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Old 11-20-2005, 02:48 AM   #51 (permalink)
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it is, because 0.3333333.... = 1/3 if it has an infinite number of 3's at the end. if it didn't, it wouldn't be exactly 1/3
therefore 3 * 1/3 = 3 * 0.3333333.......... (infinite amount of digits)
and 3 * 1/3 = 1, so 3 * 0.3333333.......... also = 1
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Old 11-20-2005, 03:18 AM   #52 (permalink)
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i agree with apokalipse...1/3= .3(repeater)
so 1/3 * 3 = 1 and .3(repeater) * 3 = 1 also...
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Old 11-20-2005, 03:43 AM   #53 (permalink)
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...ok..... 1/3 and .3(repeater) are not the same thing, so you can not then multiply 1/3 * 3 = 1 and expect that .3(repeater * 3 = 1 are the same thing.

1/3 is basically rounding it off.... 'technically' a third doesn't exist...not out of 1 anyway...there is no 'if' about the inifite repeating 3 of .3(repeater)

it IS infinite... .3(inifite 3's) would not be 'exactly' 1/3 in the first palce...1/3 is technically a rationalized number so that we can have something to add or subtract or overall get the math done.

You can't say that because .3(repeater) has infinite 3's then it's 'exactly 1/3' that makes no sense mathematically considering 1/3 is a rounded approximation rationalized whatever the hell else you want to call it kind of number.
.3(repeater) and 1/3 are two completely different things.

.3(infinite repetition) would NOT = 1/3 no way no how

.3(repeater) is the attempt at trying to make a third out of 1 which technically would never happen..... 1/3 would be more like making a third out of 9 which CAN happen...granted yes I realize 3/3 = 1 it is not the same as .3(repeater) * 3

that '1' just stands for a whole...a whole can be represented by a '9' as an example.

This is where the problem lies in what you're saying...I know I sound like a broken record but 1/3 can not = .3(infinite loop) considering one is a rationalized rounded fraction and the other is an infinite
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Old 11-20-2005, 04:10 AM   #54 (permalink)
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okay, when you divide 1 by 3, you put a 3 after the decimal place. once you do, you have to put another 3 after that, and another after that and so on.
basically, to get the answer 100% correct, you cannot end it anywhere.
that is why you have the infinite amount of 3's.

you can express 1/3 as 0.3333333... but only if you indicate that there are an infinite number of 3's
yes it is stupid to do so, but it does actually work. only by having the infinite number of 3's
it doesn't not work just because it's infinite, against a fraction with whole numbers.

but you're probabbly gonna disagree anyway, so let's just agree to disagree.
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Old 11-20-2005, 10:40 AM   #55 (permalink)
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The pre-calculus definition is:
0.333.... = 1/3

Calculus definition:
The limit of sum(3/10^n} can be made arbitrarily close to 1/3 as n tends to infinity.

When you read them, you can see a subtle difference in the definitions. The first definition "treats" infinity as something that can be quantisized, whereas teh latter treats it as a "concept" that can be used to give you an arbitrarily close approximation.

To me, it just doesn't make sense to say "when you add an infinite # of 3's, you get 1/3" - in a useful sense.. b/c well, infinity is a never ending thing. It IS a concept. When you add the infinity symbol (or a version of that) into your equations, I think your equations break down very fast. And that is why the limit notation exists.

The limit just says that (for this situation), you can make something arbitrarily close to the limit, as n approaches infinity, where n = # of 3's in this case.

SAYING something like {0.333 with infinite 3's} = 1/3 might be an "ok" "DEFINITION" for some people, but IMO, you can't really "do" anything with that definition. I mean, "how" do you work with "infinities".


What's that famous pre-school fraction approximation lesson:

x=0.333....
10x = 3.333....
10x-x = 3
x = 1/3

But, I don't think in a strict sense this is correct. Substracting infinities from other infinities is a strange thing indeed. We are essentially treating them like a finite sequence when we do this IMO. Whever we deal with situations with dividing infinities or substracting infinities (which is the case here), or deal with "infinities of different sizes", you are doing something wrong with your formulation IMO.

That is why I think mathematicians introduced very specific notation to deal with inifite sequences and such - one of them being the limit ("lim").
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Old 11-20-2005, 12:38 PM   #56 (permalink)
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Quote:
Originally posted by Qiranworms
What would happen if the number system was finite? What would happen if I...decide to add 1 to the biggest number in existance?

No matter what end you state there is, I can always add 1 to it. Thus one cannot define such an end.
on the money i'd say..........who can say what is the last number .........and yes infinite does mean the state of not being finite.........
and when you say ........a third......you are of course speakin practically.........unless of course u are talkin of a number which is an exact multiple of 3...........so just cause 3/3 is exactly one..........does not mean.........1/3 should be an exactl figure.......... cause 1/3 multiplied by 3 will give you exactly 3 (cross out the 3s)
as far as infinity goes...........yes i do agree that there is a need for one.............u will need it in in higher math.........when some function will appraoch infinity, but never gets there.
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