Timeline for Why we can analytically define $ε$ in dual numbers so to distinguish $ε$ from $-ε$ but cannot do so in complex and split-complex numbers?
Current License: CC BY-SA 4.0
21 events
| when toggle format | what | by | license | comment | |
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| S Dec 2, 2022 at 8:28 | history | suggested | J. W. Tanner | CC BY-SA 4.0 |
improved English
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| Dec 2, 2022 at 3:13 | review | Suggested edits | |||
| S Dec 2, 2022 at 8:28 | |||||
| Jun 6, 2022 at 14:17 | comment | added | Simon Henry | well I don't know what you did, but you probably did it wrong, if you're arrow point right on one side, it will point left on the other side... But anyway, as I said that's not even relevant to what we were saying. I think the difference you are seeing between $\mathbb{R}$ and $\mathbb{C}$ is that the orientation of $\mathbb{R}$ is determined by its algebraic structure (we have a convention that "1" is right of "0" that gives the orientation). While the orientation of $\mathbb{C}$ is not determined by the algebraic structure (conjugation change the order but preserve the algebraic structure). | |
| Jun 6, 2022 at 10:33 | comment | added | Anixx | "draw an arrow on a window and try it" - did it. Seems, I am right. | |
| Jun 3, 2022 at 15:16 | comment | added | Simon Henry | Ok, This is both not true (draw an arrow on a window and try it) and not relevant (the structure you are adding to $\mathbb{R}[\epsilon]$ is not the orientation of the line, but the definition of $f(a+b\epsilon)$. You are choosing to make $\epsilon$ corresponds to right, but you could exchange the role of $+\epsilon$ and $-\epsilon$). To give an example closer to what you do, I could say that I distinguish $i$ from $-i$ because the imaginary part of $i$ is $1$. Whatever you want to answer to this, my only point is that you can say the same thing about your definition. | |
| Jun 3, 2022 at 1:10 | comment | added | Anixx | Orientation in $\mathbb C$ changes if we look at the plane from the other side. Direction on real line does not. | |
| Jun 2, 2022 at 21:16 | comment | added | Simon Henry | How is defining $f(a+b\epsilon) = f(a)+b f'_r(a)$ not arbitrary ? I could equally well put the left derivative instead. Because you have somehow decided that $\epsilon$ corresponds to "right", you have added a structure that distinguish between $\epsilon$ and $-\epsilon$. That is not different from choosing an orientation on $\mathbb{C}$. | |
| Jun 2, 2022 at 16:21 | comment | added | Anixx | Hmm. Positive direction on the real line, left and right derivatives are not arbitrary but how one could define "positive orientation" unambiguously without circular referring to $i$? | |
| Jun 2, 2022 at 16:20 | comment | added | Simon Henry | Note that my point isn't that these additional structure are "natural" or not, but that deciding what is natural or not is arbitrary. And you won't be able to distinguishes $i$ and $-i$ using only algebraic properties and "analytical properties" (in the sense of using analytical functions) as these are all preserved/equivariant under the conjugation automorphisms. | |
| Jun 2, 2022 at 16:17 | comment | added | Simon Henry | The kind of objection you voice against the structure I added to distinguish $i$ and $-i$ can be said voiced about how you chose to define $f(a+b\epsilon)$ : you are chosing to use the left derivative if $b>0$ and the right derivative if $b<0$, but you could have made the other choice. By making this choice you are adding structure, which can then be used to distinguishes element that were not previously disinguishable. That is no different than chosing what will be considered a "postive orientation" for curve in the complex plan. | |
| Jun 2, 2022 at 13:08 | comment | added | Anixx | By considering directional derivatives we can even define analytically $\varepsilon i, \varepsilon j$, etc, such as $\lim_{h\to0^+} \frac{f(x+hi)-f(x)}h=f(x+\varepsilon i)$ | |
| Jun 2, 2022 at 13:05 | comment | added | Anixx | On the dual numbers we can define $\varepsilon$ implicitely assuming order but not introducing it explicitly, only via right and left derivatives. | |
| Jun 2, 2022 at 13:03 | comment | added | Anixx | "If you prefer, I can say that ∫γ1/zdz=2iπ where γ is curve around 0 parametrized in the positive direction." - this is tautology because instead of "positive direction" one simply can say "the direction o where $i$" goes after $1$. It is provable that such integral would be $2i\pi$. Even if we introduce lexicographical order on $\mathbb C$, we still could not define $i$ other than by saying "$i>-i$". | |
| Jun 2, 2022 at 6:51 | comment | added | Anixx | Exactly, but can we define difference between $i$ and $-i$ analytically and without breaking properties? What about split-complex $j$? If this is possible for duals, it could be possible for other kinds of numbers. | |
| Jun 1, 2022 at 23:29 | comment | added | Simon Henry | If you prefer, I can say that $\int_\gamma 1/z dz =2i \pi$ where $\gamma$ is curve around $0$ parametrized in the positive direction. The notion of "positive direction" involve the choice of an orientation of $\mathbb{C}$. | |
| Jun 1, 2022 at 23:26 | comment | added | Simon Henry | The exemple of the lexicographical order wasn't meant to be natural. But the point is that the notion of what is "natural" or not, is fairly arbitrary. By orientation of a vecteur space I mean as in: en.wikipedia.org/wiki/Orientation_(vector_space) This is an additional structure you put on $\mathbb{C}$ so it won't be definable in terms of the algebraic structure nor its analytical properties (given these are all preserved by complex conjugation). But this is no different from the fact that the way you distinguished $\epsilon$ from $-\epsilon$ depended on other structures. | |
| Jun 1, 2022 at 23:01 | comment | added | Anixx | I would be glad if I could see some kind of natural analytic property that could distinguish between $i$ and $-i$. I was thinking about $\ln(-1)=i\pi$ or $\int_{-1}^1 1/x dx=-i\pi$ but faced objections. | |
| Jun 1, 2022 at 22:57 | comment | added | Anixx | "But what about the structure of "being oriented"" - so far I do not see how this property can be defined... | |
| Jun 1, 2022 at 22:53 | comment | added | Anixx | For instance, if $i$ is between $0$ and $0.0001$ (as in lexicographical order), its magnitude also should be in between, but then how its square has magnitude of $1$? | |
| Jun 1, 2022 at 22:51 | comment | added | Anixx | "one could (...) put a "lexicographical order" on it" - the problem is, lexicographical order is usually considered not natural on $\mathbb C$. Not canonical, not making an ordered field, etc. On duals it is somehow more natural. | |
| Jun 1, 2022 at 22:10 | history | answered | Simon Henry | CC BY-SA 4.0 |