Yakk
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 Apr 25 comment Is there a way to write an infinite set that contains only irrational numbers without integer multiples? @NoamD.Elkies More importantly, would the integer generated by a power tower of $e$ be named after the discoverer, or called "the squealing integer"? Apr 13 comment Different arrows in set theory: $\rightarrow$ and $\mapsto$ $f:A\to B$ defines the type of the variable $f$, the kind of thing it is. $f:x\mapsto y$ defines the value of the variable $f$. You can often reason about things purely from their types, without having to think about the details of where specific values go. Mar 28 comment What books should I get to self study beyond Calculus for someone about to start undergrad mathematics? @schmidt73 Suppose you took a bunch of top 0.1% high school students in math, stuck them in a classroom at 3 hours/week, and gave them a weekly assignment of 4-10 questions from Spivak. It taking 2-5 hours to complete wouldn't be unreasonable, and that is after classroom exposure to how to prove things. 5 minutes means the problem was trivial. The reason why doing the exercises is important is you need to get good at doing proofs and solving harder problems; merely being able to solve a single hard problem isn't enough, because even harder ones are in the next chapter. Mar 14 comment Is linear algebra laying the foundation for something important? In mathematics, we divide problems into linear problems and non-linear problems. This is like dividing the universe into bananas and not-bananas. Mar 11 comment In simple English, what does it mean to be transcendental? @KRyan Well, if we describe the game that determines if a number is algebraic as "algebra" (add integers, divide integers, multiply by x), then an algebraic function is just a function whose action is some set of algebra. Gussy that up to make action and function make sense? Mar 8 revised Explain “homotopy” to me added 2 characters in body Feb 24 comment Are proofs by contradiction really logical? Everything for which we know one of A and not A to be true, only one of them is true. I am unaware of an observation of the "real world" that actually shows that always A or not A is true: that presumes there is nothing inherently unknowable about the real world, and/or that the real world exists in ways we cannot know about. Are there sub-plank gremlins who move particles following our physical laws and use QM as a joke on us? That is unknowable: stating it is either true or false is ridiculous. It has no impact on any possible observations we can make. Feb 17 comment What's the point in being a “skeptical” learner Don't you get faster at checking details the second time through, even if you "forgot" it the firs ttime? By checking and redoing the technique, you go from "sure, that seems possible" to "I am really certain I can confirm that, because I've done something almost exactly like it a few dozen times before". Feb 17 comment What's the point in being a “skeptical” learner @SimpleArt It is possible to separate a vote into two votes, or two votes into three votes, or in general some number of votes into a larger number of votes. I have discovered a truly marvellous method to do this, which this comment field does not provide enough characters to let me type it in. Feb 11 answered Explain “homotopy” to me Oct 2 comment Each of the two persons makes a single throw with a pair of unbiased dice.What is the probability that the throws are equal? Are the dice the same color? In short, what is your equals test -- add up the values? Oct 1 comment Sequence converging to different limits @Pakk It is just the usual metric on $\mathbb{R}$, with 0 and $\pi$ swapped. Sep 24 revised What are “instantaneous” rates of change, really? added 13 characters in body Sep 21 comment How many fingers do martians have? Easy. Martians have 10 fingers (in their base, naturally). Sep 16 answered How do I make a student understand contradiction? Sep 16 comment How do I make a student understand contradiction? Except when the proof is fininshed, we have established the falsity of part of the thing we assumed. This does not make the proof by contradiction invalid. You can do a proof by contradiction starting with "assume 0=1", even if you have already established "not 0=1", and it remains valid. It is useless, but valid. Sep 9 answered I roll a die repeatedly until I get 6, and then count the number of 3s I got. What's my expected number of 3s? Sep 8 comment Why are all knots trivial in 4D? Bah, tricky. Can we really have an uncountable number of crossings, and not just infinite? Continuity and compactness should buy us that, no? (been a while) I think I see an attack for a countable number of points with an infinite number of crossings within all neighborhoods. Slide them apart, and then limit as epsilon->0 untangle them. Sep 8 comment Why are all knots trivial in 4D? For "Alternatively", could there be an uncountable number of crossings, and would that block that solution? Jul 27 answered The set of all real functions of a real variable