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Does .99999… = 1?

I'm only doing this at GSCE and I'm really only asking here because of an interesting email conversation between my Grandfather and I regarding the fact that 0.(9) equals 1, so I'd appreciate it if you could make any explanation as simple as possible.

Basically, I have proven to my Grandfather that 0.(9) must equal 1, using the following method:

Let x = 0.(9)

So, 10x will equal 9.(9); 10x - x is 9x which is the same as 9.(9) - 0.(9) = 9, and therefore 9 / 9 is 1!

However, he has questioned the fact that 0.(9) * 9 equals 9, as he rightly stated that it equals 8.(9). I do remember learning in my maths lesson a rule regards to upper and lower bounds that meant that 8.(9) was actually the same as 9, or something along those lines, but I can not remember the correct statement to inform my Grandfather - so any suggestions would be appreciated.

Thanks in advance

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    the equation 10x = 9.(9) - x is false, instead, write 10x=9.(9)=9 + 0.(9) = 9+x therefore 10x=9+x2011-11-30
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    @sdcvvc I don't see why the equation is false; this is what I have been taught at school, and surely 9 + 0.(9) is the same as 9.(9)?2011-11-30
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    Andy: Sorry, my mistake, the equation is correct.2011-11-30
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    @sdcvvc: Thanks, I'm glad it is correct.2011-11-30
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    As was pointed out by @sdcvvc, the equation $10x=9.(9)-x$ is false, it leads to $11x=10$. Alternately (and you explicitly say this yourself) it gives $10x=9x$, which, if $x\ne 0$, implies that $10=9$. Of course, that's not what you *mean*. But that is precisely what is written.2011-11-30
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    @AndréNicolas: I fear you have misinterpreted my method because of my lack of formatting. I mean to say that 10x is equal to 9.(9), and then 9.(9) - x is equal to 9x which must be 9 and 9/9 is 1 - you know, the standard procedure - in which case, I do believe the equation is correct as sdcvvc later 'admitted'! PS If somebody can edit my question so it reads that way that would be great as I am unfamiliar with the formatting code on this site...2011-11-30
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    The equation is not correct. When one writes $A=B$, that means that $A$ is equal to $B$. If you intend to say that $10x=9.(9)$, *and therefore* $9.(9)-x=9x$, then say so. This kind of care is needed for accurate communication. It is also a necessity in one's own work, as soon as the work is of any complexity. It is all too easy to lose control of the logic of an argument.2011-11-30
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    Andy: My admission is incorrect - the equation *is* wrong, what you should write is 10x = 9.(9); 9.(9) - x = 9x. You should never write 2+2 = 4*5 = 20 because 2+2=4.2011-11-30
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    Point Taken; I've changed the post so it should read correctly. I fully admit this was my fault, and it probably doesn't help that I don't know anything about the formatting on Math Overflow2011-11-30

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