Math Buffs Find an Easier "e"

Any study of exponential growth--from bacterial populations to interest rates--depends on a fundamental constant called "e." Because this number (often rounded to 2.718) can't be expressed as a fraction, scientists must estimate it with an approximate formula. Now a self-taught inventor and a meteorology professor describe in the October issue of Mathematical Intelligencer several new formulas for e and uses them to calculate it to thousands of decimal places with a desktop computer.
     For both bankers and bugs, e describes a basic limit to exponential growth. For example, if you invested $1 at 100% interest, compounded monthly, you would have $2.61 at year's end. If the interest were compounded every 30 seconds, you would end with about a dime more. No matter how frequently you earned interest, you could never take home more than e multiplied by the number of dollars you first deposited.
     Economists and population biologists often treat the discrete processes of compounding interest or of dividing cells as if they were continuous, because this allows them to describe the process by simple formulas involving e. The formulas derived by Harlan Brothers and John Knox, a meteorologist at Valparaiso University, Indiana, can continuously compound down to the equivalent of a few millionths of a penny. The formulas, in effect, reduce the discrepancy between discrete and continuous compounding. They averaged a simple formula, (1 + 1/n)ⁿ, that slightly underestimates e, with another, (1 - 1/n)-n, that slightly overestimates it. This doubled the number of correct decimal places. With further tinkering they were able to improve the accuracy sixfold.
     The new formulas would require too much computer memory to challenge the most accurate estimate of e, which is already known to 50 million decimal places, says numerical analyst Simon Plouffe of Hydro-Quebec in Montreal, holder of several numerical computation records. That doesn't worry Brothers and Knox. "What we've done is bring mathematics back to the people," says Knox, by demonstrating that ordinary folks can find fresh ways of representing e. "I'd like college math teachers to add it to the curriculum" to show students that textbooks don't always have the last word.
     --Dana Mackenzie

Definitions from the AP Dictionary of Science and Technology

Related Pages on APNet

• Journal of Mathematical Analysis and Applications
• Theoeretical Population Biology
• Games and Economic Behavior

Related links from the article above:

• Valparaiso University Department of Geography and Meteorology
• Hydro-Quebec
• American Mathematical Society
• European Mathematical Society
• Mathematical Association of America
• Ask Dr. Math
• Mathematics Archives
• Frequently Asked Questions in Mathematics
• Serendipit-e by John Knox

© 1998 The American Association for the Advancement of Science
This item is supplied by the AAAS Science News Service