Try this with a slice of pi

The history of the physical sciences is filled with dead ends. No chemist today would try to create a new compound from the four ancient elements of earth, air, fire, and water. Deep space probes to the outermost reaches of the solar system have found no evidence of revolving crystalline spheres with the earth at the center. Mathematics, by contrast, is a steady progression. What was true for Euclid more than two millenniums ago is still true today. In this way, mathematics is its own history.

For evidence of this, one need only turn to a recent sample of books that attempt the unenviable task of explaining modern math to the popular reader. A good place to start is Keith Devlin's The Millennium Problems, an account of seven puzzles, the solution to which is worth a cool million each.

Before you leap to pick up a pencil and paper, though, it would be well to read this remarkable effort to explain exactly what is involved. Devlin is doubly qualified as the "Math Guy" from NPR's Weekend Edition and the author of numerous previous books on mathematics. Nonetheless, the job of describing these problems in understandable terms is far from easy. The scope of modern mathematics has become so complex and far-reaching that many professional mathematicians are not exactly clear about what each of these enigmas may mean. Devlin does a superb job trying, though, and his account is both fascinating and accessible to any reader who can remember some high school math.

For background it might be a good idea to look in Barry Mazur's Imagining Numbers, a poetic and profound meditation on the mathematical imagination. Mazur "does mathematics," as he says, at Harvard University and is an authority on (among other things) the branch of math called "number theory." This is the study of numbers and fractions, although considerably advanced from grade-school arithmetic. Mazur explores the history of imaginary numbers, that is, numbers that are some multiple of the square root of minus 1. He evocatively recreates the bewilderment of the Renaissance scholars who first grappled with this idea, and through the lens of their imagination, he leads the reader to a deeper comprehension of the realm of numbers.

The first of those seven $1 million millennium problems, by the way, is the oldest, devised by the German mathematician Bernhard Riemann in 1859. Riemann was interested in prime numbers (3, 7, 11, 13), those curious figures that cannot be evenly divided except by 1 and themselves. From the time of Euclid, the primes have presented a riddle for mathematicians trying to understand their puzzling pattern. This question has become extraordinarily significant for us because mathematical operations on prime numbers are the basis of most of the current systems that allow shoppers to send credit card numbers securely across the Internet.

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