Reverting multiple rotations of an (irrational) angle
Problem
Let $ainmathbb{C}$ , $|a|=1$ and $cinmathbb{R}$.
Let us assume that $exists{ninmathbb{N}} ;; a=exp(c i n)$
Find the value of smallest possible $ninmathbb{N}$.
Background
I am working on a computer program where the value of $a$ is generated by substituting some $n$ into the expression. Now I would like to revert the process and find out what was the value of $n$ when I am given only $a$ itself, or actually its floating representation .
geometry complex-numbers
asked Dec 3 '18 at 19:38
Andrzej GolonkaAndrzej Golonka
488
add a comment |
Problem
Let $ainmathbb{C}$ , $|a|=1$ and $cinmathbb{R}$.
Let us assume that $exists{ninmathbb{N}} ;; a=exp(c i n)$
Find the value of smallest possible $ninmathbb{N}$.
Background
I am working on a computer program where the value of $a$ is generated by substituting some $n$ into the expression. Now I would like to revert the process and find out what was the value of $n$ when I am given only $a$ itself, or actually its floating representation .
geometry complex-numbers
asked Dec 3 '18 at 19:38
Andrzej GolonkaAndrzej Golonka
488
The mathematical question you have asked is totally different from the programming problem you want to solve, since in the programming problem you are only representing the numbers to some finite level of accuracy and presumably you only want $a=exp(cin)$ to hold approximately.
– Eric Wofsey
Dec 3 '18 at 19:44
@EricWofsey, the programming problem, I believe, concerns the case where $cinmathbb{Q}$.
– Andrzej Golonka
Dec 3 '18 at 19:51
add a comment |
Problem
Let $ainmathbb{C}$ , $|a|=1$ and $cinmathbb{R}$.
Let us assume that $exists{ninmathbb{N}} ;; a=exp(c i n)$
Find the value of smallest possible $ninmathbb{N}$.
Background
I am working on a computer program where the value of $a$ is generated by substituting some $n$ into the expression. Now I would like to revert the process and find out what was the value of $n$ when I am given only $a$ itself, or actually its floating representation .
geometry complex-numbers
asked Dec 3 '18 at 19:38
Andrzej GolonkaAndrzej Golonka
488
Problem
Let $ainmathbb{C}$ , $|a|=1$ and $cinmathbb{R}$.
Let us assume that $exists{ninmathbb{N}} ;; a=exp(c i n)$
Find the value of smallest possible $ninmathbb{N}$.
Background
I am working on a computer program where the value of $a$ is generated by substituting some $n$ into the expression. Now I would like to revert the process and find out what was the value of $n$ when I am given only $a$ itself, or actually its floating representation .
geometry complex-numbers
geometry complex-numbers
asked Dec 3 '18 at 19:38
Andrzej GolonkaAndrzej Golonka
488
asked Dec 3 '18 at 19:38
Andrzej GolonkaAndrzej Golonka
488
asked Dec 3 '18 at 19:38
Andrzej GolonkaAndrzej Golonka
488
asked Dec 3 '18 at 19:38
Andrzej GolonkaAndrzej Golonka
488
asked Dec 3 '18 at 19:38
Andrzej GolonkaAndrzej Golonka
488
488
The mathematical question you have asked is totally different from the programming problem you want to solve, since in the programming problem you are only representing the numbers to some finite level of accuracy and presumably you only want $a=exp(cin)$ to hold approximately.
– Eric Wofsey
Dec 3 '18 at 19:44
@EricWofsey, the programming problem, I believe, concerns the case where $cinmathbb{Q}$.
– Andrzej Golonka
Dec 3 '18 at 19:51
add a comment |
The mathematical question you have asked is totally different from the programming problem you want to solve, since in the programming problem you are only representing the numbers to some finite level of accuracy and presumably you only want $a=exp(cin)$ to hold approximately.
– Eric Wofsey
Dec 3 '18 at 19:44
@EricWofsey, the programming problem, I believe, concerns the case where $cinmathbb{Q}$.
– Andrzej Golonka
Dec 3 '18 at 19:51
The mathematical question you have asked is totally different from the programming problem you want to solve, since in the programming problem you are only representing the numbers to some finite level of accuracy and presumably you only want $a=exp(cin)$ to hold approximately.
– Eric Wofsey
Dec 3 '18 at 19:44
The mathematical question you have asked is totally different from the programming problem you want to solve, since in the programming problem you are only representing the numbers to some finite level of accuracy and presumably you only want $a=exp(cin)$ to hold approximately.
– Eric Wofsey
Dec 3 '18 at 19:44
@EricWofsey, the programming problem, I believe, concerns the case where $cinmathbb{Q}$.
– Andrzej Golonka
Dec 3 '18 at 19:51
@EricWofsey, the programming problem, I believe, concerns the case where $cinmathbb{Q}$.
