Hearted Youtube comments on Mathologer (@Mathologer) channel.

  1. Another video of Mathologising beauty. The 4D cube rotating in space was a delight.
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  2. for the 1/x is a anti-shapeshifter if you squish the graph by 2 along the y axis, then each value of y gets mapped to y/2, while values of x stay where they are. the function that describes this graph is f(x)=(1/x)/2 = 1/(2x) now let's stretch this graph by 2 along the x axis each value of x gets mapped to 2x while y doesn't change so f(2x) = 1/(2x) and so f(x)=1/x the function of this graph is 1/x; same as before
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  3. 8:24 The magic constant is sum of all numbers in the square divided by the number of rows. For those looking for a concise formula: it should be (n²(n²+1)/2)/n or simply n(n²+1)/2. Plugging in 33 will give us the magic constant of 17985. 18:48 Go up-right one space. If you exit the board, wrap around to the other side. If you run into a space already filled in, drop straight down one space instead.
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  4. Only 20 years too late! This is the calculus lesson I wish I had while I was an engineering student. Very well done!
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  5. love to see a video about Sophie Germain's work, she doesn't get nearly enough attention
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  6. Thanks! Excellent video as always. So interesting to learn that this type of math was being done in the 1400s.
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  7. Mathematicians doesn't believe in "coincidence", these are true mathematical Facts. 😌 3³+4³+5³=6³ but not possible for further case. Pls continue doing such 10-15 min videos also, Sir. 🙏🏼 Long vid are itself goldmine. ❣️❣️❣️
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  8. 15:19 Am I the only one who thinks it's funny how he dodges the triangle? By the way, it was a great video, very interesting!
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  9. i am but a simple man i see video about Ramanujan I click
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  10. The best thing about Reve's puzzle; "We're going to have some cheese."
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  11. I've been a fan ever since your humble beginning, and I'm more excited than ever for new Mathologer vids!
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  12. As long as you don't start to follow the YouTube "shorts" trend everything is fine;)
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  13. I thought I'd know all of this already, but was pleasantly surprised when you put the circle and hyperbola together and at the geometric proof of the e^α identity. Great video!
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  14. Your videos are really calming to the mind. Pleasant music during algebra autopilot and then fascinating math explained in a natural way!
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  15. 9:23 I'm a high-school maths teacher. I have taught that dumbed-down divisibility-by-9 test without going into digital-roots and the remainder property. I shall never do it again. Props to your deliciously made videos.
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  16. 3:20 there is a tiny mistake: after the four non coincident digits you have in one case10582 and in the other10581. The "2" and the "1" should be coloured as different digits. Amazing video, many thanks.
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  17. That sounds like it would be related to the amount of ways you can permute an infinite list, which sounds uncountable to me.
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  18. I would rather have expected the Spanish Inquisition than the 1800-year-old Talmudic bankruptcy problem. Professor Polster never ceases to amaze us.
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  19. "What comes next?" Me: Almighty Lagrange's interpolation
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  20. Clearly, the highlight of the Euler-Mascheroni constant is a splendid part of the video...the sum of no 9's animation is very impressive.
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  21. 16:00 I found the area of the tree to be 5, since we have a depth of 5 and for every iteration, the new squares sum up to 1, thus a a tree with infinite iterations has an infinite surface area (still counting overlapping surfaces)
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  22. as a teacher, I'd avoid using Blue because of how easy it is to confuse with side B of the triangle.
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  23. I feel like this sort of "discrete calculus" was something I was vaguely aware might exist, but I've never seen it formalised like this before.
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  24. I'm confused, what if I take a 5d hypercube, if I pick a random point won't the 5 points connected to it form a regular pentagon? Edit: Aha, these points are not even on the same plane.
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  25. Mathologer Nice “save” from a small mistake in the video.
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  26. You are damn right
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  27. So for that first puzzle... I think my mind just got blown I started with 5 since I was basing on the fact that the first equation of both patterns end and start with 3 (1 + 2 = 3 and 3² + 4² = 5²), so why not do the same for the cube pattern. What I ended up is that 5³ + 6³ = 341, which is almost close to 7³ = 343. I was curious enough, so I tried to expand this by utilizing the same trick from the previous patterns and used 6*(1 + 2) for the next pattern and got 16³ + 17³ + 18³ ≈ 19³ + 20³ (with the difference being 18) Expanding this pattern leaves me with a list of differences and... I don't know about you, but (2, 18, 72, 200, etc.) just screams "I have a pattern"... and it does! Each difference is just twice a triangular number *squared*, and knowing that just blew my goddamn mind... because THAT'S LITERALLY HOW I STARTED working on this pattern. For each sum that uses 6(1 + 2 + ... + n) cubed as a starting point, there is a difference of 2(1 + 2 + ... + n)². How cool is that!? In honor to document this amazing pattern, here's my christmas tree for the cube pattern, with the difference added in. You can think of them like they're ornaments or something xD (the formatting might only work with monitors) 5³ + 6³ = 7³ - 2 16³ + 17³ + 18³ = 19³ + 20³ - 18 33³ + 34³ + 35³ + 36³ = 37³ + 38³ + 39³ - 72 56³ + 57³ + 58³ + 59³ + 60³ = 61³ + 62³ + 63³ + 64³ - 200 85³ + 86³ + 87³ + 88³ + 89³ + 90³ = 91³ + 92³ + 93³ + 94³ + 95³ - 450 ... and here's another one with the sum expanded, just to see the beauty :D 5³ + [6(1)]³ = 7³ - 2(1)² 16³ + 17³ + [6(1 + 2)]³ = 19³ + 20³ - 2(1 + 2)² 33³ + 34³ + 35³ + [6(1 + 2 + 3)]³ = 37³ + 38³ + 39³ - 2(1 + 2 + 3)² 56³ + 57³ + 58³ + 59³ + [6(1 + 2 + 3 + 4)]³ = 61³ + 62³ + 63³ + 64³ - 2(1 + 2 + 3 + 4)² 85³ + 86³ + 87³ + 88³ + 89³ + [6(1 + 2 + 3 + 4 + 5)]³ = 91³ + 92³ + 93³ + 94³ + 95³ - 2(1 + 2 + 3 + 4 + 5)² ...
