The arithmetic of rational numbers is now established by means of appropriate definitions, which indicate the entities meant by the operations of addition and multiplication.

A real number is a class (a, say) of rational numbers which satisfies the condition that it is the same as the class of those rationals each of which precedes at least one member of a.

This is exactly the same reason as that which has led mathematicians to work with signed real numbers in preference to real numbers, and with real numbers in preference to rational numbers.

The compactness of the series of rational numbers is consistent with quasi-gaps in it - that is, with the possible absence of limits to classes in it.

Thus the class of rational numbers whose squares are less than 2 has no upper limit among the rational numbers.