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 Dec28 comment What is the practical difference between a differential and a derivative? In the version of that joke I originally heard, the second "problem" had the faucet already running and making a mess, which the mathematician reduces to a solved problem by (of course) setting the trash on fire. Dec28 comment A new imaginary number? $x^c = -x$ Hm. Seems to me you only lose that rule about exponents half the time--if you preserve parity of the number of c exponents it should work, I think? Note that introducing an extra factor of 1^c would always suffice. You still end up with x^2c == x^2, of course, which seems peculiar. It's sort of a hypernegative unit, that constructs the inverse of two hyperoperations down, like -1 does one hyperoperation down. Unfortunately, multiplying by c doesn't give you the predecessor operation... Dec12 answered Evaluate expressions in lambda calculus Dec5 comment How can I introduce complex numbers to precalculus students? @AlistairBuxton: Consider which of the following i represents: Half of a reflection along the real line, an axis in 2D space, or a direction of rotation in 2D space? Now, is there any "trick" that will similarly represent half of a complex conjugation, an axis in 3D space, and a direction of rotation in 3D space (recall that you need a scalar component here to get a magnitude of 1!)? Nov30 comment Notation for repeated application of function Oh, nice! Unconventional, but unambiguous and nearly self-explanatory to anyone familiar with ∘ and the standard superscript notation. Nov30 comment how to prove $437\,$ divides $18!+1$? (NBHM 2012) @TimPietzcker: Also, the significance in this context is that if a and b are relatively prime, and both divide c, then their product also divides c for hopefully obvious reasons. Nov16 awarded Commentator Nov16 comment Why do some people state that 'Zero is not a number'? Yeah, that's part of the distinction I'm trying to make--the expression 0/0 has no defined value, i.e., the division operation is not defined on those arguments in the same way that division by "cucumber" is not defined. There is nothing denoted by 0/0 to be a number or not. Nov16 comment Why do some people state that 'Zero is not a number'? Strictly speaking "NaN" is distinct from "is not defined". NaN comes from the specification for floating point values, which computers generally use as an approximation of the real numbers. NaN is a well defined value (actually several, as there are distinct flavors of NaN) with specific properties (such as being unequal to all floating point values, including itself). Nov15 comment Isn't every Endofunctor an identity Functor? @drozzy: I was talking about the constant mapping example, but the same applies to both. Perhaps a more straightforward example would be an ordering on the natural numbers, and a functor that maps each number to its successor? Nov15 comment Isn't every Endofunctor an identity Functor? @drozzy: Why would it? The other objects are still there, even if the functor isn't mapping anything to them. Nov14 comment Why is the set of all real numbers uncountable? ...are you sure you understand Cantor's diagonal argument? Aug19 comment Why does this distribution of polynomial roots resemble a collection of affine IFS fractals? @Rahul Narain: Well, I've been meaning to do a similar tidy-up on a different image rendering program, which the root plotting could then be dropped into easily, but it wouldn't be an immediate priority otherwise. If I do get around to it, I'll ping you with a comment here. Aug19 comment Why does this distribution of polynomial roots resemble a collection of affine IFS fractals? @Rahul Narain: I'd be happy to share but, 1) I don't recall where the code is right now, and 2) recompilation is not the most user-friendly approach to accepting input parameters. I could probably find and/or reimplement it tonight with modifications to be slightly less inconvenient to use, if you'd like. Aug19 comment Why does this distribution of polynomial roots resemble a collection of affine IFS fractals? @Rahul Narain: It was about a dozen lines of Haskell, which simply enumerated the possible coefficients based on whatever generating scheme, passed each one at a time to a matrix calculation library that had a root finder, plotted the results on an in-memory bitmap, then saved the result as a .PNG. The plots above, with polynomials of degree 18 or so, took only a few minutes to create. Changing the polynomial parameters was done by edit-and-recompile. Jul30 awarded Nice Answer Jul28 awarded Yearling Jun29 answered Why does this distribution of polynomial roots resemble a collection of affine IFS fractals? Jun21 comment Alternative to imaginary numbers? It really is true that Clifford algebra reveals the geometric structure better, even moreso at higher dimensions--for instance, the even sub-algebra in Cl_3 gives the quaternion representation for rotations in three dimensions in a very intuitive way. On the other hand, representing N-dimensional geometry requires a 2^N-dimensional Clifford algebra, so projective 3D coordinates, common in computer graphics, would require 16 coefficients, rather than the standard 4. Jul29 awarded Teacher