Comments on 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?
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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?
Why we can analytically augment the algebraic definition of $ε$ in dual numbers so to distinguish $ε$ from $-ε$ but cannot do so in complex and split-complex numbers? Is there a theorem on this?
Algebraically one cannot distinguish between $\varepsilon$ and $-\varepsilon$. But it is possible to augment their definition analytically so to distinguish them.
In dual numbers there is a common equality: for differentiable at $x=a$ function $f(x)$, $f(a+b\varepsilon)=f(a)+b\varepsilon f'(a)$.
Now, one can define that if at point $x=a$ $f(x)$ has right and left derivatives $f'_r(a)$ and $f'_l(a)$, and they are not equal, then $f(a+\varepsilon)=f(a)+\varepsilon f'_r(a)$ and $f(a-\varepsilon)=f(a)-\varepsilon f'_l(a)$. In other words, $\varepsilon$ is defined as a positive infinitesimal, and $-\varepsilon$ is defined as a negative infinitesimal. This provides an optional analytic definition distinguishing $\varepsilon$ from $-\varepsilon$, but the algebraic structure can work just well without such additional analytic property (but with it one can evaluate more functions at more dual numbers).
Still, in dual numbers one cannot distinguish $\varepsilon$ from $a \varepsilon$ when $a>0$ even with this analytic addition.
So, my question is: why is it possible to define dual numbers in such a way so to distinguish the sign of dual unity, but the same cannot (?) be done with sign of imaginary and hyperbolic unities and with scale in dual numbers?
Is there any strong argument, why?
Is it because the lexicographical ordering in dual numbers naturally embeds into ordering of reals, while in complex and split-complex numbers it does not?
The same question on Mathoverflow
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The reason is clear enough from what you stated. $-\epsilon < 0 < \epsilon$, so the ordering properties pick out which one is which. And so when one is defining left and right derivatives, one defines them with reference to the ordering properties of $\mathbb{R}$, and so one is able to distinguish between $\epsilon$ and $-\epsilon$.
Note that in contexts where left and right derivatives are not distinguished (e.g. in smooth infinitesimal analysis), one does not have this ability to "pick out" $\epsilon$ to uniquely distinguish it from $-\epsilon$.
Basically, there are symmetries in the underlying algebra, and to distinguish between a number (like $\epsilon$ or $i$) and its conjugates under these symmetries (like $-\epsilon$ or $-i$) one has to more or less explicitly "break" the symmetry. One can prove that this is the case by simple metalogical reasoning, systematically replacing all occurrences of a number by occurrences of its conjugates and verifying that the axioms remain invariant under this transformation.
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