value of pi

[Copernicus]

Are there any values of pi where no one knows why pi is included in calculations?

[Jens]

It’s a good question. I would assume that the answer is no, because you would not put a pi into an equation unless you had a reason to do so, and that would normally be because you are dealing with a sphere somehow. So for example, I would guess that the density of a gas as it expands would have a pi component, because it is expanding into a sphere.


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[Jens]

It also turns up in things like products of infinite series, and I’m not mathematically competent enough to know why but I assume there must be a reason.


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[Strange]

It’s a good question. I would assume that the answer is no, because you would not put a pi into an equation unless you had a reason to do so, and that would normally be because you are dealing with a sphere somehow. So for example, I would guess that the density of a gas as it expands would have a pi component, because it is expanding into a sphere.

Or any kind of wave or regular change.

[Ufonaut99]

Not sure about "where no one knows why", but there are cases where it unexpectedly pops up where on first glance it doesn't belong. For example, Buffon's Needle from the 18th century :

Suppose we have a floor made of parallel strips of wood, each the same width, and we drop a needle onto the floor. What is the probability that the needle will lie across a line between two strips?

There's no immediate link to Pi, and we can imagine early experimenters spending enjoyable afternoons throwing needles around - only to find that when the needles are shorter than the width of the panels, the probability they'd have found is :

P(crossing) = 2 * NeedleLength / (Width * Pi)

So why is that Pi there ? We can see that needles would be more likely to cross when they fall perpendicular across the panels rather than in line with them - hence rotation and circles play a part.

[DonM435]

Not sure about "where no one knows why", but there are cases where it unexpectedly pops up where on first glance it doesn't belong. For example, Buffon's Needle from the 18th century :



There's no immediate link to Pi, and we can imagine early experimenters spending enjoyable afternoons throwing needles around - only to find that when the needles are shorter than the width of the panels, the probability they'd have found is :

P(crossing) = 2 * NeedleLength / (Width * Pi)

So why is that Pi there ? We can see that needles would be more likely to cross when they fall perpendicular across the panels rather than in line with them - hence rotation and circles play a part.

I've seen that concept illustrated with a bunch of toothpicks and the U.S. flag.
For a physics class, I wrote a computer program to simulate the toothpick tosses, picking a randomized angle and a randomized center-point, and showing how the number in the ratio approached pi with more and more trials.
(The tricky part was handling the randomization without using pi in the algorithm, which would be cheating.)

[Strange]

Not sure about "where no one knows why", but there are cases where it unexpectedly pops up where on first glance it doesn't belong. For example, Buffon's Needle from the 18th century :



There's no immediate link to Pi, and we can imagine early experimenters spending enjoyable afternoons throwing needles around - only to find that when the needles are shorter than the width of the panels, the probability they'd have found is :

P(crossing) = 2 * NeedleLength / (Width * Pi)

So why is that Pi there ? We can see that needles would be more likely to cross when they fall perpendicular across the panels rather than in line with them - hence rotation and circles play a part.

Surely, this result was found analytically, not experimentally? And it is fairly obvious that π is going to be involved because the probability depends on angle, and even the sine of the angle.

[Copernicus]

So I have two equations. The following

1.

2.

Equation 1 is equal to pi for all real numbers values of n

Equation 2 is equal to pi/2-1/n where is approximates pi/2 at large values of n.

When I put equation 1 into wolframalpha it won't solve the problem for me, anyone have any ideas why.

[Strange]

It worked for me, but it seems to take a while
https://www.wolframalpha.com/input/?...%3D-n+to+x%3Dn

[DonM435]

It has something to do with getting a sine curve when you plot needle center (distance from nearest line) against needle angle.

Search this text for "matches" or "flag."
(Note: No flag-burning!)

https://archive.org/stream/GeorgeGamowOneTwoThreeInfinityFactsAbOk.org/%5BGeorge_Gamow%5D_One%2C_two%2C_three--_infinity_facts_a%28b-ok.org%29_djvu.txt

or

https://books.google.com/books?id=fu...roblem&f=false

[Copernicus]

It worked for me, but it seems to take a while
https://www.wolframalpha.com/input/?...%3D-n+to+x%3Dn

Ijust wanted it to solve it for n undifined.

[Strange]

Ijust wanted it to solve it for n undifined.

I don't see how it could, if n is not defined.

You can integrate stand solve for various value of n: https://www.wolframalpha.com/input/?...%7B.5%7D%7D+dx

[Copernicus]

I don't see how it could, if n is not defined.

You can integrate stand solve for various value of n: https://www.wolframalpha.com/input/?...%7B.5%7D%7D+dx

For equation 1 the value is always pi, but is there a way to specify n for all real numbers and imaginary numbers.

[Copernicus]

It has something to do with getting a sine curve when you plot needle center (distance from nearest line) against needle angle.

Search this text for "matches" or "flag."
(Note: No flag-burning!)

https://archive.org/stream/GeorgeGamowOneTwoThreeInfinityFactsAbOk.org/%5BGeorge_Gamow%5D_One%2C_two%2C_three--_infinity_facts_a%28b-ok.org%29_djvu.txt

or

https://books.google.com/books?id=fu...roblem&f=false

Are you able to summarize how that applies here?

[Copernicus]

I was also wondering if anyone knows of any other equation where all values for n causes the equation to equal pi.

