Let $\alpha$ be a positive real number. Does it make sense to define the closest rational to $\alpha$ as the number $R(\alpha)=\frac{p_1}{p_2}$ such that $p_1,p_2$ are positive co-prime integers minimizing $p_2 \cdot |p_2\alpha - p_1|$? Clearly, there are going to be some irrational numbers for which this makes no sense, for instance $\alpha=\frac{1+\sqrt{5}}{2}$. My guess is that such numbers are rare: there are countably infinitely many of them, or their set has Lebesgue measure 0. Do we have $R(\pi) = \frac{355}{113}$? I looked at the fist 12 convergents of $\pi$, and $\frac{355}{113}$ achieves the minimum.

This is related to approximations of irrational numbers, continued fractions, and the irrationality measure of a number. More generally, we could define the closest algebraic number of degree $d$, as the number $R_d(\alpha)$ defined as follows.

$$P^*= \arg\min_P\Big(H(P)\Big)^{\mu(d)}\cdot |P(\alpha)|,\\ R_d(\alpha) = \Big(H(P^*)\Big)^{\mu(d)}\cdot |P^*(\alpha)| $$

where $P$ is any polynomial of degree $d$ with integer coefficients, with highest and lowest coefficients not equal to $0$, and $H(P)$ is the height of $P$, that is, its highest coefficient in absolute value. To make this work for most $\alpha$, how should we choose $\mu(d)$? Does $\mu(d)=d$ work? It seems to work if $d=1$.

As of now, as far as I know, all results involving $d>1$ are conjectures. Related material includes the Wirsing conjecture. See also the "Generalizations" section in the Wikipedia article on Roth's theorem, here.

**Update on Feb 8, 2021**: It is possible that the best approximation, if $d$ is an even integer, may be a complex number. Also, see my new question here, about approximations by dyadic fractions. The plan is to look at approximations using the first $n$ digits of $\alpha\in [0, 1]$ in base $b$, where (say) $b=\sqrt{2}$, focusing on values of $n$ where a long run of zeros start, leading to approximations of transcendental numbers by quadratic irrationals. This is briefly discussed in the comments in my new question.