Still not enough, or at least pi is not known to have this property. You need the number to be “normal” (or a slightly weaker property) which turns out to be hard to prove about most numbers.
> natural numbers
> rational numbers
> real numbers
> regular numbers
> normal numbers
> simply normal numbers
> absolutely normal numbers
Have mathematicians considered talking about what numbers they find okay, rather than everyone just picking their favorite and saying that one’s the ordinary one?
I mean, unironically yes. It seems the most popular stance is that all math regardless of how weird is Platonically real, although that causes some real bad problems when put down rigorously. Personally I’m more of an Aristotelian.
In the case of things like rational or real numbers, they have a counterpart that’s weirder (irrational and imaginary numbers). For the rest I’m not sure, but it’s pretty common to just pick an adjective for a new concept. There’s even situations where the same term gets used more than once in different subfields, and then they collide so you have to add another one to clarify.
For example, one open interval in the context of a small set of open intervals isn’t closed analytically under limits, or algebraically closed, but is topologically closed (and also topologically open, as the name suggests).
“Nearly all real numbers are normal (basically no real numbers are not normal), but we’re only aware of a few. This one literally non-computable one for sure. Maybe sqrt(2).”
We’re so used to dealing with real numbers it’s easy to forget they’re terrible. These puppies are a particularly egregious example I like to point to - functions that preserve addition but literally black out the entire x-y plane when plotted. On rational numbers all additive functions are automatically linear, of the form mx+n. There’s no nice in-between on the reals, either; it’s the “curve” from hell or a line.
Hot take, but I really hope physics will turn out to work without them.
Simply containing each number sequence is a significantly weaker property than having them all occur at the right frequency. Still, while nobody has proven it, it’s generally expected to be true.
Still not enough, or at least pi is not known to have this property. You need the number to be “normal” (or a slightly weaker property) which turns out to be hard to prove about most numbers.
Wikipedia for normal numbers, and for disjunctive sequences, which is the slightly weaker property mentioned.
> natural numbers
> rational numbers
> real numbers
> regular numbers
> normal numbers
> simply normal numbers
> absolutely normal numbers
Have mathematicians considered talking about what numbers they find okay, rather than everyone just picking their favorite and saying that one’s the ordinary one?
I mean, unironically yes. It seems the most popular stance is that all math regardless of how weird is Platonically real, although that causes some real bad problems when put down rigorously. Personally I’m more of an Aristotelian.
In the case of things like rational or real numbers, they have a counterpart that’s weirder (irrational and imaginary numbers). For the rest I’m not sure, but it’s pretty common to just pick an adjective for a new concept. There’s even situations where the same term gets used more than once in different subfields, and then they collide so you have to add another one to clarify.
For example, one open interval in the context of a small set of open intervals isn’t closed analytically under limits, or algebraically closed, but is topologically closed (and also topologically open, as the name suggests).
“Nearly all real numbers are normal (basically no real numbers are not normal), but we’re only aware of a few. This one literally non-computable one for sure. Maybe sqrt(2).”
Gotta love it.
We’re so used to dealing with real numbers it’s easy to forget they’re terrible. These puppies are a particularly egregious example I like to point to - functions that preserve addition but literally black out the entire x-y plane when plotted. On rational numbers all additive functions are automatically linear, of the form mx+n. There’s no nice in-between on the reals, either; it’s the “curve” from hell or a line.
Hot take, but I really hope physics will turn out to work without them.
Simply containing each number sequence is a significantly weaker property than having them all occur at the right frequency. Still, while nobody has proven it, it’s generally expected to be true.