Editing Diophantine approximation (section)

=== Approximation of a rational by other rationals ===
A rational number <math display="inline">\alpha =\frac{a}{b}</math> may be obviously and perfectly approximated by <math display="inline">\frac{p_i}{q_i} = \frac{i\,a}{i \,b}</math> for every positive integer ''i''.

If <math display="inline">\frac{p}{q} \not= \alpha = \frac{a}{b}\,,</math> we have
:<math>\left|\frac{a}{b} - \frac{p}{q}\right| = \left|\frac{aq - bp}{bq}\right| \ge \frac{1}{bq},</math>
because <math>|aq - bp|</math> is a positive integer and is thus not lower than 1. Thus the accuracy of the approximation is bad relative to irrational numbers (see next sections).

It may be remarked that the preceding proof uses a variant of the [[pigeonhole principle]]: a non-negative integer that is not 0 is not smaller than 1. This apparently trivial remark is used in almost every proof of lower bounds for Diophantine approximations, even the most sophisticated ones.

In summary, a rational number is perfectly approximated by itself, but is badly approximated by any other rational number.