Hearted Youtube comments on Mathologer (@Mathologer) channel.

  1. I'm always looking forward to these monstrously long videos from the many Math-oriented content creators on YouTube. In particular, I feel the animated approach in these videos really helps to visualize the most intricate arguments. I should also mention that your work is very enjoyable to mathematically inclined people, mathematicians and non-mathematicians alike. And even if one has a deep background in Mathematics, there is always something new to learn.
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  2. From where you got all this?
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  3. Exactly.
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  4. I don't understand any of this. But whatever it is, I'm glad it got solved...
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  5. Thank you for this Christmas gift for us all.
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  6. Given that shapes on the area-preserving transformation become more distorted the closer they are to the edge of the circle, it makes sense for that to be something boring - like a large expanse of water. What point on the Earth's surface should be chosen to minimize the amount of land near and on the perimeter of the circle? What would such a map look like?
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  7. Why is 27 not on your t-shirt?
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  8. I just logged in to leave exactly this comment.
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  9.  @cliveso  That's a fair point. But given that law would need to be modified in order to implement such fair divide, it's not far fetched to suppose that a provision for debt consolidation would be included. Especially if that's sort of obvious. Or else, it's about supposing the incompetence of the law makers, which is a far different topic that I don't want to go into. Given the debt hierarchy presented in this video, you can then proceed to fairly divide amongst a given debt type, then move on.
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  10.  @Vaaaaadim  Exactly. If the amount to be split is more than half the total claim, the largest creditors get the highest percentage of their claim. That's what he said in the video. If there's only a little to split, then the loans are all bad, but everyone gets a little. If the amount is only slightly short, then the loans are all mostly good, with only a slight loss for each creditor.
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  11. Despite being a Physics graduate, I had never seen this. And the very first time you built up to the 7th tow, at the start of the video, I actually let out a little involuntary gasp of delight. I LOVE it when this happens. Thank you, Mathologer!
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  12. Excellent as well as beautiful
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  13. no, in australia we didn't learn the formula, but luckily i came across it somewhere and was so enthralled by it, i used it in class to my teacher's dismay.
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  14. So the true genius of Leibnitz and Newton was to invent the proper symbols for expressing the ideas in their heads. So it would follow: to spread great ideas, it is not enough to 1) prove them, 2) show they work in practice but also 3) develop expressions that make it easy to parse with our brain and 4) open doors to new realms.
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  15. The benefit of this solution is that, in the case of a small estate, it makes a greater number of creditors less unhappy.
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  16.  @mr.johnson3844  Totally agree. High school teacher from USA.
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  17. I saw the first twisted square proof first, and it was from either you or numberphile lol. Am in America for context
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  18. This is awesome. Thank you!
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  19. Great video as always! Personally, I would have preferred if Torricelli's construction for the exact volume also included a top-down view (looking down the axis of the horn) after 14:34. This way we could see clearly how each point on a radius of the base circle was assigned its own circle to construct the cylinder.
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  20. i have never seen the bridge from conic to catenary explained this way! now i understand why so much effort went into trying to find the catenary as a conic section in early days of math
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  21. This is so awesome! I'd never gotten around to learning this, because (and I quote) "if anyone ever asks me to find a formula for 1^10 + ... + n^10, then, well, I know it's a polynomial starting with n^11 / 11, so I'll just calculate the first 12 sums and find the unique polynomial that fits". But I'm so glad to see such beautiful maths behind these power sums! And yes, keep the insane stuff coming! Especially on the thing about the "asymptotic expansion" at 47:11.
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  22. After a fashion, I think you have shown the nature of how slide rules work. Logarithmic scaling of two sets, and when compounded, brings to value in magnitude, similar to a pantograph.
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  23. Rewatch 100%. Liked. Already subscibed. Congrats on your integrity.
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  24. 2:42 “Why on earth would anybody be interested in doing that?” Well, one thing you can do is to show that a prime number actually can be the product of two integers — two complex integers, that is. The fact that p = a^2 + b^2 implies that (a + bi)(a - bi) = a^2 -abi + abi +b^2 = p So, when someone says, “A prime number can only be divided evenly by itself and one”, you can say, “Not so fast! Is it congruent to one, mod four?”
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  25.  @Mathologer  Please keep it that way.
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  26. Great. Both the proves ant the music.
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  27. new vid pog
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  28. This is one of the many things I enjoy about traditional geometric patchwork design quilt patterns , (like Flying Geese, Irish Chain, Feathered Star, Grandmother's Flower Basket, Disappearing 9-patch, Bear's Paw, Log Cabin and all it's variations, Pinwheels, etc.) with 1/2 square or equilateral triangles, rectangles, squares, etc. and shuffling them around to create different patterns. Now I have a name for them, and may design quilt patterns with the names of Pythagoras or Trithagoros, or the long ago Chinese mathematician's name! Great visual explorations of these relationships, thank you. (of course I enjoy even more, the random sewing together of odd shaped and sized pieces, called, "crumb quilting"! lol) (see also, the 'golden ratio' seen so much in nature and used from antiquity in architecture to produce unconsciously, naturally pleasing, building facades.)
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  29. Thank you, Mr. Mathologer. You explain the geometric meaning of mathematical formula precisely. We are happy to see more.
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  30. Guess It's time to get really interested in a topic I didn't know existed!
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  31. Yeah, it took me a while to see it from the diagram, but I started doing the French lacing since finding it in a store because it looks nice. I like to imagine that it's less stressful for the lace to be getting pulled against the edges of each side, but when has a lace ever broken there?
