The best A – A ≠ 0 paradox

Comments · 947

• MA

  @Mathologer · 3 years ago · pinned

  Eddie suggested that I ask.the keen among you the following nice question (first watch the video): How many different ways are there to rearrange a conditionally convergent series to get the sum π? Yes, of course, infinitely many. The real question is whether there are countably infinitely many or uncountably infinitely many ways?

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• AL

  @alphafound3459 · 3 years ago

  This demonstrates something non-math people don't get: Infinity is full of trap doors, subtleties, and other frustrations. The early infinity theorists like Cantor nearly lost their mind over this kind thing.

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• FF

  @FFELICEI07 · 3 years ago

  An Italian Math and CS Senior Lecturer here. Just want to share that, as it happened to the Mathologer, when my professor did the Riemann rearrangements theorem in Real Analysis I, as a freshman, was totally upset and amazed by this counterintuitive result. Congrats to The @Mathologer whose videos always make us see the things we know under new and interesting perspectives.

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• MA

  @MathFromAlphaToOmega · 3 years ago (edited)

  There's actually a sort-of-explanation for why e^π is roughly π+20. If you take the sum of (8πk^2-2)e^(-πk^2), it ends up being exactly 1 (using some Jacobi theta function identities). The first term is by far the largest, so that gives (8π-2)e^(-π)≈1, or e^π≈8π-2. Then using the estimate π≈22/7, we get e^π≈π+(7π-2)≈π+20.

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• HE

  @heuristica117 · 3 years ago

  This channel has brought me intellectual ecstasy for years

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• 左括

  @左括號 · 3 years ago

  I really love how there&apos;s subtitles for every video since I&apos;m still learning English<br>Thanks for the great content

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• HE

  @henridelagardere264 · 3 years ago

  Squeezing and stretching the snake, that sounds like lots of fun, and the result is quite beautiful indeed.

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• AB

  @ABruckner8 · 3 years ago

  As soon as you moved the negative fractions below the top line, my first instinct was &quot;Wait...isn&apos;t the top part &apos;outpacing&apos; the bottom part?&quot; Then I lost confidence when you collapsed them, lining up all pos and neg, lol. I was like &quot;but, but, but....&quot; Anyway, I love that stuff!

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• RE

  @RebelKeithy · 3 years ago

  I think it&apos;s easy to get distracted by the fact that there is a matching negative for every positive term in the sequence. A similar paradox makes it more intuitive what&apos;s wrong with rearranging terms.<br>∞ = 1 + 1 + 1 +...<br>∞ = (2 - 1) + (2 - 1) + (2 - 1) +...<br>∞ = (1 + 1 - 1) + (1 + 1 - 1) +...<br>∞ = 1 + 1 - 1 + 1 + 1 - 1 +...<br>Then we can pull out positive and negative terms.<br>1 + 1 + 1...<br>- 1 - 1 - 1...<br>So every +1 is canceled by a - 1.<br>You can even create a mapping from the nth positive 1 to the n*2 negative term, so every positive term has a negative to cancel it.<br>This, to me, intuitively shows why you can&apos;t add infinite sums by rearranging terms. You need to look at how it grows as you add terms.

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• CI

  @cipeman3498 · 1 year ago

  <a href="https://www.youtube.com/watch?v=VO2A6I3Woos&amp;t=232">3:52</a> i thought that was going to be the whole visualization and i missed that it was an april fools day upload lmao

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• НИ

  @НикитаДёмочкин-й3ж · 3 years ago

  It is always a pleasure to watch your vids. Not only because these are great educational videos, but also because your voice and wordings make them even better

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• YU

  @yummyyayyay · 3 years ago

  You&apos;ve shown me the true beauty in math. Your videos are truly intellectually stimulating

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