– Andrzej Golonka
Dec 3 '18 at 19:51
add a comment |
1 Answer
1
active
oldest
votes
Assume $cneq 0.$
Since $a=e^{itheta},$ we are looking for $n$ satisfying $;{cn}={theta}.$ If such $n$ exists, then any $$n+frac{2pi k}{c},quad kinmathbb{Z}$$ satisfies, as $$e^{ic(n+2pi k/c)}=e^{icn}=e^{itheta}=a.$$
answered Dec 4 '18 at 11:39
user376343user376343
2,9132823
That is a right observation, thank you. One could use it to find out the $n$, by taking $n_*=frac{log{a}}{ci}$, evaluating expression $n_*+ frac{2pi k}{c}$ for different values of $k$, and looking for the values that are integers. It is not a satisfying solution since it is not any better than just guessing $n$.
– Andrzej Golonka
Dec 4 '18 at 18:57
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
Assume $cneq 0.$
Since $a=e^{itheta},$ we are looking for $n$ satisfying $;{cn}={theta}.$ If such $n$ exists, then any $$n+frac{2pi k}{c},quad kinmathbb{Z}$$ satisfies, as $$e^{ic(n+2pi k/c)}=e^{icn}=e^{itheta}=a.$$
answered Dec 4 '18 at 11:39
user376343user376343
2,9132823
That is a right observation, thank you. One could use it to find out the $n$, by taking $n_*=frac{log{a}}{ci}$, evaluating expression $n_*+ frac{2pi k}{c}$ for different values of $k$, and looking for the values that are integers. It is not a satisfying solution since it is not any better than just guessing $n$.
– Andrzej Golonka
Dec 4 '18 at 18:57
add a comment |
Assume $cneq 0.$
Since $a=e^{itheta},$ we are looking for $n$ satisfying $;{cn}={theta}.$ If such $n$ exists, then any $$n+frac{2pi k}{c},quad kinmathbb{Z}$$ satisfies, as $$e^{ic(n+2pi k/c)}=e^{icn}=e^{itheta}=a.$$
answered Dec 4 '18 at 11:39
user376343user376343
2,9132823
That is a right observation, thank you. One could use it to find out the $n$, by taking $n_*=frac{log{a}}{ci}$, evaluating expression $n_*+ frac{2pi k}{c}$ for different values of $k$, and looking for the values that are integers. It is not a satisfying solution since it is not any better than just guessing $n$.
– Andrzej Golonka
Dec 4 '18 at 18:57
add a comment |
Assume $cneq 0.$
Since $a=e^{itheta},$ we are looking for $n$ satisfying $;{cn}={theta}.$ If such $n$ exists, then any $$n+frac{2pi k}{c},quad kinmathbb{Z}$$ satisfies, as $$e^{ic(n+2pi k/c)}=e^{icn}=e^{itheta}=a.$$
answered Dec 4 '18 at 11:39
user376343user376343
2,9132823
Assume $cneq 0.$
Since $a=e^{itheta},$ we are looking for $n$ satisfying $;{cn}={theta}.$ If such $n$ exists, then any $$n+frac{2pi k}{c},quad kinmathbb{Z}$$ satisfies, as $$e^{ic(n+2pi k/c)}=e^{icn}=e^{itheta}=a.$$
answered Dec 4 '18 at 11:39
user376343user376343
2,9132823
answered Dec 4 '18 at 11:39
user376343user376343
2,9132823
answered Dec 4 '18 at 11:39
user376343user376343
2,9132823
answered Dec 4 '18 at 11:39
user376343user376343
2,9132823
2,9132823
That is a right observation, thank you. One could use it to find out the $n$, by taking $n_*=frac{log{a}}{ci}$, evaluating expression $n_*+ frac{2pi k}{c}$ for different values of $k$, and looking for the values that are integers. It is not a satisfying solution since it is not any better than just guessing $n$.
– Andrzej Golonka
Dec 4 '18 at 18:57
add a comment |
That is a right observation, thank you. One could use it to find out the $n$, by taking $n_*=frac{log{a}}{ci}$, evaluating expression $n_*+ frac{2pi k}{c}$ for different values of $k$, and looking for the values that are integers. It is not a satisfying solution since it is not any better than just guessing $n$.
– Andrzej Golonka
Dec 4 '18 at 18:57
That is a right observation, thank you. One could use it to find out the $n$, by taking $n_*=frac{log{a}}{ci}$, evaluating expression $n_*+ frac{2pi k}{c}$ for different values of $k$, and looking for the values that are integers. It is not a satisfying solution since it is not any better than just guessing $n$.
– Andrzej Golonka
Dec 4 '18 at 18:57
That is a right observation, thank you. One could use it to find out the $n$, by taking $n_*=frac{log{a}}{ci}$, evaluating expression $n_*+ frac{2pi k}{c}$ for different values of $k$, and looking for the values that are integers. It is not a satisfying solution since it is not any better than just guessing $n$.
– Andrzej Golonka
Dec 4 '18 at 18:57
add a comment |
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The mathematical question you have asked is totally different from the programming problem you want to solve, since in the programming problem you are only representing the numbers to some finite level of accuracy and presumably you only want $a=exp(cin)$ to hold approximately.
– Eric Wofsey
Dec 3 '18 at 19:44
@EricWofsey, the programming problem, I believe, concerns the case where $cinmathbb{Q}$.
– Andrzej Golonka
Dec 3 '18 at 19:51