16. The continuous real variable

The ‘real numbers’ may be regarded from two points of view. We may think of them as an aggregate, the ‘arithmetical continuum’ defined in the preceding section, or individually. And when we think of them individually, we may think either of a particular specified number (such as \(1\), \(-\frac{1}{2}\), \(\sqrt{2}\), or \(\pi\)) or we may think of any number, an unspecified number, the number \(x\). This last is our point of view when we make such assertions as ‘\(x\) is a number’, ‘\(x\) is the measure of a length’, ‘\(x\) may be rational or irrational’. The \(x\) which occurs in propositions such as these is called the continuous real variable: and the individual numbers are called the values of the variable.

A ‘variable’, however, need not necessarily be continuous. Instead of considering the aggregate of all real numbers, we might consider some partial aggregate contained in the former aggregate, such as the aggregate of rational numbers, or the aggregate of positive integers. Let us take the last case. Then in statements about any positive integer, or an unspecified positive integer, such as ‘\(n\) is either odd or even’, \(n\) is called the variable, a positive integral variable, and the individual positive integers are its values.

Naturally ‘\(x\)’ and ‘\(n\)’ are only examples of variables, the variable whose ‘field of variation’ is formed by all the real numbers, and that whose field is formed by the positive integers. These are the most important examples, but we have often to consider other cases. In the theory of decimals, for instance, we may denote by \(x\) any figure in the expression of any number as a decimal. Then \(x\) is a variable, but a variable which has only ten different values, viz. \(0\), \(1\), \(2\), \(3\), \(4\), \(5\), \(6\), \(7\), \(8\), \(9\). The reader should think of other examples of variables with different fields of variation. He will find interesting examples in ordinary life: policeman \(x\), the driver of cab \(x\), the year \(x\), the \(x\)th day of the week. The values of these variables are naturally not numbers.