Hearted Youtube comments on Mathologer (@Mathologer) channel.

  1. Most memorable part: the 100 zeros sum being larger than the no nines sum.
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  2. This was one your best videos ever Mathologer; thank you. I'm curious if anyone has the answer to the puzzle about the Red Cross at 2:18? Cheers.
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  3. Es hat wie immer sehr viel Spaß gemacht. Die Zusammenhänge der Funktionen geometrisch in dieser Form zu erleben ist wirklich der Hammer.
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  4. Great content, made my day! One curiosity about Horner form for polynomials: it is used in computers to actually compute the value of a polynomial, of degree say n, because it involves n multiplications and n sums, instead of computing all powers and summing up, that involves n sums but n(n+1)/2 multiplications. Horner is much more efficient, because of lower error propagation in numerical arithmetics at multiplications, and also requires less memory.
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  5. An alternate solution to the bugs problem: Let each of the bugs have velocity 1 unit/s. Split each velocity vector into a vector that points towards the center of the square and one that points 90 degrees to that one. Using the 45-45-90 triangle created with the original velocity vector and two component vectors, we find that the two new vectors have magnitude sqrt(2)/2. Thus the bugs are going towards the center at a rate of sqrt(2)/2 units/s. The center is sqrt(2)/2 units away so it will take the bugs one second to get to the center. Distance = rate*time so distance = 1 unit/s * 1 s = 1 unit.
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  6. 7:00 yes actually. In 9th grade (or, as we call it here, Class IX), there's an entire chapter in the book called Area of Triangle and it's simply filled with good old Heron. Respect from India
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  7. You're the only math youtuber that makes me audibly gasp every video, let alone multiple times. Keep up the amazing work!
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  8. I dislike, and object to, "What comes next?" questions on IQ exams. And so I appreciate your video as you elevate this to more of a science. I still think there should be no such questions on IQ tests. Or...here is my alternative approach...Give the sequence, ask what comes next, and give 4 answers, where each answer includes 3 or more terms of continuation. In other words, you have something to check against. Otherwise you are just testing a very obscure math thing, that has zero applicability and is proof of nothing. Rant over. Summary: "too many answers are possible"
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  9. Merry Christmas to Mathologer and to all!
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  10. I studied electrical engineering and at University we learned something called symmetrical components theorem, of wich we use the specific simpler case for triangles. In reality we're not taught it as a triangle, but rather, as the three components of our three phase electrical grid (be it current or voltage). You see, in the electrical grid, the 3-phase system is not always symmetrical (with 3 phasors of the same length and 120° apart) because of asymmetrical loads and asymmetrical faults and such. So it would be very complicated to model and visualize it as a 3 phase system because of the symmetries. So instead, we model it as a sum of 3 symmetrical systems (analogous to those 5 symmetrical base pentagons). And seeing them as triangles would be like one equilateral triangle, one "mirrored" equilateral triangle and one "triangle" with all 3 vertices in the same location. It's sad the in my university we didn't go into much depth with this theorem, we just saw a glimpse of it and were immediately learning the application. But it's really nice to now see a video about it explaining everything in detail in an easy to comprehend format.
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  11. Congratulations on 100 videos. Your videos are impressive, to say the least.
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  12. Exercising my bragging right - yellow.
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  13. I think instead of being a wheel, it's really the cross section of a log. I'll see myself out...
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  14. "Time for a cat video?" lol
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  15. I was so excited when I found this pattern with the exponents when I was younger Edit: yay you went over it
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  16. If the L-shaped part was called a "gnomon", I propose the term "kutos" (κύτος), for the 3D version. As in, the hull of a ship :-)
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  17.  @Mathologer  Fantastic to hear it. On subject of "easy" books. I have Polish book from 1946, written by one of the few survivors of Polish School of Mathematics. It is on Complex numbers. In less then 30 pages it takes you from "what is complex number" to "calculus on complex numbers". It is incredibly easy to follow and it's free. The only problem is that it is in Polish.
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  18. 200 hundred different proofs. Me, an intellectual: Sorry this margin is too small to contain a proof 🤷🏻‍♂️
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  19. The 2nd of the basic geometry theorems (that the inscribed angle is half the measure of the subtended arc) can also be used to easily prove the 1st: two opposite angles of a cyclic quadrilateral subtend arcs that clearly add to the entire circle, and their measures would then add to half of a circle, or 180°.
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  20. "Logarithm explode to infinity" What a violent explosion 😂
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  21. Nathan's digit proof was so much more elegant. The faces (mathologer's) proof was just based on observation.
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  22. This ruling is actually from the Mishna, although it was reprinted in the Talmud. In fact, most of the Talmud is essentially just a commentary on the Mishna. So on the page you showed (12:19), it essentially says "The Mishna says XYZ... Now here's various Talmudic rabbis' arguments over why the Mishna said that" You may notice two boldfaced words in the central column. The bit between the two boldfaced words is the Mishna quote, and the bit after it is the Talmud's discussion on it (Rabbi Shmuel said this, Rabbi Ravina said this, etc) You can look up Ketubot 93a for more. Sefaria has a nice translation
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  23. Hey I found 2019 in your π-shirt after the 244th digit after the decimal I am a man of culture
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  24. Ok, can we take a moment to appreciate the slide transition at 25:40? It's magnificent.
