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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 $\alpha\in\mathbb{R}$.

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To answer Q1 and Q1, the answer is yes. See https://brilliant.org/discussions/thread/number-base-representation-2/ for an explanation of base $b$ representations of numbers.

For questions 3 and 4, we can write down sums like $x = \sum_i a_i b^i$ for $b \in \mathbb Q$ or $\mathbb R$ using the same idea, but the 'digits' then represent coefficients of powers of $b = r$ or $\alpha$, and won't be so nice as they are in base $b$ for $b = 2, \dots, 10$.

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