A neat survey of the major fields of math by a professor adept at writing both popularizations and textbooks.

Strogatz (Applied Mathematics/Cornell Univ.; *Sync: The Emerging Science of Spontaneous Order*, 2003, etc.) begins with counting and a reference to a *Sesame Street* video called *123 Count with Me*, “the best introduction to numbers I’ve ever seen.” Throughout the book, the author never loses sight of the mystique and charm of numbers, and at the end, he explores concepts of infinity. There, Strogatz includes a classic proof of why some infinite systems of numbers contain more numbers than other infinite systems, an idea that shocked 19th-century mathematicians as much as the concept of imaginary numbers (the square roots of negative numbers) shocked their peers a century earlier. What’s remarkable about the author’s approach is that he conveys so much of the basic essences of the topics he covers, including algebra, geometry, trigonometry, calculus, probability theory, vector analysis and group theory, and he often stresses intuition and visualization. His discussion of the Pythagorean theorem expertly shows how the squares on the sides of a right triangle can be added to make the square on the hypotenuse. Not surprisingly, Strogatz also emphasizes the utility of math, quoting the physicist Eugene Wigner on “the unreasonable effectiveness of mathematics in the natural sciences.” In many cases, however, the author states the utility as a matter of fact rather than something to be proved—e.g., the ripples on a pond, the ridges of a sand dune and the stripes of a zebra, which reflect “the emergence of sinusoidal structure from a background of bland uniformity.” To learn why, readers should dig into the math more deeply.

A great book for the bright and curious, including even kids at grade school level up to college and beyond.