Deciding which number is bigger Suppose that $a$ and $b$ are two algebraic numbers with $0<|a-b|\approx 10^{-50}.$ Suppose further that a calculator can evaluate $a$ and $b$ up to say 12 digits. Are there some general transformation I can do such that my calculator will output if $a<b$ or not?

Suppose that $f(x)\in \mathbb{Q}[x]$ has roots $\{\alpha_1,\ldots,\alpha_k\}$ and $g(x)\in \mathbb{Q}[x]$ has roots $\{\beta_1,\ldots,\beta_m\}$.  Let $m(x)$ denote the minimal polynomial for $\alpha_1-\beta_1$ (e.g.) over $\mathbb{Q}[x]$, which can be found as a factor of the polynomial
$$F(x)=\prod_{i,j} (x-\alpha_i+\beta_j).$$
Thus $m(x)$ has a root very near $0$, the sign of which can be determined (heuristically) by looking at $m(0)$ and the sign of $m'(0)$.  To make the answer obtained here precise, we just need to bound the other roots of $m$ away from zero, which is not too hard (the bound comes from the discreteness of the roots of $f$ and $g$.
