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
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.
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