Hearted Youtube comments on Mathologer (@Mathologer) channel.

  1. For the second problem: the bogdanov-belski painting. I already knew it was 2. -) But yeah following the video: It's 2x(10²+11²+12²). And 100+121+144=365, goes indeed fast. (memorized) - ) 13²,14² (never memorized those for some reason): I tend to do 10x16+9 and 10x18+16, which is also still fast. (explanation: 13² = (13-3)x(13+3)+3²). So for instance if I have to square 62: 60x64+4=3844, goes decently fast. It's somewhat similar I think to what was described in the video: 7² = 6x8+1 = 5x9+4 = 4x10+9 = 3x11+16 = 2x12+25 = 1x13+36. Note that the added numbers are all squares. Each time you take a column of a squared number and you turn it into a row, you'll lose a square.
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  2. I did this in class in first semester. I was a kind of dizzy, thank you for helping me understanding and appreciating this after so many years.
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  3. Hearing "difference equation" just gave me flashbacks to my second year electrical engineering courses, where we worked with discrete signals and needed to use difference equations and z transforms
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  4. Excellent point, and an easy way to remember this characteristic of cyclic quadrilaterals!
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  5. So glad I stumbled on some of your videos today. I remember when we both worked at The University of Adelaide. Your enthusiasm and humour have inspired many, many people for several decades. I am thrilled to see you are still actively exciting people about Mathematics. I will have to view all your videos when I can! Best Wishes, DB
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  6. Wow ... sincere thanks ... wasn't aware of this at all ... Great video, as usual ... Cheers
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  7. I stumbled on something similar to this in my last year of high school, 31 years ago... but my analysis of the nth powers resulted in a reduction to n! with some additional terms that I couldn't figure out what to do with at the time. I put my expanded tables into a spreadsheet that I still have... and I'm going to have to have another look at my data with this video in mind.
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  8. This wasn't discovered too soon because notation wasn't very precise and polynomials were not useful those times.
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  9. Awesome! Your visual sum of Ln(2) @22:04 can be also be Ln(m) = \sum_{i=n}^{m*n}1/n for large n
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  10. This is such a brilliant video. I am so happy I watched it. Initially I wanted to watch it in two sittings but I could not take my eye off it.
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  11. Last puzzle: The area of the middle square is 0.2, since you can divide the big square of area 1 in five equal squares of the desired area. 1/5=0.2
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  12.  @Mathologer  😢
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  13. Bad ones? Why not naughty?
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  14. ​ @Mathologer  cos(π/7) is transcendental or not
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  15. As a colorblind, can you not color like you did with good and bad numbers? It may be very clear to people without vision problems, but green/red is the most common colorblind problem, using blue/red, blue/green, yellow/red is not a problem, even a strong red with green is generally okay. Great video again, nice job!
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  16. Yeah! When we got to the sum of 5 pentagons I started thinking a lot about how it might relate to the Fourier series.
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  17. Wow!! It's always a great pleasure to see your animated proofs! Challenge myself as a teacher to improve my skills and imagination for my lessons. Thank you a lot!!
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  18.  @Mathologer  He must have had an amazing childhood in mathematical wonderland.
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  19. "Oh , I know about this puzzle" "how tf I never thought about multiple pegs"
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  20. 6.10. : Me: What if you cut out 2 black and 2 green squares? 6.20. : Mathologer: What if you cut out 2 black and 2 green squares? It's like he read my mind.
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  21. Taylor series expansion of tan inverse x........then put x=1
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  22. Laughted a lot with the t shirt
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  23. Cool shirt as always doc!
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  24. "YOU'RE GODDAMN RIGHT I AM"
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  25.  @Mathologer  I think it is because when we saw the West with all these technological advances and compared to our own country, We just see that we have nohope for the future with ever increasing population. So nobody even bother to look at the history, because we never had any kind of industrial achievement. But we forgot that it is just the brain and knowledge which is a necessary recipe for advancement. Can you describe the mathematics and engineering required to build Ellora caves in India. We just have a feeling of inferiority based on the current condition of India in comparison to the country like U.S.
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  26.  @kinshuksinghania4289  this one was one of my favorites for sure. Completely peaked my interest in the pythagorean theorem as I was already interested because it's basically the foundation of trigonometry which also makes waves and fourier series and a lot of stuff in physics ofcourse. And I have a particular interest recently in the connection between Fourier series, Taylor series, ect, and Calculus. I mean have u noticed that the Fourier transform is basically using an infinite number of curved lines to make a straight line, and an integral is using an infinite infinitesimal straight lines to make a curved line?
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  27. Great ways to conceptualize and visualize properties of logarithms. One of my favorite videos.
