i forgot for a second that the winters and summers get flipped in the southern hemisphere
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
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.
people joined a cult because of this theorem. that must be awkward
it will only be the strongest material in the universe until it gets boiled. trust me on this one
if they invent some new kind of fucked up math to do it then there could be far reaching consequences
"shittitest alchemist currently alive" has got to be one of the most challenging titles to hold onto for any serious length of time
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
being a prompt engineer is so much more than typing words. you also have to sometimes delete the words and then type new ones
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"
they won’t even turn off the ads if you pay them. what a joke
edit: oops i just saw that these are the “free benefits”
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.
what if you just attach a second magnet to the car so that it pulls the first magnet forwards?
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.
once you get enough diseases they just start attacking each other and you end up being healthy again