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Mathematical oddity (5)
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Jun 20, 2006
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Does 0.99999... (0.9 repeating) equal 1? Most people think so. And this guy proves it. I think the best proof he provided was this: 10x = 9.9999999999...
- x = 0.9999999999...
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9x = 9
x = 1 = 0.9999999999...
The awesome part about this little mathematical oddity is that people get really mad about it and ferociously fight to explain their point of view. I personally think we should burn at the stake anyone who doesn't agree. At least I'm not angry about it. (via Digg) #math
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Comments:
2006-06-21 13:33:25
mad? not so much... intrigued? most definately.
OK, well, me and the maths, we were never such good buddies, and while I'm not good enough to point out a counter-proof or anything, I think I can find a teeny hole in the logic. It's all about the ellipses. On his site, when showing why it's not false (mathematically distinct from true), he asks what 1-0.9999...=? And the response people tend to come up with is something like 0.0000...1. Which is of course impossible because you would have an infinite number of zeros, followed by a 1, which would be infinity plus 1 digits, which is no good, right? Wouldn't the same be true if we added digits on the other side? When we multiply 10 by 0.9999..., we get 9.9999..., which has how many digits: infinity to the right of the decimal, plus 1 on the left, for infinity plus one! When you multiply by 10, you promote the decimal, which with a non-repeating number keeps the same number of digits. (10 x 0.234 = 2.34: 3 digits) It would be sillly to think that with repeating decimal numbers that suddenly multiplying by 10 will give you an extra digit, so that would mean that there are infinity minus one digits past the decimal in 10x=9.9999..., so its not really infinitely repeating. At least, not as infinately as the original x=0.9999... Anyway, you can't have infinity plus one, or infinity minus one of anything, but I think you can say that 10x has one less digit on the right side of the decimal than x, so that subtraction would involve a whole lot of regrouping (an infinite amount in fact). So if we could see the end (which doesn't exist because infinite things don't end) of these decimals in the subtraction, it would look something like 9....99990-0....99999 = 8.9999...99991. (I'm letting bold periods be the decimal to keep them clear from elipses, which i'm using here to mean more semi-infinite than truly infinite, since this whole argument hinges on degrees of infinity) Which we then say is equal to 9x, and who am I to argue with that? No matter how you look at this, in order to handle it correctly, you have to deal in degrees of infinity, which I've only heard about in conversation (yea, I know I'm a nerd) but never actually had to deal with. The material point to all of this is that 10x is less infinitely repeating than x. Therefore when he subtracts like that, he's missing this difference.
I think.
Or at least that sounds good to me. I still feel strangely unsatisfied, though... I'm hoping that Britt will weigh in on this one. She always had a better head for math than me.
2006-06-21 15:01:02
At first, I was painfully confused by what you said. But then I read it a second, third, fourth, and fifth time, and it made sense. I see your point, but this is getting into the really theoretical side of things, and I've never been a fan of that mumbo jumbo. And based on Britt's career choice, I don't know that she's all that in to math anymore. She took the high road. You obviously chose a different one. And I chose the one that passed the Quizno's and the Dunkin Donuts.
2006-06-26 10:57:42
Heh, I'm flattered, but as Dave has pointed out, the math part of my brain has been shriveling up and dying since about 2000. Also, while I used to be good at solving problems, you were always better than me at the thinking-outside-the-box stuff, Rich. So I think we will be in the same position we were in the first time we saw this proof at KRHS, where Rich says "No, that can't be right," and I say "Okay, there are like 20 proofs, it has to be right, look: Proof, Proof, Proof" and Rich says "I don't care about the damn proofs, something is just not right."
But to tackle the issue a little more directly: "Anyway, you can't have infinity plus one, or infinity minus one of anything, but I think you can say that 10x has one less digit on the right side of the decimal than x, so that subtraction would involve a whole lot of regrouping (an infinite amount in fact)." This is the heart of your argument, right? And I disagree. I'd like to see what your logic is for this statement because I'm not sure exactly how to argue against it. If you can't have infinity minus one, why can you have an infinite number of digits minus one? You say "It would be sillly to think that with repeating decimal numbers that suddenly multiplying by 10 will give you an extra digit." Kinda yes, kinda no-- you're applying the logic for finite numbers to infinite numbers, which doesn't really work. You still have an infinite number of digits on the right, the concept of "extra" doesn't really make sense here. You're not getting an extra digit, but you're not losing a digit either, because that's not how infinity works. Multiplying by 10 just takes a tiny piece off the infinite string and brings it back into the world of finite numbers on the left side of the decimal point.
I think the real issue here is just a problem with the way our system of mathematics tries to capture concepts like infinity that are hard to describe. It doesn't feel right because it doesn't look right, but really there's the exact same problem with 1/3 = .333... or 5/9= .555... but those don't jump out at us. (Do you agree that those are correct, Rich?) It's basically mostly that it's hard to wrap our mind around infinity and we want to keep treating it like a really really big finite number.
(Disclaimer: I don't understand fancy theories of levels of infinity. If there's something in there that explains this better, maybe I'm wrong. But as far as I know, this is how it works using basic math.)
Also: mmmm, donuts.
2008-03-23 18:01:15
so lets see
if x=0.888... 10x=8.888...
-1x=0.888...
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9x=8
x=8/9=0.888...
elegant
2012-06-19 01:46:50
Doesn't really matter whether you believe it or not, .999999....never ending has been DEFINED to be = 1, because it makes sense and the math works. No proof necessary.
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