It is easy to prove that the series of segments of any series is Dedekindian. For, given any set of segments, their boundary will be their logical sum, i.e. the class of all those terms that belong to at least one segment of the set.[18] [18]For a fuller treatment of the subject of segments and Dedekindian relations, see Principia Mathematica, vol. II. * 210-214. For a fuller treatment of real numbers, see ibid., vol. III. * 310 ff., and Principles of Mathematics, chaps. XXXIII. and XXXIV. The above definition of real numbers is an example of "construction" as against "postulation," of which we had another example in the definition of cardinal numbers. The great advantage of this method is that it requires no new assumptions, but enables us to proceed deductively from the original apparatus of logic. There is no difficulty in defining addition and multiplication for real numbers as above defined. Given two real numbers and , each being a class of ratios, take any member of and any member of and add them together according to the rule for the addition of ratios. Form the class of all such sums obtainable by varying the selected members of and . This gives a new class of ratios, and it is easy to prove that this new class is a segment of the series of ratios. We define it as the sum of and . We may state the definition more shortly as follows:— The arithmetical sum of two real numbers is the class of the arithmetical sums of a member of the one and a member of the other chosen in all possible ways. [Pg 73] We can define the arithmetical product of two real numbers in exactly the same way, by multiplying a member of the one by a member of the other in all possible ways. The class of ratios thus generated is defined as the product of the two real numbers. (In all such definitions, the series of ratios is to be defined as excluding 0 and infinity.) There is no difficulty in extending our definitions to positive and negative real numbers and their addition and multiplication. It remains to give the definition of complex numbers.