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  25. Nooooo!! It's a nice way to spend your time indoors... and humanity needs you right now... keep it up!
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  26. I was taught Heron's formula in high school, in Greece. I found it rather interesting, but considered it a curiosity, mostly, as it wasn't connected to anything else I was taught.
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  27. Awesome, Mathologer! Another ab-so-lute-ly delightful journey. Apart from its important and most beneficial applications, Mathologer never ceases to amaze with another revelation on how maths has just this amazing beauty and harmony in itsself. This channel is such a gem on YT. ✨Thank you very much, indeed, Sir. 🙏
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  28. "Why are manhole covers circular?" "Because manholes are circular." I wish I had the balls to give that answer.
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  29. I gasped out loud when he pointed out that the windmills pair up with each other. That was amazing
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  30. You remove all the tedium, replacing it with slick graphics, perfect music and your happy personality. Your love of math is the "answer" for us. Thank you!
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  31. oh just realised that's what you show at the end
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  32. nvm just found it in the description 😅
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  33. I learned Heron's formula in school in the US state of New Jersey in the 1990s. Unfortunately, it was not proven, though I did derive my own proof using Law of Cosines similar to the one in the video. I am always on the lookout for better proofs, and the one in the video is definitely beautiful. I love the way the exposition on the 345 triangle gently leads the viewer into the appropriate conceptual space. Well done.
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  34. Having the third part is silly. The second and third is just a log rule, but even worse the third gives up the joke
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  35. Given that there are 7! = 5040 possible orderings of the 7 proofs and a subscriber count in the hundreds of thousands, I can say with certainty that multiple viewers of this video will have the same ordering of most liked to least liked :)
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  36. Undergraduate mathematician here. The better I get at math, the more I appreciate your videos. These videos give a great visual experience which is generally not taught in proof courses. My favorite chapter was probably Chapter 5, reminded me of some of the concepts discussed in my analysis course.
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  37. "I can't be bothered to do the hard cases" in my day = "Left as an exercise." "I can't be bothered to do the hard cases" today = "Leave your thoughts in the comments."
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  38. The iris was interesting, but I'm here for Conway!
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  39. Made it to the very, very end - as usually.
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  40. I made it to the very end... ...and I liked it. I know a decent bit of recreational math and most Mathologer videos contain "something old, something new, something borrowed, something blue". But this one - apart from the concept of the partition numbers - open a new part of the math world. Thanks Burkard for coming up with these amazing and very followable adventures! 👍
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  41. I find it particularly interesting how a square grid can give rise to a circle... basically you get rotational symmetry out of something that is not... And why does it have to be L2 symmetry not some other Lp?
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  42. Speaking of Sophie Germain, my son and wife were looking up facts about the number 47, when I mentioned that 47 is the end of the Cunningham chain of the first kind with the smallest Sophie Germain prime as the initial number.
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  43. Shhhhh.... they don't want us knowing these things, hahaha. Thank goodness you are here!!!! What a time to be alive!!!
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  44. The reading at 6:14 is 3.18 not 3.14, but the magical thing really understood you were multiplying by pi :)
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  45. This is also why the two obvious ways to pack spheres are the same.
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  46. I feel like I watch this guy 20% for his amazing math demonstrations and 80% for him laughing at his own jokes <3 thanks for the incredible content! Math is an amazing thing!
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  47. Wunderbar, diese Wiederbelebung der fast in Vergessenheit geratenen Entdeckung von Steinbach! Wie immer, Vergnügen pur und besten Dank!
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  48. Yeah, ancient Indians loved converting everything and anything into Sanskrit verses. If you don't know how to properly decode it (like most people today), there is a high chance that you will mistake it to be a flowery story or poem. I think they did this so that the students could memorize easily. Edit: Oh, and being a Malayali myself I am surprised I couldn't completely read the manuscript. It is so fascinating that the script changed a lot with time!
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  49. Second challenge: it is obvious that in this hexagon there is the same number of dominoes of each color because transposing this 3D volume(from isometric axonometry) into projects on the XOYZ axes we obtain identical squares on OX , OY and OZ planes so an identical number at any scale of dominoes ;(and whenever we place the dominoes , the projections will always be squere).
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  50. A fun intuition for why the frozen sections show up with high probability: Imagine the left corner of the n'th Aztec Diamond has a horizontally-oriented block on it. If you draw it on paper, you will see that this completely determines that the entire left side of the diamond is only horizontally-oriented blocks, what remains undetermined is nothing more than the (n-1)'th Aztec Diamond. So the number of configurations with a horizontal domino in the left corner is equal to A(n-1), which is fairly intuitively a small fraction of A(n).
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