The above comparison of the domain R of rational numbers with a straight line has led to the recognition of the existence of gaps, of a certain incom… - Richard Dedekind

If a is any definite number, then all numbers of the system R fall into two classes, A₁ and A₂, each of which contains infinitely many individuals; the first class A₁ comprises all numbers a₁ that are < a, the second class A₂ comprises all numbers a₂ that are > a; the number a itself may be assigned at pleasure to the first or second class, being respectively the greatest number of the first class or the least of the second. In every case the separation of the system R into the two classes A₁, A₂ is such that every number of the first class A₁ is less than every number of the second class A₂.

The way in which the irrational numbers are usually introduced is based directly upon the conception of extensive magnitudes—which itself is nowhere carefully defined—and explains number as the result of measuring such a magnitude by another of the same kind. Instead of this I demand that arithmetic shall be developed out of itself.

That such comparisons with non-arithmetic notions have furnished the immediate occasion for the extension of the number-concept may, in a general way, be granted (though this was certainly not the case in the introduction of complex numbers); but this surely is no sufficient ground for introducing these foreign notions into arithmetic, the science of numbers.