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Pi and e ar irrational numbers and cannot have a periodic representation in a fixed base number (binary, decimal, hex, etc). However, if you choose variable base like 1!, 2!, 3!,.. e becomes 1.1111..1.. there is a similar trick to select a “variable base” to make pi=3.222..2... (or was it 2.2222...2... ?) but I don’t remember the definition of the base. What works of be the rule for the base and how to find such rules for arbitrary numbers (say Euler constant 0.577...)?

Stepan
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    Don't quite understand what you want to do (or why), but anyway here is one method to write $\pi$ as a sum of '$1/$ integer'. Start with $\lfloor \pi \rfloor = 3$. Then compute $\lceil 1/(\pi - 3) \rceil = 8$ (so $\pi \approx 3 + \frac{1}{8}$). Then compute $\lceil 1/(\pi - 3 - 1/8) \rceil = 61$ and so on and so on to get $\pi = 3 + \frac{1}{8} + \frac{1}{61} + \frac{1}{5020} + \ldots$ or if you will $\pi = 3 + \frac{2}{16} + \frac{2}{122} + \frac{2}{10040} + \ldots$. – Winther Aug 06 '18 at 03:18
  • @Winther It was something along these lines, but the denominators formed a meaningful sequence. Not as beautiful as 1/(n!), but something like products of two neighboring primes or ((2^n) * n-th prime). It was converging reasonably fast as well. – Stepan Aug 06 '18 at 16:05
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    Ok, what you seem to be asking for is some special infinite series for $\pi$ or a continued fraction? The "variable base" concept is not even defined and only serve to be confusing IMO. I would try to figure out what you really want and then edit the question. As written this should be closed a unclear and missing details. – Winther Aug 06 '18 at 17:20
  • https://en.wikipedia.org/wiki/Euler%27s_continued_fraction_formula#A_continued_fraction_for_%CF%80 was the answer I was looking for. Please make it an answer, so I can accept it. – Stepan Aug 09 '18 at 13:04

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According to this logic:

Since $e=1+\frac{1}{2!}+\frac{1}{3!}+\cdots$ we can choose "base" $1!$, $2!$, etc to write $e$ as $1.11111\ldots$.

Then $\pi=4-\frac{4}{3}+\frac{4}{5}-\frac{4}{7}+\cdots$. So choose "base $4$, $\frac 43$, $\frac 45$, etc., to write $\pi$ as

$$1.010101010\cdots-(0.101010\ldots)$$

pancini
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  • This is almost an answer, but there should s a way to represent pi as a single “number” with fractional part .2222...2.. that I am looking for. – Stepan Aug 06 '18 at 12:42