Hunting e

Of the irrational, transcendental numbers, pi seems to get all the attention. Web sites and books celebrate its quirks and quandaries. Its digits have been computed to 51,539,600,000 decimal places (see A Passion for Pi, 3/11/96).

Lagging far behind in the celebrity sweepstakes is the number known as e. Carried to 20 decimal places, e is 2.71828 18284 59045 23536. Only 50,000,817 of its decimal digits have been computed so far--though there are unconfirmed reports that 1 billion digits were once calculated. People can't even agree on its proper name. It has been called the logarithmic constant, Napier's number, Euler's constant, and the natural logarithmic base.

One way to define e is as the number that the expression (1 + x)^(1/x) approaches as xgets smaller and smaller. Thus, when x is 1, the expression equals 2; when xis .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!

In graphic terms (see below), 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.

The number e comes up in a wide variety of mathematical contexts. The constant also plays a key role in descriptions of phenomena such as radioactive decay and population growth and, in the financial world, in calculations of compound interest.

Now, two amateur mathematicians have discovered new, amazingly simple formulas for calculating e. Harlan J. Brothers, an inventor in Branford, Conn., and John A. Knox, a meteorology professor at Valparaiso University in Indiana, describe their findings in the October Mathematical Intelligencer.

Brothers began 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, Pam, 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 says. Some of those 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 says. "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 adds. For example, try out [(2 + 2^-x)/(2 - 2^-x)]^2^x.

"The logarithmic constant e is famous for turning up whenever natural beauty and mathematical elegance commingle," Brothers and Knox conclude. "Our work provides a new glimpse of its austere charm."

And perhaps an improved image for this venerable, underappreciated number.

References:

Brothers, H.J., and J.A. Knox. 1998. New closed-form approximations to the logarithmic constant e. Mathematical Intelligencer 20(No. 4):25.

Conway, J.H., and R.K. Guy. 1996. The Book of Numbers. New York: Copernicus.

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.

John Knox has a Web page at http://www.giss.nasa.gov/research/intro/knox.03/.

Additional information about e can be found at the following Web site: http://www.astro.virginia.edu/~eww6n/math/e.html.

For a fascinating glimpse at where the symbols for mathematical constants, including e, originated, take a look at http://members.aol.com/jeff570/constants.html.

Comments are welcome. Please send messages to Ivars Peterson at ipeterson@maa.org.