Commensurability/Q and R/Equivalence relation/Section

The side length and the diagonal in a square are not commensurable.

Definition

Commensurability/Definition

Definition

Commensurability/Real numbers/Not 0/Definition

This is the equivalence relation for the subgroup

${}(\mathbb {Q} ^{\times },1,\cdot )\subseteq (\mathbb {R} ^{\times },1,\cdot )\,$

in the sense of definition.

Remark

Commensurability/Historic/Motivation for real numbers/Remark

The subgroup relation

${}(\mathbb {Q} ,0,+)\subseteq (\mathbb {R} ,0,+)\,$

(which might be considered as a relation of linear subspace of ${}\mathbb {Q}$-vector spaces) yields also an equivalence relation on the real numbers. Here, two real numbers are equivalent if their difference is a rationale number.