Hearted Youtube comments on Mathologer (@Mathologer) channel.

  1. Lets see if I can finish this one before the end of 2021 =) Thanks Burkard
    14
  2. I liked the motivation and the path it took us down, but now I want to know if there are more of these anti-shapeshifters, besides the class we explored!
    14
  3. Such a lovely insight! This explains the error term in any base, but certainly easier to notice in base 10. I wonder if Madhava also considered series for Pi based on the series expansion for arctan(1/sqrt(3)) = Pi/6, which converges faster. It is hard to imagine how they discovered these results without using our present day notation.
    14
  4. The most memorable for me has to be Kempner's proof, just due to how counterintuitive it is after seeing so many divergent series, but how intuitive the proof is.
    14
  5. Since 2020/4 = 505 which has prime factors 5 and 101, we just have the power set of {5,101} which is {1,5,101,505}. Since both 5 and 101 are congruent to 1 mod 4, they are both good and so are all their products. You definitely gave us an easy one. So 4 good - 0 bad gives us 16 ways of writing 2020 as the product of squares. To find them, we'll first use the usual trick of dividing out the 4's. We'll find the ways to write 505 as squares and multiply each component by the square root of 4. True confession time: I'm adapting this from work by Dario Alejandro Alpern, whose fsquares program I ported to Gnu GMP. We can find the ways to write 505 by finding the way to write its factors and using the fact that (a^2+b^2) (A^2+B^2) = (aA+bB)^2 + (aB-bA)^2 We'll find the solutions in positive integers, and then convert each such solution (a,b) into 4 solutions, {(a,b)(-a,b)(a,-b)(-a,-b)}. We know that every prime congruent to 1 mod 4 is the sum of two squares. For 5 this is easy: 5 = 1^1 + 2^2 and that's all. For 101 it's not hard either: 101=1^2+10^2, and this confirms that you are pitching us your softest softball. A quick check vs {49, 64, 81} confirms that this is the only way to write 101. Again, this is just the positive/positive solutions. So we have: (1^2+2^2)(1^2+10^2) = (1+20)^2 + (10-2)^2, which gives us: 505 = 21^2 + 8^2 2020 = 2*21^2 + 2*8^2 = 4*441 + 4*64 = 1764 + 256 and consequently 3 other solutions, 2020 = (-42)^2 + (16)^2 = (42)^2+(-16)^2 = (-42)^2+(-16)^2. We also have: (1^2+(-2)^2)(1^2+10^2) = (1-20)^2 + (10+2)^2 = (-19)^2 + 12^2 = 361 + 144 Thus we find 8 ways of writing 2020: 2020 = 42^2+16^2 = -42^2+16^2 = 42^2+-16^2 = -42^2+-16^2 = 19^2+12^2 = -19^2+12^2 = 19^2+-12^2 = -19^2+-12^2 According to the formula, there must be 8 other solutions out there, but I'm not seeing the permutation of these equations that gives them.
    14
  6. I learned Heron's formula when I was in middle school in the state of California (4 years ago) and I'm pretty sure it's still being taught, but I've never used it outside of math competitions. Lots of great information in the video that I didn't know before though. Thanks!
    14
  7. I have a habit every time you release a video, of taking a cold bath, then I turn my air conditioner on to remain cold to make the video more fun, and then I watch the video and try to do every challenge in it, just to make the best out of this video.
    14
  8. 4k+1, 4k-1
    14
  9. Saturday morning fun with Mathologer! What a pleasant surprise! šŸ˜Ž
    14
  10. Fermat invented it first, Newton's fluxions make the most physical sense. Leibnitz's notation is way the best.
    14
  11. A bit late on the lower bound for gamma, but... You can take every blue piece, place it into a rectangle of dimensions 1 x 1/(2^n), and split that rectangle in half with a diagonal from the top left to the bottom right. If you were to take the upper triangle from every one of these divided rectangles, you would get an area of one half of the square. Because every piece has a convex curve, it will stick slightly outside of the upper half of its rectangle. This means that every piece has an area greater than half of the rectangle, and the sum of all the pieces is greater than one half of the square. Because the square is 1x1, the area of the blue pieces is greater than 1/2.
