Some thoughts on Pi (Day)
a little something I wrote up Saturday morning for Pi Day:
I've been a bit skeptical about the rise of \pi day, b/c (a) obviously it's a function of our particular calendar, and on top of that it depends on our mm/dd convention, and obviously \pi is much more fundamental than that; (b) it refers to the decimal expansion of \pi (see
http://en.wikipedia.org/wiki/Pi#Approximate_value
for the initial digits of \Pi in binary, hexadecimal, and sexagesimal (i.e., base 60))
Finally, and most significantly in my mind, 3/14 sells \pi seriously short: (c) 3 (digits, i.e., 3.14), or even 10 (3.141592653, if you in fact observed 9:26:53 this morning) is a very poor approximation to \∞
All that said, I'm not going to hate on people spending a day giving respect to \pi. So here are a few facts to ponder:
As alluded to above, and as you probably are aware of, \pi is irrational: a real number with infinite non-repeating decimal expansion, or equivalently one that can't be expressed as a fraction of integers (i.e., is not a rational number). But that wasn't proved until the 18th century (cf http://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational), whereas the ancient Greeks discovered (in some sense) that \sqrt(2) is irrational--a discovery which apparently caused them (the Pythagoreans) something of a philosophical crisis;
You probably know \pi as the ratio of the circumference of a circle to it's diameter: \pi = c/d = c/(2r) ... for any circle. But you probably also know that the area of a circle is \pi*r^2. (So in particular, the unit circle (circle of radius 1) has area \pi.) It's not obvious (at least to me) that those two properties, area and c/d ratio, should involve the same (irrational) constant. I consider it an example of the unexpected beauty of mathematics (cf this Strogatz column for more on these aspects of \pi, and for how they naturally lead to the concept of limits: [Update: a friend pointed out that the connection between area and c/d is revealed by the integral by which one computes the area of the circle. Doh!]
Finally, another example of the unexpected beauty of mathematics: if you learned about infinite series (usually covered in 2nd semester calculus), you most likely learned that the harmonic series 1 + 1/2 + 1/3 + 1/4 + ... + (1/n) + ... diverges (i.e., goes off to infinity), whereas the very similar-looking series 1 + 1/4 + 1/9 + 1/16 + ... + (1/n^2) + ... actually converges (i.e., adds up to finite number; the precise definition of this involves limits of partial sums).
But most likely you didn't see what finite number it converges to. This was called the Basel problem, and was solved by the singular mathematician Leonard Euler in 1735, at the age of 28. Remarkably, he showed that
1 + 1/4 + 1/9 + 1/16 + ... = (\pi^2)/6
From there Euler calculated a bunch of fantastical results, such as
(2*2*4*4*6*6*8*8*...)/(1*3*3*5*5*7*7*..) = \pi/2
and
1 + 1/16 + 1/81 + 1/256 + ... + 1/n^4 + ... = (\pi^4)/90
and
1 + 1/64 + 1/729 + 1/4096 + ... + 1/n^6 + ... = (\pi^6)/945
(cf http://en.wikipedia.org/wiki/Basel_problem
&
https://plus.maths.org/content/infinite-series-surprises
& Ch 9 of the excellent book "Journey Through Genius", which adds:
A question immediately comes to mind: What is the sum of the reciprocals of odd powers of the integers? For instance, can we evaluate the infinite series
1 + 1/(2^3) + 1/(3^3) + 1/(4^3) + ... = 1 + 1/8 + 1/27 + 1/64 + ...
as well? Here even Euler was mute, and the past 200 years of mathematical research have advanced our knowledge of such odd powers very little. It is easy to conjecture that the sum in question is of the form (p/q)*\pi^3 for some fraction p/q, but to this day no one knows if this conjecture is valid.
