Why Is Pi Fascinating in Science and Mathematics?

[The_Imagizer]

I'm doing this paper on pi and the question popped up in my mind: why is pi interesting to us?

ok, it might be interesting to find new, faster ways to calculate it and stuff like that, but does it have any use, function?
what progress are we making?
I know it's ralated to the strings in the superstring theorie for example, but since that's a bit over my leage, and I do not know the exact use of pi in that either, I would like to hear what you guy's think:

Why is pi interesting to us?

 

[matt grime]

Most peculiar - the last paper I read on string theory didn't use pi once, at least not the number pi.

Pi is the ratio of the circle's circumference to its diameter. It helps us calculate areas, volumes; it arises in trigonometry, probability and analysis all the time. It also satisfies

$$\pi^2 = 6 \sum_{n \geq 1}n^{-2}$$amongst other elegant relations.

 

[The_Imagizer]

well, it might not have that big of a part in the theory, I just picked this up somewhere...

but what I was really wondering is: what progres are we making by studying pi?

 

[matt grime]

Pi is a transcendental number and therefore has many interesting properties. I believe it is still an open question as to whether there infinitely many occurences of 7 in its decimal expansion.

Personally I don't know what you mean by "studying pi" because I don't study pi, but there are, I beleive, people using this pseudo-randomness of its digits for various things. An analogy might be - the primes are distributed pseudo-randomly (I'm not using that in a technical sense) and any property that is true for a suitably random selection of integers (which can be made precise) will be true for the primes. I think Kolmogorov used this idea. Anyway, perhaps looking for things like that for pi might be useful in your search.

There's a start, perhaps.

 

[Bob3141592]

I'm doing this paper on pi and the question popped up in my mind: why is pi interesting to us?

Pi, interesting? No, I don't think so. Personally, I have no interest in pi whatsoever.

 

[deltabourne]

Maybe because it pops up in the most random places.. who would have thought that the minimum uncertainty of a particle's position/speed would involve pi? I think e^(pi*i)+1 = 0 is such a beautiful formula, as it includes the 5 most important numbers in mathematics.. all working in a simple equation. This includes pi.

Well, I find it interesting anyway :)

 

[Janitor]

Who would have thought that the minimum uncertainty of a particle's position/speed would involve pi?

Actually, I think that is something of a historical accident. Planck was the person who first determined the need for a fundamental action constant h. I believe that he was working with cycle frequency rather than with radian frequency in his derivation of black body radiation. As a result of this, he chose to define h in such a way that when, later on, people worked with it and used radian frequency (a somewhat more elegant choice, from a mathematical point of view), they found themselves invariably dividing h by 2 pi. In fact, they did this division so much that they got tired of having to show it explicitly, and they shortened the notation to h-bar, i.e. the letter 'h' with a bar through it. So in the context of the uncertainty principle the presence of pi really means nothing more than that a complete cycle/circle consists of 2 pi radians, another way of saying that the circumference of a circle is 2 pi times its radius.

EDITED dumb spelling error- Planck, not Plank.

 

[1/137]

Without pi, we wouldn't have mind-blowing equations like this one:

$$\frac{1}{\pi}\ = \frac{\sqrt{8}}{9801}\sum\limits_{n=0}^{\infty}\frac{(4n)!\cdot(1103+26390n)}{(n!)^4\cdot396^{4n}}$$

Which would be Ramanujan's Method for Pi.

 

[matt grime]

We would, but there would just be a different letter...

 

[Math Is Hard]

e

I think Pi is interesting because it's very useful, but probably what holds more fascination for me is Euler's number. It seems to pop up in a lot of places (sometimes unexpectedly) in science as well as finance. Euler's number is kinda like that joker you went to school with who had to sneak into every club picture in the yearbook whether he was a member or not.

 

[Janitor]

Funny analogy, Math Is Hard.

Since the solution to dy/dx - y = 0 is ke^x, and the solution to the second-order LDE you get by replacing the first derivative in that equation with the second derivative can be written in terms of e to imaginary exponents, e tends to turn up in the solutions to linear differential equations of the sort that crop up in describing a wide range of natural phenomena. (And I am not restricting this comment to just first-order and second-order LDEs; often you can factor higher-order LDEs down.)

 

[Integral]

Maybe because it pops up in the most random places.. who would have thought that the minimum uncertainty of a particle's position/speed would involve pi? I think e^(pi*i)+1 = 0 is such a beautiful formula, as it includes the 5 most important numbers in mathematics.. all working in a simple equation. This includes pi.

Well, I find it interesting anyway :)

Not only does it include 5 important numbers in an unexpected relationship it contains all of the basic mathematical operators, exponentiation, addition, multiplication and equality.

 

[h2]

Without pi, we wouldn't have mind-blowing equations like this one:

$$\frac{1}{\pi}\ = \frac{\sqrt{8}}{9801}\sum\limits_{n=0}^{\infty}\frac{(4n)!\cdot(1103+26390n)}{(n!)^4\cdot396^{4n}}$$
Which would be Ramanujan's Method for Pi.

.. and Comte de Buffon needle problem (1777):"Suppose a number of parallel lines, distance one unit apart, are ruled on a horizontal plane, ans suppose a homogeneous uniform rod of length 1/2 is dropped at random onto the plane. The probability that the rod will fall across one of the lines in the given plane is 1/pi"

 

[The_Imagizer]

thanks, I read several of the things you all posted in this little book about pi that I got from my math-teacher, and although they might be quite interesting, I was wondering if pi also has any practicle uses?
like most math has it's use in encryptions and such, is there such a use for pi?

 

[Jack Martinelli]

...and they shortened the notation to h-bar...

And according to the Encyclopedia of Physics, h-bar was called Dirac's constant. Nobody calls it that anymore. Why I don't know.

 

[matt grime]

"most math has its uses in encryption"? Erm, no it doesn't. Perhaps most of the maths you know can be used in encryption, but that doesn't mean that remains true in general.

One place where pi is used: MRI scanners.
One area where pi is used: would Engineering count?

 

[Janitor]

Thanks Jack

I didn't know that h-bar had been called Dirac's constant.

 

[The_Imagizer]

"most math has its uses in encryption"? Erm, no it doesn't. Perhaps most of the maths you know can be used in encryption, but that doesn't mean that remains true in general.

One place where pi is used: MRI scanners.
One area where pi is used: would Engineering count?

Well, when we discuss quite advanced math in class, and the question pops up what use it has, it very oftenly turns out to be something like that, or at least something to do with ICT... :)

thats sortof why I asked you this question aswell, I just know the question what use it has is going to pop up, so now I can do some research so I can answer their question... thanks ;)

 

[matt grime]

I think the reason why encryption is often cited is because it is relatively easy to explain, interesting to people in general and relelvant to their everyday lives (now) in a way they care about.

pi is a constant of integration is a useful slogan that I may have just invented. Things like Fourier Series and Fourier transfoms etc all use some kind of normalizing factor that is often related to pi, and often comes from integrals. For instance, switching tack slightly, the shape of the normal distribution, which arises so naturally in biology or even mechanics (one chooses the intitial velocities of particles in models of ideal gases using the normal distribution) and the shape of that curve is e^{-x^2}, if you work out the constants, so that the integral over R is 1 then pi appears.

 

[The_Imagizer]

Pi also seems to pop up quite frequently in quantum mechanics as well, or at least in chapter one of "introduction to quantum mechanics" isbn:0-582-44498-5

 

[chroot]

Pi is a rather elementary entity; it'll appear in any system that involves periodicity. It would be rather silly to list all the places where pi appears; the list of places where pi doesn't appear would be much shorter.

- Warren