Pi -- it sure gets around

Factual Questions
1

I couldn’t search for this due to the 3 character restriction.

I have a reasonable math background - college level.

I’m curious as to why PI shows up in such diverse places. I know it can be defined as the sum of an infinite series, or informally as the ratio of a circle’s circumference to its diameter. I’ve also seen PI crop up in the solutions to some probability puzzles. What have these things got to do with each other? What has the geometry of a cirlce got to do with pulling colored balls from a bag? (Can’t remember any examples of probability questions that have answers involving PI right now.)

2

Examples please?

When come back … oh never mind.

3

Pi is not informally defined as the ratio of the circumfrence to diameter. That is its definition.

As for a probability, I don’t think you’d get pi from any urn picking problem except possibly as a limit. I don’t see how you could get anythign but rational probabilities with discrete objects. You do get answers involving pi from geometry based problems. Th esimplest adn most obvious is. If you pick a point at random from inside a unti square, what’s the probability it is inside an inscribed circle? The answer is proportion to the area of the inscribed cirlce pi*(1/2)^2. You can disguise this problem in various ways.

4

I seem to recall, for instance, that given two large numbers, the probability that they will be relatively prime is 1/pi. There’s also Buffon’s Needle Problem, but there the connection to circles is a little more obvious.

It’s not surprising that it’s the sum of various infinite series, since you can make an infinite series to sum to pretty much whatever you want. And a lot of places where it comes up, it’s due to its relationship with e (the base of natural logs, approximately 2.718281828). Euler famously found that e[sup]ipi[/sup] = -1, or more generally, e[sup]ix[/sup] = cos(x) + isin(x). And then you just need to explain why e shows up in so many places.

5

53 ways to make Pi :slight_smile:

http://functions.wolfram.com/Constants/Pi/06/01/

6

The Normal distribution and the Gamma integral are related and GAMMA(1/2) = SQRT(Pi).

http://mathworld.wolfram.com/NormalDistribution.html

7

Lots of books out there that explain the many, many associations that pi has with other numbers:

A History of Pi, by Petr Beckmann

The Joy of Pi, by David Blatner

Pi: A Biography of the World’s Most Mysterious Number, by Alfred S Posamentier and Ingmar Lehmann

and for math majors:

The Number Pi, by Pierre Eymard et al.

Pi: A Source Book, by Lennart Berggren et al.

8

I strongly recommend that if anyone comes back to this thread (after reading it once and going elsewhere), they not hijack it! :wink:

Something I’ve always wondered, myself, is the significance of e – the base of natural logarithms. I gather it has pretty much as pervasive a placement in math as pi, and for equally abstruse reasons. Why was that particular transcendental number chosen as the base for the natural logs? What else is it significant in? And is there any fundamental reason for the strange collection of seemingly unrelated places in which pi and/or e crop up?

9

The trig functions relating to triangles, circles, and graphs are basic elements in math, they are used over and over again to define and analyze systems in higher math. It should be obvious that trig functions and Pi go hand in hand. So, where ever you roam in the world of math, don’t be surprised to see trig functions and Pi show their pretty little faces again.

The fundamentals behind trig functions can be extended to hyperbolic trig functions, which are defined using powers of e. So by the same token, e shows up endlessly in higher math also.

Trig functions, Pi, and e are enormously versatile fundamentals of mathematics.

10

A must read: e: The Story of a Number, by Eli Maor

11

The beauty of this makes me want to cry.

From: http://mathworld.wolfram.com/e.html

The special case of the Euler formula

(14) e^(i * x) = cos x + i * sin x

with x = Pi gives the beautiful identity

(15) e^(i * Pi) + 1 = 0

an equation connecting the fundamental numbers i, pi, e, 1, and 0 (zero).

12

Pi typically shows up when whatever you’re looking at has a relation to circles or spheres. For example, electromagnetism formulae contain pi because, among other reasons, abstract point charges have a spherically symmetric electric field. Pi also shows up when there’s a relationship to e. Recall the Euler identity that Chronos pointed out, particularly the general form e^(iy) = cos(y) + isin(y). This arises from the Taylor series of the functions e^x, cos(x) and sin(x).

http://home.ecn.ab.ca/~jsavard/math/ide01.htm

e^(i pi) = -1 is just a specific case of this.

e is so ubiquitous, and is so simply related to pi, that pi just happens to occur in a lot of real-world phenomena*.

