page 45 note * It will be realized that there are an infinity of numbers between 0 and 1 which cannot be expressed as a vulgar fraction (rational number), e.g. any square root such as 1/ and any exponential number such as e ⁻¹.

page 46 note * Bernoulli's Theorem, which the student will meet in the Statistics course, is a fundamental theorem about the clustering of relative frequencies as the number of trials gets larger.

page 53 note * There is, however, another mathematical difficulty that should be mentioned but which the student who has not specialized in Pure Mathematics should not worry over. We have confined attention so far to intervals, but we have not discussed the difference between closed and open intervals (ie intervals excluding one or both of the end-points). We must assume that the smaller (or larger) end-point is included in an interval and the other excluded, otherwise the end-points (except a and b) will all be included twice. By processes of addition and subtraction of intervals it is possible to arrive at sets of points which are not intervals and strictly a rigorous system requires all such cases that can arise to be covered. By means of the advanced mathematics of measurable sets a complete system on these lines can be rigorously developed. The student's work will not however extend to problems involving sets of points which are not intervals, except perhaps occasional cases of isolated points for which the probability obviously tends to zero, e g. the probability of choosing any particular number (e g. 5), if a number between 0 and 1 is chosen at random, is zero, although strictly such a result is not ‘impossible’ To illustrate the need in a rigorous theory to cover sets of points which are not intervals, it is perhaps sufficient to mention that, although the number of rational numbers between 0 and 1 is infinite, there is an infinite number of real numbers between any two rational numbers however close together these two rational numbers may be.