1. Funnily, that's actually a valid deduction in first order logic.

  2. Yes, they do. Which rights do they not have? There are practically two levels to legalizing homosexuality: decriminalizing homosexual sex, which both countries have done long ago, and legalizing homosexual marriage, which neither has done.

  3. I mean it shouldn't be a valid argument in the first place, this is so young.

  4. Most arguments revolving around the AOC are based on this principle though. You're just using, say, the US as a reference and they're using Japan.

  5. You have an error in your first line. Why are you adding 73 to the second product? In your original attempt, you were correctly multiplying by it.

  6. To answer the second question, I'd say single and multivariate calculus, complex analysis, linear algebra. Those will get you quite far. Abstract algebra (specifically group theory) will get you further.

  7. Don't know much physics. What is group theory good for?

  8. Not to take anything away from Gilbert Strang, but I heavily dislike his teaching style. In my opinion, mathematics can't be properly taught with such a handwavy approach.

  9. Ah. So at most, n2 - 7 is divisible by 2 once - this explains in part why there is no square root of 7 in the 2-adic numbers, right? Since the distance from n2 - 7 to zero is always at least 1/2.

  10. This is an incredibly pretty insight! Does anyone know if it is formalizable? If so, I'd love to see a formalization.

  11. And there's another puzzle along the same lines: suppose any black square and any white square are removed. Can you always cover the board?

  12. I dislike this question. It violates the basic principle of combinatorics, that is: "the answer to every question is 'no'".

  13. I tried to word it as vaguely as possible, but it's impossible to phrase problems without hinting one way or the other!

  14. Oh no, your phrasing wasn't what I was commenting on. It's just a joke on the fact that the expected answer to a combinatorial question is usually "no".

  15. Illegal doesn't mean immoral. If someone managed to scam a multimillion dollar business out of $500, I'd be very happy to see that here. Robbing somebody's parents, though, is being a total dickhead.

  16. Usually any analysis counterexample proof. I'm looking at you: prove or disprove so and so function exists.

  17. In Baby Rudin's second chapter on topology, every exercise which asks you to find a counterexample has the same dumb solution: the set {0, 1, 1/2, 1/3, 1/4...}. It's the ugliest set ever, I guess.

  18. Yeah, what do I think about yet another brother war being started just so that a bunch of elites in Russia and America could get filthy rich.

  19. So it was predictable, you cannot recognize Ukrainians die because russians attacked then and are killing them even civilians hundred km away from the frontline

  20. Unless you are an open nationalist (i.e. right-winger), you shouldn't be offended at the concept of your home country being criticized. It's a very odd phenomenon that Americans are so stupidly nationalistic that they get offended when their government is mocked by foreigners.

  21. Against... Americans? Not really an oppressed class.

  22. The strangest thing about this article for me is him quoting On Proof and Progress in Mathematics by Bill Thurston, whilst trying to learn high school algebra!

  23. Can this sub please stop pretending like the answer to everything is an animated YouTube video? Elementary algebra can be learned and understood through conventional means and at a much deeper level than it could be from watching pretty graphs draw themselves on the screen. If your experience consisted of "plugging (into formulas?)" and nothing else, that's a failure of your teachers, not of the whole idea of learning math by actually doing it.

  24. But something that i realized about complex numbers much later is that a real function staying in it's domain will give you a real number but any function input with a number outside it's domain gives a complex number. We all know about the negative square root, but input a real number greater than 1 for the inverse sin or cos and you'll get a complex number as well.

  25. Functions don't do anything outside of their domains. There is no inherent property of the principal root or the inverse trigonometric functions that makes them output complex numbers for inputs outside of their typical domains. The domains and functional values were simply extended in a convenient way.

  26. I don't think most people have actually tried to solve a trig function outside of it's domain but if you do using the euler equations you will definitely get a complex number. A log function outside of it's domain also gives you a complex number.

  27. I know that, but that's not what I'm talking about. For example, the trigonometric functions were initially defined geometrically (i.e. through triangles) and there is no way to apply that definition to a number outside of the domain of the function to get a complex output.

  28. Most of the time people are told they're assholes while trying to "debate" is when they're trying to "debate" others from bigoted standpoints. If you're going around trying to debate trans people on the validity of their identities and self-expression, you're acting like an asshole.

  29. Raising teachers' wages is not socialist at all. If anything, it is antisocialist, as teachers are one of the main upholders of capitalism.

  30. 15 years difference at that age is quite a bit. Their brain hasn’t even fully formed yet (takes 25 years). It’s not a big deal, plenty couples with that age difference albeit I wouldn’t recommend it with someone under 25, but it’s not unexpected for people to comment.

  31. For a quick demonstration, I counted to 100 using these rules:

  32. d(61) = 6 + 1 = 7. I think you missed that one.

  33. Interesting thought process. You can post images as part of text posts, it's only image posts that are disallowed (that is images without any accompanying text). That probably shouldn't matter to you, though, because just posting a white circle on a black background without any explanation would probably not get you very far in terms of explaining autism.

  34. I think you're right. "A's" in this context is incorrect. Sometimes apostrophes are used when referring to letters like this:

  35. Now I'm not sure if you saw the formatting of my comment lol

  36. Oh. My mobile app didn't show that.

  37. Such a typical American comment. Everything boils down to the standard domestic dichotomy of Democrat vs Republican. If you just vote for Democrats more, everything will fix itself.

  38. That’s not why. It’s IQ and cultural attitudes that cause wealth, education, and also political values.

  39. Disgusting. What a horribly racist attitude. I guarantee to you that people of the Third World aren't any less intelligent than you are, but it's difficult to build skyscrapers and superhighways when all of your nation's wealth was plundered by American imperialists.

  40. Currently the only comment on YouTube (emphasis mine):

  41. I was honestly quite stunned when Rudin managed to use MVT on almost every proof for two chapters in a row. I had never thought of it as being so consequential before.

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