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The infinite sum makes sense when using the p-adic number system [1] -- and measuring distance using the p-adic metric -- which forms an ultrametric space [2].

• cderwin 10y ago
  I think the problem is that the introduction of the p-adic metric was poorly motivated. It's not clear why we would want 1+p+p^2+p^3+... to converge to -1, and introducing the p-adic metric to do so doesn't show why the p-adic numbers are useful in general (all it's really saying is that the partial sums are 2^(n+1)-1). I'm sure you can derive all sorts of weird metrics so that various weird identities are true; that alone fails to make them interesting. Based on this video alone it's not clear that either the identity or the p-adic norm are interesting in any non-trivial sense of the word. The result is that the introduction of the p-adic metric is hard to follow and the resulting identity seems arbitrary, even if you manage to follow the bit about the metric. (And these combined with a lack of rigor where it's needed seem to be recurring problems in 3Blue1Brown videos.)
  • t-ob 10y ago
    > I'm sure you can derive all sorts of weird metrics so that various weird identities are true On the rational numbers, at least, the p-adic metrics are more or less your whole lot, according to Ostrowski's Theorem [1]. There is a kind of cognitive hurdle everyone who studies these numbers has to clear, in that things that should be "large" turn out to be very small indeed, when viewed under a p-adic lens. I think it's more instructive to build up the ring of p-adic integers first [2, chapter 2], and construct the p-adic numbers from there. I can assure you they are very useful, though! A general theme in number theory is to take a "global" problem, defined over the integers, and to translate it into infinitely many "local" ones (over the p-adics, for each prime p). These are sometimes easier to solve and, if you're lucky, offer insight into the global solution you're looking for. [1]: https://en.wikipedia.org/wiki/Ostrowski's_theorem https://en.wikipedia.org/wiki/Ostrowski's_theorem [2]: http://www.springer.com/gb/book/9780387900407 http://www.springer.com/gb/book/9780387900407
  • posterboy 10y ago
    >doesn't show why the p-adic numbers are useful in general well, you certainly are neither more help either ;)
  • stablemap 10y ago
    My short advertisement is that it has proven very useful to study Diophantine equations by reducing mod n. The Chinese remainder theorem [1] tells you to focus on reducing mod p^r, where p is a prime. If r = 1 then you are working in a field but in the ring Z/4, for example, I know 2 ≠ 0 and yet 2·2 = 0. To make the p-adics Z_p I stitch all of these Z/p^r together: an element is a choice of a_r in each Z/p^r and these have to be compatible: a_2 reduces mod p to a_1 and so on. The resulting Z_p has no "zero divisors", and if I allow myself to invert p I get a field Q_p. This is a huge improvement, a foundation on which to build analysis and geometry as we did over R. [1] https://en.wikipedia.org/wiki/Chinese_remainder_theorem https://en.wikipedia.org/wiki/Chinese_remainder_theorem