Actually tbh, I think I’ve erred: the binary expression “0.1010010001…” is algebraic, but the deciml expression “0.1010010001…” is almost certainly transcendental. Proving such a number is algebraic/irrational revolves around transcendence theory
A countable union of measure zero sets has measure zero. So once you’ve proven that, for each b, almost all values in [0, 1] are normal in base b, you’ve automatically established that almost all values in [0, 1] are absolutely normal.
When people say that pi’s decimal expansion continues infinitely without repeating, they don’t mean that “There’s never a point at which some initial part of the sequence appears again.” After all, there are plenty of 3s in the decimal expansion of pi.
Instead, what they mean is that “There’s never a point after which the sequence just cycles through the same values over and over forever”. As opposed to, say, 46.2815412412412412412412412…[repeating 412 forever]…
And this sort of thing, indeed, provably doesn’t happen for pi. (The easy part of the proof is showing that this sort of thing happens just for numbers which are ratios of integers. The hard part of the proof is showing that pi cannot be a ratio of integers.)
The number 4 must be very odd indeed. But the only number that is both even and odd is infinity, therefore 4=infinity.
Infinity is not a number.
All numbers are interesting, especially 1729.
Except for large values of ‘number’
My bugaboo: I generally don’t find it useful to say (or think!) things like “X is not a number”. Often, it’s just a way to shout down what would otherwise be a fruitful mathematical investigation, out of reflexive inertia. There’s not just one number system, and not just one sense of the term “number”.
It depends on what you take “number” to encompass. It seems perfectly ordinary to me to take it to encompass various senses of the term “infinity”, given that people already so standardly take it to encompass fractions, negations, imaginaries, and the like. Why draw the line here (and then chastise others for not happening to draw it in the same place)?
(I will concede that it’s not often useful to take 4 to equal infinity…)
That’s cause you’re not a statistician.
Especially since we’ve already just decided that ∞ = -1/12 
This. Sort of. When I was a wee infinitesimal, just learning algebra in Junior High, it was standard pedagogy to beat into our heads that infinity is NOT a number!!! This usually came up when some student asserted something like “Infinity is the largest number.”
Problem with that, as I saw it: Nobody teacher ever really discussed was infinity IS, if it’s not a number. (Unless, maybe, some students connected enough dots to understand infinity as a size that a set can be.) I got the sense that a lot of students got the sense that infinity is an invalid concept that we aren’t ever supposed to think about at all, because it’s just total baloney or something.
I’m interpreting this expression as the binary number with 1s at the triangular-number positions (positions 1, 3, 6, 10, 15, …) — do you have a cite or explanation for the statement that this is algebraic? My understanding (which is not very deep) is that there are no known non-normal irrational algebraic numbers (e.g., Wikipedia gives this as a conjecture). This expression also can be written as a Jacobi theta function, and I think these are supposed to be transcendental at algebraic arguments.
I thought statisticians took infinity to be 30…
This statement is irrational.
What definition of number are you using here?
I know someone else has already said that it depends on the definition. I’m just wondering what definition you use. I’m wanting a definition that is consistent with everything else we consider a number, but does not include infinity (or, I assume, negative infinity).
The real numbers.
Ah, so it’s not a real number. It must be a fake number, then. 
There is actually at least one calculation in physics where 3 is taken as being approximately equal to infinity. I can’t remember the details, but there’s a result that holds only in the limit of a large number of particle generations, and it’s used as an approximation to our world of three particle generations. Mind you, physicists don’t particularly like using this approximation, but it’s the only way anyone has yet found of even approximating that problem, and so it’s (reluctantly) accepted.
After that post, I tried to contrive some examples in which it is useful to identify 4 with infinity, in some sense, for some purposes. Only to realize, hell, there are many non-contrived contexts in which even 1 and infinity ought be identified [e.g., when one doesn’t care exactly how many elements a set has, but just whether it is inhabited or not, one may carry out the ordinary cardinality calculation in an arithmetic where 1 = 2 = 3 = … = infinity].
Perhaps all I should have conceded was that it’s not often useful to identify “odd (as in peculiar)” with “odd (as in 1, 3, 5, and perhaps others)”…
314159 shows up 21 times in the first 33 million digits. My phone number shows up, too.