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does 0.33333... = 1/3 also seem inelegant?
No it doesn't. The reason that 0.33333333…=1/3 doesn't seem inelegant is that there is no other way to give the decimal representation of 1/3. 1.00000000… is vastly superior to 0.99999999… as a way of representing 1.
There is also the minor problem that there seems to be a conception that adding 9s to 0.99999, even an infinite number of 9s doesn't get to 1. The difference when you get the infinite number of 9s on the 0.9999999… is 0.0000000…0001.
You run into another communication problem using the term integer with computer types who may not have the math background. In computer parlance, 0.99999999… cannot be the representation of an integer. Neither can 1.0000000… There is no provision for representing a fractional part of an integer since there can be no fractional part.
Teach me to take a half hour to finely craft a response
OK, twenty minutes
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old hand
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The difference when you get the infinite number of 9s on the 0.9999999… is 0.0000000…0001.
No, it can't be. If you know where to put that last "1" then you don't have an infinite number of digits anymore. That is the difference between the infinite and finite number of digits.
1.00000000… is vastly superior to 0.99999999… as a way of representing 1.
Is it? That's an opinion. When you're talking math, you're not really dealing with opinions. Therefore, as I said to tsuwm above, it may be ugly to use 0.999999..., but that doesn't make it untrue.
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Pooh-Bah
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Pooh-Bah
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Bean's arguments are compelling and elegant but nobody seems to have an answer as to why, if (2.9999.....), the integer 3, and (3.00000...0001) [that is, an infinite number of zeros before the final one]) are the same and truly equal, they produce different answers when plugged into certain functions.
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Carpal Tunnel
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>(3.00000...0001) [that is, an infinite number of zeros before the final one])
you lose me with this. how can you have a *final one after an *infinite number of zeros??
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old hand
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I think there has been some confusion here. (Which explains why I feel like I am not being understood.) 3.000....00001 is not the same as 3. I meant that 3.000...000...000...000000... is the same as 3, the integer. (no final nonzero digit, just zeros to hell and back and then some more) Just as 2.9999999... is another way to write 3.
I think the crucial thing here is the infinite digits; you simply can't allow yourself to imagine the digits having an ending. Which is super-difficult because we as humans are on the whole pretty bad at imagining infinity. I mean really, really trying to imagine infinity.
I'm not very good at imagining infinity but I can work with it fairly well. Just as, say, a doctor can't really sit down and visualize everything that happens in the human body, all at once in real-time like some kind of bizzarre movie, but they will believe in it anyway and then work with the part(s) that needs working on.
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Carpal Tunnel
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a *final one after an *infinite number of zeros?
It ain't easy, but when has Aint Easy stopped a determined mathematician?
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Pooh-Bah
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Pooh-Bah
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In reply to:
>(3.00000...0001) [that is, an infinite number of zeros before the final one])
you lose me with this. how can you have a *final one after an *infinite number of zeros??
Well it seems as easy to visualize (4 minus 0.99999...) as it does to visualize (2.99999999...). They are each numbers that are very close to three, but on opposite sides of the integer three. I was expressing (4-0.99999...) as 3.000...0001. Maybe it would have been more precise to have expressed it as simply (4-0.99999...).
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Carpal Tunnel
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blergh.
ya, I'm stoopit, and the ekwazhuns are real purty, but I jus don't see how .999...… anything could be equal to 1
seems to de-feat the purpose of the dessimal point. but then, maybe it ain't no dessimal point neither. could be some other ar-cane cymbal.
I think it's all hocus-pocus.
formerly known as etaoin...
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Pooh-Bah
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In reply to:
a *final one after an *infinite number of zeros?
Oh yeah I forgot to mention. You just add them on at that middle, right behind the decimal. More seriously, you could do it this way:
1/10^n where n is infinity = 0.0000000000000000...0001
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1/10^n where n is infinity
Is that equal to zero?
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