In 1913, an amateur mathematician named Srinivasa Ramanujan sent several pages of his formulas for approximating π (the number 3.14159..., the ratio of the circumference of a circle to its diameter) to three mathematicians. Two of them rejected his work, probably thinking he was a crackpot. The third, revered English theorist G.H. Hardy, decided that Ramanujan's ideas were original and correct. The rest is history; Ramanujan taught Hardy a thing or two and became the most famous mathematical prodigy of our century.
Can history repeat itself, in a small way? In 1997, an inventor and amateur mathematician named Harlan Brothers discovered some simple algebraic expressions that always seem to yield the number 2.71828..., the universal constant e, which, like π turns up in an incredible range of mathematical and physical settings. In March of 1997, Harlan mailed several pages of his formulas to a well-known mathematician, and also to the host of National Public Radio's "Science Friday" program, a fellow inventor.
The mathematician ignored his mail. However, by chance my wife Pam was at that time working as an unpaid intern at "Science Friday", where she answered the mail that came to the show. Upon receipt of Harlan's paper, Pam asked me (a math major in college) to look it over and determine its worth.
|A comparison of traditional ("Classical" and "Direct") and two new methods ("MIM" and "BK") for calculating e (the dashed line). The Classical method is (1 + 1/x)x; the Direct method is 1 + 1/1! + ... + 1/(x-1)! + 1/x!, where the exclamation point defines the factorial operator. MIM is defined as [(2x + 1)/(2x - 1)]x, and BK is [(2 + x-x)/(2 - x-x)]x^x. For small values of x, MIM is more accurate than the traditional methods; for all x, BK is more accurate than the traditional methods.|
To my growing astonishment, I decided that Harlan's formulas were correct and didn't turn up in any textbook I could lay my hands on. Over the next several months, Harlan and I pooled our collective wits to take his ideas and flesh them out. Together, using no more mathematical knowledge than is taught in college calculus, we discovered and formally proved more than two dozen new algebraic expressions that yield e to extraordinary accuracy.
For example, using two Maclaurin series expansions (mathematical tools widely used since the 18th century), you can show that the expression [(2x + 1)/(2x - 1)]x yields e for any choice of x greater than 1, with the accuracy of the result increasing the larger the value of x (see MIM in Fig. 1).
Try it and see for yourself: if x = 10, [(2x + 1)/(2x - 1)]x is equal to (21/19)10 or 2.72055..., which is e accurate to two decimal places. Try it with x = 1000 and you get e accurate to 6 decimal places. We've even used a version of this expression to obtain e correct to 30,000 decimal places! Not bad for an expression an eighth-grader could understand, yet one that eluded the founding fathers of calculus and all their successors. What's more, we have discovered other new expressions for calculating e that are even better (see BK in Fig. 1)!
"Serendipity" is defined by Webster's on-line dictionary as "the faculty or phenomenon of finding valuable or agreeable things not sought for." This one word sums up how I, as a meteorologist with research interests in the stratosphere and in climate modeling, found myself collaborating with an inventor to discover new and elegant methods to calculate e. While these discoveries don't necessarily have practical applications, they do shed more light on the remarkable properties of e. And while Harlan Brothers may not be the next Ramanujan — and I'm probably closer to being Laurel than G.H. Hardy — perhaps tomorrow's college calculus students will be more motivated to do their homework when they hear how we discovered formulas for e that the founders of modern mathematics — from Sir Isaac Newton on down — had overlooked!
Brothers, H.J., and J.A. Knox 1998. New closed-form approximations to the logarithmic constant e. Math. Intelligencer 20, 4:25-29.
Knox, J.A., and H.J. Brothers 1999. Novel series-based approximations to e. College Math. J. 30, 269-275.