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?

Every number which can be written in a decimal notation in such a way that from some point on, there exists a subsequence that repeats itself, is rational.
For example, $1=1.0000000000000000000000...$ is rational as the zero repeats until infinity. $1/3=0.33333333333333333333...$ is also rational, as the three repeats on and on.
Also rational is the number $0.4872823482342934765656565656565656565656$, because the $56$ begins to repeat at some point.

In your specific case, it isn't hard to see that $0.234343434...$ is rational, since, if $t=0.234343434...$, then $100t = 23.434343434...$, so $$99t = 100t - t = 23.2000000000000\dots = \frac{242}{10} = \frac{116}{5}$$
meaning that $$t=\frac{\frac{116}{5}}{99} = \frac{116}{495}$$ which is rational.
