Timeline for Can there be a numerical system in which logarithms can be expressed in terms of exponentials in closed form?
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15 events
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| May 15, 2020 at 10:16 | comment | added | YCor | Related subsequent question: mathoverflow.net/questions/297454/… | |
| Apr 13, 2017 at 12:19 | history | edited | CommunityBot |
replaced http://math.stackexchange.com/ with https://math.stackexchange.com/
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| Nov 13, 2016 at 21:47 | comment | added | Timothy Chow | I think the question is whether the log function can be expressed using a finite number of compositions of arithmetic operations and the exponential function, starting with a finite number of constants ($w$ in Anixx's notation). The answer is "obviously" no but I think it is not trivial to prove it. | |
| May 3, 2016 at 13:08 | history | edited | Emil Jeřábek | CC BY-SA 3.0 |
fix misleading typo
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| May 3, 2016 at 12:10 | answer | added | Mikhail Katz | timeline score: 1 | |
| May 3, 2016 at 11:50 | comment | added | Loïc Teyssier | "people there just do not understand the question". Well, I don't mind confessing my thickness, but I don't understand the question either. I don't know what a "numerical field" is, and I don't know how you're going to extend the exponential/logarithm to that set. Except tautological answers like BenMcKay's, I just don't know what new "numbers" you'd expect. Just because you state "I wish I knew some system in which such and such formula is true" does not mean you ask a well-formed question and that everybody is supposed to understand its meaning. | |
| May 3, 2016 at 11:49 | review | Close votes | |||
| May 3, 2016 at 18:08 | |||||
| May 3, 2016 at 11:37 | history | edited | Anixx | CC BY-SA 3.0 |
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| May 3, 2016 at 11:08 | history | edited | Anixx | CC BY-SA 3.0 |
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| May 3, 2016 at 11:08 | comment | added | Ben McKay | It is a field: commutative, associative, distributive, and every nonzero element has a multiplicative inverse. | |
| May 3, 2016 at 10:53 | comment | added | Anixx | @Ben McKay fine, I would not expect it to be an algebraic extension. By "numerical field" I understood a field that would have the majority of properties of real/complex numbers (that is commutativity and associativity of multiplication etc). At least what allows to call say, hyperreal numbers still "numbers". | |
| May 3, 2016 at 10:48 | comment | added | Ben McKay | I don't know. What is a numerical field? It won't be a number field, or an algebraic extension field. | |
| May 3, 2016 at 10:41 | comment | added | Anixx | @Ben McKay what you are saying is interesting. Will this extension satisfy the usual notions of a "numerical " field? Is it possible to somehow derive other algebraic properties of such extension? | |
| May 3, 2016 at 10:38 | comment | added | Ben McKay | Anything is possible: take the set of solutions $z,w$ to your equation, as a Riemann surface inside $\mathbb{C}^2$. Then on that set, $w$ is a function satisfying your equation, by definition. The field of meromorphic functions of the $z$ variable pulls back, from projection $(z,w) \mapsto z$ to live inside the meromorphic functions on that Riemann surface. Not a satisfying or explicit solution, more like a solution by definition. | |
| May 3, 2016 at 9:59 | history | asked | Anixx | CC BY-SA 3.0 |