting to the core of things with beautifully clear geometric arguments in both these cases. The other chapters are less innovative, although we are happy with the initiative to derive the infinite series for pi (essentially by the power series for the arctangent) only ten pages after the idea of infinitesimals is introduced (again relying on geometric methods rather than, as others would have it, abstract theories like Taylor's theorem). The role of the impossible in mathematics is pointed out along the way, and Stillwell offers some rewarding reflections on this subject; these are highly retrospective, however, and if we were to take this topic seriously we would have wished for greater insights into the historical mathematicians' thoughts on these supposedly impossible things.
Many of the mathematical ideas once considered impossible October 15, 2007
Charles Ashbacher (Marion, Iowa United States(cashbacher@yahoo.com))
16 out of 16 found this review helpful
There are many great ideas in mathematics and what makes them unique is that many of them were considered impossible for a long period of time. In this book, Stillwell presents many of those ideas using an expository style that is both understandable and complete. The chapters are:
*) The Irrational - where the discovery of irrational numbers and how it shocked the Pythagoreans is explained. It forever destroyed the idea that everything could be completely expressed using only the integers. This discovery also made it clear that some things would forever remain unknown.
*) The Imaginary - this section describes the development of the "imaginary" numbers, where the impossible task of taking the square root of a negative number became routine.
*) The Horizon - where converging parallel lines allowed artists to perform what was considered impossible, give two-dimensional paintings a three-dimensional perspective.
*) The Infinitesimal - where splitting a figure into extremely small sections made it possible to easily solve an enormous number of complex problems.
*) Curved space - where the natural world of Euclid was suddenly overturned by the creation of curved worlds that are even more natural.
*) The Fourth Dimension - where the impossibility of structures having more than three dimensions is proven false. Along the way, imaginary numbers are made even more so by the development of the quaternions.
*) The Ideal - in this case, the impossibility of numbers having more than one fundamental factorization is overturned only to be partially restored.
*) Periodic Space - among others, M. C. Escher demonstrated that it is easy to place impossible objects on a canvas.
*) The Infinite - where it is demonstrated that not all infinities are alike, it is the case that some infinities have more elements than others.
Stillwell does an excellent job in pointing out that "impossible" is a difficult word to use in mathematics, as it is relative to the definitions of the object being examined. While there is absolute truth in mathematics, something lacking in many other areas of human endeavor, the truth is also often relative to how imaginative we are in our definitions.
Published in Journal of Recreational Mathematics, reprinted with permission
Excellent July 18, 2007
Mark Shapiro (USA)
14 out of 14 found this review helpful
This book, which can be viewed as a prequel to Stillwell's "History of mathematics", is an excellent resource for someone who wishes to get a view of mathematics as a field of inquiry driven by the need to solve problems as much as by creative desire to uncover connections between seemingly unrelated ideas by people who made mathematics, such as Gauss, Hamilton, Abel, Euler, Riemann. There are lively short essays about these and other great mathematicians. When read along with regular (good) textbooks on, e.g., complex variables, geometry, the two Stillwell's books will lead to a much better understanding of mathematical ideas.
Beyond Common Sense May 29, 2007
Lewis H. Robinson
10 out of 10 found this review helpful
I liked this book. I particularly liked Chapter 1, The Irrational, Chapter 5, Curved Space, and Chapter 6, The Fourth Dimension.
Chapter 7, The Ideal, is also excellent and alone worth the purchase price, albeit the reader needs to follow closely the notational details and diagrams. In fact Chapter 7 is the reason I bought the book in the first place. I had always struggled with this important concept and was pleasantely surprised upon finding a book--Stillwell's--that devoted a whole chapter to the subject at an introductory as well as historical level. The author follows the development of the notion of the ideal concept from Gauss, to Kummer, to Dedekind's final generalization, where the payoff comes in Section 7.8. "Ideals, or Unique Prime Factorization Regained".
Excellent overview of many less "traditional" topics August 11, 2007
Kenneth Knowles
7 out of 7 found this review helpful
It is very nice to see a book that treats topics other than irrational and complex numbers (though they are important to understand first, of course!) like quaternions and prime ideals, not to mention all the geometrical connections. This book gives a great historical and motivational perspective; the author may be augmenting the personalities in the book to add to the suspense and mystery, but overall the effect is beautiful.
I would recommend this book for anyone interested in Mathematics, including advanced students (I am a PhD student hovering near the border of Computer Science and Math). It is a welcome inspirational supplement to the tragedy of axioms and formalism that is modern mathematics education.
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