Hearted Youtube comments on Mathologer (@Mathologer) channel.

  1. Most memorable: The intuitive geometric proof that γ is finite, hadn't seen that one before!
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  2. Thank you, At 71 years of age I thought I would never see anything about slide rules again. When the video started I thought to myself - this could be about slide rules. It was surprisingly satisfying that my suspicion was correct. When I was a senior in HS calculators were just becoming available. True engineers keeping slide rules on hand is like current day sailors keeping and using sextants at sea. Young people are going to learn the hard way that all of this electronic technology is going to fail them at some point.
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  3. Yet another Mathologer gem. It's definitively my preferred channel! Thx Mr. Polster, for give us such a wonderful presentation in all senses.
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  4. 28:11 Hey! Euler is your Patreon patron! 28:25 And so is Mandelbrot!
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  5. The locus of points with AB=A+B is the hyperbola XY=1 shifted up and right 1. This gives nice pairs like (3/2,3), (4/3,4), (1+1/n,n+1).
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  6. Hi Mathologer, could you please explain why the perimeter of the ellipsis has no finite formula?
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  7. ​​ @Mathologer  I went through a few minutes of the Hindi subtitles (another Indian language), it's not perfect but it's pretty decent in my opinion. Definitely better than no subtitles.
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  8. Wow, this is a great application using math. I also like math modeling myself, and this video is in the same context of my interest so much. Great job and nice explanation!
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  9. And I thought train spotters were strange. Now I'm aware there are triangle spotters, too.
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  10. Uhh, a new Mathologer video! I'm soo excited!
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  11. Most memorable: Tristin’s visual proof of the finiteness of the no-9 series.
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  12. This approach is awesome! We covered the law of quadratic reciprocity in number theory class, but the proof was omitted, and it came down to uninspired manipulation and flipping of Legendre symbols. But now I'm 95% convinced that I understand why it all works :) I suppose something that could have been expanded upon for the benefit of other viewers is what the Legendre symbol is used for (which was briefly mentioned in the video), such as an example of solving a quadratic congruence over Z/pZ. Maybe that would make the whole LQR/Legendre business feel more motivated, but it's great as it is. And maybe another visualisation of the quadratic residues/non-residues mod a larger prime (like 17 maybe?) so we can get a feel for the numbers that appear along the diagonal, in particular what to expect (somehow it is easy to forget obvious things like (49/p) = 1 for any prime p, no need to find remainder first, as the symbol can lose attachment to its definition when purely evaluating by rule).
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  13. On a scale of 1-10 for how beautiful that recursive argument is, because 10 is just a special case of 4, the rating I give is 4.
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  14. Personally I find the limit definition of calculus very unwieldy and intuitive. Working with Leibniz notation like this is always much preferred in my eye. Turning your differential operations into statements about the convergence of sequences doesn't seem like a particularly natural step and is one that I don't tend to see people do when working with calculus in physics or dealing with them numerically. In physics and while working with numerical methods I feel like what we do most resembles non-standard calculus, especially ERNA and ERNA^A. Sam Sanders 2010 article "More infinity for a better finitism" gives a really nice write up of this. Being able to manipulate infinitesimals symbolically without fearing that one of your implied sequences stops converging is great peace of mind. In particular their example of pushing sums through integrals without fear (as long as your result is sensible) is one of those things I bet thousands of people do without thinking about whether it is justified.
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  15. Witchcraft!
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  16. Feedback: Mechanical engineering student with interest in math . Think I got almost everything. Will try to do the bits you left for the viewer. Thank you for awesome videos like this. :D
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  17. Puzzle at the end: Conway's circle is larger. The weird curve's diameter is precisely the length of the chords since the chords are perpendicular to the curve's tangent at the intersection (this can be trivially shown from the curve's construction), so the diameter of the circle is necessarily larger (specifically, the square of the diameter of Conway's circle is the square of the incircle's diameter plus the square of the sum of the three sides of the triangle).
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  18. 31:05 I believe I read, was mentioned in one of my Logic classes, or ???: For every Provable True Theorem, there are an infinite number of True Unprovable theorems. "There are more things in heaven and earth, Horatio, than are dreamt of in your philosophy.".
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  19. Every so often I think to myself, "It sure has been a while since Mathologer put out a new video..." and it seems that more often than not, you come through with something lovely in a day or two :)
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  20. Programming challenge (did this around 13:41): because all of the coins evenly divide a dollar, it must be the case that a sum of coins to a multiple of a dollar can be separated into some groups of coins in which no single denomination adds up to a dollar, and the rest, in which each single denomination adds up to a multiple of a dollar. The former cases can be enumerated by doing the product trick without the exponent of 100, and taking the coefficient of each exponent divisible by 100, including 0. There are ways to make change like this for zero through four dollars inclusive. This gives us one part of the solution. The other part is to determine how many ways the remainder could be made. But this is a simpler problem, because it's equivalent to asking "How many ways are there to add five (number of denominations - 1) boundaries to a set of a given size" which is equivalent to asking for (size of set + 5) choose 5, which can be hard-coded as a sequence of multiplications followed by a division. (I could have tried expanding out the polynomial explicitly, but that sounds like effort). So, the final answer is to, for each amount of dollars that can be made without using a dollar's worth of a single denomination, multiply that by the number of combinations for the remaining dollars. I coded this up in Python, and it's pretty fast. I just started added zeroes to the exponent on the ten, and it was pretty acceptable performance up to 2 * 10 ^ 100,000. I'll post the value for 2 * 10 ^ 1,000,000 when it finishes, because that's much slower. Okay, yeah, that took a few minutes. One sec... Oh geez, it overflowed the buffer. I'm not pasting five million digits in here. See https://pastebin.com/MA2a9p3R EDIT: At 23:02 I'm seeing the same coefficients in the center row that I had my program calculate for "ways to make dollars without a single denomination adding up to a dollar" So I guess this is going in the same direction.
