Applicability of Maths to the Universe

[polymath257]

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?

Well, pi can also be defined as twice the value of the first zero of the cosine function or as the value of certain integrals. In more formal mathematics, that is more often the definition.

The number e comes up most naturally in the solution of differential equations (y'=y) and is thereby related to 'constant growth or decay rates'. That is why it 'comes up naturally'. That particular differential equation is picked out not only for simplicity, but because it is translation invariant.

But, for example, we could study Bessel functions and ask for the value of the first zero of J_0(x). That will also be a transcendental value. Such functions come up naturally when looking at certain spherically symmetric situations (as the radial function). and the values of the roots of the Bessel functions show up 'naturally' in nature as well for such cases.

And, of course, all of these are related to each other. The cosine is the real part of the complex exponential, so pi and e are related via Euler's formula. The Bessel values are related to Fourier integrals, which involve either complex exponentials or trig functions.

And, Fourier integrals and series very naturally have both pi and e arise as scaling factors.

As for your last question, I'm not sure how to answer a 'why' question when it comes to math. The value of pi is what it is because we define trig functions the way we do or we need a circumference or surface for a symmetric figure (which naturally involves trig functions and hence complex exponentials). Finally, exponentials are homomorphisms from the additive reals to the multiplicative positive reals, so we selected algebraically.