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.
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.
Complex numbers, though capable of a geometrical interpretation, are not demanded by geometry in the same imperative way in which irrationals are demanded. A "complex" number means a number involving the square root of a negative number, whether integral, fractional, or real. Since the square of a negative number is positive, a number whose square is to be negative has to be a new sort of number. Using the letter
for the square root of
, any number involving the square root of a negative number can be expressed in the form
