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Having good “small” rational approximations is actually a characterization of transcendental numbers: there are upper bounds for how good rational approximation
by mananaysiempre 4y ago
Having good “small” rational approximations is actually a characterization of transcendental numbers: there are upper bounds for how good rational approximations to an algebraic number can be as the maximum allowed denominator grows, so by proving that this bound is violated for your number you can prove it’s not algebraic; that’s how people initially went about constructing transcendental numbers and—later—proving π and e were such.
(Nowadays, it’s easy to construct a transcendental number because it’s easy to construct a noncomputable one; constructing a transcendental computable number still requires additional ideas such as those bounds—I don’t really know of a simple way to do it.)
- actually_a_dog 4y agoHere's a computable transcendental number for you: https://en.wikipedia.org/wiki/Liouville_number#The_existence_of_Liouville_numbers_(Liouville's_constant https://en.wikipedia.org/wiki/Liouville_number#The_existence...)
- mananaysiempre 4y agoProving it’s transcendental still requires the bound I was talking about, which to my admittedly very ignorant eyes looks like it doesn’t lead to anything interesting outside number theory. This is unlike the undecidability of halting needed to construct a noncomputable number, or Cantor’s theorem used to show they exist; both are also quite simple conceptually, containing a grand total of one idea each and basically no computations. (The Lindemann–Weierstrass theorem you’d usually use for proving e or π is transcendental at least looks like interesting algebra.)