In Chapter 6 of THE BOOK OF NUMBERS, which is brilliant, I don't quite know
what is going on at the bottom of page 151--the first page of the chapter.
To enable the reader to multiply fractions via the golden rule for fractions
multiplying numerator & denominator by the same number does
not change the value
they give this example:
2/3 x 1/4 = 4/6 x 1/4 = 1/6.
Thus they believe that the reader knows, and the reader no doubt does, that
we can combine fractions in the way that Euclid and the other ancient Greek
mathematicians combined ratios of magnitudes. This is usually translated
into English as compound ratio, and the way one does it is to find a common
middle term--in this case 4.
What is the logic of this? Do Guy and Conway think that the readers have
learned that the definition of the product of a/b and c/d is ac/bd and that
they have forgotten it? Do they think that the reader knows or will
investigate the justification for compounding the ratios of numbers, or
what?
I like it, by the way, but I don't get it. I think that is because it is
too easy.
My main point is that the chapter is brilliant.
Best wishes from Annapolis,
Sam Kutler