You cannot represent 1/3 as a finite number of decimal places, but you *CAN* represent it as an *infinite* number of decimal places, that's the big step to realise :]
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Anonymous2006-06-03 18:43
Well you can't really, because you'd never finish writing them down.
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Anonymous2006-06-03 20:01 (sage)
>>82
Sure you can, just write every 3 half as wide as the one before it.
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Anonymous2006-06-03 22:32
>>82 It's not about being able to write it down, though
Correct, but 0.999... is 1, so long as the ...'s are there.
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Anonymous2006-06-04 13:54
stop trolling. 0.999... isn't 1
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Anonymous2006-06-05 1:34
You people are fucking retarded. There have been multiple proofs posted and you still don't understand that .999... = 1. You idiots obviously have NO concept of infinity.
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Anonymous2006-06-05 1:50
>>84
It is if you use the word "represent", which is what I was responding to. You'd never really show all of the digits, because you can't.
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Anonymous2006-06-05 1:53
Anyway, chew on this: According to the same argument, in binary, 0.0000000000000... equals 1. WTF?!
>>90 Yes, but represent does not mean write down. I could represent something with an infinite number of 9s, even though I can't write that down. Anyway this is kind of getting pointless, even by 4chan standards (4chan has standards?)
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Anonymous2006-06-05 10:30 (sage)
>>91
No. If the base is 10, the repeating digit is 9. If the base is 2 (binary), the repeating digit is 1:
0.111... = 1.
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Anonymous2006-06-05 14:52
>>95
Yeah, I figured that out myself when I was in bed for the night, but didn't feel like getting up to correct myself.
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Anonymous2006-06-05 14:54
>>94
I'd like to see you try to represent it with an infinite number of 9s. Better get started, you have much work to do.
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Anonymous2006-06-05 15:11
0.999... = lim(n -> inf, 1-(1/10^n))
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Anonymous2006-06-05 16:12 (sage)
>>80,97
You can actually extend the 'decimal point'-system for representing non-integers in several ways. One such extension is to append an infinitely repeating group of digits with '...' instead. In that system, 1/3 can be exactly represented by '0.333...'.
(Another (more logical) method is to introduce a second decimal point, preceding a group of repeating digits. 1/3 would then be represented as 0..3. In that system, eg 7/23 = 0.3.18 (and 1 = 0..9, etc).)
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Anonymous2006-06-05 16:25
>>98 >>99
All you are doing is pretending that there are infinitely many digits by using a symbolic notation for the idea. You never accomplish showing the infinite number of digits, because you can't.
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Anonymous2006-06-05 16:41 (sage)
by using a symbolic notation for the idea.
lol, that's pretty much the only way you can communicate ideas, no?
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Anonymous2006-06-05 16:57
>>101
There is a difference between actually showing a set of digits and indicating through symbols that a larger set of digits is implied. You can only imply the infinite set, you can't actually show all of the digits.
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Anonymous2006-06-05 17:05 (sage)
>>102
Obviously ideas involving infinity force you to find economical representations of them, but how the hell do you go from that to saying "1/3 != 0.333..."? Those are just different ways of representing the same thing.
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Anonymous2006-06-05 17:28
Well, the symbolism of 0.333... implies a process that can't be finished. I'd prefer to say that the limiting value of that process is what equals 1/3.
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Anonymous2006-06-05 17:39 (sage)
No, the '…' represents the "limiting value" directly. Otherwise the whole notation would be rather useless, wouldn't it?
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Anonymous2006-06-05 20:05
>>Fraction proof
>>The standard method used to convert the fraction 1⁄3 to decimal form is long division, and the well-known result is 0.3333…, with the digit 3 repeating. Multiplication of 3 times 3 produces 9 in each digit, so 3 × 0.3333… equals 0.9999…; but 3 × 1⁄3 equals 1, so it must be the case that 0.9999… = 1.
If we have a fraction for 0.3333... which is 1/3,
then we MUST have a fraction that = 0.9999...
Oh wait, we don't have a fraction that = 0.9999..., but we do have a similar one that = 1 !!!!
Amirite or am i wrong? Is there a fraction that = 0.9999... ???
106 asked a dumb question, how can you expect a good answer? I fraction that's equal to 0.999... is 3*(1/3)=3/3. I could say 9/9 = 0.999... too, or pi/pi = 0.999... for that matter.
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Anonymous2006-06-09 1:52
>>105
The ellipsis ... doesn't have a formal definition in mathematics. It merely stands for "and so forth". Their meaning is derived from conventional use. I recognise that in practice, a limiting value is understood by agreement among certain academics, but it is not necessarily so.
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Anonymous2006-06-09 2:08
I should revise the above statement to include the fact that the more official ways to show the repetition of digits (bars, dots) are widely understood to mean merely that: the continued repetition of digits without end. The interpretation that they must symbolize a limiting value of that repetition is an interpretation after all, and those that do not share the interpretion will not readily accept the assertions of equality.
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Anonymous2006-06-10 14:12
>>110,111
This is irrelevant. The problem is not in the interpretation of the ellipses, it's on the definition of the number system you're using. On the real number line, the '...'s don't have to mean limiting value; 0.999... is still equal to 1.
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Anonymous2006-06-10 14:46 (sage)
the '...'s don't have to mean limiting value
What else could they mean?
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Anonymous2006-06-11 20:03
>>113
They merely mean that the 9s are repeated forever. The fact that the ellipses imply convergence to the limiting value is a consequence of the continuity of the real number line.
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Anonymous2006-07-01 8:16
>>39
But...but .99999 is .33333 X 3
.333333 is 1/3, thus .99999=1
Not infinitely close, SAME
OR ELSE PARADOX
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Anonymous2006-07-01 10:26 (sage)
As a result of not having infinitesimals, it cannot be 'infinitely close', thus must be equal.
However .99---> infinite places will never be equivelently equal to 1 because of the extra 0.0(infintite)0001 needed to make it one, so therefore is will always be approaching 1
It can be described as a limit LIM(n -> infinity)[.99~~]-> 1 where n is the decimal places