Can somebody explain this to me?

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cristo
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There are lots of threads on here discussing "0.99...=1." Try searching!
 
mathwonk
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the point is there is a difference bewteen a real number and the numerals used to represent it.

just as 1/2 = 2/4, there are more decimals than there are reals, and1.000, and .99999 represent the same real.

one way to see this is to define the real number represented by an infinite decimal as the smallest number not smaller than any of the finite decimal approximations. then one eventually sees that the smallest number not smaller than.9, .99, .999, ..... is 1.00000.

if one does not eventually see this, one continues to post it here ad infinitum.
 
HallsofIvy
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The definition of the decimal representation 0.a1a2a3... is the limit of the sequence 0.a1, 0.a1a2, 0.a1a2a3, ...
In particular, 0.999... means the limit of the sequence 0.9, 0.99, 0.999, 0.9999, .... Since that is a geometric sequence, it is easy to show that its limit is [itex]\frac{0.1}{1- .9}= 1.
 

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