We are now in a position to define a real number and an irrational number.
A "real number" is a segment of the series of ratios in order of magnitude.
An "irrational number" is a segment of the series of ratios which has no boundary.
A "rational real number" is a segment of the series of ratios which has a boundary.
Thus a rational real number consists of all ratios less than a certain ratio, and it is the rational real number corresponding to that ratio. The real number 1, for instance, is the class of proper fractions.
In the cases in which we naturally supposed that an irrational must be the limit of a set of ratios, the truth is that it is the limit of the corresponding set of rational real numbers in the series of segments ordered by whole and part. For example,
is the upper limit of all those segments of the series of ratios that correspond to ratios whose square is less than 2. More simply still,
is the segment consisting of all those ratios whose square is less than 2.