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  21. The number of ‘toys’ in your office is astounding. Always amazed at your presentations.
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  22.  @arikwolf3777  yes. Nowhere do any of the equations depend on the digits of the numbers
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  23. These are some of your most visually satisfying animations to date, which is really saying something!
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  24. I had to teach myself about digital roots when i was in middle school, although the deepest I came up with was 9 or 18...didnt even think about 1+8 equaling 9 because I only went one sequence in. To me, the fact that 99 equaled 18 was enough. Glad to be getting deeper 26 years later. Edit: wouldnt this also be considered modular number maths? Edit: aaaaand I got to 16:50
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  25. This is almost click bait. Actually, I was curious if it was a female or just a male mathematician with long curly hair and feminine features. So I actually clicked to check the name. Does that constitute click bait?
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  26. But since the horn is an ‘open’ surface as 1/x never ‘closes’ by crossing zero then wouldn’t your paint just run out the other end?
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  27. I've now realized how easy it is to miss the edge cases when using visual proof. It's a crucial lesson in mathematical proof. I'll keep in mind when I'm dealing with visual proof in the future. Thank you for the beautiful insight professor. Yet another gem you've shared with us 🎉
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  28. youtube: HERE’S SOME MATH me: okay
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  29.  @Mathologer  I'm just picturing you stuck in a perfect sphere where the internal boundary is a blackboard.
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  30.  @ravenecho2410  In fact some actuaries are fantastic at pure maths but most are what you would say “good at maths”. It’s prerequisite to study highest level at school to enter courses. As you’d know, but many wouldn’t, actuarial studies is a discipline of applied maths combining a diverse range of fields such as demography, finance, investment, commerce, marketing, accounting, risk analysis, modelling to problems in insurance, banking, health, pensions etc. So the emphasis is on applied maths to real world problem solving. Being brilliant at calculus or number theory is not of much use. if your friend chose another calling where such maths fields are useful, that’s fine. My son is an aeronautical engineer, who despite all the theoretical maths work at uni, discovered real world work generally didn’t require it.
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  31.  @Macieks300  page 21 of the book I mentioned. As a source it cites: D. R. Heath-Brown: Fermat's two squares theorem, Invariant (1984), 2-5. latex version, with appendix on history, January 2008, at eprints.maths.ox.ac.uk/677/1/invariant.pdf The URL is archived at: https://web.archive.org/web/20110606154228/eprints.maths.ox.ac.uk/677/1/invariant.pdf
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  32. I have the most excellent documentation of who came up with the windmill interpretation of this proof, but there isn't enough space to place it into this youtube comment.
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  33. In 1983 I got a Commodore 64 for my 8th birthday. My dad got some 5-1/4" floppy disks full of public domain BASIC games and one of them was Tower of Hanoi. That was the last time I heard of Tower of Hanoi. 🙂
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  34. Surprisingly fun for such a short video! Also, as a big fan of empty operations, I was quite happy to see the "0 =" and "0^2 =" at the tops of the trees :)
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  35. This was an amazing video. Ptolemy theorem is one of my fav theorems now!
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  36. TheOneThreeSeven. I love the fine structure of his name
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  37. My brain ain't braining
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  38. 20:58 All this time, I never knew that ln stands for logarithmus naturalis always thought that it's just *natural* to use ln (pun intended) Also, another beautiful visual explanation from Prof. Polster Thank you.
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  39. For the challenge at 13:00, I got the sequence n!(n-1)!
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  40. Is this it? Is this the hemisphere turned inside out? That wasn't easy to follow, was it?
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  41. Pure mathematics at its finest Thanks for revealing this bag of gems
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  42. Simple is King.
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  43. 6:32 The beauty of the movement looks like a 4-dimensional hyper cube / Tesseract. So many great videos on this channel. Thank You!
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  44. Mathologer Then your students are very lucky. I am an. 83 year old Maths lover, who got as far as calculus but would love to have had a greater understanding of Maths. But, which, with you help, I now have.
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  45. For the trick to convert pascal's triangle to our game traingle, I originally thought of multiplication by 2. Which is the same as multiplication by (-1) anyway since we are doing modular arithmetic here :)
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  46. I only realized it was a 50 min video when I read this comment after the video
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  47. colouring proof is the one i came up with after you pointed out the incircle. also, the shape of constant width has width equal to one of the chords of the iris, definitely less than the diameter of the circle.
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  48.  @JMairboeck  Yes. As he mentions at the end of the video, any odd number can be written as x^2-y^2. So any odd prime p has p=x^2-y^2=x^2+(iy)^2
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  49. Would you like to know more?
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  50. I'm mathematically challenged, but this was one of the most entertaining videos I've seen in a while! Job well done! 👍
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