Hearted Youtube comments on Mathologer (@Mathologer) channel.

  1. I always liked to describe differentiation as just a bunch of rules you have to apply and it's usually straight forward how to do it. Integration on the other hand consists of either knowing the answer or trying to manipulate the function until you do.... with the optional third step of giving up and looking it up on a table. Also I like that the music got way more epic as soon as you got to the chain rule.
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  2. Instant classic. This just feels like the perfect “Mathologer-y” topic! It also had the possibility for some great animations, which I’m so happy you delivered on flawlessly. I’ll probably watch this a few times throughout the future to enjoy it again!
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  3. I'm homeschooling my children, and interestingly I just taught them Heron's formula yesterday as an example of where square roots can be very practical in real life problems. Specifically, we talked about how convenient it can be if you are trying to determine the size of a plot of land you want to buy when all you can measure are distances and no idea if the land is square. Just walk the perimeter, walk the diagonal, apply Heron's formula to the 2 triangles, and you know immediately how big the plot is. For 7th graders who don't yet know trigonometry, the example worked beautifully. It should absolutely be taught in schools. It is a crime that it is not considered essential these days.
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  4. I think the shirt in this might be one of my top five favorite shirts I've seen him wear so far. Impossible triangle made of rubik's cubes, perfect hexagram in the middle, and it almost looks 3D. This shirt is a winner.
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  5. Some carnival guy right now: scribbling furiously
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  6. Given that you can do the operation on the final card without having to turn a card to the right (since it doesn't exist). The largest possible number of turns to terminate the sequence is equal to [n*(n+1)]/2. edit: If you cannot perform the operation on the last card then the most turns possible to terminate the sequence is given by x=(n-1)*n/2 though the sequence may then terminate with one card left.
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  7. Another gem from Mathologer. It's because of Mathematicians like you out there, Maths is still beautiful and elegant.
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  8. Haha, was just wondering why you didn't cover partitions and then seen this. Very interesting and an intriguing topic with the contributions of several important people like Euler and of course Ramanujan .
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  9. I once performed a version of the trick at 26:18 where the AUDIENCE got to choose the card to keep, and the only choice the magicians had was the arrangements of the 4 cards. Of course that’s normally be impossible, so we cheated a little and used an extra bit, a right-hand vs left-hand choice. It goes pretty smoothly if you’re quick at mental math, and the audience never suspects a thing!
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  10. Nice :)
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  11. Once again I am rewarded for my procrastination: I never saw the original. "Hard work often pays off after time, but laziness always pays off now".
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  12. all values that have paradoxical meaning are just new planes to fly our imaginary numbers on.
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  13. I don‘t recall seeing any of these proofs in school. Most relevant formulas were simply introduced to us without ever being proven before college.
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  14. Some of us will recognise Donald Knuth as the creator of the typesetting language TeX. For that alone, he has been responsible for the professional quality appearance of thousands of PhD theses.
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  15. He might not be your typical name, which you could give and everyone would recognize it, such as its with Newton. Despite that however, Ramanujan is a genius, who sadly didn't live for long and would probably be one of the most important mathematicians, should he have lived and published more papers, perhaps even be an advisor to some future mathematicians :)
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  16. At 28:42, your generalization is wrong- it should be that cbrt(ABC)<=(A+B+C)/3 rather than that sqrt(ABC)<=(A+B+C)/2. As stated, (3,3,3) (for example) breaks your generalization. At 35:50, it’s because the bug that each bug is walking towards is always moving orthogonally to the first bug’s path, never getting closer or farther outside of the first bug’s motion. So it has to go exactly the same distance as it was at the beginning. At 36:30, it’s 1/5. You can rearrange the square pieces to get five squares of equal area, summing to area 1.
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  17. I studied the infinite generating function for high school math competitions. They are very useful in combinatorics problem!
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  18. The cicadas :) Nice and crisp video as always. And big thanks to Andrew!
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  19. Yet again, you manage in 30 mins to better explain something than my maths teachers could over a year. Bravo, sir!
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  20. 1:36 "circle gets turned inside out" *"smiles and frowns" flashbacks intensify*
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  21. "...The Euclidean Algorithem, an ancient mathematical superweapon" perfect description!
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  22. Let's all just forget about yesterday shall we? Great to see the videos are back and I really hope they work their situation out (IN PRIVATE)!
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  23. Red Cross solution: Align the centre of one of the smaller crosses to the centre the big cross, matching their orientation. Rotate the smaller cross until its vertices touch the edges of the big cross. The 4 segments produced form the 2nd smaller cross.
