Precision mathematics and approximation mathematics: the conceptual and educational role of their comparison

The relationship between applied and pure mathematics is of utmost concern for Klein. Examples from Volume III of his “Elementarmathematik” illustrate how, starting from an intuitive and sometimes practical approach, Klein develops abstract concepts working in rich “mathematical environments”. The examples concern the concept of empirical function and its comparison with an idealised curve, point sets obtained through circular inversion that lead to compare rational numbers and real numbers, and the “continuous” transformation of curves with the help of a point moving in space.