[Copernicus]

It looks like, when n is between 1 and -1 the value is a complex number or at zero I think the value is zero. Otherwise it is pi

n = 2 ; integrate (x/n)\frac{(x/n)^2 - ((x-1)/n)^2}{(1-(x/n)^2)^{.5}} from x=-n to x=n

[21st Century Schizoid Man]

I was also wondering if anyone knows of any other equation where all values for n causes the equation to equal pi.

Shouldn’t be hard to find other examples. For example, take a definite integral that has value of pi. Change the variable of integration with a monotonic function that depends on a parameter n.

[Copernicus]

Shouldn’t be hard to find other examples. For example, take a definite integral that has value of pi. Change the variable of integration with a monotonic function that depends on a parameter n.

Thanks, I had thought of that too, but I was thinking of any other way, but that way.

[21st Century Schizoid Man]

for all , , , , and , unless I've made a careless error somewhere. Those five parameters can be any real numbers, with the indicated restrictions, not just integers.

But, it is essentially generated by taking



and changing variables.

[Copernicus]

for all , , , , and , unless I've made a careless error somewhere. Those five parameters can be any real numbers, with the indicated restrictions, not just integers.

But, it is essentially generated by taking



and changing variables.

Hi 21CenturySchizoidman,

Thanks, I saw a list of equations on Wikipedia for equations equal to pi. I was looking for another, where every number one enters, in the equation I made, n can be any value of n greater than or equal to plus or minus one or plus or minus i, the answer is pi.

[21st Century Schizoid Man]

There, I fixed my post, which had the last part of the final equation up in the exponent

[Copernicus]

There, I fixed my post, which had the last part of the final equation up in the exponent

I appreciate your help 21st Century Schizoid Man. What I am looking for is an equation like this

where any value I put in for n except between one and minus one, it still returns the value of pi. If I put in 1.1 or 200,000 it still returns pi as an answer. Between one and minus one it is a complex number, which should be expected by the way I derived the equation.

[Shaula]

I appreciate your help 21st Century Schizoid Man. What I am looking for is an equation like this.

And he has given you:
a) A simple way to generate as many examples of equations like that as you like
b) An example with even more parameters you can set to anything you like

Demonstrating that what you have there is not anything particularly special or unusual. I'm not sure why you are arbitrarily dismissing what he has provided.

[Copernicus]

And he has given you:
a) A simple way to generate as many examples of equations like that as you like
b) An example with even more parameters you can set to anything you like

Demonstrating that what you have there is not anything particularly special or unusual. I'm not sure why you are arbitrarily dismissing what he has provided.

Thanks I didn't look at it well enough. Would you know what the physical meaning to the equations?

[21st Century Schizoid Man]

Would you know what the physical meaning to the equations?

If you divide the integrand in my original equation by , it is the probability density function for random variables and with a bivariate Gaussian (or normal) distribution, with means and , standard deviations and , and correlation coefficient . In other words, if and are random variables with this joint distribution, then the probability that and is

.

If we set the means to zero, the standard deviations to one, and the correlation to zero, this simplifies enormously, to

.

Extending the range to and , we get

,

since probability density functions have to integrate to one. Multiply both sides by , we get

.

It may not be real obvious where the comes from, but the may suggest the equation for a circle. And if you change from Cartesian coordinates and to polar coordinates and , the connection becomes more obvious.

[21st Century Schizoid Man]

If you prefer a single variable,

.

You get this by applying Fubini's theorem to the double integral in the last equation in my previous post, separating the two integrals, and then changing the variables of integration to be the same.

[Copernicus]

If you divide the integrand in my original equation by , it is the probability density function for random variables and with a bivariate Gaussian (or normal) distribution, with means and , standard deviations and , and correlation coefficient . In other words, if and are random variables with this joint distribution, then the probability that and is

.

If we set the means to zero, the standard deviations to one, and the correlation to zero, this simplifies enormously, to

.

Extending the range to and , we get

,

since probability density functions have to integrate to one. Multiply both sides by , we get

.

It may not be real obvious where the comes from, but the may suggest the equation for a circle. And if you change from Cartesian coordinates and to polar coordinates and , the connection becomes more obvious.

You are very good at this. I wish I was the same. I did recognize the equation as from a circle.

[Copernicus]

I was also wondering if anyone knows of any other equation where all values for n causes the equation to equal pi.

Is there a way to graph the value of this equation as n goes from 0 to 1?

[21st Century Schizoid Man]

Is there a way to graph the value of this equation as n goes from 0 to 1?

By evaluating the integral explicitly, I am getting it is a constant value of for all real values of . So that should be fairly easy to graph.

* I've assumed you have your limits of integration backwards. If this is really what you meant, then the value is .

[Copernicus]

By evaluating the integral explicitly, I am getting it is a constant value of for all real values of . So that should be fairly easy to graph.

* I've assumed you have your limits of integration backwards. If this is really what you meant, then the value is .

When using Wolframalpha, between -1 to 1, I get some values to be complex numbers and some numbers to be close to . I am wondering if Wolframalpha is spitting out some wrong answers, or if this is actually what is happening.

For example the following yields

n = .9991 ; integrate (x/n)\frac{(x/n)^2 - ((x-1)/n)^2}{(1-(x/n)^2)^{.5}} from x=-n to x=n

3.14159 - 3.23009×10^-16 i

I think it might have something to do with significant digits.