 The class of real numbers, Θ, is defined in Volume III of Principia Mathematica at *310.01 as consisting of “Dedekindian series” of rational numbers, which are in turn relation numbers of “ratios” of natural numbers. Whitehead and Russell follow the account of real numbers as Dedekind cuts of the rational numbers, and only differ from more standard developments of the numbers in contemporary set theory by treating rational numbers as relation numbers of a certain sort, rather than ordered pairs of and integers (the “numerator” and “denominator”). Like the construction of relation numbers as classes of similar classes, the “logical construction” of real numbers differs from the theory of definite descriptions and classes in general in not defining “incomplete symbols” or by showing that these numbers are really “fictions”. They are best characterized as definitions that allow for the proof of theorems about these numbers that would otherwise have to be postulated as axioms. They are the product of the “honest toil” that Russell prefers.