Questions on Pi

[Beerina]

Actually that's the definition of an irrational number. Transcendental numbers like pi also have the property that they are not the roots of any polynomial of finite order with rational coefficients (unlike e.g. the square root of 2, which is the root of x^2 - 2).

Ouch! Thank you sir, can I have another?

I did not realize the distinction, either.

I note there's still another class: irrationals that cannot be expressed with a formula, ala Turing and Goedel and all that.

But that would not be the same as transcendentals, since pi has a formula, and thus this class would be even more exclusive*.



* If, of course, by "exclusive", you mean a group so large that all the numbers from simple integers through transcendentals form an infinitely small set compared to them. In fact, nothing could be less exclusive than that set.

 

[ddt]

1) The Pi from geometry (or "physical" Pi) which is not constant. This Pi is the ratio of the diameter to the circumference and as you get closer to a black hole, this ratio varies because the curvature of space varies. So this geometrical Pi has not the same value depending of where you are in the universe.

Pi is the relation between diameter and circumference of a circle in Euclidean geometry. That is a constant. That actual physical space is not Euclidean, is irrelevant.

2) The Pi from number theory is a constant :

[qimg]http://googolplex.net/temp/euler.gif%3C/a%3E%3Ca%20href=http://googolplex.net/temp/euler.gif%20target=_blank%3E[/qimg][qimg]http://googolplex.net/temp/euler.gif[/qimg]

and has the same value everywhere in the universe.

Number theory? Number theory deals with integer numbers. This is calculus.

 

[JanisChambers]

That burning smell is my brain overheating. I knew there was a reason why I went into art instead of science. And I thought the problem of god was a tough question...

 

[JanisChambers]

Be careful Diagoras, You know the Discovery Institute is going to be quoting you as a source now.

No, actually the value of pi changed yesterday. Mathematicians are still trying to figure out why, not to mention the philosophers, for whom this sudden change in the seemingly fixed laws of mathematics has caused quite an existential and epistemological conundrum.

 

[TrueSceptic]

That burning smell is my brain overheating. I knew there was a reason why I went into art instead of science. And I thought the problem of god was a tough question...

"God" is actually a very simple idea, but discussing that idea belongs elsewhere.

 

[GodMark2]

Never done that kind of formula formatting, but I always wanted to learn it. Let's see now...

[qimg]http://formula.s21g.com/?%5Cfrac%7B1%7D%7B%5Cpi%7D%20%3D%20%5Cfrac%7B2%5Csqrt%7B2%7D%7D%7B9801%7D%5Csum_%7B0%7D%5E%7B%5Cinfty%7D%5Cfrac%7B(4n)!(1103%20+%2026390n)%7D%7B(n!)%5E4%7B396%5E%7B4n%7D%7D%7D.png[/qimg]
[/indent]


By the way, I take sqr(8) to be sqrt(8) = 2 sqrt(2), which seems to be the way it's usually shown in this formula.

BTW, this uses one of many online utilities to generate an embeddable picture. Is there some kind of direct input to the BB code here? BTW, the above formulae link to more info at the web site, but the links are broken, even though I cut and paste the "Here's your BB code version!" text directly. BTW, I love to use BTW.

Yes, there is.
[latex]$$\frac{1}{\pi} = \frac{2\sqrt{2}}{9801}\sum_{0}^{\infty}\frac{(4n)!(1103 + 26390n)}{(n!)^4{396^{4n}}}$$[/latex]

[latex]$$\frac{1}{\pi} = \frac{2\sqrt{2}}{9801}\sum_{0}^{\infty}\frac{(4n)!(1103 + 26390n)}{(n!)^4{396^{4n}}}$$[/latex]

 

[quarky]

How many zeros in a row would it take before we decided that Pi resolves?

 

[NobbyNobbs]

"How I want a drink, alcoholic, of course, after the heavy chapters involving quantum mechanics."
Mnemonic, from who?
And this:
3.14159265358979323846264338327950

"How I need a drink, alcoholic of course, after all those problems involving quantum mechanics. Not to say problems won't bother me; rather they are too involved for an English professor today."

 

[sol invictus]

Maybe. But my point is that if you do geometry near a black hole then that "geometrical Pi" is not equal to the "number theory Pi". So I speculate that math may not be the same everywhere in the universe.

That doesn't mean math isn't the same - math is math. It just means the space is curved.

Take a circle on the surface of a sphere (a latitude line on earth, for example). What's the ratio of its circumference to its diameter - with the rule that the diameter line must stay on the surface and isn't allowed to go through the earth?

But of course on earth both Pi are equal.

Actually they're not. The earth's gravitational field warps the space - just not very much. The mathematician Carl Friedrich Gauss was smart enough to realize space might be curved, and so he used surveying equipment and three mountain tops to see whether the sum of the angles of a big triangle was 180 degrees (this was about 100 years before Einstein). It was - but only because his equipment wasn't accurate enough to detect the deviation.

 

[69dodge]

How many zeros in a row would it take before we decided that Pi resolves?

