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Right, it is a funny way to say it. Computationally, we all only ever work with approximations, but when doing mathematics, pi is exact all the way out to the
by ncmncm 4y ago
Right, it is a funny way to say it.
Computationally, we all only ever work with approximations, but when doing mathematics, pi is exact all the way out to the infinity-th digit. To multiply by pi (or any irrational, but particularly transcendental) in a pure mathematical context is to audaciously specify an infinitely long computational process. It is dizzying to contemplate, almost mystical. Sort of like modular arithmetic with an infinitely-precise irrational modulus.
- api 4y agoIs any number mod infinity equal to itself?
- tromp 4y agoIn the surreal numbers [1], the definition of division and hence modulo applies to all numbers, finite and infinite alike. And for any finite number -omega < x < omega you would indeed have x mod omega == x. [1] https://en.wikipedia.org/wiki/Surreal_number https://en.wikipedia.org/wiki/Surreal_number
- SAI_Peregrinus 4y agoYou might already know about it, but Ehrlich claims[1] that the Surreals form an ordered field which is a maximal hyperreal number system in NBG, and therefore all the results of non-standard analysis "work" (up to isomorphism) in the Surreals. [1] Philip Ehrlich. "The absolute arithmetic continuum and the unification of all numbers great and small." Bull. Symbolic Logic 18 (1) 1 - 45, March 2012. https://doi.org/10.2178/bsl/1327328438 https://doi.org/10.2178/bsl/1327328438 Theorem 20
- gspr 4y agoWhat do you want mod infinity to mean?