This is one of those, surprisingly, rare mathematics books that is accessible to a general audience, without talking down or glossing over the mathematics. The book records a series of presentations by Serge Lang to high school students in Toronto. The talks cover primarily geometry and algebra, and in this context readers are introduced not only to geometric ideas but concepts also relevant to calculus and infinity. The first chapter is nominally about pi, but while introducing pi the author presents numerous concepts including the use of regular polygons inscribed in a circle to approximate pi. The number of sides of the inscribed polygons are increased until they approach infinity, with future relevance to integration (although not explicitly covered in this chapter). Later chapters extend the idea of using various geometric concepts and figures, some multi-dimensional, to find the volume of a ball, the length of a circle, and the area of a sphere. The book closes with chapters on Pythagorean triplets, infinities, and how mathematics should be taught and learned.
This book should be accessible to readers with a modest mathematics background, such as obtained in high school. Topics covered are presented with unique insights and perspectives that frequently differ from typical classroom texts. While mathematics is a not a spectator sport, readers can gain real insights and understanding by thinking about the concepts as they are presented, and without necessarily taking pencil to paper. The title may suggest that this is a book primarily for high school students and their teachers. However, this would be inappropriately limiting its audience. Even recognizing the relatively high list price for this small paperback, this unique and relatively short book (it can be completed in less than a day) should prove informative and interesting to both a general audience and those with more extensive mathematics backgrounds - highly recommended.
