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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, rather than the much-more-informative continued fraction representation...) distinguish transcendentals from irrational algebraics -- and yet \(\sqrt2\) seems to mystify no one, despite also plausibly encoding all possible variations of Hamlet, going on forever, holding all the universe's secrets, &c.

For more, the ever-wonderful Vi Hart:

Right so. Scott Aaronson has a recently-published essay titled "Why isn’t it more mysterious?", in response to the prompt "Q: Is there something mysterious about mathematics?":

Granted, not all mathematical mysteries have the character of "rigorously proving what common sense would predict." In 1978, John McKay noticed that the number 196,883 showed up in two completely