1998/05/08 by Timothy Y. Chow, Chow, Timothy Y.
Mathematics · #03B25 (Secondary) #11J81 (Primary) 12F20 #68Q40 #Combinatorics (math.CO) #FOS: Mathematics #History and Theory of Mathematics #Logic (math.LO) #Mathematics and Applications #Number Theory (math.NT) #math.CO #math.LO #math.NT #msc:03B25 #msc:11J81 #msc:12F20 #msc:68Q40
paper · pdf · doi:10.48550/arxiv.math/9805045
11 pages; submitted to Amer. Math. Monthly
arxiv created 1998/05/08 · openalex publication_date 1998/05/08 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
If a student asks for an antiderivative of exp(x2), there is a standard reply: the answer is not an elementary function. But if a student asks for a closed-form expression for the real root of x = cos(x), there is no standard reply. We propose a definition of a closed-form expression for a number (as opposed to a *function*) that we hope will become standard. With our definition, the question of whether the root of x = cos(x) has a closed form is, perhaps surprisingly, still open. We show that Schanuel's conjecture in transcendental number theory resolves questions like this, and we also sketch some connections with Tarski's problem of the decidability of the first-order theory of the reals with exponentiation. Many (hopefully accessible) open problems are described.