# Rational numbers in irrational bases

If you take the base-$$b$$ expansion of a rational number where $$b$$ is irrational, do you get a non-terminating sequence of digits (assuming you pick the right(?) digits)? More informally, do rational numbers look irrational in irrational bases?

• What is a base-$b$ expansion when $b$ is not a positive integer? Commented Oct 21, 2018 at 9:38
• I assume something like base-$\phi$ en.wikipedia.org/wiki/Golden_ratio_base Commented Oct 21, 2018 at 9:39

Rational numbers in decimal or binary or similar cases can have non-terminating recurring representations, for example $$\frac13 = 0.3333333\ldots_{10}= 0.\overline{3}_{10}$$. Presumably you want to include these
Your golden ratio base link shows that there are irrational bases in which all rationals can be represented with terminating or recurring expressions. For example $$2 = 10.01_{\phi}$$ and $$\frac12 = 0.\overline{010}_{\phi}$$
• @EyobTsegaye Suppose I had something like $x=12.1\overline{02}_e$. Then I would know that $x=e^1+2e^0+e^{-1}+2e^{-3}+2e^{-5}+2e^{-7} +\ldots = e+2+e^{-1}+2e^{-3}/(e^{2}-1)$ so $e^6-e^4+(2-x)e^5+e^4-(2-x)e^3-e^2+2 =0$ and if $x$ were rational this would make $e$ algebraic which it is not Commented Oct 21, 2018 at 10:10