[Math] Why aren’t there more numbers like $e$, $\pi$, and $i$? This is based on looking through the Handbook of Mathematical Functions and online.

big-picturesoft-question

This is kind of a big picture question. I just counted up all the symbols used in normal mathematics and, give or take, there are probably around 150 of them, tops. And that's really stretching things. I am including:

  • the transpose symbol in linear algebra,
  • the gamma function symbol,
  • the direct sum symbol for two vector spaces,
  • the tensor product symbol,
  • …

I am including a lot of seemingly esoteric stuff! Even so, no matter what formula spits out of Wolfram Alpha, or no matter where you look in the Handbook of Mathematical Functions, for practical applied math purposes there really aren't that many symbols. That got me to thinking: we have these famous numbers $e$, $i$, and $\pi$. They are related by the famous Euler formula which blows everybody's mind when they first see it, except for, reportedly, Gauss.

$\pi$ relates to the circle. $e$ relates to rate of change — relates to integration and differentiation somehow. $i$ gives us an extra number dimension to solve problems.

Question: Why aren't there more of these numbers? Is it the case that $99$% of all important mathematics in practice is covered by the rationals, operations like taking rational roots, the vast swath of anonymous, non-name-worthy irrational & transcendental reals, and $e$, $i$ and $\pi$?

What is it about these three numbers that makes them, in effect, practically the only important numbers in mathematics other than those that can be expressed in terms of regular numbers? Is it because this circle relationship, this extra dimension thing with $i$, and this rate of change thing with $e$, covers all the really important relationships between the dimensions? I really want someone to break this down for me and tell me why this is the case.

Best Answer

Have a look at http://en.wikipedia.org/wiki/Mathematical_constants where you will find any number of important mathematical constants which have no known expression in terms of $e$, $i$, and $\pi$. The Euler-Mascheroni constant, $\gamma$, is a biggie in analytic number theory. $\zeta(3)$ was immortalized by Apery. Feigenbaum's constant $\delta$ is super-important in dynamical systems and transition to chaos. Khinchin's constant $K$ is big in continued fractions. And so on.

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