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May 15th, 2017, 10:17 AM   #61
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These are the sort of questions that people ask, but we should be telling them that they are the wrong sort of questions. Real numbers are not a product of the decimal system, the decimal system is a product of real numbers. Real numbers are limits. Nobody who understands limits questions that $$\sum_{n=0}^n \frac1{2^n} =2$$
So there is no reason why anyone who understands limits should fail to understand that $$0.999\ldots = \sum_{n=1}^n \frac9{10^n} =1$$
Perhaps, as Maschke says, the best solution for those that don't understand limits is to say:
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Real numbers are not a product of the decimal system, the decimal system is a product of real numbers. Real numbers are limits of sequences of rational numbers. You have to understand the theory of limits to understand any formal proof that $0.999\ldots=1$, but there are informal demonstrations that might convince you...
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