Can you obtain $pi$ using elements of $mathbb{N}$, and finite number of basic arithmetic operations +...
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Is it possible to obtain $pi$ from finite amount of operations
${+,-,cdot,div,wedge}$ on $mathbb{N}$ (or $mathbb{Q}$, the answer will still be the same), note that set of all real numbers obtainable this way contains numbers that are not algebraic (for example $2^{2^{1/2}}$ is transcendental)
Bonus: If it happens that the answer is no, is it a solution to some equation generated that way (those $5$ operations performed finitely many times on elements on $mathbb{N}$) ?
algebra-precalculus
edited Jan 1 at 11:32
Eevee Trainer
7,04321337
asked Jul 28 '18 at 13:29
MathemagicianMathemagician
103116
$endgroup$
add a comment |
$begingroup$
Is it possible to obtain $pi$ from finite amount of operations
${+,-,cdot,div,wedge}$ on $mathbb{N}$ (or $mathbb{Q}$, the answer will still be the same), note that set of all real numbers obtainable this way contains numbers that are not algebraic (for example $2^{2^{1/2}}$ is transcendental)
Bonus: If it happens that the answer is no, is it a solution to some equation generated that way (those $5$ operations performed finitely many times on elements on $mathbb{N}$) ?
algebra-precalculus
edited Jan 1 at 11:32
Eevee Trainer
7,04321337
asked Jul 28 '18 at 13:29
MathemagicianMathemagician
103116
$endgroup$
$begingroup$
Related: math.stackexchange.com/questions/2611084/…
$endgroup$
– user202729
Jul 28 '18 at 13:34
1
$begingroup$
(this one is more general. If the answer to this one is "no" then the answer to the other one is "no" as well)
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– user202729
Jul 28 '18 at 13:35
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Of course not, but I can't prove it. There are only countably many finite expressions of this sort, so you can only express countably many real numbers this way. Unless a real has some reason to be expressible it almost certainly can't be.
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– Ross Millikan
Jul 28 '18 at 14:36
1
$begingroup$
Yes, there are only countably many of them, so most real numbers do not fall into this category. Still, not much can be deduced about any specific number from that fact. π is arguably somewhat special.
$endgroup$
– Mathemagician
Jul 28 '18 at 14:57
add a comment |
$begingroup$
Is it possible to obtain $pi$ from finite amount of operations
${+,-,cdot,div,wedge}$ on $mathbb{N}$ (or $mathbb{Q}$, the answer will still be the same), note that set of all real numbers obtainable this way contains numbers that are not algebraic (for example $2^{2^{1/2}}$ is transcendental)
Bonus: If it happens that the answer is no, is it a solution to some equation generated that way (those $5$ operations performed finitely many times on elements on $mathbb{N}$) ?
algebra-precalculus
edited Jan 1 at 11:32
Eevee Trainer
7,04321337
asked Jul 28 '18 at 13:29
MathemagicianMathemagician
103116
$endgroup$
Is it possible to obtain $pi$ from finite amount of operations
${+,-,cdot,div,wedge}$ on $mathbb{N}$ (or $mathbb{Q}$, the answer will still be the same), note that set of all real numbers obtainable this way contains numbers that are not algebraic (for example $2^{2^{1/2}}$ is transcendental)
Bonus: If it happens that the answer is no, is it a solution to some equation generated that way (those $5$ operations performed finitely many times on elements on $mathbb{N}$) ?
algebra-precalculus
algebra-precalculus
edited Jan 1 at 11:32
Eevee Trainer
7,04321337
asked Jul 28 '18 at 13:29
MathemagicianMathemagician
103116
edited Jan 1 at 11:32
Eevee Trainer
7,04321337
asked Jul 28 '18 at 13:29
MathemagicianMathemagician
103116
edited Jan 1 at 11:32
Eevee Trainer
7,04321337
edited Jan 1 at 11:32
Eevee Trainer
7,04321337
edited Jan 1 at 11:32
Eevee Trainer
7,04321337
7,04321337
asked Jul 28 '18 at 13:29
MathemagicianMathemagician
103116
asked Jul 28 '18 at 13:29
MathemagicianMathemagician
103116
asked Jul 28 '18 at 13:29
MathemagicianMathemagician
103116
103116
$begingroup$
Related: math.stackexchange.com/questions/2611084/…
$endgroup$
– user202729
Jul 28 '18 at 13:34
1
$begingroup$
(this one is more general. If the answer to this one is "no" then the answer to the other one is "no" as well)
$endgroup$
– user202729
Jul 28 '18 at 13:35
$begingroup$
Of course not, but I can't prove it. There are only countably many finite expressions of this sort, so you can only express countably many real numbers this way. Unless a real has some reason to be expressible it almost certainly can't be.
