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I'd expect this question to be asked here before, but I've not been able to find it.

The generalized definition of the multiplication operator for complex numbers is simple:

The product of the lengths and the sum of the angles (relatively to the $X$-axis).

But what is the generalized definition of the $<$ and $>$ comparators for complex numbers?

barak manos
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2 Answers2

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There isn't one. There is no ordering compatible with the algebraic structure as there is for the case of real numbers. (Eg because everything is a square.)

Occasionally it is useful shorthand to write $z>0$ or something similar when referring to a condition on a complex number $z$, but this means that $z$ is real and greater than 0.

Jonas Meyer
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One interesting partial order on $\mathbb{C}$ is the following: $z \leq z'$ if $| z | = 0$ or ($| z | \leq | z' |$ and $\arg z = \arg z'$).

polmath
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  • I'm curious, is this useful? It is potentially confusing because of things like $1>0$ and $-1>0$ conflicting with the usual real order. – Jonas Meyer Sep 17 '14 at 12:17
  • It extends the natural order of $\mathbb{R}_+$ but not that of $\mathbb{R}$. With this partial order $\mathbb{C}$ is thought as a union of half-lines intersecting at $0$. There is a link with the notion of inverse semigroup. – polmath Sep 17 '14 at 12:26
  • Could you please indicate what the link to inverse semigroups is? – Jonas Meyer Sep 17 '14 at 12:44