The Real Numbers—in which we find holes in the number line and pay the price for repairs

As advertised in the previous chapter, we shall take for granted the elementary arithmetic of fractions and all the generally illogical manoeuvrings it takes to get there. What we now need is a working definition of the set ℚ of rational numbers and a specific interpretation or model of them. Both are easy: the model is the familiar number line in which the rational number x is represented as a distance x along the line from 0, to the left or right, depending on whether x is negative or positive; and our (semi-formal) definition of a rational number is any number which can be expressed as the ratio between two integers, n/m, with the proviso that m is not zero.