0
$\begingroup$

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.

  • 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 1

1

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.