Proving that $π^e$ is irrational [closed]
2
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I tried for a few hours to come with a proof that $π^e$ is irrational. I mainly tried with the method "proof by contradiction" and didn't use calculus at all, but couldn't come up with a proof.
Can anyone give a proof that $π^e$ is irrational? I would appreaciate if it was without using calculus. because that's more fun, but it's completly fine using calculus too :) Just want to see a proof.
Thanks in advance!
exponential-function pi
asked Jan 19 at 18:26
Casimir RönnlöfCasimir Rönnlöf
1154
$endgroup$
closed as off-topic by Xander Henderson, mrtaurho, Arthur, Andrés E. Caicedo, Mark Bennet Jan 19 at 18:45
- This question does not appear to be about math within the scope defined in the help center.
If this question can be reworded to fit the rules in the help center, please edit the question.
|
show 3 more comments
2
$begingroup$
I tried for a few hours to come with a proof that $π^e$ is irrational. I mainly tried with the method "proof by contradiction" and didn't use calculus at all, but couldn't come up with a proof.
Can anyone give a proof that $π^e$ is irrational? I would appreaciate if it was without using calculus. because that's more fun, but it's completly fine using calculus too :) Just want to see a proof.
Thanks in advance!
exponential-function pi
asked Jan 19 at 18:26
Casimir RönnlöfCasimir Rönnlöf
1154
$endgroup$
closed as off-topic by Xander Henderson, mrtaurho, Arthur, Andrés E. Caicedo, Mark Bennet Jan 19 at 18:45
- This question does not appear to be about math within the scope defined in the help center.
If this question can be reworded to fit the rules in the help center, please edit the question.
6
$begingroup$
This is unknown
$endgroup$
– Peter
Jan 19 at 18:28
$begingroup$
@Peter What? Seriously? Without calculus even? Ooo that's a fun challenge for a 14 year old math lover :DD
$endgroup$
– Casimir Rönnlöf
Jan 19 at 18:34
4
$begingroup$
I'm voting to close this question as off-topic because it involves speculation about an open problem in mathematics. I don't think that such speculation is on-topic here (though I suspect that a question about what is known about the ir/rationality of $pi^mathrm{e}$ might be both (1) on-topic and (2) already answered here somewhere).
$endgroup$
– Xander Henderson
Jan 19 at 18:35
2
$begingroup$
Good hazing for a young mathematician!
$endgroup$
– mathcounterexamples.net
Jan 19 at 18:36
1
$begingroup$
Note that $e^pi$ is proven to be transcendental on the same wiki page. However, the proof is only easy if we can use the Gelfond-Schneider theorem and apply it to $(-1)^{-i}$.
$endgroup$
– I like Serena
Jan 19 at 19:22
|
show 3 more comments
2
2
2
$begingroup$
I tried for a few hours to come with a proof that $π^e$ is irrational. I mainly tried with the method "proof by contradiction" and didn't use calculus at all, but couldn't come up with a proof.
Can anyone give a proof that $π^e$ is irrational? I would appreaciate if it was without using calculus. because that's more fun, but it's completly fine using calculus too :) Just want to see a proof.
Thanks in advance!
exponential-function pi
asked Jan 19 at 18:26
Casimir RönnlöfCasimir Rönnlöf
1154
$endgroup$
I tried for a few hours to come with a proof that $π^e$ is irrational. I mainly tried with the method "proof by contradiction" and didn't use calculus at all, but couldn't come up with a proof.
Can anyone give a proof that $π^e$ is irrational? I would appreaciate if it was without using calculus. because that's more fun, but it's completly fine using calculus too :) Just want to see a proof.
Thanks in advance!
exponential-function pi
exponential-function pi
asked Jan 19 at 18:26
Casimir RönnlöfCasimir Rönnlöf
1154
asked Jan 19 at 18:26
Casimir RönnlöfCasimir Rönnlöf
1154
asked Jan 19 at 18:26
Casimir RönnlöfCasimir Rönnlöf
1154
asked Jan 19 at 18:26
Casimir RönnlöfCasimir Rönnlöf
1154
asked Jan 19 at 18:26
Casimir RönnlöfCasimir Rönnlöf
1154
1154
closed as off-topic by Xander Henderson, mrtaurho, Arthur, Andrés E. Caicedo, Mark Bennet Jan 19 at 18:45
- This question does not appear to be about math within the scope defined in the help center.
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by Xander Henderson, mrtaurho, Arthur, Andrés E. Caicedo, Mark Bennet Jan 19 at 18:45
- This question does not appear to be about math within the scope defined in the help center.
If this question can be reworded to fit the rules in the help center, please edit the question.
6
$begingroup$
This is unknown
$endgroup$
– Peter
Jan 19 at 18:28
$begingroup$
@Peter What? Seriously? Without calculus even? Ooo that's a fun challenge for a 14 year old math lover :DD
$endgroup$
– Casimir Rönnlöf
Jan 19 at 18:34
4
$begingroup$
I'm voting to close this question as off-topic because it involves speculation about an open problem in mathematics. I don't think that such speculation is on-topic here (though I suspect that a question about what is known about the ir/rationality of $pi^mathrm{e}$ might be both (1) on-topic and (2) already answered here somewhere).
