It maps quadratic irrationals to non-dyadic rational numbers. Thus quadratic irrationals are precisely those real numbers in this form that are not rational. A homeomorphism between Baire space and the irrationals can be constructed using continued fractions. Restricting the Euclidean distance function gives the irrationals the structure of a metric space. Furthermore, the set of all irrationals is a disconnected metrizable space. Hence, irrationals could only be handled geometrically; and indeed Greek mathematics was mainly geometrical. If such a function existed, then the irrationals would be an set.