Aleph ω+4

@rzeta0@mathstodon.xyz just as the order of formal quantifiers creates a dependence of the later quantified variables on the earlier ones, so is there a dependence in the natural language... it wouldn't make sense for a variable to depend on one that hasn't yet been introduced imo. One informal mathematical habit I don't like is something like "there exists such that x^2=y for all y" - again putting the introduction of a variable after its use. It's ok for simple sentences but ambiguous for more complicated.

@rzeta0@mathstodon.xyz definitely the first one!

@rtn yeah, I mean I'm assuming you wanted one book for convenience and cost; there's no need to worry with textbooks available for free (and if you read them on a computer or tablet). Old high school texts just went surprisingly far (to my modern eyes) into some topics, and were surprisingly broad. I don't count trigonometry or geometry to be part of algebra but it appears in such books, for example. I guess I'd call it "school pure maths".

@rtn echoing @AdrianRiskin as far as university level texts go. In times past you would get some school level textbooks that went quite far in many areas (and they'd be labelled "algebra") but they wouldn't teach you any *abstract* algebra like group theory, no proper number theory, etc. The following has some very good textbooks. I suspect the copyright situation might be dubious and it may get pulled, but who knows. https://github.com/enilsen16/The-Math-Group/

@ottaross a tasty casserole

@ottaross for FUCKS sake

@shia with a strong induction stove you're allowed to assume you can cook n meals for all n ≤ k

@LiseAndreasen@mastodon.world then there are some more digits which may also differ. But at each n+i, there are 4 possibilities (with equal probability): 0, 0; 0, 2; 2, 0; 2, 2. The identical pairs add nothing to the overall difference, and pair 0,2 adds 2/3^(n+i), whilst the pair 2,0 subtracts the same amount. Thus everything beyond n on average contributes 0 difference. So the probability is the sum of 1/[2^(n-1) * 3^n], which is a geometric series with limit 2/5 = 0.4 :)

@LiseAndreasen@mastodon.world let a and b be the two random numbers on the Cantor set and assume without loss of generality that a <= b. Divide into cases according to which ternary digit is the first to differ. In 1/2^n cases, that digit is the nth (there is a 1/2 chance that the first pair is the same, 1/2 that the second is, and so on, and finally we have a 1/2 chance that the nth pair differs). In that case, the difference truncated at the nth digit is 2/3^n,

@NanoRaptor@bitbang.social and the B paper sizes are the geometric mean in-between A sizes!