TITLE: Why is $\pi $ equal to $3.14159...$?
QUESTION [68 upvotes]: Wait before you dismiss this as a crank question :)
A friend of mine teaches school kids, and the book she uses states something to the following effect: 

If you divide the circumference of any circle by its diameter, you get the same number, and this number is an irrational number which starts off as $3.14159... .$

One of the smarter kids in class now has the following doubt: 

Why is this number equal to $3.14159....$? Why is it not some other irrational number?

My friend is in a fix as to how to answer this in a sensible manner. Could you help us with this?
I have the following idea about how to answer this: Show that the ratio must be greater than $3$. Now show that it must be less than $3.5$. Then show that it must be greater than $3.1$. And so on ... . 
The trouble with this is that I don't know of any easy way of doing this, which would also be accessible to a school student.
Could you point me to some approximation argument of this form?

REPLY [6 votes]: First off, notice that the kernel of your question is obscured by focusing on the decimal digits of an irrational (indeed, transcendental) number. We could ask the same kind of question about a plain vanilla positive integer. For example, why is it that 1729 ("Hardy's taxicab number") is the smallest positive integer representable as the sum of two cubes in two different ways? Why isn't it some other number instead? Why is it this specific number? However, this new perspective puts us on the road to the answer to your question. Consider this question: Why is it that, for convex polyhedra, the number of vertices minus the number of edges plus the number of faces always equals 2? However, by stereographic projection, this is equivalent to asking why it is that, for connected planar graphs, the number of vertices minus the number of edges plus the number of regions (counting the "infinite" region as one of the regions) always equals 2. However, the proof for the case of connected planar graphs typically involves dismantling the graph in a canonical manner that reduces to a triangle, for which the value of 2 is "obvious". This brings us to the answer to your question: it is not a matter of mathematics, or even of metamathematics, but a question of psychology. The more steeped you are in Mathematics, the more obvious things seem to you. For example, for Euler, who had, among other things, committed many primes to memory, the value of Hardy's taxicab number would perhaps seem self-evident, as it did to Ramanujan. To an Infinite Intelligence (God), everything would be self-evident, but for finite minds there always comes a boundary where they wonder why a certain value is what it is. Nor is Mathematics unique in that regard. In Physics they still haven't figured out why the so-called "fine structure constant" is the value that it is (approximately 1/137), and the reasons for its value is the subject of on-going debate.
Anyway, perspective might be gained and retained by bearing in mind the following famous joke:
Mathematics is really Psychology.
Psychology is really Biology.
Biology is really Chemistry.
Chemistry is really Physics.
Physics is really Mathematics.