Rational number ($\mathbb{Q}$)

A rational number is a quotient of two integers with nonzero denominator.

The increasing list of reduced rational numbers in $[0,1]$ with denominators at most its level $Q$, including zero and one.

A member of a Farey sequence. Two distinct level-$Q$ fractions have ordinary distance at least $Q^{-2}$. On the quotient circle identify zero and one before forming a set; the shorter wrapped distance of two remaining distinct fractions obeys the same bound.

A reduced fraction $a/b$ has coprime integers $a,b$ and positive denominator $b$. Distinct reduced fractions with denominators at most $Q$ differ by at least $Q^{-2}$, since their difference has a nonzero integer numerator over a denominator at most $Q^2$.

Between any two distinct real numbers lies a rational number.

A real number is rational exactly when its expansion in any integer base is eventually periodic.

A positional expansion in an integer base represents a rational number exactly when its digit sequence is eventually periodic. Long division proves the forward implication because only finitely many remainders occur; summing a periodic tail as a geometric series proves the converse.