Forgive me if I have set this up wrong, I haven't done proofs in a long time. I was thinking at lunch today what it would be like if we could write binary behind a decimal point. Imagine
$$110.11 = 6.75$$
This made me think, are there numbers in base-10 decimal that cannot be represented rationally in base-2 decimal? Maybe, but since 2 evenly divises 10, maybe not. How about 3? $n$?
This is my conjecture:
$$\forall ( x_1,l_1,n \in Z | x_1 < 10^{l_1} ) \exists i_2,l_2 (\frac{x_1}{10^{l_1}} = \frac{x_2}{n^{l_2}} ) $$
How would I go about proving it?