    14
  12. Same here. Am a teacher myself, but not of this level - not by a long shot. Burkhard and his team are something else šŸ˜Š
    14
  13. Great stuff! I did find myself looking at the zig-zag, and thinking - ugh, that's the one that gradually de-centres the lace each time you tighten it until it's moved so far out of centre you have to take the time tease it back into position. It can also easily look wonky, as you can pull it out of shape pulling on the long diagonal. Enter what you called the French lacing, which I use on boots, Doc Marten's, that kind of thing - horizontals are always on top, diagonals always underneath; you just see the horizontal pieces neatly perpendicular to the seam since the diagonals are hidden, but at the same time solving both the de-centring problem and the skew problem. 8-)
    14
  14. Archimedes' bunch of bananas šŸ˜€
    14
  15. 8:58- I think it is 292 (one less the result you gave us due to the fact we exclude the possibility of using the 100 cent coin).
    13
  16. This is perfect! I have a phd in applied mathematics, and still have to stop and ponder quite a bit during the last few parts. Also I would love more videos on group theory/Galois theory :)
    13
  17. I love playing with differences and sums of sequences! (I had taken to calling them "discrete derivatives" but that might be sort of a backwards name.) So cool to see a proper mathematician talk about them!! You went much deeper than I did, but I did use this on 2^x and x^2 to see how it worked similar to a derivative, and on the Fibonacci sequence to note that it is exponential in nature :) That the number of rows is the same as the degree of the polynomial feels so nice to me
    13
  18. Great video! I recently had a project related to hyperbolic trigonometry and it was really nice to see a completely different approach.
    13
  19. The {7/2} animation looks like a rotating pentagonal prism to me.
    13
  20. My goodness! These people were so smart. This is crazy. Wonder is essential to beauty, and I am feeling both wonder and beauty right now. It is a travesty that I was unaware of Madhava's existence.
    13
  21. Love your stuff and how much fun you seem to have presenting it....I get lost pretty easily because I'm old but enjoy the journey....anxious to see what you have up your towel next
    13
  22. That was sum homework assignment.
    13
  23. I was looking forward to this whole december :). Merry Christmas from Czechia!
    13
  24. 48:47 my feedback: i'm a undergrad engineer, and loved the video. The explanation was very good, and very gradual. please make more :)
    13
  25. Interesting video! I ran into vortex maths videos some years ago. I drew the number circle in photoshop and played around with it awhile, trying to understand what I was doing. I'm not great at maths so in the end I didn't properly understand what you taught in this video, but I did come to the conclusion back then that the vortex diagram was just an emergent feature of our arbitrary numbering system and you could make any number of them and make them seem mystical by highlighting patterns and features. It does make pretty patterns. Was Tesla really fixated to 3,6,9? What I was left wondering then and now is what kind of number theories Tesla was really thinking about! I don't think I've seen... objective video on the topic, and he must've had better stuff in his genius brain than what the youtube videos on vortex math are showing! This is a topic for a different channel, but did Tesla really believe that 3,6 and 9 were the key to the universe, and if so, surely not because of the "vortex diagram"? He's a smart guy so he must've understood that our numbering system is just one of many. What do we actually know about his fixation with numbers? Is the whole vortex math thing even in anyway linked to Tesla, other than by coincidence through youtube conspiracy mythos? Anyone know good sources that try to understand how Tesla thought by referencing his notes and such?
    13
  26. I absolutely agree with you, it is a crime that schools don't introduce proofs. Even the most simplistic proofs, like the ones you showed. The only time they are put into my curriculum is if you take the hardest math course, which only 5% of the cohort takes. 95% of students aren't even introduced to proofs.
    13
  27. It's a good thing King Solomon didn't follow through with his apportioning of the contested baby itself between the two candidate mothers, and went for a conservative "winner takes all" approach šŸ¤£
    13
  28. Ā @MathologerĀ  Don't erase everything on it, or else it will become a black hole, and suck you in !
    13
  29. I feel like a kid who just found a $100 bill on the roadside. He will now burn his weekend to think about all the toys he can get ;)
    13
  30. The closed formula for the mortal enemy's sequence is: E_n = 0.8469 (1.5747)^n + 0.2636 (1.3802)^n cos(1.7805n + 0.9507). All the decimals are rounded so this won't be very accurate past n = 15 or so, but we can use this to get the asymptotic formula for E_n. Since 1.3802^n << 1.5747^n for large n, we can drop the second term and say E_n ~ 0.8469 (1.5747)^n. So for large enough n, each term in the mortal enemy's sequence will be 57.47% larger than the previous term.
