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


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 $b = 2, \dots, 10$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.