One is (almost) normal in base π

See previous posts if you’re confused:

Part 1: Normal numbers in base ten
Part 2: Numbers in base φ and π
Part 3: One is normal in base π

This post is also available on LessWrong.

This wasn’t supposed to be a whole series of posts, so we’re going to speed through the punchline:

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.

One response to “One is (almost) normal in base π

  1. Barbara Scherlis

    Dear Adam,

    Thank you for the mathematical information. I used to be good a math, but not that good! Keep it up — Enjoy it! love, Grandma

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