0.9_ = 1?

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People seem to think that 0.9_ (0.9999999999...) = 1, but I'm not too shore. At http://en.wikipedia.org/wiki/0.999_%3D_1 Wikipedia says it is and brings up some evidence for it but Uncyclopedia has evidence to prove that 0.9_ doesn't equal one and proves Wikipedia's evidence wrong at http://uncyclopedia.wikia.com/wiki/0.999... . I know Uncyclopedia specialises in making stuff up but it has come up with some good evidence, like the fact that anything minus itself always equals 0, but 1-0.9_ equals 0.0_1, not 0. Do you think 0.9_ = 1?

Note that Uncyclopedia's algebraic proof first assumes .999... equal to 1 and then proves it is equal to .999..., so it is actually proving the statement true, not false. And their fraction proof first proves .999...=1, and then they say "Now that can't be right!" and change it so that they are proven inequal.

Yes, but Uncyclopedia also says Bill Gates like Ubuntu!

But if X-X=0 then 0.9_ can't =1, because 1-0.9_=0.0_1.

The real numbers (that is all the usual numbers) are whats called the 'completion' of the rational numbers (the fractions), what this means is if we have a sequence of fractions and start adding infinitely many fractions together what the result may be is not a fraction. For instance 1, 1.4, 1.41, 1.412... is a sequence of fractions that converge to a limit (root 2) that is not rational. The way that one constructs a completion is to consider the space of all sequences and define whats called equivalence classes on those sequences, the way we do this is to say that two sequences are equivalent if and only if the terms between the sequences become closer as you progress through the sequence.

In short, when we write the number 0.99999.... we don't mean 'infinitely many 9's' as thats not useful. We mean that 0.999 is the limit of the sequence 0.9, 0.99, 0.999, 0.9999... and by '=' we mean that the terms between the sequences get close together. So now considering the relation 0.999... = 1 in the sequential format:

0.9, 0.99, 0.999, ...
1, 1, 1, ...

You can see that they get close together and in terms of sequences are 'equivalent' (This alternative definition of equivalence for sequences follows from verify that the relation is symmetric, reflexive and transitive)

I can get references if your interested futher.

You could also see it this way:
If 0.9_ does not equal 1, then there has to be a real number A 'in between' 0.9_ and 1 ( 0.9_ < A < 1 ). It is quite easy to see that for any real number A < 1 you find a number B = 0.9...9 with a finite number of '9's such that B > A.
E.g. if you chose A=0.99999999243569 then B=0.999999999 would be greater than A. This means that there is no number A that fits 'in between' 0.9_ and 1, which implies that 0.9_ and 1 must be 'infinitely close' together, i.e. they are identical.

I will treat it as if it were 1.

it depends on how important the difference is. If you're measuring something, just round it off...

Sorry, been out of school too long; I tend to get practical with math...

It equals 1. I proved it in an interesting, roundabout and unintentional way a while back while working on something else.

Here is a pseudo-proof:

Let S = 0.999...
10*S = 9.999......

10*S - S = 9*S = 9 (the .999.... is subtracted off)
divide by 9 and get S = 1.

This is not a proper proof but a hint.

ruveyn

The skepticism with which people approach .999... = 1 is truly amazing. I was on a forum one time where an argument went on for dozens of pages, complete with PHDs chiming in to tell the OP he was wrong, where still he and others adamantly refused to accept it. It's astounding. Wikipedia has a subsection devoted to it in .999...=1

"Close" is never "Perfect."