Heuristically, the below theorem/conjecture makes sense to me, but I do not have a formal mathematical proof to it; what I am actually looking for. May you perhaps, help with a formal proof or disproof?
Definition
We call a number definite, if it has a terminating decimal (primitively followed by only and only nil digits); otherwise, we call the number indefinite. For example: $8/11=0.727272...$ and $\sqrt2=1.41421356237...$ are indefinite, however, $124.234[0000000...]$ is definite.
Theorem
Let $q$ be a definite rational number and $\tau$ a transcendental number.
If
$$q \; x \; = \; \tau $$
then $x$ is always an indefinite real number. $\blacksquare$
Hint: Show that all definite numbers are rational. Then go by contradiction, if $x$ was definite, then you would have that $qx$ is the product of two rational numbers and hence rational, but then $\tau$ would be rational which contradicts it being transcendental.