Is there any way to finitely represent all the information in pi? [closed]
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Of course, we can represent it as 10 in base pi but that won't be much useful. Think of pi as a length from 0 to some unique point on the real line. A length which cannot be finitely expressed in any integer base system because integer base systems are all about rationals while pi is inherently irrational. It's a no-brainer why all representations of pi have been infinite. It's because we've chosen natural numbers to study this more complex phenomenon known as pi. Base systems are all about strings of natural numbers. Pi goes beyound natural numbers.So how about we change the system? Is there any system you know of where pi can be represented finitely?
UPDATE More specifically, Do you know of any number system (besides base pi, base root pi, etc) in which pi can be represented finitely? It need not be a base-system.
Also, the current answer by @HenningMakholm is technically a finite representation of pi, but I don't think it qualifies as a number system.
pi number-systems
asked Aug 15 at 13:06
Ryder Rude
354110
closed as unclear what you're asking by Peter, Clayton, Henning Makholm, Daniel Fischer♦ Aug 15 at 14:42
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.
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Of course, we can represent it as 10 in base pi but that won't be much useful. Think of pi as a length from 0 to some unique point on the real line. A length which cannot be finitely expressed in any integer base system because integer base systems are all about rationals while pi is inherently irrational. It's a no-brainer why all representations of pi have been infinite. It's because we've chosen natural numbers to study this more complex phenomenon known as pi. Base systems are all about strings of natural numbers. Pi goes beyound natural numbers.So how about we change the system? Is there any system you know of where pi can be represented finitely?
UPDATE More specifically, Do you know of any number system (besides base pi, base root pi, etc) in which pi can be represented finitely? It need not be a base-system.
Also, the current answer by @HenningMakholm is technically a finite representation of pi, but I don't think it qualifies as a number system.
pi number-systems
asked Aug 15 at 13:06
Ryder Rude
354110
closed as unclear what you're asking by Peter, Clayton, Henning Makholm, Daniel Fischer♦ Aug 15 at 14:42
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.
@Peter I addressed that in the very first sentence of my post. It's like we know only two ways to represent pi. One way using strings of natural numbers, which is infinite, can never give us the complete information. In the other way, like in base pi or radians, we make pi itself the unit and that's just self-referential, doesn't tell us how much pi is exactly.
– Ryder Rude
Aug 15 at 13:13
1
What do you say about $pi = 4 ;arctan(1)?$
– gammatester
Aug 15 at 13:21
2
@RyderRude: If you want to get an infinite amount of information (bits?) out of the representation you want, but do not want to spend an unbounded amount of effort to extract that information, then I think your desire is inherently unreasonable.
– Henning Makholm
Aug 15 at 13:24
1
Since $pi$ is transcendental it cannot be represented by a finite expression $sum_k=0^n a_k$ with rational $a_k$. Please specify more exactly what you mean with finitely representing a number.
– gammatester
Aug 15 at 13:37
1
@RyderRude: No -- it remains open for you to claim that whatever anyone proposes does not qualify as a "number system" for you.
– Henning Makholm
Aug 15 at 15:21
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show 27 more comments
up vote
-3
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up vote
-3
down vote
favorite
Of course, we can represent it as 10 in base pi but that won't be much useful. Think of pi as a length from 0 to some unique point on the real line. A length which cannot be finitely expressed in any integer base system because integer base systems are all about rationals while pi is inherently irrational. It's a no-brainer why all representations of pi have been infinite. It's because we've chosen natural numbers to study this more complex phenomenon known as pi. Base systems are all about strings of natural numbers. Pi goes beyound natural numbers.So how about we change the system? Is there any system you know of where pi can be represented finitely?
UPDATE More specifically, Do you know of any number system (besides base pi, base root pi, etc) in which pi can be represented finitely? It need not be a base-system.
