Comments by "Lawrence D’Oliveiro" (@lawrencedoliveiro9104) on "Numberphile" channel.

  1. You mean the way the digits add up to 9? Imagine a planet where they use hexadecimal, and some little alien child discovers a similar pattern in their F-times table. Yes, maths is universal in that way.
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  2. Ah, binomial theorem, my old nemesis, we meet again.
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  3. 10:23 That’s OK, we salute you for trying ... a 21-gon salute. Thank you, thank you, I’m here all week.
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  4. Peeling quaternions can apparently do that to you.
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  5. 5:57 Presumably it’s actually a “geodesic”, i.e. the shortest distance between those two points.
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  6. 6:20 By “most” he means “100%”. The ones inside that outermost circle make up the remaining 0%.
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  7. 4:51 Every time something gets called an “atom”, it turns out to be divisible into smaller components. So far this hasn’t happened with primes ...
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  8. Nature doesn’t divide itself into compartments, after all. The compartments exist only inside our heads.
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  9. 2:09 So only incompressible fluids? That would leave out all gases.
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  10. By “forever”, do you mean ℵ₀ seconds or something greater, like, say, ℵ₁ seconds?
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  11. e by gum. Actually I think he’s from Manchester?
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  12. I can think of something worse: infinitely many mimes.
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  13. 10:41 So towers are clearly made out of timber, since you can take them apart log by log. ∗Ahem∗
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  14. 4:57 See, the definition I learnt, many years ago, was that the naturals were 1, 2, 3 ... excluding 0, that if you included 0, you got the whole numbers.
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  15. There’s another, even wackier formalism, called λ-calculus, where everything is a function. It’s often quite useful in computer science.
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  16.  @PattyManatty  How does that answer my point?
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  17.  @PattyManatty  But it can only show that when the mapping has been completed. Which takes an infinite number of steps. Conversely, adding the new number to the list only takes one step.
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  18. Very edgy. Did you get the point?
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  19. Is “Chaitin” named for “Gregory Chaitin”, by any chance? I remember seeing a lecture he gave some time ago where he discussed his project of reworking physics to get rid of the real numbers, and only use computable ones.
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  20. For some reason I want to read “ß” with that really bad Daffy Duck-style lisp. As in: “You’re deßßßßpicable!”
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  21. 5:33 After he said at 2:38 that such cutting and rejoining was something that “mathematicians would never do” ...
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  22. Because it covers less than 12 facets!?
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  23. 3:10 You did say “number”, not “integer”, though.
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  24. He’s just a numberizer and a philnumberer.
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  25. 1:23 Fun fact: there was no universal rotary-dial layout. For example, here in 🇳🇿, the numbers increased clockwise, with the 1 down at the bottom next to the 0. Further fun fact: This meant that the number of pulses generated for each digit no longer matched up with the digit.
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  26. That was my guess, too. Those spiky bits are probably not rigid. Wonder if they would bend or break? Or both?
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  27. I can’t imagine being allergic to oranges. To me, citrus fruit are such happy fruit.
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  28. 2:58 What kind of programmer computes an absolute value from the square root of a square? Must be a maths major.
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  29. Fermi was the one who pointed out that, almost regardless of what numbers you plugged into the Drake Equation, over a very wide range of assumptions, if it was possible for just one or two civilizations in our galaxy to achieve interstellar travel, they would likely be all over the galaxy by now.
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  30. Real mathematicians call it the “hexahedron”. Definitely not the “sexahedron”, though. And never put it in the “Large Hedron Collider”, nooooooo ...
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  31. In terms of countable versus uncountable infinity, I think the usual proof of this, Cantor’s famous diagonal construction, has a flaw in it: the numbers that his construction produces all seem to belong to the set of computable numbers. But the cardinality of the computable numbers is known to be the same as that of the integers. So how does that prove the existence of a set with a larger cardinality?
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  32. Go look up the definition of a “computable number”, and you should see what I mean.
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  33.  @PattyManatty  Show how the Cantor construction encompasses all real numbers, then.
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  34. It takes an infinite number of steps to construct that number. It is possible to make up the list so that, after the first n steps of the Cantor construction to produce the first n digits of the number, the result will always provably lie within the first 10**n elements of the list. So you can never actually come up with a number not on the list!
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  35. Kind of begs the question: if their tests showed this layout was better than the calculator layout, why did calculators adopt the calculator layout?
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  36. Most of them actually fit inside your head.
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  37. Rest In Peace, John.
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  38. 11:43 But do all of these other instances form infinite regressions of gaps that cannot be completely plugged, as in Gödel’s original construction?
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  39. Here’s a more imaginative answer: 101 (prime) as the difference of two squares: 101 = 10² - i² Or in other words, the factors are 101 = (10 + i)(10 - i) Are these called “Gaussian integers”?
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  40. Electrical/electronics engineering uses “j” to avoid confusion with “I” for current, I believe. By the way, Python also uses “j”, but you have to use it as a suffix on a numeric literal:
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  41. Fractals seem like a very convincing illustration of infinity within a finite space.
