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This tweet contained this image which is of course complete nonsense but it got me thinking -- is there such a thing where the distance of two points is a complex number?

enter image description here

Ps. it seems such questions are fit for this SE for example Can there be a geometry where angles can be infinite?

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    $\begingroup$ If by "distance" you mean metric, then no: a metric is by definition real valued $\endgroup$ – G. Chiusole Apr 5 at 18:06
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Akbar Azam, Brian Fisher & M. Khan (2011) Common Fixed Point Theorems in Complex Valued Metric Spaces, Numerical Functional Analysis and Optimization, 32:3, 243-253, DOI: 10.1080/01630563.2011.533046

We introduce complex valued metric spaces and obtain sufficient conditions for the existence of common fixed points of a pair of mappings satisfying contractive type conditions.

While I can't find this paper freely available online, this PDF cites it and defines " XxX→C is called a complex valued b-metric":

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  • $\begingroup$ Fair enough, I deleted my comment. That's the trouble with words, you get these "eats shoots and leaves" problems. $\endgroup$ – Guy Inchbald Apr 5 at 20:04
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There can be no such metric, as it would not obey the requirements of a metric. Specifically, you must be able to have the "$>$" operation (for the triangle inequality: $d(a,b) + d(b,c) \geq d(a,c)$), and that does not hold for complex numbers.

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  • $\begingroup$ -1: the question doesn't even contain the word "metric". And you could apply your answer to the word "measure", yet complex and even operator-valued measures are a thing (and a useful one!). $\endgroup$ – Martin Argerami Apr 6 at 4:06
  • $\begingroup$ @MartinArgerami: "distance" implies the metric, for instance "Euclidean distance" is a direct result of a "Euclidean metric." Any distance must obey the rules of a metric: $d(x,x) = 0$, $d(x,y) = d(y,x)$, and $d(x,y) + d(y,z) \geq d(x,z)$. Look it up. $\endgroup$ – David G. Stork Apr 6 at 4:51
  • $\begingroup$ @MartinArgerami is there a complex valued measure on the one dimensional Euclidean space ? $\endgroup$ – chx Apr 6 at 6:04
  • $\begingroup$ Yes, lots.$\ \ $ $\endgroup$ – Martin Argerami Apr 6 at 6:43
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A distance is always a real number, for reasons given in other answers. It is only when the angle is also introduced to make it a vector that things get complex. For example the shortest distance from one point to another is a real number, but the shortest path is a vector and may prove to be complex.

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