Also, the current answer by @HenningMakholm is technically a finite representation of pi, but I don't think it qualifies as a number system.
pi number-systems
asked Aug 15 at 13:06
Ryder Rude
354110
Of course, we can represent it as 10 in base pi but that won't be much useful. Think of pi as a length from 0 to some unique point on the real line. A length which cannot be finitely expressed in any integer base system because integer base systems are all about rationals while pi is inherently irrational. It's a no-brainer why all representations of pi have been infinite. It's because we've chosen natural numbers to study this more complex phenomenon known as pi. Base systems are all about strings of natural numbers. Pi goes beyound natural numbers.So how about we change the system? Is there any system you know of where pi can be represented finitely?
UPDATE More specifically, Do you know of any number system (besides base pi, base root pi, etc) in which pi can be represented finitely? It need not be a base-system.
Also, the current answer by @HenningMakholm is technically a finite representation of pi, but I don't think it qualifies as a number system.
pi number-systems
asked Aug 15 at 13:06
Ryder Rude
354110
edited Aug 15 at 15:20
asked Aug 15 at 13:06
Ryder Rude
354110
asked Aug 15 at 13:06
Ryder Rude
354110
asked Aug 15 at 13:06
Ryder Rude
354110
354110
closed as unclear what you're asking by Peter, Clayton, Henning Makholm, Daniel Fischer♦ Aug 15 at 14:42
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 Peter, Clayton, Henning Makholm, Daniel Fischer♦ Aug 15 at 14:42
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.
@Peter I addressed that in the very first sentence of my post. It's like we know only two ways to represent pi. One way using strings of natural numbers, which is infinite, can never give us the complete information. In the other way, like in base pi or radians, we make pi itself the unit and that's just self-referential, doesn't tell us how much pi is exactly.
– Ryder Rude
Aug 15 at 13:13
1
What do you say about $pi = 4 ;arctan(1)?$
– gammatester
Aug 15 at 13:21
2
@RyderRude: If you want to get an infinite amount of information (bits?) out of the representation you want, but do not want to spend an unbounded amount of effort to extract that information, then I think your desire is inherently unreasonable.
– Henning Makholm
Aug 15 at 13:24
1
Since $pi$ is transcendental it cannot be represented by a finite expression $sum_k=0^n a_k$ with rational $a_k$. Please specify more exactly what you mean with finitely representing a number.
– gammatester
Aug 15 at 13:37
1
@RyderRude: No -- it remains open for you to claim that whatever anyone proposes does not qualify as a "number system" for you.
– Henning Makholm
Aug 15 at 15:21
 |Â
show 27 more comments
@Peter I addressed that in the very first sentence of my post. It's like we know only two ways to represent pi. One way using strings of natural numbers, which is infinite, can never give us the complete information. In the other way, like in base pi or radians, we make pi itself the unit and that's just self-referential, doesn't tell us how much pi is exactly.
– Ryder Rude
Aug 15 at 13:13
1
What do you say about $pi = 4 ;arctan(1)?$
– gammatester
Aug 15 at 13:21
2
@RyderRude: If you want to get an infinite amount of information (bits?) out of the representation you want, but do not want to spend an unbounded amount of effort to extract that information, then I think your desire is inherently unreasonable.
– Henning Makholm
Aug 15 at 13:24
1
Since $pi$ is transcendental it cannot be represented by a finite expression $sum_k=0^n a_k$ with rational $a_k$. Please specify more exactly what you mean with finitely representing a number.
– gammatester
Aug 15 at 13:37
1
@RyderRude: No -- it remains open for you to claim that whatever anyone proposes does not qualify as a "number system" for you.
– Henning Makholm
Aug 15 at 15:21
@Peter I addressed that in the very first sentence of my post. It's like we know only two ways to represent pi. One way using strings of natural numbers, which is infinite, can never give us the complete information. In the other way, like in base pi or radians, we make pi itself the unit and that's just self-referential, doesn't tell us how much pi is exactly.
