Interesting Esoterica

What is a closed-form number?

Article by Timothy Y. Chow
  • Published in 1998
  • Added on
If a student asks for an antiderivative of exp(x^2), 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.

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BibTeX entry

@article{Whatisaclosedformnumber,
	title = {What is a closed-form number?},
	abstract = {If a student asks for an antiderivative of exp(x^2), 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.},
	url = {http://arxiv.org/abs/math/9805045v1 http://arxiv.org/pdf/math/9805045v1},
	year = 1998,
	author = {Timothy Y. Chow},
	comment = {},
	urldate = {2019-01-09},
	archivePrefix = {arXiv},
	eprint = {math/9805045},
	primaryClass = {math.NT},
	collections = {notation-and-conventions,the-act-of-doing-maths}
}