Saturday, September 3, 2011

Asymptotes: Closer But Never QuiteTouching...

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Imagine trying to get to a finish line where with each leap you halve the distance remaining between you and your objective. Sounds good at first – but after a short period, you will realize that you can never actually get all the way there. You get closer and closer, halving the distance with each leap, but you won’t quite make it. Close, but no cigar. Your approach to the finish line is asymptotic. A mathematical limitation makes the intuitively simple task into the impossible conundrum.

No matter how assiduously you proceed, you can merely cut the distance in half – even after a (theoretically) infinite number of leaps you cannot bridge that gap. The early leaps are the most productive…however, with each successive leap, your dilemma becomes clearer, and you become more frustrated. You have come up against a limiting mathematical constraint.

The irony of this predicament is that although the objective is fixed, it might just as well be a moving target… retreating in smaller increments as you approach it.

This asymptote paradox is serious business, unlike when we were kids and we would (if there were only two of us) “each want the bigger half of the cake”; it was even more hysterical when there would be three or more of us each requesting an equal half of the cake. It wasn’t like the Miracle of the Loaves and the Fishes (in the New Testament); it was more like a rag-tag confederation of pre-pubescent imbeciles battling for territory with no concept of either the basic Laws of Physics or of rudimentary division or multiplication. As a youngster, I personally helped to put the “ass” in “asymptote.”

Here are two questions to ponder, with answers (I promise – I really do) to follow within the next three days.

A) In adding up the following series of numbers, what whole number (integer) will you ultimately approach asymptotically?

1 + ½ + ¼ + 1/8 + 1/16 + 1/32 + …. + 1/ 2* [where 2* represents 2 to the infinite power]

B) In performing the following arithmetic operation (looks like division) with an increasingly larger number substituted in for ‘N’, what number will you ultimately approach asymptotically?

(N-1) / N

Graphical Illustration Of Asymptotes - Vertical And Horizontal

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