The Art of the Infinite: The Pleasures of Mathematics - ISBN:0195176065

The Art of the Infinite: The Pleasures of Mathematics

Date: 15 April 2011, 16:57

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Here's human imagination at work. The flights of fancy the Kaplans show us are not about dragons and wizards, but about imaginary numbers, square roots, triangles, and infinite series.
I bought this book to mine for ideas to use in the notes I am writing to accompany the Third Edition of Geometry by Harold Jacobs, and I struck a rich lode. My professional interests made me look at material of a more technical nature, such as the proof of the theorem of Pappus. Pappus noticed that if you take six points A, B, and C on one side of an angle and a, b, and c on the other side of this angle and join each point to the two points labeled by *different* letters, then the three points of intersection of these six segments lie on a straight line. I knew this as a fact since my high school days, but it is not easy to give a proof that is reasonable at that level. The Kaplans have a beautiful explanation of this result, putting it in context and giving a gentle proof. Very nice indeed.
They have found just the right diagram or line of argument for many things I have seen before. Those of us who have suffered through the terrors of trigonometry will remember that there are some angle sum formulas, though we may not remember exactly what they are. The diagram at the top of page 187 tells you why these formulas are true and will make them unforgettable, if you decide to remember it. The path to this figure is made easy and natural in the book. What was new to me was the idea of adding a box around the tipped triangle --- suggested in the throw away line at the top of page 186. This gives us just what we need, neither too much nor too little.
One virtue of this book is that you can leaf through it and dive into the text wherever you see an interesting illustration or some idea you have been wondering about. The topics are mostly self-contained and there is always a nice story or bit of historical context to give you a sense of where you are and how this fits into the larger picture.
Buy this book, browse it, read it, and now and then get out your paper and pencil and puzzle through whatever tickles your fancy. This book is not just *about* mathematics, it gives you the real stuff.

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