My Foolish π Call of Duty
A few days ago, I broke my Twitter.
Let me explain.
I saw this tweet which links to this image.
Knowing that normal numbers are not necessarily the same as irrational numbers, I thought, “Oh! How interesting, there must be some proof that π is normal! I wonder if they have a link, because I’d love to read it!”
The image, of course, presumes that an infinite pseudorandom sequence of digits will contain every digit. (That is, that any irrational number will be normal in base-10, so long as it contains every digit.)
Foolishly, perhaps, I replied, and in the last few days, I’ve been getting about 10 responses per day saying, roughly, DURRR!!! DONT YOU KNOW INFINITE MEANS HAS EVERY NUMBER LIKE THE MONKEYS WRITE SHAKE SPEAR?
As a simple proof, consider a number, very close to π, but different
in the respect that any time a
41 appears, it is replaced by
Let’s call it P. It would start
and continue on forever.
If π is indeed normal in base 10, then clearly P would not be.
Because any digit sequence that goes
41 would instead go
you could never find the first 4 digits of π within the digits of P.
But P is also infinite, irrational, and pseudorandom. It would contain exactly the same ratio of digits as π does, but not have every combination.
For the first few days, I explained this over and over again. Since then, I’ve simply accepted that my “replies” tab on Twitter is temporarily useless.
As it turns out, π has NOT been proven to be
though it passes all known tests for normality, meaning that it has
not been proven to not be normal either. The same is true of
e, the square root of 2, and many other natural constants.
What makes matters worse, there is not (yet?) any proof that these
things can or cannot be proven, so it may be a wild goose chase as
infinitely irrational as π itself!
As one might expect, Vi Hart covered this math question in the most entertaining fashion possible.
If there was a proof that π is normal, then this would be a groundbreaking mathematical insight of historic proportions.
It saddens me deeply that the internet is so full of fools who think that such a revolution has not only already occurred, but is obvious and foolish to even wonder about.
Sigh. Duty calls.