In the last module we introduced the idea of thinking about lengths as quantities which can be added and multiplied by whole numbers.
Once we have multiplication by whole numbers we can also ask about division. For counting numbers we say that “n can be divided by m with quotient q” if n=mq, and in that case we often write q as n/m. But of course it doesn’t always work out (eg 3 can’t be divided by 2 with a whole number quotient). Applying the same idea to multiplication of lengths by whole numbers we should say that length a can be divided by counting number m with quotient length q if mq=a. And in this case it always works out! Given any length a and any counting number m we can always find a length q (which we will call a/m) for which mq=a exactly. To see this we’ll use a bit of geometry.[then just go to insert Peter’s Ch2 Module_1.pdf] [And if we want to introduce rationals at this point rather than after defining the general product (about which I am still not entirely convinced)
Eg/Ex Check that n(a/m)=(na)/m=(n/m)a
i.e. insert Peter’s Ch2 Module_2.pdf=New_Module_3.pdf]
So far we have shown how to add lengths and multiply and divide them by arbitrary counting numbers (or, if you like, to multiply them by arbitrary positive rational numbers).
When two lengths are related by a whole or fractional number, say b=(n/m)a, and one of them is similarly related to a third, say c=(q/p)b, then the relationship of the first to the third is obtained by multiplying the factors. ie c=(q/p)(n/m)a.
A natural question at this point might be whether every pair of lengths is related this way – or in other words, given two lengths a and b, is it always possible to match one by putting together many fractional parts of the other? Perhaps surprisingly, the answer is “No”, so the set of all possible relative length relationships is larger than just the set of relative counting number relationships.
[Insert geometric proof that diagonal of square is incommensurable with side] If we pick some particular length u as a “unit” (eg a metre, a millimetre, or an inch), then starting along a line from some point O we can measure off multiples of the unit and label points on the line according to how many units they are from O. By dividing the unit into equal parts we can also label points that are fractional numbers of units from O. And by choosing one direction as positive and the other as negative we can assign every rational number to a position on the line. (But because of the geometric fact discussed above there are actually many points on the line which can’t be so labelled. They may have absolutely specific distances from the origin – such as the length of the diagonal of a unit square – but not ones which can be expressed as rational multiples of the unit.)
For any two points on the line (whether or not they are rational) there is a point whose distance from O is the geometric sum of their distances, and if their distances do happen to be rational multiples of the unit, then so is their sum with number of units just being the numerical sum of their two multipliers.
But what about multiplication? Can we define a way of multiplying that works even for points that do not have rational labels? (and which agrees with rational number multiplication when they do)
The answer to this question is yes, and we shall see one way to do it in the next section.