– Ryder Rude
Aug 15 at 13:13
@Peter I addressed that in the very first sentence of my post. It's like we know only two ways to represent pi. One way using strings of natural numbers, which is infinite, can never give us the complete information. In the other way, like in base pi or radians, we make pi itself the unit and that's just self-referential, doesn't tell us how much pi is exactly.
– Ryder Rude
Aug 15 at 13:13
1
1
What do you say about $pi = 4 ;arctan(1)?$
– gammatester
Aug 15 at 13:21
What do you say about $pi = 4 ;arctan(1)?$
– gammatester
Aug 15 at 13:21
2
2
@RyderRude: If you want to get an infinite amount of information (bits?) out of the representation you want, but do not want to spend an unbounded amount of effort to extract that information, then I think your desire is inherently unreasonable.
– Henning Makholm
Aug 15 at 13:24
@RyderRude: If you want to get an infinite amount of information (bits?) out of the representation you want, but do not want to spend an unbounded amount of effort to extract that information, then I think your desire is inherently unreasonable.
– Henning Makholm
Aug 15 at 13:24
1
1
Since $pi$ is transcendental it cannot be represented by a finite expression $sum_k=0^n a_k$ with rational $a_k$. Please specify more exactly what you mean with finitely representing a number.
– gammatester
Aug 15 at 13:37
Since $pi$ is transcendental it cannot be represented by a finite expression $sum_k=0^n a_k$ with rational $a_k$. Please specify more exactly what you mean with finitely representing a number.
– gammatester
Aug 15 at 13:37
1
1
@RyderRude: No -- it remains open for you to claim that whatever anyone proposes does not qualify as a "number system" for you.
– Henning Makholm
Aug 15 at 15:21
@RyderRude: No -- it remains open for you to claim that whatever anyone proposes does not qualify as a "number system" for you.
– Henning Makholm
Aug 15 at 15:21
 |Â
show 27 more comments
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$pi$ can be represented by a finite formula in standard mathematical notation, such as:
$$ pi = 4 sum_n=0^infty frac(-1)^n2n+1 $$
This contains all information there is to get about the value of $pi$.
(This formula, known as the Leibniz series, has the primary benefit that it is short to write down. It converges infeasibly slowly in practice, but we can use it to prove that other, longer but more efficient, formulas describe the same number).
More precisely, this formula gives you a concrete procedure for deciding, for each rational number and in finite time, whether your rational number is smaller or larger than $pi$. Simply start summing the series, and as soon as you reach a point where the difference between the partial sum so far and your target rational is smaller than the next term in the series, you're done.
(That this works depends on the fact that the series is an absolutely decreasing alternating series, and that your target rational is not $pi$ itself. The latter is because we know $pi$ is not rational, which is not obvious but definitely known).
answered Aug 15 at 13:11
Henning Makholm
228k16294525
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
$pi$ can be represented by a finite formula in standard mathematical notation, such as:
$$ pi = 4 sum_n=0^infty frac(-1)^n2n+1 $$
This contains all information there is to get about the value of $pi$.
(This formula, known as the Leibniz series, has the primary benefit that it is short to write down. It converges infeasibly slowly in practice, but we can use it to prove that other, longer but more efficient, formulas describe the same number).
More precisely, this formula gives you a concrete procedure for deciding, for each rational number and in finite time, whether your rational number is smaller or larger than $pi$. Simply start summing the series, and as soon as you reach a point where the difference between the partial sum so far and your target rational is smaller than the next term in the series, you're done.
(That this works depends on the fact that the series is an absolutely decreasing alternating series, and that your target rational is not $pi$ itself. The latter is because we know $pi$ is not rational, which is not obvious but definitely known).
answered Aug 15 at 13:11
Henning Makholm
228k16294525
add a comment |Â
up vote
2
down vote
$pi$ can be represented by a finite formula in standard mathematical notation, such as:
$$ pi = 4 sum_n=0^infty frac(-1)^n2n+1 $$
This contains all information there is to get about the value of $pi$.
