Mathematical proof demonstrates the partition of rational numbers into disjoint monotone sequences using an integer lattice, highlighting a hidden geometric order in dense sets.
SummaryWe show that the rational numbers can be partitioned into disjoint, strictly increasing sequences so that each rational number appears as the limit of exactly one sequence. This uniqueness requirement imposes a rigid global structure on a set that is usually viewed as highly entangled. Our construction is geometric: by representing rational numbers as slopes in the integer lattice and applying a simple vertical shift, the partition emerges naturally from the geometry of the plane. This viewpoint makes both the monotonicity of the sequences and the uniqueness of their limiting values transparent.
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Aleams Barra (2026) studied this question.
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