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  42. 6:56 A “milliard” could also be a “sesquillion”, couldn’t it -- a million to the power 1½.
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  43. 7:11 “Frezznell” and “Yoo-ler” ... tisk-tisk.
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  44. In Python you can use a generator function to emit the successive candidate positions for each queen. And make it a recursive generator to handle the remaining queens. On modern machines, even a “slow” language like Python can find all the solutions very quickly.
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  45. Actually even back in the days of 32-bit machines, language compilers would offer software support for 64-bit integers.
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  46. Don’t believe the hype. Rbola.
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  47. Ah, but they didn’t specify the rotation axis, did they? That meant they were only thinking in 2 dimensions.
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  48. Yes it is. Because reductio ad absurdum (look it up).
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  49. Actually, that was never in doubt. Simple thought experiment: 1) Is the Universe too complex to have arisen by itself? 2) If so, something or someone had to have created it. 3) Was this something or someone more complex than the Universe? 4) If so, how could it have arisen by itself? Doesn’t really matter whether you answer “yes” or “no” at step 3, you still end up with the conclusion that “yes, no matter how complex it is, the Universe could indeed have arisen by itself”.
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  50. Being an upside-down delta, It could be a Southern Hemisphere Greek letter. You know, Greek from Australia.
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  51.  @ythehunter755  There were several varieties of Ancient Greek. Consider Classical Greek versus Cretan Linear B Greek.
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  52. Then in India they have “lakh” and “crore”.
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  53. This has got to be the most pointless and beautiful thing I’ve seen today. 🏆👏👍
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  54. 5:59 That first term is in fact 1 ÷ 0!
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  55. 6:06 Convolutions are commonly used in digital signal processing. Interesting to see them pop up in epidemiology as well. What next -- Fourier transforms? (The mathematics of time series is a slippery slope...)
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  56. But what if each component of the quaternion was not a real number, but a complex number? In other words, introduce a second level of complexification, if you like, where you could have real/complex/quaternion/etc at each level, independently. What properties would the resulting numbers have? And then, of course, what happens if you have 3 or more levels?
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  57. 1:26 No chance you could get that exact answer with logs to only 4 decimal places.
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  58.  @JobBouwman  So far 5 people have disagreed with you. Likely more, except YouTube doesn’t show downvotes.
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  59.  @JobBouwman  Up to 8 and counting.
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  60.  @JobBouwman  Maybe you should just delete your comments and move on.
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  61.  @JobBouwman  Comments on what? Try considering that question before hitting that keyboard again.
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  62.  @JobBouwman  So you made an irrelevant comment on something you thought was an irrelevant comment? Remember, it’s not too late to delete your comments.
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  63. So in higher dimensions, the regular polytopes consist of the simplexes, the hypercubes, and the duals of the hypercubes. Q: Why not the duals of the simplexes as well? A: Brpnhfr gur fvzcyrkrf ner gurve bja qhnyf.
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  64. I wonder what bait you could use for a trap. “Here, virus virus ...”
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  65. That’s pretty cool. Do you have a stationery storeroom or cupboard? Something like 50 reams pf paper would have 1000 times as many sheets as that, and that would be how much you would need to print a billion digits. Still not too big to exceed the school’s entire paper supply, I would imagine. ;)
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  66. 9:12 The Wile E Coyote guide to drying a Klein Bottle.
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  67. 1:39 So the number of crossings in the not knot is ... nought.
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  68. 11:44 There is only approximate equality between this formula and the construction. Remember the construction consists of a sequence of quarter-circles, so the radius only decreases in discrete steps when going to the next quarter-circle, whereas this formula produces a continuously-decreasing radius of curvature.
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  69. 4:17 That’s Clifford Stoll. You know, The Cuckoo’s Egg ? Yeah, that Clifford Stoll.
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  70. 2:17 8:17 Actually, no, the Wankel rotor doesn’t need to be a Reuleaux triangle. The curvature of the sides can be shaped to give a particular compression ratio. Remember, the rotor slides around the casing while spinning, it doesn’t roll, so the Reuleaux property is irrelevant. In fact, you can see the discrepancy between the curvatures of the Reuleaux and the actual rotor in that diagram at 2:17.
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  71. 0:22 Pigments or dyes. The difference is, pigments are inorganic compounds, dyes are organic. I think.
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  72. 4:39 You could just say it’s a differential equation, and density is the differential of mass.
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  73. What’s different about the pentagon is that it cannot make a periodic tiling of the plane.
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  74. Seems on a par with ancient Cretan writing, don’t you think? That is, primarily for accounting, not for anything literary.
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  75. Common programming languages nowadays let you express numbers in binary, octal, decimal and hex bases, and that’s it. Ada, on the other hand, let you use any base from 2₁₀ to 36₁₀, inclusive. E.g. 9 * 6 = 13#42#
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  76. 8:18 I’m thinking that the common cinematic screen aspect ratio of 2.35:1 isn’t too far off the silver ratio.