So, this raises the question, Why is e all over the place? I really don’t know. It has some very simple definitions, so I guess it’s pretty much bound to show up in some places. When I’ve done physics problems, the formulae we derived typically involved inadvertantly taking derivatives of 1/x, yielding the natural log of x.

Wherever e shows up, you can probably expect pi to follow. e plays a prominent role in growth and decay, compound interest, normal and gamma probability distributions**, the distribution of prime numbers, catenaries (the shapes wires assumes when suspended from both ends), blah blah blah
*The fact that a formula with ‘imaginary’ numbers has any relation to the real world may be puzzling, but actually there is nothing imaginary about imaginary numbers. ‘Real’ and ‘imaginary’ are misnomers. Many physical phenomenon require more numbers to describe than we have on the real number line. The solution is to take a second number line, called imaginary, and orient it perpendicular to the real number line so that the two intersect at the origin so that we now have an plane of numbers rather than a line.

**Since these were mentioned earlier: the normal distribution is a bell-shaped curve given by 1/Sqrt(2 pi) * e^(-z^2/2). The pi-containing factor appears because the area under f(z) = e^(-z^2/2) is equal to Sqrt(2 pi), and we intuitively want the total probability of something to equal one.

13

I can’t speak for the historical reasons, but if you solve the equation integral( 1/t, 1 < t < x ) = 1 for x, you get x = e.

14

Not only that but it uses each arithmetic operator (+, , ^, =) once and only once in the form 0 = e^(ipi) + 1.

15

Hmmm… the relationship between pi and e. Pi and e. Pi + e = pie.

May not be beautiful, but it’s delicious.

16

I love Euler’s equation (identity?) too. I opine it’s the most beautiful thing in all mathematics.

I still can’t get an intuitive grasp for what circles and geometry have to do with probability though, the needle experiment notwithstanding.

17

IIRC “e” like “Pi” is a “natural” number. As such it crops up in all sorts of functions.
As a base for logarithms it greatly simplifies certain calculations.

18

It’s more important than ?, actually. It’s the base for the natural logarithm because e[sup]x[/sup] is its own derivative.

More specifically, asking for solutions to f’(x) = f(x) gets you a unique function (and an arbitrary multiplicative constant). This function is shown to behave like an exponential, and e is defined as f(1), so f(x) = e[sup]x[/sup].

? also comes from differential equations: the two main trigonometric functions are the basic solutions of f’’(x) = -f(x). They are shown to be periodic, and ? is defined as half that period.

By studying slight variations on these equations through either power series or formal algebraic techniques, one finds that

sin(x) = (e[sup]ix[/sup]-e[sup]-ix[/sup])/2i
cos(x) = (e[sup]ix[/sup]+e[sup]-ix[/sup])/2

so the famous relation between e and ? follows.

From here to probability is (relatively) easy: the “error function”, defined as the integral of e[sup]-t[sup]2[/sup][/sup] from 0 to x has a limit as x goes to infinity: sqrt(?)/2. The proof of this comes back to properties of exponential functions (and thus trigonometric functions) in the complex plane. Since repeating an experiment and averaging makes probability distributions approach a Gaussian curve, this shows up everywhere in probability.

Elsewhere, the exponential function (defined algebraically as a power series since we don’t always have “exponentiation”) has amazing uses in Lie theory and differential geometry. Consider the fact that e[sup]x+y[/sup] = e[sup]x[/sup]e[sup]y[/sup]: addition and multiplication are interchanged. Lie theory studies curved spaces with a notion of what it means to multiply two points, and these spaces are approximated by vector spaces (which have a notion of addition). The exponential map sends the vector space to the curved space, and if the curved space is a set of matrices (with the multiplication being matrix multiplication), then the exponential map is given by exactly the same series formula as gives the exponential function we’re all familiar with.

19

Think more about the graphs of the trig funtions rather than circles. If you look at the graphs for the Normal distribution and the cosine function, they look very similar.
http://mathworld.wolfram.com/Cosine.html

http://mathworld.wolfram.com/NormalDistribution.html

20

Yeah, and looks are enough to get by in math…

It comes from the fact that the integral over the real line of the Gaussian distribution is ?. That comes from complex analysis, which is just loaded with exponentials, which are related to circles as has been pointed out.