it is possible to rigorously say that 1/0 = ∞. this is commonly occurs in complex analysis when you look at things as being defined on the Riemann sphere instead of the complex plane. thinking of things as taking place on a sphere also helps to avoid the “positive”/“negative” problem: as |x| shrinks, 1 / |x| increases, so you eventually reach the top of the sphere, which is the point at infinity.
from a topological perspective, wraps and tacos are two different beasts.
in a wrap, the bread completely surrounds (and encloses) the other ingredients, so theres a 2-dimensional hole involved (which basically means the inside is hollow).
in a taco, no such wholes are present.
you can also distinguish sandwiches from tacos and wraps (since sandwiches involve two pieces of bread, like you said). but unfortunately, you can’t topologically distinguish a burger from a sandwich
you can always add an empty room without changing the total number of rooms, so there should be plenty of room for sisyphus and his boulder at the hotel
i think this is a fairly reasonable gut reaction to first hearing about the “unnatural” numbers, especially considering the ways they’re (typically) presented at first. it seems like kids tend to be introduced to the negative numbers by people saying things like “hey we can talk about numbers that are less 0, heres how you do arithmetic on them, be sure to remember all these rules”. and when presented like that, it just seems like a bunch of new arbitrary rules that need to be memorized, for seemingly no reason.
i think there would be a lot less resistance if it was explained in a more narrative way that explained why the new numbers are useful and worth learning about. e.g.,
negative numbers were invented to make it possible to subtract any two whole numbers (so that it’s possible to consistently undo addition).
rational numbers were invented to make it possible to divide any two whole numbers (so that it’s possible to consistently undo multiplication, with 0 being a weird edge-case).
real numbers were invented to facilitate handling geometrical problems (hypotenuse of a triangle, and π for dealing with circles), and to facilitate the study of calculus (i.e. so that you can take supremums, limits, etc)
complex numbers were invented to make it possible to consistently solve polynomial equations (fundamental theorem of algebra), and to better handle rotations in 2d space (stuff like Euler’s formula)
i think the approach above makes the addition of these new types of numbers seem a lot more reasonable, because it justifies the creation of all the various types of numbers by basically saying “there weren’t enough numbers in the last number system we were using, and that made it a lot harder to do certain things”
the standard (set theoretic) construction of the natural numbers starts with 0 (the empty set) and then builds up the other numbers from there. so to me it seems “natural” to include it in the set of natural numbers.
it’s mathematically provable that the shortest path between any two points on a sphere will be given by a so-called “great circle”. (a great circle is basically something like the equator: one of the biggest (greatest) circles that you can draw on the surface of a sphere.) i think this is pretty unintuitive, especially because this sort of non-euclidean geometry doesn’t really come up very frequently in day to day life. but one way to think about this that on the sphere, “great circles” are the analogues of straight lines, although you’d need a bit more mathematical machinery to make that more precise.
although in practice, some airlines might choose flight paths that aren’t great circles because of various real world factors, like wind patterns and temperature changes, etc.
do people actually fall for this stuff? it seems like so many business management types are working overtime to make sure they’re always up to date with the latest in corporate jargon. why? do people actually think these people are saying anything?
For me the first thing that comes to mind is Tales from Earthsea. I don’t think it’s excellent or anything and has plenty of problems but people act like it killed their dog. While it has its problems that have been covered extensively, I think it has a beautiful atmosphere and art....
watching oppenheimer felt like watching a 3 hour compilation of trailers for oppenheimer. everything was so dramatic, there was no downtime, and they always had some kind of music playing. it felt like every scene wasn’t allowed to last more than 5 minutes
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