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  77. Has anybody mentioned that the Brits kept the flaw secret, so they could sell Enigma machines to unsuspecting victims other countries after the war?
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  78. 0:15 >>> import math >>> print(math.isnan(math.inf)) False Python seems to disagree with the notion that infinity is not a number.
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  79. 0:23 Positive integer ℤ⁺. Aka a natural number ℕ.
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  80. 3:02 If all the possible moves are recursively enumerable, then it is still possible to conduct an exhaustive search to any given finite depth. That won’t be enough to prove impossibility, but it could still find a solution if one exists, at least not too far along.
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  81.  @Om_1337  To show that the holy book was wrong!?
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  82. 17:06 It makes sense for finite-sized particles, not infinitesimally-small ones. Do I sense a repeat of the whole Planck-oscillator business? That if you take the classical formula down to the zero limit the numbers blow up, but if you stop at a nonzero size, you can get answers that make sense? That are quantized, even? Just suggesting that you stop thinking like a mathematician, and start thinking like a physicist ... ;)
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  83. He’s really Wallace, isn’t he? But where’s Gromit?
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  84. 7:03 A patch of nonzero area, in other words.
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  85. >>> 1j * 1j
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  86. (-1+0j)
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  87. 1:30 It’s the infinity of the counting numbers, which is probably why it’s called “countable”.
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  88. 1:07 Low-poly fingernails? Could a new fashion.
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  89. Fun fact: Don Knuth’s first published article was in MAD Magazine, a spoof item on the “Potrzebie system of weights and measures”.
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  90. 10:18 There seems to be an asymmetry in the diagram, in that 1 and 2 occur three times each, the others only twice.
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  91. 6:46 Not ASCII, but Unicode! ASCII has no code for “ö”, while Unicode has a code for everything.
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  92. I think it’s called a “perfect information” game in game theory.
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  93. The way I learned it, you had this hierarchy of mathematical structures, e.g. groups, rings, fields, number systems. A “group” starts out with one operation defined on its elements (you might call it “addition” or “multiplication” or whatever), while a “number system” has two separate operations (so you call one “addition” and the other “multiplication”), and they have to interact in certain ways: the operations must be closed on the set of elements, every addition and (nearly) every multiplication must have an inverse, and so on.
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  94. 2:26 But air, being compressible, means that more mass might be entering a point than leaving it, if the pressure (and hence density) at that point is rising, and conversely if it is falling.
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  95. Nice animations.
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  96. 5:28 Should really use “×” instead of “x”.
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  97. 18:58 It seems to me you are confusing the physics with the mathematics. The mathematics predicts physically-impossible situations, and you feel that means something is missing in the maths. But it’s not missing there, it’s missing in the physics. Understanding the discrepancy isn’t going to come from any deep insight into Navier-Stokes as it stands, but in altering the model in some way (like introducing the quantization limit I mentioned earlier) to be more physically accurate. Will this lead to new maths? Maybe ... maybe not.
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  98. 6:18 There is a problem with that construction: it never terminates. Remember what an “algorithm” is: it’s supposed to produce an answer after a finite number of steps. Therefore, the Cantor diagonal construction is not an algorithm. So your proof that the infinity of the reals is greater than the infinity of the integers can never actually be completed.
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  99.  @MuffinsAPlenty  The specification of each decimal place allows you to define convergence. That still doesn’t allow for Cantor’s diagonal proof.
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  100.  @MuffinsAPlenty  But the Cauchy construction only gives you one real. At each step, you get closer to the number you are approximating. Cantor’s construction is infinite and does not converge: after any number of steps, you are no closer to the final answer than when you started.
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  101. You could flip the proof on its head: you could say that, every time the diagonal construction produces a new number, you can always add it to the list. Or, look at it this way: it takes an infinite number of steps to produce a new number, but only one step to add it to the list. Or, to put it another way, in an infinite construction, who gets the last word?
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  102. I don’t know why. His theorems are so straightforward. It’s like basic knowledge that every programmer should have absorbed into their DNA.
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  103. 1:55 If the board wrapped around, then that open tour would become a closed tour.
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  104. 7:03 Is the natural language of growth only in the real numbers. Once you go into the complex plane, you see that it is the natural language of trigonometry as well. Where you have periodic functions, rather than infinitely growing ones. And Euler’s famous identity.
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  105. Maybe it should have been called “Tralfamadorean” arithmetic. The Moon is not that far from the Earth, after all.
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  106. 15:48 Don’t we all, JHC, don’t we all ...
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  107. 10:37 But can you in fact formally prove that you can “keep on playing the same game”?
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  108. 7:17 In other words, if you pulled apart the bits and randomly stuck them back together, you only have a 1 in 12 chance that the result can be solved.
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  109. 0:47 See, I was taught that the “natural” numbers ℕ did not include zero, while the “whole” numbers 𝕎 did.
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