I was trying to analyze the following statement: "Any integer can be expressed as a function of any irrational number". For which I got:
\begin{equation} a=\lfloor{10^b \alpha}\rfloor-\lfloor{10^{b-s(a)} \alpha}\rfloor10^{s(a)} \end{equation}
where a, b and $\alpha$ represents the integer, the position of the integer in the irrational and the irrational, respectively. S is size of the integer (amount of digits). As an example:
\begin{equation} 15=\lfloor{10^4 \pi}\rfloor-\lfloor{10^{2} \pi}\rfloor10^{2} \end{equation}
Expanding the floor functions I got:
\begin{equation} a=\frac{1}{2} (10^s-1)+\frac{1}{\pi} \sum_{k=1}^{\infty} \frac{1}{k} (\sin(2\pi k 10^b \alpha)-10^s\sin(2\pi k 10^{b-s} \alpha)) \end{equation}
There are infinite values for b that solve the relation. Now, it is tempting to solve for b and find out the position of every number inside an irrational. It sounds too good to be true and I probably have something conceptually wrong. Anyway, every approach of mine results with b getting cancelled out of the relation.
Could someone show how to isolate b or explain me what is the flaw in my idea?
I would like to apologize for any dumb mistake, I am just an engineer venturing in math for fun.