Example 1

Every Hardy field contains the field of rational constants (the smallest field of characteristic 0, cf. Alg., V, §1), which one can identify with the field $\mathbf{Q}$; moreover, since two constants cannot be congruent modulo $R_\infty$ unless they are equal, $\mathbf{Q}/R_\infty$ is identical to $\mathbf{Q}$. The real constants also form a Hardy field, which one can identify with $\mathbf{R}$.