(June 15, 2020 at 6:57 pm)polymath257 Wrote: 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.
Makes sense.
Quote: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.
Bolded mine. Yeah, this is what I was trying to understand, not why certain numbers show in "nature", but what determines the values of these numbers. So it seems to me, based on the bolded, that if we really wanted a clean value for pi, we could invent a system that leads to that, right?
