Thomas (boggyb) wrote,

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The link I posted earlier today was inspired by an old school TV program I once saw, about infinity. Throughout the program they kept on showing segments where someone would say:

"'Twas a dark and stormy night, and the storyteller said to his friends, "Gather round and I'll tell you a tale." So the listeners gathered round, and the storyteller said:"

At this point the speaker walked off to one side, revealing another person behind him who repeated the same phrase. Given enough people you could continue this for hours.

Infinity is a strange concept, when you think about it. Many of you will have at some point heard the tale of explaining infinity by picking larger numbers. You think of the largest number you can come up with ("one thousand"), but the other person will always be able to pick a larger number ("one thousand plus one"). This can continue forever, as there's always a larger number ("two thousand! three thousand! one million! one million plus 1! one squillion! one squillion plus 1! ..."). There's a similar trick that can be done with prime numbers, to prove that there is no such thing as an absolute largest prime: multiply all the primes you know, and add 1. Now, if you then divide that number by any of your primes there will always be a remainder of one. Therefore, either this number is itself prime, or you missed out a smaller one. And, of course, this can be applied to the new number as well.

Once you get close to infinity, other stuff becomes strange as well. Here's another example from that TV program:

Start by imagining an infinitely long straight line. That's reasonably easy to come up with, and to comprehend. Let's look at a bit in the middle of it, say a yard long (no particular reason). Now imagine a circle just above the line, with a diameter of a yard. It's clearly a circle: you can easily see the curvature of it.

Now make that circle bigger, but still concentrating only on a section of it about a yard across. Twice as big? Still a circle. Ten times as big: well, it's less curved than it was but it's definitely circular.

As the circle gets larger and larger the small bit you're looking at becomes flatter and flatter. At some point, the small section you're imagining will be so flat and straight that it looks just like the infinitely long straight line we came up with earlier.

Now, do you have two infinitely long straight lines, two infinitely large circles, or one of each? Do both even exist as separate things?

Tags: mathematics

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