Periodic representation of pi in varying base system [closed]

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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...)?




irrational-numbers




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closed as unclear what you're asking by Winther, José Carlos Santos, Taroccoesbrocco, Shailesh, max_zorn Aug 8 at 6:21


Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.










  • 1




    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 $pi approx 3 + frac18$). Then compute $lceil 1/(pi - 3 - 1/8) rceil = 61$ and so on and so on to get $pi = 3 + frac18 + frac161 + frac15020 + ldots$ or if you will $pi = 3 + frac216 + frac2122 + frac210040 + ldots$.
    – Winther
    Aug 6 at 3:18










  • @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 6 at 16:05







  • 2




    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 6 at 17:20










  • en.wikipedia.org/wiki/… was the answer I was looking for. Please make it an answer, so I can accept it.
    – Stepan
    Aug 9 at 13:04














up vote
1
down vote

favorite












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...)?




irrational-numbers




share|cite|improve this question













closed as unclear what you're asking by Winther, José Carlos Santos, Taroccoesbrocco, Shailesh, max_zorn Aug 8 at 6:21


Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.










  • 1




    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 $pi approx 3 + frac18$). Then compute $lceil 1/(pi - 3 - 1/8) rceil = 61$ and so on and so on to get $pi = 3 + frac18 + frac161 + frac15020 + ldots$ or if you will $pi = 3 + frac216 + frac2122 + frac210040 + ldots$.
    – Winther
    Aug 6 at 3:18










  • @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 6 at 16:05







  • 2




    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 6 at 17:20










  • en.wikipedia.org/wiki/… was the answer I was looking for. Please make it an answer, so I can accept it.
    – Stepan
    Aug 9 at 13:04












up vote
1
down vote

favorite









up vote
1
down vote

favorite











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...)?




irrational-numbers




share|cite|improve this question













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...)?





irrational-numbers





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Aug 6 at 16:05
























asked Aug 6 at 2:43









Stepan

444311




444311




closed as unclear what you're asking by Winther, José Carlos Santos, Taroccoesbrocco, Shailesh, max_zorn Aug 8 at 6:21


Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.






closed as unclear what you're asking by Winther, José Carlos Santos, Taroccoesbrocco, Shailesh, max_zorn Aug 8 at 6:21


Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.









  • 1




    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 $pi approx 3 + frac18$). Then compute $lceil 1/(pi - 3 - 1/8) rceil = 61$ and so on and so on to get $pi = 3 + frac18 + frac161 + frac15020 + ldots$ or if you will $pi = 3 + frac216 + frac2122 + frac210040 + ldots$.
    – Winther
    Aug 6 at 3:18










  • @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 6 at 16:05







  • 2




    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 6 at 17:20










  • en.wikipedia.org/wiki/… was the answer I was looking for. Please make it an answer, so I can accept it.
    – Stepan
    Aug 9 at 13:04












  • 1




    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 $pi approx 3 + frac18$). Then compute $lceil 1/(pi - 3 - 1/8) rceil = 61$ and so on and so on to get $pi = 3 + frac18 + frac161 + frac15020 + ldots$ or if you will $pi = 3 + frac216 + frac2122 + frac210040 + ldots$.
    – Winther
    Aug 6 at 3:18










  • @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 6 at 16:05







  • 2




    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 6 at 17:20










  • en.wikipedia.org/wiki/… was the answer I was looking for. Please make it an answer, so I can accept it.
    – Stepan
    Aug 9 at 13:04







1




1




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 $pi approx 3 + frac18$). Then compute $lceil 1/(pi - 3 - 1/8) rceil = 61$ and so on and so on to get $pi = 3 + frac18 + frac161 + frac15020 + ldots$ or if you will $pi = 3 + frac216 + frac2122 + frac210040 + ldots$.
– Winther
Aug 6 at 3:18




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 $pi approx 3 + frac18$). Then compute $lceil 1/(pi - 3 - 1/8) rceil = 61$ and so on and so on to get $pi = 3 + frac18 + frac161 + frac15020 + ldots$ or if you will $pi = 3 + frac216 + frac2122 + frac210040 + ldots$.
– Winther
Aug 6 at 3:18












@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 6 at 16:05





@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 6 at 16:05





2




2




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 6 at 17:20




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 6 at 17:20












en.wikipedia.org/wiki/… was the answer I was looking for. Please make it an answer, so I can accept it.
– Stepan
Aug 9 at 13:04




en.wikipedia.org/wiki/… was the answer I was looking for. Please make it an answer, so I can accept it.
– Stepan
Aug 9 at 13:04










1 Answer
1






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up vote
1
down vote













According to this logic:



Since $e=1+frac12!+frac13!+cdots$ we can choose "base" $1!$, $2!$, etc to write $e$ as $1.11111ldots$.



Then $pi=4-frac43+frac45-frac47+cdots$. So choose "base $4$, $frac 43$, $frac 45$, etc., to write $pi$ as



$$1.010101010cdots-(0.101010ldots)$$






share|cite|improve this answer





















  • 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 6 at 12:42

















1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
1
down vote













According to this logic:



Since $e=1+frac12!+frac13!+cdots$ we can choose "base" $1!$, $2!$, etc to write $e$ as $1.11111ldots$.



Then $pi=4-frac43+frac45-frac47+cdots$. So choose "base $4$, $frac 43$, $frac 45$, etc., to write $pi$ as



$$1.010101010cdots-(0.101010ldots)$$






share|cite|improve this answer





















  • 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 6 at 12:42














up vote
1
down vote













According to this logic:



Since $e=1+frac12!+frac13!+cdots$ we can choose "base" $1!$, $2!$, etc to write $e$ as $1.11111ldots$.



Then $pi=4-frac43+frac45-frac47+cdots$. So choose "base $4$, $frac 43$, $frac 45$, etc., to write $pi$ as



$$1.010101010cdots-(0.101010ldots)$$






share|cite|improve this answer





















  • 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 6 at 12:42












up vote
1
down vote










up vote
1
down vote









According to this logic:



Since $e=1+frac12!+frac13!+cdots$ we can choose "base" $1!$, $2!$, etc to write $e$ as $1.11111ldots$.



Then $pi=4-frac43+frac45-frac47+cdots$. So choose "base $4$, $frac 43$, $frac 45$, etc., to write $pi$ as



$$1.010101010cdots-(0.101010ldots)$$






share|cite|improve this answer













According to this logic:



Since $e=1+frac12!+frac13!+cdots$ we can choose "base" $1!$, $2!$, etc to write $e$ as $1.11111ldots$.



Then $pi=4-frac43+frac45-frac47+cdots$. So choose "base $4$, $frac 43$, $frac 45$, etc., to write $pi$ as



$$1.010101010cdots-(0.101010ldots)$$







share|cite|improve this answer













share|cite|improve this answer



share|cite|improve this answer











answered Aug 6 at 2:53









Elliot G

9,75121645




9,75121645











  • 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 6 at 12:42
















  • 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 6 at 12:42















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 6 at 12:42




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 6 at 12:42


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