what is general theory for those type of problem: To think about the condition must be exist if a decimal with infinite digit in base 10 also have infinite digit in both base 3, base 4? please prove the theory please.
what is general theory for those type of problem: To think about the condition must be exist if a decimal with infinite digit in base 10 ...
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algebra-precalculus
number-theory
number-systems
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5$\frac{p}{q}$ has an infinite decimal expansion in base $b$ if and only if there exists a prime dividing $q$ which does not divide $b$. An irrational number has an infinite decimal expansion with respect to every base. – 2012-05-30
1 Answers
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Qiaochu has answered the question in general in the comments. Applying this to the specific cases mentioned in the question, we have the following:
If $x$ is a real irrational, it has a non-terminating expansion in base 10, base 3, base 4, base whatever-$n$-you want (so long as you want an integer 2 or greater).
If $x$ is rational, and has a non-terminating expansion in base 10, then it also has a non-terminating expansion in base 3, unless its denominator is a power of 3; it is guaranteed to have a non-terminating expansion in base 4.