The formula: $$y=\lfloor x\sqrt2\rfloor$$ is expressible in first-order PA, as: $$y^2<2x^2<(y+1)^2$$ So, even though $\sqrt2$ isn't a natural number, we can still represent a formula with $\sqrt2$ in it.
My question is: For what real numbers $r$ can the sentence: $$y=\lfloor rx\rfloor$$ be expressed in first-order PA? (Does $\pi$ work? $e$? What about noncomputable numbers?)