    13
  31. I remember using this "unusual" formula around 1995 in a math olympiad (16 years old at the time). I picked it up in a textbook, and managed to memorise it to use it "blindly"
    13
  32. Iris-free Proof: join the six exterior points consecutively, calling them A' A'' B' B'' C' and C''. Note the numerous isosceles triangles in this diagram and mark the relevant congruent pairs of base angles. Next, observe that A'A''C'C'' is a quadrilateral with supplementary opposite angles. Thus it is a cyclic quadrilateral and A'A''C'C'' (circle 1) are concyclic. The same reasoning shows the sets of points A''B'B''C' (circle 2) and B'B''C'C'' (circle 3) are concyclic. Circle 2 and 3 have three points in common, and so A''B'B''C'C'' are all concyclic, since three non-collinear points determines a circle. This circle has three points in common with circle 1, and so A'A''B'B''C'C'' are all concyclic.
    13
  33. @praveenb9048 You have it backwards. Even today the definition of the parabola is the conic section you get when you cut the cone parallel to a side. A defining property of the parabola is that all lines parallel to the axis of symmetry of the parabola cross at a certain point when they're reflected on the parabola. Ancient Greeks knew this too (as it's a purely geometric property). The fact that second degree polynomials' graph is a parabola is not a definition. You have to prove it using the defining property above.
    13
  34. Torricelli's argument about the volume of the horn equaling the volume of lots of circular discs reminds me of Archimedes' argument about the area under the parabola, except his argument also used mechanics like moment of inertia.
    13
  35. When he put the stack leaning over Oresme I lost it.
    13
  36. There's a simple process to get the first step answer without having to count all of the boxes. The pattern is made up of 4Ɨ4 rows of boxes that equal 16 boxes in total. Divide 16 in half to get 8. Divide 8 in half to get 4. Divide 4 in half to get 2. 16+8+4+2=30
    13
  37. I like how the words "unit" and "Fourier" are interchangeable. Just like "Lie" and "differentiable".
    13
  38. Wonderful proof. The incentre is slowly becoming my favourite triangle centre! (overtaking the orthocentre as my previous favourite)
    13
  39. ā€‹Ā @royalninja2823Ā he says that when he was younger, he used to get upset that people only know his game of life, because it was a quick and draft invention. But I also only know his game of life.
    13
  40. Pre pandemic I used coins everyday, I can't wait for the day the markets reopen and I can use them again!
    13
  41. In Cā†‘Dā†“, the reason it only mixes up the rows and not the columns is because both Cā†‘ and Dā†“ use the same vertical sequence (i.e., top, middle, button, top middle, bottom...). Only the horizontal sequence changes.
    13
  42. Sum, product, exponent, factorial... how does this pattern keep growing? Is the next step a factorial exponent? Where does product summation fall within this pattern? What do we do with the levels that haven't been named, let alone conceptualized yet? What can we do with them once we've figured that out?
    13
  43. You did WHAT??? You scrapped and reuploaded your whole episode to fix an error? You got my enormous respect, sir! I wish every YouTuber was as diligent about their content accuracy as you are. Meryy Christmas and all the best wishes for the New Year! Gregorian or Julian doesn't matter ā€“ your uploads will definitely be on my calendar.
    13
  44. I'd love to see this channel growing big. All the immesureasurable effort put into your videos is just awesome. Merry Christmas :)
    13
  45. Glad you remembered password after 3 months
    13
  46. I never learned this at school or university, but now I'm in love! It seems like a way to teach/reach much richness of calculus without the (potentially) intimidating infinite limits and infinitesimals. I mean, we want to learn those too, but I remember it was a lot to adjust to at once, and it made calculus seem magical for a while, rather than logical.
    13
  47. Ā @MathologerĀ  It was a long time ago. Tempus fugit! I think learning a little Latin has a long lasting effect of helping us understand the syntax, grammar, and vocabulary of other languages, including our own.
    13
  48. For chapter 6, I think we're still missing a small but important detail that the collections still have to be nonempty when we remove the overlap. Since they add to the same number and are not the same, neither can be a proper subset of the other, which proves that they will not be empty when we remove overlap.
    13
  49. At 0:18:33 it should be ...N+9/(4N), and at 0:21:16 the two "4n+" should be swapped with the two "n+" (also: I would replace the "n"s with "m"s in this yellow box, since they don't need to be equal to the "n"s above). However: this video made my day; I've been waiting for the new Mathologer video for 2 months! Great work Burkard, most of your videos are in my "Favorites" list, and Mathologer is definitely my favorite channel of all. šŸ¤—šŸ»
    13
  50. 3:24 oof... Again with the ai art? You definitely received considerable backlash last time, and i dont really see the point if doing it again! It also just looks really wonky! im sure it cant be difficult to put clip art of faces onto clip art if vessels. Great video otherwise as usual though :)
    13