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Re: .999~ = 1
Mathematicians would agree that .999... and 1 represent the same real number. In fact, from my real analysis book, "Elementary Analysis" by Kenneth Ross, it says:
[short proof, similar to above]
"Thus 0.9999... and 1.0000... are different decimal expansions that represent the same real number!"
Later on the book proves that the only case where this can occur is when the two expansions for the same real number end in an infinite series of nines and infinite series of zeros.
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