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  24. I remember watching 3B1B's video on Hanoi and feeling such awe, that a seemingly simple puzzle could weave recursion and binary counting together. I never thought anything could replicate that awe again! But damn you! Thank you soo much for making me experience that awe again, and this time much muchh more! I just can't wait to get on my computer and try to animate all the animations that you showed today (and perhaps even a generalised version for small n using Frame-Stewart algorithm?). But seriously lot's of love and respect man for making me fall in love with Maths over and over again!
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  25. I'd say the Axiom of Choice is true as long as you don't have an Electoral College :(
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  26. 9:45 12 million "and change"?? I'll take your change then, thank you!
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  27. 15:52 "There's no hidden trickery" I'm not complaining, but I'd say that counting faces is quite tricky, as it relies on topology. To see when we're removing a face and when we're not, is visually obvious and yet non-trivial. And also one needs to be careful not to disconnect the network (as you said, "starting from the outside" should guarantee this).
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  28. The most memorable proof is the original proof of the harmonic series' divergence simply for the fact that this probably the only proof I could present to my year 10 math class and most of them would understand it.
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  29. Nicolas Bourbaki will be furious. 😉
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  30. Ich würde wirklich gerne ein Mathologer Video auf deutsch sehen. Greetings to Australia! Great Video!
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  31. You had me going at the beginning. Because of the particular choice of original triangle, you briefly had me wondering whether the "folded" triangle might be (geometrically similar to) the mirror image of the original. But no, not in the general case.
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  32. I really love the graphical intuition added onto that one sentence proof. It makes it a lot clearer WHY that function is an involution and has exactly 1 fixed point. Also, you misspoke. At 28:54 you said "b squared" instead of "c squared." >.< Gotta be tough to get through that stuff without any mistakes. At least it's clear what you meant cause of the written equations.
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  33. So happy to see a new video. I was really upset when you didn't upload in February.
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  34. I am a simple man. I see mathologer upload Einstein, I click!
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  35. Engineers: Hold my Taylor Series
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  36. Mathologer’s Theorem: π is the sum of two squares. 21:19
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  37. As an Indian and a student of mathematics it is ridiculous that I came to know about Madhava and his discoverys through you sir, although I am familiar with that series but don't heard Madhava's namae before Thank you sir for this video .🙏🙏
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  38. Just like the Indian mathematician Madhava, there are other Indian mathematicians who did advanced mathematics way before western mathematicians
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  39. 21:15 "and therefore pi is a sum of two squares" 🤔 now that is some mathologer magic I missed in between the lines
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  40. One nit: the binary argument demonstrates that the sequence of moves must terminate, however, there is an additional step required to say that it terminates specifically in 00000 == all cards face up. Consider for example if one were in the state 00001, and we know every move decreases the value, but perhaps they decrease it by 10 or something. We'd overshoot 0 and be stuck at 00001. It's important to call out that as long as there is a 1 present, there always exists a legal move that does not overshoot 0
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  41. 8:30 You can still use the exponential function in the following way: Let y = ce^f(x) y' = f'(x)ce^f(x) Therefore f'(x) = x and f(x) = ½x² + const So in the case of y' = xy we have y = ce^½x²
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  42. What makes Quadratic Reciprocity so special, and in particular, why was it so important to Gauss? Why is squaring integers so important in Number Theory? To understand the answers to these questions, you need to appreciate the work Gauss did in the theory of quadratic forms and their application to differential geometry. Gauss's "fundamental theorem" of differential geometry is about how the 2nd order partial derivates of a function that defines a "surface", e.g. `f(x,y)=height`, depends on a specified quadratic form called the "metric tensor" of the surface (and its first derivates), which gives a rule for how to measure distances "on the surface". This shows a profound and deep insight into the nature of the relationship between a large number of otherwise seemingly unrelated abstract concepts. Squaring numbers plays a fundamental role in both Number Theory and Geometry. After watching this video twice, going on a few wild goose chases, and not being able to stop thinking about it: I finally think I understand why Gauss placed such a high degree of importance on this particular theorem, and feel like I have a lot of reading to do now!
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  43. The Auto Algebra Song™ makes me so happy.
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  44. Imagine having to use 360° because the formulas look weird with 2pi. If only there was a better constant to represent full circles...
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  45. log(1+2+3)=log(1)+log(2)+log(3)=log(1*2*3)
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  46. I think I'd honestly prefer the first proof, but I was too busy shouting at the screen about the second proof to enjoy it.
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  47. Wow, I had ventured into speedcubing back in 10th grade and I conjectured to my friend that no matter what repetitive steps you use you'll get back to solved state and we discussed about it at great length. This video has brought those memories back and how I never thought about it afterwards even after doing a course on group theory but now you have opened my third eye. Thank you!
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  48. I always get a 'WOW!' when I watch your videos. Thanks for bringing this stuff to YouTube.
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  49. Most memorable moment was the cat going "μ".
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  50. Sequence Calculus is just low resolution calculus.
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