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  28. behold: new Amazon Prime service translating in 4k+1 resolution :)
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  29. "I actually never gave much thought to bankruptcy problems before this video, nor did I question the idea that proportional distribution is the best solution. But now, I actually see the Talmudic approach as superior in many real life situations." As an Orthodox Jew who has studied Talmud for years, both inside and outside of a yeshiva setting, and as someone obsessed with mathematics, I found this to be an incredibly touching statement. And, I'm incredibly flattered that you chose to make a video on this subject, and thrilled that you did it such justice and improved my understanding of my own religion! You rock, Mathologer!! ❤
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  30. 15:32, it does have a turning property but you must rotate around a cardinal axis. This does mean it is useless for finding other grid points though.
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  31. "Is the no 9 series finite? You life depends on this!" Me: suspicious, has to be finite! "Believe it or not, it is finite!" Me: YAY
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  32. It wouldn't be discrete if it was working in the foreground.
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  33. 11:54 “It’s always 2 pluses, followed by 2 minuses, followed by 2 pluses and so on” How do you just expect me to know where this sequence is going? Hahaha
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  34. Proof of the length of the wheel being ln(10) : When the number x reaches the wheel, the rubber band has stretched by a factor 1/x (we consider that the rubber band is numbered from 0 to 1 like in the video). If you now wind the band just a bit more so that you reach x+dx (with dx infinitesimal), then the length added on the wheel is dl=a(x)dx. So the total length of the wheel is the integral of dl for x=0.1 to 1 that is int(dx/x, x=0.1..1), which is equal to ln(10).
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  35. The off-by-one counter-intuitive logic reminded me of (and is clearly related to) how there are only 11 places on the face of a 12-hour clock where the hands point the same direction. One of my favourite weird facts is that if either one of the two hands on that clock went backwards (what a strange clock that would be!) the hands would point in the same direction in 13 places instead.
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  36.  @Mathologer  Rest assured that your efforts are greatly appreciated. This is one of the (if not the) best mathematics channels on youtube and the reason is that people recognize quality and hard work when they see it. Happy π-day!
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  37. This has to be a record - two minutes in and I'm already pausing the video to pick exploded bits of my mind out of the carpet. How did I never learn this before? In school, Heron's formula was presented as a side note, and I never really understood the derivation (meaning I'd have to look it up again any time I wanted to use it). Given a little bit of reflection it all seems crystal clear, now. Nice!
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  38. So the "2" in "x+2" came from the fact that an edge/vertex is defined by two points and a line/edge between them. Is there a mathematical entity where you would use "x+3" or any arbitrary "x+N"?
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  39. 1 - trivial 1 1 - still trivial 1 2 1 - the middle ones have two as their greatest common factor 1 3 3 1 - GCF = 3 1 4 6 4 1 - GCF = 2, NOT 4 1 5 10 10 5 1 - GCF = 5 1 6 15 20 15 6 1 - GCF = 1 1 7 21 35 35 21 7 1 - GCF = 7 1 8 28 56 70 56 28 8 1 - GCF = 2 So, when P is prime, doing Pascal mod P makes the triangles of size P^k + 1 special because the GCF of the middle numbers in the P'th row is P itself
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  40. Most memorable: the fact that gamma is the Ramanujan summation of the harmonic series.
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  41. My only regret is that I can never watch 19:57-23:03 for the first time again.
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  42. Euler's formula for polyhedra can easily reach #1 if you realise it's actually d0-d1+d2-d3+d4...dn=1 where di is the number of i-dimensional objects that form an n-dimensional polyhedron
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  43. After checking a feel cases, I think that to get the tower from A to B, you start by moving the smallest disc to: • C, if there are 2n discs; • B, if there are 2n+1 discs.
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  44. Your video is a treasure. I hope it is incorporated in early STEM learning. My experience watching, caused me dynamic insights in three separate domains. In my opinion, Mathologer is a "miracle" math educator.
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  45. I watched this video 5 times and I have to watch 10 times more in order to really get it, but I just wanted to tell you how much I love your videos and the way you teach. Your are exceptional!
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  46. In India we have a whole chapter by the name Herons Formula . After that I decided to derive my very own formula for finding area of a triangle, but I accidentally deriverd Herons Formula. I was so Happy at that time. Those were the days. 🙂🙂
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  47. Doing seperation of variables in Diff Eqs was the first time it really hit me how nice Leibniz notation is. It's really just the chain rule, but still, super nice.
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  48. One of Mathologer best videos. The great graphic design, The clear explanation, The general idea, all are simply perfect!
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  49. Didn't learn about Ptolemy's theorem until my late 20's, well after having learned calculus and other pretty sophisticated algebraic properties up through grade school and all. Should have been the first thing I learned.
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  50. That example at the end, where you are owed 1 dollar and the bank is owed a million, where in a "logical" proportional system you would get a millionth of a dollar (ie. nothing) and the bank gets everything, while in this system you get your dollar and the bank gets the rest, really clarified why this actually sounds a more fair system than the proportional system, even though the latter sounds a lot more logical at first.
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