If $x$ is a positive real number then there exist at least one decimal (10-base) representation for $x$ in the form: $$x=\sum_{i=m}^\infty a_i\cdot \bigg(\frac{1}{10}\bigg)^i,$$ where $m\in\mathbb{Z}$, and $a_i\in\mathbb{Z}\cap[0,10)$.
Since $\sum_{i=1}^\infty(\frac{1}{2})^i=1$, I think $x$ has at least one $2$-base representation $$x=\sum_{i=m}^\infty a_i\cdot \bigg(\frac{1}{2}\bigg)^i,$$ where $m\in\mathbb{Z}$, and $a_i\in\{0,1\}$. Q1: Is that correct?
Q2: Does $x$ have a $n$-base representation for every $n\in\mathbb{N}$? (show idea or reference)
Q3: Does exist a $r$-base representation of $x$ for some $r\in\mathbb{Q}, r>0$? For every $r\in\mathbb{Q}, r>0$?
Q4: Does exist a $\alpha$-base representation of $x$ for a positive, irrational, algebraic number $\alpha\in\mathbb{R}$.