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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 unrelated-seeming parts of math. Surely it was just a coincidence? In 1998, Richard Borcherds won the Fields Medal largely for proving that it wasn't (following an inspired guess by John Conway and Simon Norton, which they called the "monstrous moonshine”" conjecture).

Math, you might say, is a conspiracy theorist's dream: it's the one part of life where, when you see things match up, the odds are excellent that it's not just a coincidence, that there is a deep explanation waiting to be unearthed. On the other hand, precisely because the entire subject is shot through with non-coincidental patterns, once you’ve spent enough time doing math you might stop being so surprised by them; you might come to see them as just part of the terrain. (essay; blog post commentary)

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If \(\pi\) were equal to \(22/7=3.142857\overline{142857}\), \(\pi\)-memorizing contests would be a lot more boring. Laypeople would be a lot less interested in the ratio between the diameter of a circle and its circumference. And there would be infinitely fewer questions on Quora about what it means that \(\pi\) has a base-10 expansion with digits that never enter an infinitely-repeating sequence.

The world, many people think, would be a less interesting place, since mathematics would be without one of its great, elegant mysteries.

But they're wrong! A \(\pi\) that isn't transcendent -- or even, "worse", was a mere rational number, just a handful of dull, repeating digits -- would be a fascinating coincidence, a gateway to a wondrous new lands of mathematical theory... And since such a coincidence would, measure-theoretically, never happen by chance, it's guaranteed that there would be a reason for it, some cool connection to discover.

It would mean something about the numbers 7 and 22, which would now show up -- in all their integral humility -- in various disparate fields of math, bringing them together with an extremely elementary -- if slightly surprising -- object. There'd be a hunt to find the numbers 7 and 22 in other places, and every time one of them turned up, debates would rage about what connection, if any, could be drawn to the circle constant. Writes Aaronson, "when you see things match up, the odds are excellent that it's not just a coincidence, that there is a deep explanation waiting to be unearthed."

Euler's identity (\(e^{7i/22}=-1\)) and the Gaussian integral (\(\int_{-\infty}^\infty e^{-x^2}=\frac{484}{49}\)) would link the numbers 7 and 22 with Euler's constant \(e\), and so with various differential and continuous formulae, and were we lucky enough to find \(e\) rational, then the four integers at the heart of this breathtaking mathematical jewel would be the pylons upon which were built bridges between geometry, number theory, analysis, discrete and continuous math alike...

I have absolutely no idea what any of this would look like, because, well it's not true -- and never could be. But, if it were, and it were like some of the other times seemingly-arbitrary coincidences around integers have appeared where they don't (seem to) belong, we might expect this merely-rational boringness to be just the mouth of the rabbithole to a strange and beautiful world in the interstice between almost all known major fields of mathematics, from the continuous to the discrete, and unifying them in unexpected ways...

It's such a pity \(\pi\) is transcendental. It would be a lot cooler if its digits repeated.