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October 29th, 2011, 10:36 PM   #1
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Uncountable

Can anyone explain why this function ins't countable?

Real #'s with decimal representation of all 1's or all 9's.
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October 30th, 2011, 05:53 AM   #2
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Re: Uncountable

These are rational numbers evident by their repeating decimal digits. As a subset of algebraic numbers they are countable.

Here is the algorithm to convert any repeating decimal number to a fraction of integers.

-a = the initial number
-b = No of first unique decimals
-c = No of repeating decimals

e.g. For a = 3.78123123... we have b = 2 and c = 3

1) Compute d = 10^(b+c) - 10^b
2) Find integer e = d*a
3) The fraction expressing a is then e/d

This does not yield the fraction in lowest terms though, but it can be so converted if we know their prime factorization.
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