$endgroup$
– Ross Millikan
Jul 28 '18 at 14:36
1
$begingroup$
Yes, there are only countably many of them, so most real numbers do not fall into this category. Still, not much can be deduced about any specific number from that fact. π is arguably somewhat special.
$endgroup$
– Mathemagician
Jul 28 '18 at 14:57
add a comment |
$begingroup$
Related: math.stackexchange.com/questions/2611084/…
$endgroup$
– user202729
Jul 28 '18 at 13:34
1
$begingroup$
(this one is more general. If the answer to this one is "no" then the answer to the other one is "no" as well)
$endgroup$
– user202729
Jul 28 '18 at 13:35
$begingroup$
Of course not, but I can't prove it. There are only countably many finite expressions of this sort, so you can only express countably many real numbers this way. Unless a real has some reason to be expressible it almost certainly can't be.
$endgroup$
– Ross Millikan
Jul 28 '18 at 14:36
1
$begingroup$
Yes, there are only countably many of them, so most real numbers do not fall into this category. Still, not much can be deduced about any specific number from that fact. π is arguably somewhat special.
$endgroup$
– Mathemagician
Jul 28 '18 at 14:57
$begingroup$
Related: math.stackexchange.com/questions/2611084/…
$endgroup$
– user202729
Jul 28 '18 at 13:34
$begingroup$
Related: math.stackexchange.com/questions/2611084/…
$endgroup$
– user202729
Jul 28 '18 at 13:34
1
1
$begingroup$
(this one is more general. If the answer to this one is "no" then the answer to the other one is "no" as well)
$endgroup$
– user202729
Jul 28 '18 at 13:35
$begingroup$
(this one is more general. If the answer to this one is "no" then the answer to the other one is "no" as well)
$endgroup$
– user202729
Jul 28 '18 at 13:35
$begingroup$
Of course not, but I can't prove it. There are only countably many finite expressions of this sort, so you can only express countably many real numbers this way. Unless a real has some reason to be expressible it almost certainly can't be.
$endgroup$
– Ross Millikan
Jul 28 '18 at 14:36
$begingroup$
Of course not, but I can't prove it. There are only countably many finite expressions of this sort, so you can only express countably many real numbers this way. Unless a real has some reason to be expressible it almost certainly can't be.
$endgroup$
– Ross Millikan
Jul 28 '18 at 14:36
1
1
$begingroup$
Yes, there are only countably many of them, so most real numbers do not fall into this category. Still, not much can be deduced about any specific number from that fact. π is arguably somewhat special.
$endgroup$
– Mathemagician
Jul 28 '18 at 14:57
$begingroup$
Yes, there are only countably many of them, so most real numbers do not fall into this category. Still, not much can be deduced about any specific number from that fact. π is arguably somewhat special.
$endgroup$
– Mathemagician
Jul 28 '18 at 14:57
add a comment |
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$begingroup$
Related: math.stackexchange.com/questions/2611084/…
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– user202729
Jul 28 '18 at 13:34
1
$begingroup$
(this one is more general. If the answer to this one is "no" then the answer to the other one is "no" as well)
$endgroup$
– user202729
Jul 28 '18 at 13:35
$begingroup$
Of course not, but I can't prove it. There are only countably many finite expressions of this sort, so you can only express countably many real numbers this way. Unless a real has some reason to be expressible it almost certainly can't be.
$endgroup$
– Ross Millikan
Jul 28 '18 at 14:36
1
$begingroup$
Yes, there are only countably many of them, so most real numbers do not fall into this category. Still, not much can be deduced about any specific number from that fact. π is arguably somewhat special.
$endgroup$
– Mathemagician
Jul 28 '18 at 14:57