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Does .99999… = 1?

I was thinking about this the other day...

if 10/3 = 3.33333... (series)

why doesn't 3.333... * 3 = 10 it can never be 10 it's always almost 10 or 9.9999... (infinity)

I have read about this but then no one has an answer yet we all accept the fact that it is true when the statement is fundamentally false.

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marked as duplicate by Ross Millikan, 5PM, Ittay Weiss, Micah, Ayman Hourieh Jan 24 '13 at 22:34

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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$10=9.9999999...$ –  Amr Jan 24 '13 at 21:21
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(Abstract) duplicate of Does .99999… = 1? –  Zev Chonoles Jan 24 '13 at 21:22
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I smell a troll... –  Mike Jan 24 '13 at 21:34
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@Val either you are trolling or you arent comprehending what infinity is. Subtract 9.9999... from 10. What happens? Lets see. $10-9.9=0.1$, $10-9.99=0.01$,... $10-9.999...999=0.000....001$ to infinity $\Rightarrow 10-9.9999...=0.00...$. The 1 never appears! Therefore, $10-9.999...=0 \Rightarrow 10=9.999....$ –  CBenni Jan 24 '13 at 21:47
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@Val and Indeed, there are theories of mathematics (particularly hyperreal numbers) where 9.999... is INFINITESMALY smaller than 10. Which means, the difference is smaller than any real number. What we mean when we say $10=9.9999...$ is $St(10)=St(9.\overline{9})$ –  CBenni Jan 24 '13 at 21:50

3 Answers 3

If $9.9999...$ and $10$ are not the same number, then name a number between them.

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Doesnt mean they are the same ;) (Of course it does, but that would have to be shown at least) –  CBenni Jan 24 '13 at 21:24
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But of course. But almost everybody knows two unequal numbers have a number strictly between them, and those that don't can be shown it fairly quickly using elementary inequalities. Better than trying to explain limits to someone who likely has a pre-calculus level. @CBenni –  Thomas Andrews Jan 24 '13 at 22:52

$10/3=3.\overline{3}$

$3\cdot 3.\overline{3}=9.\overline{9}=9+\dfrac{9}{9}=10$

This should answer your question pretty much.

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When we say that $\frac{10}{3} = 3.333\dots$

We have it like this:

$$ \text{Let } p = 3.3333\dots $$

$$10p = 33.33333 \dots$$ or

$$ 9p = 30 $$ or

$$ p = \frac{30}{9} = \frac{10}{3} $$ That's how I've always convinced myself of the proof. :|

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I believe since we can't never have truly 0 (nothing), something must give way, e.g: 3.3333 * 3.00000=10 maybe because 0 may not mean absolutely 0 if that makes sense? there must be one point on the serious where something must give way? –  Val Jan 24 '13 at 21:32

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