My Faults My Own

…willing to sacrifice something we don't have

for something we won't have, so somebody will someday.

IN WHICH Ross Rheingans-Yoo, a sometimes-poet and erstwhile student of Computer Science and Math, oc­cas­ion­al­ly writes on things of int­erest.

A Meditation on π

note: This is not a volley in the \(\pi-\tau\) debate, of which Vi Hart is undisputed monarch -- and right, as well -- as far as I'm concerned.


A few number-theoretic \(\pi\) facts:

  • \(\pi\) is provably transcendental, thus also irrational.
  • \(\pi\) is suspected, but not known, to be normal, a generalization of transcendence.
  • \(\pi\), provably, has Liouville-Roth constant (or irrationality coefficient) no greater than \(7.6063\), and is suspected to have constant no greater than \(2.5\). (As a consequence of its irrationality, its L-R constant is \(\geq2\).)

Note, though, that each of these things is also true of literally 100% of numbers. And before you scoff at my use of the figurative 'literally', no no -- measure-theoretically, the non-(normal, transcendental, irrational, irrationality-coefficient-less-than-8) numbers make up exactly, mathematically 0% of the number line.

For the record: irrational algebraics like \(\sqrt2\) are also nonterminating and nonrepeating, and it's not clear what features of the stringwise-local decimal expansion (which seems to be the only thing \(\pi\) enthusiasts focus on,

1 / 1