Mixing
Periodic representation of pi in varying base system [closed]
Clash Royale CLAN TAG#URR8PPP
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
asked Aug 6 at 2:43
Stepan
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.
add a comment |Â
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
asked Aug 6 at 2:43
Stepan
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.
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
add a comment |Â
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
asked Aug 6 at 2:43
Stepan
444311
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
asked Aug 6 at 2:43
Stepan
444311
edited Aug 6 at 16:05
asked Aug 6 at 2:43
Stepan
444311
asked Aug 6 at 2:43
Stepan
444311
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
add a comment |Â
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
add a comment |Â
1 Answer
1
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
$$1.010101010cdots-(0.101010ldots)$$
answered Aug 6 at 2:53
Elliot G
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
add a comment |Â
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
$$1.010101010cdots-(0.101010ldots)$$
answered Aug 6 at 2:53
Elliot G
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
add a comment |Â
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
$$1.010101010cdots-(0.101010ldots)$$
answered Aug 6 at 2:53
Elliot G
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
add a comment |Â
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
$$1.010101010cdots-(0.101010ldots)$$
answered Aug 6 at 2:53
Elliot G
9,75121645
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
$$1.010101010cdots-(0.101010ldots)$$
answered Aug 6 at 2:53
Elliot G
9,75121645
answered Aug 6 at 2:53
Elliot G
9,75121645
answered Aug 6 at 2:53
Elliot G
9,75121645
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
add a comment |Â
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
add a comment |Â
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