Organizing a complex sample
DOI:
https://doi.org/10.37069/3154-8229-2026-40-4Keywords:
complex plane, complex-valued sample, order relation, trichotomy lawAbstract
The problem of ordering the complex plane is highly relevant to research in various fields. This article provides a detailed review of known results regarding various methods for ordering sets of complex numbers. For a thorough investigation, we have chosen a method for ordering a sample (or subset) of complex numbers, taking into account the positions of the complex points on the plane. Since the complex plane is geometrically isomorphic to two-dimensional real space, the relative positions of complex points can be determined in a similar manner. All nine possible ways of relating two complex numbers are presented. For each case, a definition of the order relation is introduced, and it is proven that only one of the proposed relations can be established between any two complex numbers. In particular, a lemma is proven regarding the partition of the entire complex plane into a union of nine disjoint subsets. Based on the proven lemma, an analogue of the trichotomy law for complex numbers is formulated. It is shown that although the analog of the trichotomy law holds for the obtained partition of the complex plane, this does not yet imply a complete ordering of the complex plane. We analyze the possibility of partial (non-strict) and complete (strict) ordering of a complex-valued sample selected from the set of complex numbers. Cases are presented that are questionable and require further study. These are cases for which the properties of antireflexives, asymmetry, and transitivity hold, but this is insufficient to determine, for a complex-valued sample defined by these relations, whether one complex number is strictly less (strictly greater) than another. The results obtained can be used in future studies of the ordering of the complex plane.
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