Applicability of Maths to the Universe

[GrandizerII]

This brings me to a somewhat related question I've had on my mind for a while. Why is it that such special numbers "found in nature" such as pi and e are irrational (and seemingly "arbitrary") as opposed to "clean" rational numbers? Is it to do with the particular number system being used (the decimal)?

Rationality is not affected by the number system used. A number is either rational or irrational. It is rational if it is a 'ratio' of two integers and irrational if not.

So, sqrt(2) is irrational. You cannot write it as ratio of two integers. The ancient Greeks knew this well before the decimal system for *describing* numbers.

Now, pi and e are more than just irrational; they are transcendental. In other words, they are not solutions of any polynomial with integer coefficients.

Well, it turns out that there are only countably many numbers that are *not* transcendental and uncountably many that are. So, in a sense, the vast majority of real numbers are transcendental. An 'arbitrary' real number is likely to be transcendental (and hence, irrational).

Right, a number is more likely to be transcendental and therefore irrational. Fair enough. But when we speak of numbers like pi and e, we speak of numbers that are very special numbers that can be "captured in nature". If you divide the circumference of a circle by its diameter, you get this special number called pi. Why is such a special number not a "neat" rational number? More specifically, why is pi the exact value as it is?