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¥ @ ? Utilizing Infinite Loops to Compute an Approximate Value of Infinity

Jim Stanfield, co-founder, The Institute for Further Research

The concept of infinity has intrigued mathematicians for millennia. Cantor made some progress towards a value with his Aleph null and Aleph one, etc, but this thought-to-be inapproachable quantity remains the holy grail of mathematicians seeking a Fields medal (there is no Nobel prize in math).

But now, with the advent of powerful computers, the author thinks there may be a way of arriving at, if not the exact value, at least an approximate value of infinity.

Programming in Basic, on a Commodore 64, a simple infinite loop was set to iteratively increment a variable by one. The initial run of Infinite Loop v1.0 was unsuccessful. After running for three days with not and end in sight, the author realized that some test of the variable was needed to determine if it was, indeed, approaching an approximation of infinity. The author was unable to devise such a test. Also, my son wanted the computer back so he could play Pong.

In a rare mood of cooperation, my colleague and co-founder of the Institute agreed to extend the use of his newly acquired and much more powerful computer, the Commodore 128. He also suggested that I rewrite the code to increment the variable by one million instead of one. This computational trick, together with the more powerful computer, meant that I would be able to arrive at an approximation a million times faster!

The author was still plagued by the problem that I still didn’t have a suitable test of the incremented variable, which would terminate the infinite loop and spit out the desired approximation. This time, the author decided to go on the supposition that I may not know where infinity was, but that I would know it when I got there. This meant periodically checking in on the computer and manually observing its progress.

After running the program, dubbed Infinite Loop v9.0, for almost seven days, my colleague announced that he wanted his computer back so he could play Sargon on it. Rats! My scientific endeavor thwarted again on the personal whim of a short-sighted colleague. But I digress.

But just to give you an idea of the closed-minded, shortsightedness that still persists in science, when I mentioned what I was trying to do, one of my colleagues, a mathematician (he is a high school algebra teacher who took a few summer school courses at Harvard) actually laughed in my face and then imperiously proclaimed, “Sir that just can not be done!” I had to remind myself that they also laughed at Emmett Kelly.

Through an inter-Institute agreement1, the use of a state of the art desktop computer running the ‘popular’ operating system was secured. Programming in visual basic, the author was able to spiff up Infinite Loop v23.6 with a progress bar at the bottom of the screen. At this point the flying toaster screen saver was thought to be redundant and was removed from the code.

Further research of the mathematical literature on the topic led me to what I thought might be a fertile area of exploration: convergent and divergent infinite series.

One of the big, big problems with this approach is that for some functions, such as pi, the series converges. A reasonably close answer to this type of function can be obtained by calculating a small number of terms. Values to these functions can be obtained to any arbitrary degree of precision by simply evaluating more terms in the series. But with divergent functions such as infinity… so tired, that’s irrelevant -- they just seem to go on and on and never settle in on a value. What I need is a convergent-divergent infinite series that keeps on going then stops when it gets there. Boy, thinking about this has given me such a headache. I think I’ll take a couple of extra strength placebos and wash them down with a can or three of caffeine free diet Jolt. That should keep me awake for a while longer.

People criticize Al Gore for claiming to have invented the ‘Information Superhighway’ but we do owe him a debt of gratitude for inventing one of the most important concepts in all of computing, the Algorithm. But again, I digress.

Where was I? Oh, yes. The really big number goes on the top and the really small number goes on the bottom. That ought to do it. This together with the reprogramming of Infinite Loop v34.2 to increment the variable by a googolplex (10 ** googol) might finally achieve the desired result.

The author is currently writing a grant proposal to the NSF requesting sufficient funds (plus a small additional sum for incidental expenses, wink, wink) to buy time on a Cray super computer.

1. The author will be mowing the lawn at the public library next weekend.

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