Consider the serial arrangement of the **rationals** in their order of magnitude.

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.

Note that the class of **rationals** less than or equal tò 2r is not a real number.

Now, owing to the necessary inexactness of measurement, it is impossible to discriminate directly whether any kind of continuous physical quantity possesses the compactness of the series of **rationals** or the continuity of the series of real numbers.

For example, the class of **rationals** whose squares are less than 2r satisfies the definition of a real number; it is the real number A I 2.