Many of the mathematical ideas once considered impossible

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	• Yearning for the Impossible: The Surprising Truths of Mathematics John Stillwell AK Peters, Ltd., 2006 - 244 pages average customer review:based on 9 reviews view larger image
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highly recommended

Excellent overview of many less "traditional" topics

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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Many of the mathematical ideas once considered impossible

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

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Ideal Book for understanding Ideal

While browsing this book on the shelf of the Singapore National Library one sunday, I was shocked and delighted to spot on the chapter 7 of "Ideal". Being a Mathematics student of Abstract Math 30 years ago, I could not find a satisfactory answer from my French University Math professors or any Math books who could tell me what does this "Ideal" concept really mean beyond its arcane definition, where did it derive from, and why most textbooks insert "Ideal" in the chapter of Ring Theory?
Prof John Stillwell did a beautiful job by explaining in simple layman language the historical background of Kummer's work on FLT (Fermat Last Theorem), who encountered the controversy of Fundamental Law of Arithematics with Algebraic Number extended Field [a+ SQRT(-b)]. So instead of giving up, Kummer 'faked' the 'Ideal' number which he guessed could resolve the conflict. It was after his death that Dedekind discovered prime ideal, principal ideal's existence hidden in the compound of other numbers (Greatest Common Divisors of Rationals and Algebriac numbers, to be exact). Beautiful fake and discovery story in Mathematics!

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Excellent

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.

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Good but not super

If you look at the other reviews you'll see they are all full of praise. I really expected very much from this book, but the more I read in it the more I got disappointed. The material presented is indeed interesting, but the author's way of explaining things is quite often less than ideal. If this book had been written by someone like Paul Nahin, William Dunham or Adrian Paenza it would have been much better. Knowing much is one thing, explaining it the best way is another and unfortunately Stillwell isn't particularly good in the last thing.

reviews: page 1, 2

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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