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Thomas

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Infinity [Saturday 8th November 2008 at 11:05 pm]
Thomas

boggyb
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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?

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Comments:
[User Picture]From: omgimsuchadork
Sunday 9th November 2008 at 4:07 am (UTC)
Oh, gosh, this reminds me of calculus (instead of imagining large, we tried to look at smaller and smaller spaces. Same idea). So, no. If you've got an infinitely large circle, and an infinitely long straight line (which is the definition of a line, BTW), then that's exactly what you've got. Just because they look similar doesn't mean that they are; while at any given point, the line and the circle may look as though they're parallel, the circle will eventually curve away and the line will continue on.

What I like about infinity is that it's the quotient of anything divided by zero. And because "anything divided by zero is undefined" is drilled so hard into people's heads in class, they find it hard to believe that infinity is also undefined -- probably because of the game you mentioned, with trying to name bigger numbers than the person before you -- and that anything divided by zero is infinity. ... Infinite. Even with a graphical representation, some people still don't get it. It baffles me!
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[User Picture]From: boggyb
Sunday 16th November 2008 at 9:34 pm (UTC)
As in the infinite sequences? I've got a book somewhere which had a go at explaining these with infinite fractal blocks. You have an infinitely long board (owned by a pair of giants, naturally), which contains one 1/1 piece, two 1/2 pieces, three 1/3 pieces, and so on. There were various sort of thought challenges set with it, and one I remember is trying to fill in 1/1's worth of space when all you have is one of each fractional power of two. So the sequence goes 1/2 + 1/4 + 1/8 + 1/16 + 1/32... and as you put in smaller and smaller pieces you get closer and closer to 1, but you never quite reach it. Even with the infinite sequence you *still* can't reach 1.
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[User Picture]From: sarshin
Sunday 9th November 2008 at 5:46 am (UTC)
ffffffffff I have the idea in my head for what I want to say, but not the words. damn you but this post is ♥
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[User Picture]From: boggyb
Sunday 16th November 2008 at 9:28 pm (UTC)
Any luck putting your idea into words?
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