$endgroup$
– Xander Henderson
Jan 19 at 18:35
2
$begingroup$
Good hazing for a young mathematician!
$endgroup$
– mathcounterexamples.net
Jan 19 at 18:36
1
$begingroup$
Note that $e^pi$ is proven to be transcendental on the same wiki page. However, the proof is only easy if we can use the Gelfond-Schneider theorem and apply it to $(-1)^{-i}$.
$endgroup$
– I like Serena
Jan 19 at 19:22
|
show 3 more comments
6
$begingroup$
This is unknown
$endgroup$
– Peter
Jan 19 at 18:28
$begingroup$
@Peter What? Seriously? Without calculus even? Ooo that's a fun challenge for a 14 year old math lover :DD
$endgroup$
– Casimir Rönnlöf
Jan 19 at 18:34
4
$begingroup$
I'm voting to close this question as off-topic because it involves speculation about an open problem in mathematics. I don't think that such speculation is on-topic here (though I suspect that a question about what is known about the ir/rationality of $pi^mathrm{e}$ might be both (1) on-topic and (2) already answered here somewhere).
$endgroup$
– Xander Henderson
Jan 19 at 18:35
2
$begingroup$
Good hazing for a young mathematician!
$endgroup$
– mathcounterexamples.net
Jan 19 at 18:36
1
$begingroup$
Note that $e^pi$ is proven to be transcendental on the same wiki page. However, the proof is only easy if we can use the Gelfond-Schneider theorem and apply it to $(-1)^{-i}$.
$endgroup$
– I like Serena
Jan 19 at 19:22
6
6
$begingroup$
This is unknown
$endgroup$
– Peter
Jan 19 at 18:28
$begingroup$
This is unknown
$endgroup$
– Peter
Jan 19 at 18:28
$begingroup$
@Peter What? Seriously? Without calculus even? Ooo that's a fun challenge for a 14 year old math lover :DD
$endgroup$
– Casimir Rönnlöf
Jan 19 at 18:34
$begingroup$
@Peter What? Seriously? Without calculus even? Ooo that's a fun challenge for a 14 year old math lover :DD
$endgroup$
– Casimir Rönnlöf
Jan 19 at 18:34
4
4
$begingroup$
I'm voting to close this question as off-topic because it involves speculation about an open problem in mathematics. I don't think that such speculation is on-topic here (though I suspect that a question about what is known about the ir/rationality of $pi^mathrm{e}$ might be both (1) on-topic and (2) already answered here somewhere).
$endgroup$
– Xander Henderson
Jan 19 at 18:35
$begingroup$
I'm voting to close this question as off-topic because it involves speculation about an open problem in mathematics. I don't think that such speculation is on-topic here (though I suspect that a question about what is known about the ir/rationality of $pi^mathrm{e}$ might be both (1) on-topic and (2) already answered here somewhere).
$endgroup$
– Xander Henderson
Jan 19 at 18:35
2
2
$begingroup$
Good hazing for a young mathematician!
$endgroup$
– mathcounterexamples.net
Jan 19 at 18:36
$begingroup$
Good hazing for a young mathematician!
$endgroup$
– mathcounterexamples.net
Jan 19 at 18:36
1
1
$begingroup$
Note that $e^pi$ is proven to be transcendental on the same wiki page. However, the proof is only easy if we can use the Gelfond-Schneider theorem and apply it to $(-1)^{-i}$.
$endgroup$
– I like Serena
Jan 19 at 19:22
$begingroup$
Note that $e^pi$ is proven to be transcendental on the same wiki page. However, the proof is only easy if we can use the Gelfond-Schneider theorem and apply it to $(-1)^{-i}$.
$endgroup$
– I like Serena
Jan 19 at 19:22
|
show 3 more comments
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6
$begingroup$
This is unknown
$endgroup$
– Peter
Jan 19 at 18:28
$begingroup$
@Peter What? Seriously? Without calculus even? Ooo that's a fun challenge for a 14 year old math lover :DD
$endgroup$
– Casimir Rönnlöf
Jan 19 at 18:34
4
$begingroup$
I'm voting to close this question as off-topic because it involves speculation about an open problem in mathematics. I don't think that such speculation is on-topic here (though I suspect that a question about what is known about the ir/rationality of $pi^mathrm{e}$ might be both (1) on-topic and (2) already answered here somewhere).
$endgroup$
– Xander Henderson
Jan 19 at 18:35
2
$begingroup$
Good hazing for a young mathematician!
$endgroup$
– mathcounterexamples.net
Jan 19 at 18:36
1
$begingroup$
Note that $e^pi$ is proven to be transcendental on the same wiki page. However, the proof is only easy if we can use the Gelfond-Schneider theorem and apply it to $(-1)^{-i}$.
$endgroup$
– I like Serena
Jan 19 at 19:22