One of the more famous Greek puzzles is called "Zeno's paradox," named after the philosopher who conceived it. The problem goes something like this:"Achilles, the great warrior, was to compete in a footrace with a tortoise. Because Achilles is so much faster, the tortoise was given a large head start. Supposing that Achilles started 100 meters behind, and is ten times faster than the tortoise--well, he should run 100 meters, but in that time the tortoise has moved 10. So Achilles runs another 10 meters, but in that time, the tortoise has pulled ahead by 1. So he runs 1 meter, but in that time, the tortoise has stayed ahead by a tenth of a meter. As he runs a tenth, the tortoise holds his lead by a hundredth. Achilles must cover an infinite number of such smaller distances; and since no mortal can complete an infinite number of things, surely it is impossible for him to catch the tortoise!"
It's an interesting paradox. The thing that's so interesting about it, however, is that it also has quite a simple solution. To begin, this paradox tells a lie: "No mortal can complete an infinite number of things." But if the infinite number of things is ever smaller and smaller, then sometimes we mortals can do precisely that. Suppose I try to solve this problem first:
"I wish to sum up all of the distances which Achilles will travel: First the hundred meters, then then 10, then the 1, then the tenth, then the hundredth, and so on, forever. How large a number shall I have when I am done?"
Well, the answer will look very much like this:
111.11111111111111...
except of course that the string of ones shall go on forever. But this number is not infinite. In fact, consider the decimal representation of 111 and 1/9 (one-hundred-eleven and one-ninth). If you do the long division, you will soon realize that it is 111 followed by an endless string of 1's after the decimal point. Thus, the sum of the earlier infinite series is precisely 111 and 1/9 meters, which is the point in the race where Achilles will pass the tortoise.
And there you go: How to answer Zeno's paradox. I hope I said it clearly enough.
5 comments:
The tortoise photograph was taken by Moise Nicu, who released it under a Creative Commons Attribution license. Thus, my attribution :).
So 0.99999... might as well be 1.
Yup, 0.999... is equal to 1. This is provable because there exists no number in-between 0.999... and 1, and all non-equal numbers have a unique number between them :). There are probably other ways of stating it.
You can also think of it as 9/9 = 1.
I'm going to invent a number that exists between .999... and 1. Just you wait.
You're too smart for your own good. Or maybe just smart enough.
Zeno's paradox reminds me of the movie IQ. I don't know what that says about me...
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