This book is intended to explain the nature of irrational numbers, and those parts of Algebra which depend on what is usually called The Theory of Limits. Many of our text-books define irrational numbers by means of sequences; but to the author it has seemed more natural to define a number, or at least to consider a number as determined, by the place which it occupies among rational numbers, and to assume that a separation of all rational numbers into two classes, those of one class less than those of the other, always determines a number which occupies the point of separation. Thus we have the definition of Dedekind, which is adopted by Weber in his Algebra. Without attempting to inquire too minutely into the significance of this definition, we have endeavored to show how the fundamental operations are to be performed in the case of irrational numbers and to define the irrational exponent and the logarithm.
Defining the irrational number by the place which it occupies among rational numbers, we proceed to speak of its representation by sequences; and when we have proved that a sequence which represents a number is regular and that a sequence which is regular represents a number, we are in complete possession of the theory of sequences and their relation to numbers.