I guess that by "resolves" you mean "is rational"?

Pi has been proven to be irrational. So, it doesn't matter how many zeros in a row we might happen to see. We're sure that they won't go on forever, that sooner or later a nonzero digit will appear.

 

[TrueSceptic]

That doesn't mean math isn't the same - math is math. It just means the space is curved.

Take a circle on the surface of a sphere (a latitude line on earth, for example). What's the ratio of its circumference to its diameter - with the rule that the diameter line must stay on the surface and isn't allowed to go through the earth?

Indeed. If you map a 2D plane onto a 3D surface the 2D rules do not apply. Example: equilateral triangle with one vertex at the N pole and the 2 others on the equator. This will have an angle of 90° at each vertex, totalling 270°.

Actually they're not. The earth's gravitational field warps the space - just not very much. The mathematician Carl Friedrich Gauss was smart enough to realize space might be curved, and so he used surveying equipment and three mountain tops to see whether the sum of the angles of a big triangle was 180 degrees (this was about 100 years before Einstein). It was - but only because his equipment wasn't accurate enough to detect the deviation.

I wonder, though: would it be possible to measure this at all? Wouldn't any measuring device suffer exactly the same deviation?

 

[TrueSceptic]

I guess that by "resolves" you mean "is rational"?

Pi has been proven to be irrational. So, it doesn't matter how many zeros in a row we might happen to see. We're sure that they won't go on forever, that sooner or later a nonzero digit will appear.

Does anyone know what is the greatest number of consecutive repetitions of the same digit found so far?

 

[sol invictus]

Indeed. If you map a 2D plane onto a 3D surface the 2D rules do not apply.

The surface of a sphere is not a "3D surface", whatever that would mean. It's a (2D) surface which just happens to be curved.

I wonder, though: would it be possible to measure this at all? Wouldn't any measuring device suffer exactly the same deviation?

No. The deviation is only significant when the size of the triangle (or circle or whatever) is large compared to the radius of curvature. A tiny triangle on the surface of a sphere has angles that almost add to 180, but a large one (as in your example) is very far off. So your measuring device can measure the three angles with arbitrary accuracy, and then you just add them up.

 

[TrueSceptic]

The surface of a sphere is not a "3D surface", whatever that would mean. It's a (2D) surface which just happens to be curved.

You are right. I should have been more careful with my terminology.

No. The deviation is only significant when the size of the triangle (or circle or whatever) is large compared to the radius of curvature. A tiny triangle on the surface of a sphere has angles that almost add to 180, but a large one (as in your example) is very far off. So your measuring device can measure the three angles with arbitrary accuracy, and then you just add them up.

I'm talking about the (invisible) curvature of space here, not the curvature of, say, the surface of a sphere. Can we measure that in the same way? I'm having a job visualising it.

 

[sol invictus]

I'm talking about the (invisible) curvature of space here, not the curvature of, say, the surface of a sphere. Can we measure that in the same way? I'm having a job visualising it.

Yeah - in almost exactly the same way. Gauss' experiment (with three surveying tripods on three mountain tops, or if it's easier imagine three laser beams forming a closed triangle) would work, if it could be done accurately enough. However I don't think that's possible even now - the effect is really small.

 

[arthwollipot]

Does anyone know what is the greatest number of consecutive repetitions of the same digit found so far?

I know there's a bit a couple of hundred digits in where it goes "999999".

 

[orange31]

By the way, I recently have re-read Sagan's Contact. In the book the heroine, Ellie Arroway, director of the SETI project, receives a message from Vega that directs the world to create a machine in which 5 people can be conducted to Vega. It directly discusses the nature of PI:
.

does the math described above actually do that? I scanned the web and couldn't see an actual answer.

 

[shadron]

No, as I said, it's just a story. I think if it were true we'd be having lots of different discussions here than we do.

However, I believe that the theory of transcendental numbers requires that the randomness of the digits will indeed eventually have to create such a drawing, in any (and all) bases, and any (and all) sizes - it may be far too far down for any size greater than, say, 5x5, though, to be computable (and in fact probably is) in the time we have left.

Ye gods, what a sentence. Beg pardon.

I also could be wrong. You can read about this phenomenon at this webpage: http://sprott.physics.wisc.edu/pickover/pimatrix.html . Wallow!

 

[arthwollipot]

However, I believe that the theory of transcendental numbers requires that the randomness of the digits will indeed eventually have to create such a drawing, in any (and all) bases, and any (and all) sizes - it may be far too far down for any size greater than, say, 5x5, though, to be computable (and in fact probably is) in the time we have left.

If the series is truly infinite, then any and all such drawings are not only possible, but inevitable. An infinite series of digits must by necessity contain all possible combinations.

 

[shadron]

Ummmmm, maybe not necessarily. On the page I sited, a bunch of mathematicians (OK, probably math students) debate that. One argument is that, for instance, it cannot infinitely mimic the square root of two. Whether that really means anything is beyond me. Perhaps it only requires the word "finite" before combinations.