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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