 The last section of PM studies the rational numbers and real numbers. They are constructed from relations between entities, such as being longer than, or being heavier than, that might be measured with a ruler or balance scale. Contemporary measurement theory studies the relations among entities in order to determine which scales, or systems of independently characterized numbers, might be assigned to them as representing the “quantities” of various properties, such as length or weight, that they possess. Notice that the real numbers are not constructed as classes of rational numbers, but are of a uniform type as “Dedekind cuts” in a series of classes of ratios. In PM, as in contemporary mathematics, because the class (segment) of rational numbers \( \{ r | r^2 \leq 2 \} \) will have no rational number as a least upper bound , that class itself will be identified with the irrational number \( \sqrt{2}\). The rational number \( 1/2 \) is identified with its (lower) segment of rationals, \( \{ r | r \lt 1/ 2 \} \). 