See previous posts on my blog if you’re confused:
I'm only posting Part 3 (the punchline) here on LW; Parts 1 and 2 explain things in much more detail.
One is a normal number.
“No it’s not.”
In base π, I mean.
“No, it’s still just 1.“
Don’t be so greedy. Start with 0. and go from there.
“So it’s 0.222… or 0.333… or something?”
Nope, it’s 0.3011021110…
“That’s… bad. Does it repeat?”
I think it’s normal, but that seems hard to prove.
“So each digit occurs equally often?”
No, normal numbers in base π are about a 37%-30%-29%-4% split of 0, 1, 2, 3.
“Where did that distribution come from?”
Integrals of this weird function:
(Specifically, integrals over [0, 1/π], [1/π, 2/π], [2/π, 3/π], and [3/π, 1].)
“And where did that come from?”
It’s the distribution of remainders when computing a random number in base π.
“What are the x coordinates of those discontinuities?”
Sequence of remainders when computing 0.3011021110…
which are dense in [0, 1], by the way.
“Can you prove that?”
No, and neither can you.
“…And the y coordinates?”
The discontinuities get smaller by a factor of π each time.
“And almost all numbers have this distribution of remainders?”
“Including 1, if you use the 0.3011021110… representation.”
“So the histogram of x-coordinates of the N largest discontinuities in this function approaches… this function itself, as N goes to infinity.”
“Which has derivative zero almost everywhere, but has a dense set of discontinuities.”
“Any other neat facts about it?”
It’s a fractal.
Stretch out the colored regions by a factor of π horizontally, shrink by π vertically, and add them to each other. You get the original function back:
(Some imagination required.)
"And this is all because π is normal, or transcendental, or something?"
Nope, I think analogous statements are all true for base 3.5.