Ivars Peterson's MathTrek
February 16, 2004
One way to define e is as the number that the expression (1 + x)1/x approaches as x gets smaller and smaller. Thus, when x is 1, the expression equals 2; when x is .5, the expression is 2.25; when x is .25, the expression is 2.4414. . . , and so on.
You can also obtain an approximate value of e by summing the following terms, where n! represents the product of all integers from 1 to n: 1 + 1/1! + 1/2! + 1/3! + 1/4! + . . . + 1/n!
This expression is sometimes described as Newton's series approximation for e. It's also known as the Maclaurin series expansion or the Direct Method.
In graphic terms, e is the number (greater than 1) for which the area below the curve y = 1/x, above the x axis, to the right of the line x = 1, and to the left of the line x = e is precisely equal to 1.
In 1998, two math buffs, inventor Harlan J. Brothers and meteorologist John A. Knox, discovered new, amazingly simple formulas for calculating e.
Brothers had begun his search for new formulas for e in early 1997. He mailed his first results to the National Public Radio program "Science Friday," which is based in New York City. At that time, Knox's wife was an intern at "Science Friday," and she happened to open the letter that Brothers had sent. A climatologist with a background in mathematics and physics, she passed it on to her husband, who was then at Columbia University and the NASA/Goddard Institute for Space Studies in New York City. Knox, who had been a college math major, confirmed that Brothers had found a novel, correct approach to calculating e.
The two men started collaborating. "Together, using no more mathematical knowledge than is taught in college calculus, we discovered and formally proved more than 2 dozen new algebraic expressions that yield e to extraordinary accuracy," Knox said at the time. Some of these formulas outperform conventional methods used to approximate e to a large number of decimal places.
Here's one example. As x gets larger, the expression [(2x + 1)/(2x 1)]x gets closer and closer to e. For x equal to 10, the expression yields (21/19)10, or 2.72055. . ., which is e accurate to two decimal places. For x equal to 1,000, you get e to 6 decimal places.
"We've even used a version of this expression to obtain e correct to 30,000 decimal places," Knox said. "Not bad for an expression that 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," he added. For example, try out [(2 + 2x)/(2 2x)]2^x.
Brothers has continued to investigate e's quirks. In the January College Mathematics Journal, he presents new series expressions for approximating e. Here's one example of such a series:
1/0! + 3/2! + 5/4! + 7/6! + 9/8! + 11/10! + 13/12! + 15/14! + . . .,
where each term has the form (2k + 1)/(2k)!, with k starting at 0. Other series approximations use terms of the following forms: (2k + 2)/(2k + 1)!; (3 4k2)/(2k + 1)!, or [(3k)2 + 1]/(3k)!. These new series include the fastest known methods for computing e. They converge to the value of e much more quickly than Newton's original expression, which has long been the basis for computing the value of e.
Brothers describes the so-called compression techniques he used to develop series approximations that improve upon Newton's original method in his College Mathematics Journal article. "Using techniques familiar to any first-year calculus student, it is indeed possible to derive series that converge more rapidly," he says.
"It is hoped that the inherent symmetry and numerical beauty of these newly derived expressions might provide inspiration to students, educators, and all who are drawn by the allure of numbers," Brothers concludes.
"Students should also feel encouraged to explore other well-known series, such as those associated with pi, to examine which of them might lend themselves to effective compression," he suggests. In the meantime, the spotlight is on e.
"The logarithmic constant e is famous for turning up whenever natural beauty and mathematical elegance commingle," Brothers and Knox concluded in an article in the Mathematical Intelligencer describing their 1998 discoveries. "Our work provides a new glimpse of its austere charm."
The continuing fascination with e may indeed signal an improved, updated image for this venerable, under-appreciated number.
Originally posted: 11/9/98
Copyright © 2004 by Ivars Peterson
Brothers, H.J. 2004. Improving the convergence of Newton's series approximation for e. College Mathematics Journal 35(January):34-39. Available at http://www.brotherstechnology.com/docs/icnsae_(cmj0104-300dpi).pdf.
Brothers, H.J., and J.A. Knox. 1999. Novel series-based approximations to e. College Mathematics Journal 30(September):209-215. Available at http://www.brotherstechnology.com/docs/cmj_paper1.pdf.
______. 1998. New closed-form approximations to the logarithmic constant e. Mathematical Intelligencer 20(No. 4):25-29. Available at http://www.brotherstechnology.com/docs/mi_paper1.pdf.
Gardner, M. 1969. The transcendental number e. In The Unexpected Hanging and Other Mathematical Diversions. New York: Simon and Schuster.
Maor, E. 1994. e: The Story of a Number. Princeton, N.J.: Princeton University Press.
Harlan Brothers has a Web site at http://www.brotherstechnology.com/math/.
Information about the mathematical constant e and its computation is available at http://mathworld.wolfram.com/e.html, http://numbers.computation.free.fr/Constants/E/e.html, and http://pi.lacim.uqam.ca/piDATA/expof1.txt.
For a fascinating glimpse at where the symbols for mathematical constants, included e, originated, take a look at http://members.aol.com/jeff570/constants.html.
Comments are welcome. Please send messages to Ivars Peterson at email@example.com.
A collection of Ivars Peterson's early MathTrek articles, updated and illustrated, is now available as the Mathematical Association of America (MAA) book Mathematical Treks: From Surreal Numbers to Magic Circles. See http://www.maa.org/pubs/books/mtr.html.