[Sam doesn't know what's going on here ...]
> they give this example:
>
> 2/3 x 1/4 = 4/6 x 1/4 = 1/6.
>
[... the rules being a la Euclid ...]
>
> 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?
Guy and Conway DO think, unfortunately, that readers have learned that
that is the definition of the product, which it really isn't, except in
formal axiomatic contexts. Suppose one stick is three-quarters as long
as another, whose length is two-and-a-half inches. Then do you really
think that the reason the shorter stick's length is 15/8 inches is
because the DEFINITION of a/b times c/d is ac/bd? It ISN'T! It's
because what actually happens is that if you cut the longer stick into
4 equal quarters and throw one of them away, the total length of what's
left will actually be 15/8 inches.
The "Golden Rule" (that kn/kd has the same meaning as n/d) helps
one to understand just WHY multiplication and division of fractions work
the way they do. What it does is reduce the general cases of these
operations to those of multiplication or division of a fraction n/d by
an integer N, for which the answers Nn/d and n/Nd are much easier to
understand.
> I like it, by the way, but I don't get it. I think that is because it is
> too easy.
It IS easy, that's the point. One doesn't have to just LEARN,
for instance, that a/b divided by c/d just IS ad/bc - one can
UNDERSTAND WHY.
John Conway