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Joined: Jan 2001
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Carpal Tunnel
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Carpal Tunnel
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said the Queen. "When I was younger, I always did it for half an hour a day. Why,
sometimes I've believed as many as six impossible things before breakfast."
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Joined: Dec 2000
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Carpal Tunnel
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Carpal Tunnel
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no matter how many 9s you tack onto the end
I guess that's true only as long as the number of 9s you tack onto the end is finite.
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Pooh-Bah
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Pooh-Bah
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I disagree that 0.9999999999... is equal to the integer 1. All integers are understood to be exactly 1 or exactly 2, etc, as if they had an infinite number of zeros after a decimel point. Of course, they are practically the same in any conceivable everyday context, but they differ by that tiny amount. For example, 1/(1-1) is not defined because one cannot divide by zero. 1/(1-0.9999999.....) approaches infinity asymptotically the more nines you tack on at the end.
This is basic calculus. You'd express the above as the limit of 1/(1-x) as x approaches 1 from zero.
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Joined: Jan 2001
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old hand
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old hand
Joined: Jan 2001
Posts: 1,156 |
I disagree that 0.9999999999... is equal to the integer 1Why not, if you will accept the infinite number of zeros, also accept the infinite number of nines?  Something to think about. But I do remember learning, and being upset by, and finally agreeing with, what emanuela said above. Edit: To clarify: you have to differentiate between how we write down a number, and what it is. What emanuela is saying is that for the number which we commonly understand to be the integer 1, there are two ways to write it: 0.9999999999999999... or 1.00000000000...
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Carpal Tunnel
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Carpal Tunnel
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"Math is not an exact science."
-ron obvious
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Joined: Sep 2001
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Carpal Tunnel
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Carpal Tunnel
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Pooh-Bah
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Pooh-Bah
Joined: Jan 2001
Posts: 1,819 |
In reply to:
We would still say we had one couch.
...and that is why I say that they are the same in any conceivable, everyday aspect. Of course if you are talking about real couches, 0.99999999999999 of a couch is macroscopically indistinguishable from 1.00000000 of a couch.
What I am saying is that strictly speaking, in the abstract world of mathematics, there is a difference between the following numbers:
2.99999
3
3.00001
To prove my point, here is a function which will return a different value depending on which of the three above numbers you plug in: f(x)=3/(x-3)
Now if those three numbers were truly the same, then you should get the same result from the function above, but you don't.
3/(2.99999-3) = 3/(-0.00001) = (-300,000)
3/(3 -3) = 3/(0) = not defined/infinity/or, you go to jail for dividing by zero and your mother is very disappointed
3/(3.00001-3) = 3/(0.00001) = 300,000
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Joined: Dec 2000
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Carpal Tunnel
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Carpal Tunnel
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f(x)=3/(x-3)
Which is certainly true if we have a finite number of 9s after the decimal point. If we have an infinite number of 9s, there is an infinite number of 0s before we get to that pesky 1 in the .000000…0001 meaning that we'll never get there.
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Joined: Jan 2001
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old hand
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old hand
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Yeah, that's the difference here, which maybe wasn't clear. The ellipsis after a number in math means "going on to infinity" so 0.99999999... means nines going on to infinity, that is, you never run out of digits. That is emphatically NOT the same as 0.99999999. Which means any kind of numerical example given with a finite number of nines isn't talking about the same thing.
Anyway, I looked it up in my analysis book last night, and you can prove pretty straightfowardly (if you've read the first three chapters of the book) that both decimal representations (that is, 0.999999.... and 1.000000...) are, indeed, EQUAL to the number 1.
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Joined: Jan 2001
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old hand
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old hand
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Another thing I thought of, Alex, is If you don't believe that 0.9999999... is equal to 1, then please tell me how far from 1 it is. That is, what is 1 - 0.9999999... = ???? If you are right, the difference must be equal to some probably quite small but finite number. 
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