November 26th, 2012, 12:23 PM   #24 (permalink)
jhawkkw
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I could give you the real reason, but would require the use of real analysis which is usually a 4th year mathematics course. As I stated earlier, it has to deal with the fact that there is no non-zero infinitesimal between .999... and 1.0. In other words, there is no number between .999... and 1.
For the purpose of a proof, lets assume they are distinct numbers. For any number not ending in a infinite sequence of 9's, you can always find another number in between two different numbers by simply finding the midpoint (x1+x2)/2. This doesn't work with numbers ending in the infinite sequence of 9's because the midpoint between .9 and 1 is .95. so if you could take the midpoint of the two, following that idea, the last number would have to be a 5, which contradicts the whole notion because .999 is an infinite sequence of 9's and would actually be greater than our midpoints. This creates an absurdity, which many mathematicians call a proof by contradiction. So since our original idea about them being two different numbers let to a contradiction, that means that assumption must be false, and thus means they have to be the same number.

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