(This formula, known as the Leibniz series, has the primary benefit that it is short to write down. It converges infeasibly slowly in practice, but we can use it to prove that other, longer but more efficient, formulas describe the same number).
More precisely, this formula gives you a concrete procedure for deciding, for each rational number and in finite time, whether your rational number is smaller or larger than $pi$. Simply start summing the series, and as soon as you reach a point where the difference between the partial sum so far and your target rational is smaller than the next term in the series, you're done.
(That this works depends on the fact that the series is an absolutely decreasing alternating series, and that your target rational is not $pi$ itself. The latter is because we know $pi$ is not rational, which is not obvious but definitely known).
answered Aug 15 at 13:11
Henning Makholm
228k16294525
add a comment |Â
up vote
2
down vote
up vote
2
down vote
$pi$ can be represented by a finite formula in standard mathematical notation, such as:
$$ pi = 4 sum_n=0^infty frac(-1)^n2n+1 $$
This contains all information there is to get about the value of $pi$.
(This formula, known as the Leibniz series, has the primary benefit that it is short to write down. It converges infeasibly slowly in practice, but we can use it to prove that other, longer but more efficient, formulas describe the same number).
More precisely, this formula gives you a concrete procedure for deciding, for each rational number and in finite time, whether your rational number is smaller or larger than $pi$. Simply start summing the series, and as soon as you reach a point where the difference between the partial sum so far and your target rational is smaller than the next term in the series, you're done.
(That this works depends on the fact that the series is an absolutely decreasing alternating series, and that your target rational is not $pi$ itself. The latter is because we know $pi$ is not rational, which is not obvious but definitely known).
answered Aug 15 at 13:11
Henning Makholm
228k16294525
$pi$ can be represented by a finite formula in standard mathematical notation, such as:
$$ pi = 4 sum_n=0^infty frac(-1)^n2n+1 $$
This contains all information there is to get about the value of $pi$.
(This formula, known as the Leibniz series, has the primary benefit that it is short to write down. It converges infeasibly slowly in practice, but we can use it to prove that other, longer but more efficient, formulas describe the same number).
More precisely, this formula gives you a concrete procedure for deciding, for each rational number and in finite time, whether your rational number is smaller or larger than $pi$. Simply start summing the series, and as soon as you reach a point where the difference between the partial sum so far and your target rational is smaller than the next term in the series, you're done.
(That this works depends on the fact that the series is an absolutely decreasing alternating series, and that your target rational is not $pi$ itself. The latter is because we know $pi$ is not rational, which is not obvious but definitely known).
answered Aug 15 at 13:11
Henning Makholm
228k16294525
edited Aug 15 at 13:34
answered Aug 15 at 13:11
Henning Makholm
228k16294525
answered Aug 15 at 13:11
Henning Makholm
228k16294525
answered Aug 15 at 13:11
Henning Makholm
228k16294525
228k16294525
add a comment |Â
add a comment |Â
@Peter I addressed that in the very first sentence of my post. It's like we know only two ways to represent pi. One way using strings of natural numbers, which is infinite, can never give us the complete information. In the other way, like in base pi or radians, we make pi itself the unit and that's just self-referential, doesn't tell us how much pi is exactly.
– Ryder Rude
Aug 15 at 13:13
1
What do you say about $pi = 4 ;arctan(1)?$
– gammatester
Aug 15 at 13:21
2
@RyderRude: If you want to get an infinite amount of information (bits?) out of the representation you want, but do not want to spend an unbounded amount of effort to extract that information, then I think your desire is inherently unreasonable.
– Henning Makholm
Aug 15 at 13:24
1
Since $pi$ is transcendental it cannot be represented by a finite expression $sum_k=0^n a_k$ with rational $a_k$. Please specify more exactly what you mean with finitely representing a number.
– gammatester
Aug 15 at 13:37
1
@RyderRude: No -- it remains open for you to claim that whatever anyone proposes does not qualify as a "number system" for you.
– Henning Makholm
Aug 15 at 15:21