The Art of the Infinite: The Pleasures of Mathematics [Hardcover]

As you might expect, things start simple and get very complicated, and this is true right off in the first chapter, considering more and more complicated numbers. The Natural Numbers are introduced with patterns, as if you had stones to position on a table. 1, 3, 6, and 10 stones make pleasing equilateral triangles, and 1, 4, 9, and 16 make pleasing squares. We move from these to zero and negative numbers: "Certainly zero and the negatives have all the marks of human artifice: deftness, ambiguity, understatement." Are these numbers invented or discovered? The profundity of this question is plumbed throughout the book. Rationals, irrationals, and finally the complex numbers are all included. As the numbers mount up, the irregularity and regularity of the primes is considered, one of the most fruitful arenas of number theory. Euclid had to make an assumption about the infinite, his famous fifth postulate; but it is only an assumption; assuming that parallel lines meet eventually produces also a worthy geometry that tells us much about how the Einsteinian universe works. But there is no need to look into these strange worlds to find wonders; before leaving Euclid's terra firma, we are reintroduced to the triangle, and are presented with some astonishing revelations of secret points within and around the simple three sides that will remind you that no matter how simple things look, or even how simple things are, everything is more complicated than you can imagine.

And if you want your infinities more complicated still, the final chapter has to do with Cantor's work. Common sense tells us there must be half as many even numbers as there are whole numbers, but Cantor showed that the infinity of both was equal. He showed that the infinite number of points in a line as long as your finger was equal to the infinite number in a line as long as from here to the Sun. In fact, the number of points on a line is equal to the number of points in a plane. And yet, some infinities are bigger than others. This is strange territory indeed, and requires some concentration to understand and enjoy, even with the Kaplan's literate, witty, and clear explanations. This is a fine introduction to different aspects of serious mathematics; true to its subtitle, it is a book full of pleasures.

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.

Highly recommended.