Consider two irrational numbers $a$ and $b$. Are there well known sufficent conditions on the relation between $a$ and $b$ that would allow me to conclude that there is $c \in \mathbb{R}$, $c \neq 0$, such that both $ca$ and $cb$ are rational?
2026-04-01 19:14:25.1775070865
Condition on two irrational numbers
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If $a=qb$ for some $q\in\mathbb{Q}$ then the condition clearly holds. For the other implication, say there exists $0\neq c\in\mathbb{R}$ with $ca$ and $cb$ are rational. Then let $q=ca/cb$. Then $a=qb$.
Therefore there exists a $0\neq c\in\mathbb{R}$ with $ca$ and $cb$ rational iff $a=qb$ for some $q\in\mathbb{Q}$.