Wednesday, December 14, 2005

Computers and Mathematics

Back in the early seventies of the last century I spent an academic year as a visitor at the Ohio State University College of Law in Columbus, Ohio and ate many of my lunches at the Faculty Club. Several times I had as a luncheon companion a physicist from the Battelle Memorial Institute, which is headquartered in Columbus.

One day the physicist and I got to talking about early computers--which meant computers back in the 1950's--and he told me this story.

Battelle had an early computer--it may well have been one that was programmed by moving plugs and jumpers around on a so-called ``breadboard''--that had only a few kilobytes of memory. And researchers at Battelle also had an equation that they wanted to solve, but the equation turned out to be a bit too large to fit into the memory of their computer. So the researchers--who I guess were physicists--asked one of the mathematicians at the Institute to see if he could simplify the equation enough to squeeze it into the computer's memory.

The next day the mathematician reported that he had been able to simplify the equation and that the result was: e.*

I assume that if the original equation had fit into the computer's memory, the computer would still be grinding away trying to solve the problem since the string of digits representing e is unending.

I suspect that there is some sort of moral here.


e is one of those special numbers in mathematics, like pi, that keeps showing up in all kinds of important places. For example, in Calculus, the function f(x) = c(ex) for any constant c is the one function (aside from the zero function) that is its own derivative. It is the base of the natural logarithm, ln, and it is equal to the limit of (1 + 1/n)n as n goes to infinity....

Like Pi, e is an irrational number....

A Proof that e is Irrational

e is approximately equal to: 2.7182818284590452353602874713526.


Blogger Seth Finkelstein said...

Regarding: "I assume that if the original equation had fit into the computer's memory, the computer would still be grinding away trying to solve the problem"

Umm ... Huh? The program would stop as soon as it calculated the result to the limits of the computer's memory, or storage allocated for the result. As physicists, they probably wouldn't be interested in more than a handful of the result's digits anyway.

This genre of story is actually a common legend - the mighty computational labor which turns out to have a simple result (which, it must be admitted, sometimes does have a basis in fact).

10:42 PM  
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1:42 PM  

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