Is the number 0.2343434343434.. rational? Consider the following number:
$$x=0.23434343434\dots$$
My question is whether this number  is rational or irrational, and how can I make sure that a specific number is rational if it was written in decimal form.
Also, is $0.234$ rational or irrational?

$\begin{array}{r}-\\\ \end{array}
\begin{align}
1000x & = 234.3434...\\
10x   & = \phantom{00}2.3434...\\
\hline
990x  & = 232
\end{align}$
$x = 232/990=116/495$. Thus $x$ can be represented as a quotient of two integers with denominator not equal to zero, so it is a rational number.