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  32. What does 2 being prime have to do with Pluto not being a planet lol
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  33. The reason all even numbers are not prine is because 2 is! You're elementary brain wants to look at even vs odd but you don't understand the structure
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  34. What a coincidence. Tomorow, 1 december is the national day of Romania and the flag is red yellow blue
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  35. In the 'Second Proof' you showed that the folded triangles are all the same, but not that the fold of the fold is similar to the original. However this is a simple extension when you see that the black line in the center is one of the medians of the folded triangle and it is also half of a side length of the original triangle. Since all three foldings of the original are the same, we know that it's medians are all half the length of one of the sides of the original triangle. Then folding again and applying the 'Second Proof' again, we get a triangle with side lengths are the medians of the folded triangle, ie. a triangle who's sides are half the side lengths as the original. In the 'First Proof' we did not know what the ratio between the original and folded folded triangle was, but the 'Second Proof' gives us that.
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  36. I love watching these videos of advanced math and I am inspired to get into it, could you guys make great tutorials that would help beginners to learn basic key math concepts, that would allow to understand bigger concepts easier?
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  37. I had requested long back as comment in one of the earlier videos, an intuitive graphical explanation for why the Heron's formula for area of the triangle works, without needing to use trigonometric formulae. Thanks a ton for releasing a video that exactly answers to my request. I haven't found such a great shortcut to reach to the Heron's formula anywhere else in internet.
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  38. I vote for the second one.
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  39. @24:20 is a little tougher I think if you try to take it’s parent you get a square of 1,0,1,0 So I suspect the Feuerbach centre is directly on the incircle. In other words you get a 0,0,0 pythagorean triplet. Which... works? For some definition of work!
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  40. How do you solve the integral in your shirt, btw?
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  41. For the puzzle at 11:37, I was too tired yesterday to get it, but now that I had time today I gave it another shot. My square was defined as: ┌──┬──┐ I defined the layout of any pythag.. triple as:. │B1│B2│ X^2 + Y^2 = Z^2 . ├──┼──┤ . │B4│B3│ . └──┴──┘ . I started out pretty standard with the main three equations of: B1*B4 = X 2*B2*B3 = Y B1*B3 + B2*B4 = Z I also included these: B1 + B2 = B3 B2 + B3 = B4 To define the innercircle, I used r_i to denote its radius. The equation is then: r_i = B1*B2 I started by creating an equation to find r_i. To do this, I used the slope of the hypotenuse (Y/X) to solve for an angle, divide it by two, and convert it back to a slope. This gives a slope which bisects the angle XZ. The equation is: y=tan(1/2 * arctan(Y/X)) * (x+X) (my origin is placed at the corner XY so x+X places the lines origin at the corner XZ) Using trig identities, namley half angle identity and inverse trig identities, I could simplify the expression: tan(1/2 * arctan(Y/X)) => (sin(arctan(Y/X))) / (1 + cos(arctan(Y/X))) => (((Y/X)/sqrt[1 + (Y/X)^2]) / (1 + (1/sqrt[1 + (Y/X)^2])) => (Y/X) / (sqrt[1 + (Y/X)^2] + 1) => Y / (sqrt[X^2 + Y^2] + X) This means that the equation is: y = (Y / (sqrt[X^2 + Y^2] + X))(x+X) Which is our equation for the first bisector line. Since the angle XY is 90°, it has a bisector with a slope of -1. This results in the equations: y=-x Setting them equal to find the centre of the incircle results in a simplified equation of: x = -XY / (X + Y + sqrt[X^2 + Y^2])) Since X^2 + Y^2 = Z^2, this becomes: x = -XY / (X + Y + Z)) OR: r_i = (XY) / (X+Y+Z) Plugging in the values, we get: r_i = 36 since r_i = B1*B2 = 36, this means B1 = 36/B2. plugging into the equation: B1 + B2 = B3 yields: 36/B2 + B2 = B3 => 36 = B2*B3 - B2^2 using Y = 2*B2*B3 = 104, we know B2*B3 = 52. 36 = 52 - B2^2 => B2^2 = 52-36 => B2 = 4. So, using B2*B3 = 52 we find: B3 = 13 Since B1 + B2 = B3: B1 + 4 = 13 Or: B1 = 9 Since B2 + B3 = B4: 4 + 13 = B4 Or: B4 = 17 To Conclude, my guess as to the numbers in the square are: ┌──┬──┐ │ 9│ 4│ ├──┼──┤ │17│13│ └──┴──┘
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  42. The magic square construction at 18:45 was the method I was shown many years ago. You start with one in that location, then go up and to the right one space for the next number. You pretend the grid wraps on itself, the top going to the bottom and the right to the left extremes, and if the square it is headed to one that is already occupied you go down one. This construction works for all odd grids as far as I know
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  43. Here’s my answer for @12:00 9,4,13,17 And you go Right, right, right, left
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  44. I found the hidden 2019 on your t-shirt in 11th row (whole row is 165271*2019*)
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  45. This is one of the most beautiful things I’ve ever seen in my life.
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  46. That’s more than a bit dismissive.
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  47. they didn't obviously otherwise they would have given a general solution not 4 cases like they gave, the idea is to achieve contested garment type solution for more than 2 people for any type of distribution and any amount of garment, they obviously couldn't do it in general sense hence they gave ones which they were able to solve as the cases they presented.
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  48. I am curious if this visualization can be attributed to @Mathologer. The limit is well-known, but I never saw the limit presented in this way before.
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  49. The question in the thumbnail: Do you have a hair doppelganger? The answer: No idea. I just know that someone has.
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  50. I was looking for an Intuitiy proof of this some time ago. Excited to see you explain it. Thank you :)
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