2021-12-26

Irrational numbers and copyright

One of the fun concepts in mathematics is that every (finite) sequence of digits exists somewhere in π.

There are webs sites that explain this (e.g. here) and point out that it has not been proven. It sort of makes some sense, as π is irrational, meaning the digit sequence in infinitely long and never starts to repeat. It seems intuitive that it must therefore contain all finite sequences of digits.

If true, then yes, every possible sequence exists. and the idea is Don't work out π in binary. If you do, you breach copyright (not true, just creating the same thing originally rather than copying it is not a breach of copyright), trademark infringement, and a whole load of other issues.

Of course the same argument applies to creating a near infinite random number :-)

A rational number will always start to repeat - that is pretty easy to prove, and even that the repetition is going to be no longer than the denominator. π, being irrational, does not repeat.

My first reaction to this idea is that just being irrational (and so infinitely long), does not mean it has every finite digit sequence. My argument is simple, take π, and add a 0 between every digit. You still have an infinite sequence, and it still won't start to repeat and so is irrational, but does not even have the sequence 123 in it.

But then it occurred to me - is what I am suggesting a sane thing to do? Someone else pointed out (on twitter) that they do not know of any mathematical function that could insert a 0 every other digit. OK, but does what I am doing have to be a mathematical function?

Today it occurred to me that maybe what I am saying, even though it sounds simple, may not actually be a reasonably thing to suggest. That maybe the number I am creating is not "genuine" in some way, a number that could not be created. So maybe my argument is not sound.

Bear with me - it is not as daft as it sounds to dismiss a number I have created by description as not genuine.

E.g. take the number ⅓, in decimal, and add a 7 to the end of it. Now it is not rational, is it?

That clearly is nonsense. You cannot add a 7 to the end of an infinite sequence of digits (well, not that end anyway). So that concept is easy to dismiss as not "genuine".

But suggesting a 0 is added between each digit of π is equally impossible - I am adding infinite digits. If ever I stop doing so, π goes on as normal and maybe contains every finite sequence of digits.

So is what I suggested not sensible. Could there by a genuine irrational number that does obviously not contain every finite sequences of digits?

3 comments:

  1. Feels like you've got this going on

    https://www.youtube.com/watch?v=mZBwsm6B280

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  2. Liouville's number is a beautiful thing, created by description:
    https://www.youtube.com/watch?v=c9nUAXUSuII

    But if you're really uncomfortable with numbers created by description, then you're heading into territory inhabited by only a rather strange minority of mathematicians, and you'll have to face the consequences. If, for example, you extend the arguments of someone like Norman Wildberger, you might end up having to exclude not only the irrationals but also the rationals beyond some arbitrary denominator.
    It all boils down to what you mean by "exist". The philosophy of numbers includes various interpretations: Platonism, constructivism, formalism, etc. It's an absorbing rabbit-hole to explore...

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  3. 0.2 is infinitely recurring when written in base-2 but simple to write in base-10. Therefore you can "add something to the end" of all rational numbers by representing them in the "correct" base.

    It might be useful to represent numbers as a series of Continued Fractions rather than in a particular base. I used a similar representation in an encoder I wrote that can represent *any* rational number in such a way that if you lexicographically sort numbers in that representation you end up with them sorted numerically.

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