by James Madden
The idea that a ratio is a pair of magnitudes is in Euclid (fl. 300 BC), Elements, Book V. It is interesting to note that Euclid says that a ratio is the relationship between two magnitudes, not the pair itself.
Greek mathematics did not have the explicit concept of equivalence relation, but the “conceptual grammar” of Greek math is most easily understood by us if we describe it using the idea that they recognized certain equivalence relations “implicitly.” Consider Euclid’s definition of angle:
“the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line”
Clearly, Euclid means something that can be the same in two different pairs of lines. This pair of lines and that pair of lines may have the same inclination, and if so, they are equivalent. The same thinking applies to ratio: two different pairs of magnitudes may stand in the same ratio. The genius of ancient Greek mathematics is to produce an operational definition for sameness. Book V, Definition 5 states:
“Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever are taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order.”
One thing that I find interesting about the discussion on the notations of ratios—and it’s an issue that occurs often in the analysis of school math—is the contrast between the thing itself, and the means by which we make reference to it. The most basic example is the distinction between the written symbol and the thing the symbol refers to: the number 6 (which, according to Plato, the soul observes before birth—but you can take its ontological status to be what you prefer) and the numeral 6, which is a mark on paper.
As for ratio, is it a number? A pair of numbers—or pair of quantities—by means of which we refer to a number? Or an equivalence class of pairs of numbers? One can find advocates for each of these conceptual images. The Progressions Document on Ratios and Proportional Relationships has chosen to view a ratio as an ordered pair of real numbers, not an equivalence class of such pairs, nor any of the other options.
Because of history, the word “ratio” has many meanings. I haven’t even mentioned the “Rule of Three” and the way that has shaped the way we talk about proportion. We can acknowledge that there have been many different traditions. But if we don’t agree on a simple, direct language, we will wind up like the poor servant in the comical fairy tale of the “Master of All Masters.” What we need is a firm grasp of an idea and the ability to provide an account of how we are using our words. There is no need to preserve odd notions from Textbook School Mathematics.
For the purposes of presenting math clearly, we must attach meanings to words in only one way. We can control our own classrooms and the language used there, but we cannot do anything about the fact that in the world there is a mixture of habits, traditions and perspectives, and everyone will surely encounter that at some point in some way or another. After students have mastered one way of talking about things, they may find it convenient or even necessary to consider or use other ways.
[Please feel free to leave comments about anything said here in the comment section below.]
CHANNEL: Engineering School Mathematics
© 2015 James Madden