Irrational numbers/Structural properties/Exercise

Let ${}u\in \mathbb {R}$ and ${}v\in \mathbb {Q}$.

1. Show that ${}u$ is an irrational number if and only if ${}u+v$ is irrational.
2. Suppose now also that ${}v\neq 0$. Show that ${}u$ is irrational if and only if ${}u\cdot v$ is irrational.

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