Why, in the context of p-adic numbers, do we have the convention $$\mathbb{Q}_\infty = \mathbb{R} \quad$$ ?
It must have something to do with the generalization of the Legendre-symbol for $\mathbb{R}$: $$\left(\frac{\cdot}{\infty}\right) = \mathrm{sgn}(x)\, ,$$ but why does "$\infty$" make sense here?
The archimedean absolute value is denoted $|\cdot|_\infty$ for notational conveniency, so one can write for instance $$ \prod_{p\leq\infty}|q|_p=1\qquad\forall q\in\Bbb Q, q\neq0. $$ Thus, as $\Bbb Q_p$ denotes the completion of $\Bbb Q$ with respect to $|\cdot|_p$, it is more than natural to denote $\Bbb Q_\infty$ the completion of $\Bbb Q$ with respect to $|\cdot|_\infty$.