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  28. One of the best teacher of mathematics.❤️❤️❤️❤️❤️ u sirji
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  29. I've been recently working on Tower of Hanoi animations for my channel and in the process discovered some of the stuff mentioned in the video, for example the algorithm to solve Tower of Hanoi without recursion by moving the smallest disk every second turn. I also proved why it works. Answer to challenge questions: Direction to move the smallest disk is: if (n is even) then Right else Left That's because to move (N)-tower in some direction, you first have to move (N-1)-tower in the opposite direction, so the direction to move the smallest disk changes every time and it starts with Left for N=1 disk The number of turns is (as mentioned in the video) equal to the number of reachable states, which is = 3^n, because: there are 'n' disks, each disk can be placed in one of 3 towers, so there are 3^n valid states. Each valid state is reachable with a standard recursive algorithm, so there are 3^n reachable states. Another interesting puzzle for you: given a number of disks (N) and a number of turns (T), how do you quickly find the state of all disks in the optimal solution of N disks after making T turns? I had to solve this puzzle while making my Tower of Hanoi animations, so I can instantly preview my animation at any moment in time (instead of re-playing it from the start), which becomes increasingly important with several-hour animations. Check out my channel for these animations: I've already published the shorter ones and longer once, including 10-hour 16-disk solution animation, coming soon:)
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  30. I have a pet peeve about math videos that claim to discuss inherent properties of numbers but are really discussing representations of numbers in base 10, without generalizing. We have here a video in which the unmentioned special treatment of base 10 appears to come from the numbers themselves, not from the person describing them. Incredible! This marks the first time I decided to pay a creator using this donation system, as well as the first time I noticed this option exists. Imagine my frustration when I learned that donations are limited to values defined apparently by the prime factors of 10. When I was an engineer at Google, at least half my coworkers had some kind of degree in math. I guarantee you someone there is annoyed by this as well :)
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  31. Just dreaming here: Mthologer creates a full course on class field theory. With historical references and proofs. That book you briefly flashed might be a backbone.
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  32. The formula shown at 24:00 works for any dollar amount if you let n choose m = 0 for n<m, which is a natural definition. It only falls apart when you expand the formula.
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  33. Good idea
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  34. Like the short videos. Call then "mathologer quick bois"
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  35. I have not verified this myself, but I was told that the fact that R^n has exactly one smooth structure for all n except n=4 (where it has uncountably infinitely many) is related to the fact that 2*2 = 2+2 = 4
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  36. snooker fans are horrified
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  37. The are of the tree @16:00 I’m assuming the angle is 45 degrees. The side length of the trunk is 1. (1x1=1) Focusing on just the first branch we know that the side length is sqrt(0.5). (sqrt0.5 ^2 + sqrt0.5 ^2 =1) So the area of one first branch is 1/2 (sqrt0.5 x sqrt0.5). But there are 2 of them so the total area of first branches is 1! By similar logic the area of all second branches is 1, and so on. Total area of tree =5.
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  38. 31:38 For a long time I was hoping the pi^2/6 identity and its relatives could be held up as a counterpoint to "everything is easier with tau" zealots. Then, wanting to know if there was a general formula for the multiples of the powers of pi in those identities (whose numerators and denominators have OEIS sequences, btw), I learned about Euler's general form for those sums and how it contains 2*pi as a unit. 😐
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  39. "On two occasions I have been asked, — "Pray, Mr. Babbage, if you put into the machine wrong figures, will the right answers come out?" In one case a member of the Upper, and in the other a member of the Lower, House put this question. I am not able rightly to apprehend the kind of confusion of ideas that could provoke such a question." Charles Babbage.
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  40. Wow, ich hätte nie geglaubt, dass etwas das mit Analysis zu tun hat auch Spaß machen kann... :) Sehr cool!
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  41. For the challenge near the end related to the Wallis product, if you simply eliminate the 1 in the denominator then each factor is < 1 converging to 1, meaning the overall product is finite.
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  42. Amazing story. Makes one wonder if those old European guys ever caught wind of and (perhaps) were "inspired by" the much earlier, related mathematical work in India. ...seems unlikely but still fun to ponder
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  43. First challenge: Π(1 + x^2^i) from 0 to n = Σ(x^i) from 0 to 2^(n+1) - 1. Nice! :D
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  44. Yeah, infinite sets make sophistry an easy task, because standard logic dictates that there are no greater or lesser infinities. Problems like this one, among others, prove that this is not the case; you just have to add to the infinity in a different direction.
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  45. It will NOT reduce to proportional splitting. You can only game it if you are the only one splitting your loan, if everyone else divides their loans too, the distribution of money will not change.
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  46. Most memorable : Euler comes in all fascinating formulaes !!! And ur math relating shirts😁 Pls can u pls make a video on Euler's line???
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  47. i can't help but smile whenever self-similarity pops up out of seemingly nowhere!
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  48. Most memorable: Kempner's Proof. Just pipping the 100 zeros weirdness.
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  49. I love it! What a great video! Your deep dives into maths are amazing and I can't imagine just how much editing goes into all these presentations.
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  50. You've shown me the true beauty in math. Your videos are truly intellectually stimulating
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