Actually it only needs to be a disjunctive (or rich) number which is a weaker condition.

We don't know whether pi is that either for any integer base.

> We don't know if pi is a normal number.

Sure we do. There are plenty of proofs out there that pi is an irrational number.

Irrational does not imply normal. For example, 1.01001000100001... is irrational but it's certainly not normal.

Technically, 1.01001000100001... can be normal depending on what ... stands for. :)

Well, obviously. But presumably the ... is meant to imply that this is the summation of 1/(10^(x(x+3)/2)).

Or what 1 or 0 or . stands for.

Actually I'd argue the example you provided is normal, as long as you authorise a particular encoding where every number n you're looking for is encoded as a string of n zeros.

It's then trivial to see that every number you can think of is encoded in there, and therefore any data, piece of music or movie that ever existed.

(I'm not sure we're allowed to fiddle with the encoding, but since we allow ourselves to represent a piece of music into a number, we're already talking about encoding anyway, so it doesn't seem like cheating to me...)

Normality of a number is with respect to number bases, so your trick with encoding is invalid. Otherwise, every computable number could be considered normal - take an algorithm for generating of it, supply a random string (this is the encoding), disregard the random string, and you have a perfectly valid normal representation of your number. So it is cheating.

I agree that normality is a specific formalized concept, but you could always require that an encoding function like this is injective.

Encoding doesn't count. Normality is a very specific mathematical concept: https://en.wikipedia.org/wiki/Normal_number

Also, 1.01001000100001... is a good example of a number that is both irrational and transcendental but not normal.

Normal in this sense means that all the frequency of all digits approaches a uniform distribution as the length of the sample increases towards infinity. Basically if we could see "all of" π and count all the 0s, 1s 2s, 3s, &c to 9 all the counts would be equal.

That on its own can't be right, because 0.12345678901234.....

According to wikipédia, you gave a definition for "simply normal", and for normal numbers the distribution of any sequence of digits is uniform. So 00, 01, ..., 99 each occur uniformally too.

Moreover you need to consider it with regards to all other bases than 10 too.