I agree with every word of what you say in the message below. Part of my
difficulty is that I make a distinction between ratio and fraction, whereas
you write
a rational number is defined to be the ratio of two whole numbers.
For me, ratios are combined, or compounded, but not added--see Euclid's
ELEMENTS Book 5. They are essentially two things. For me, fractions are
essentially one thing, but that thing is usually expressed in two parts.
I know that your book is a short book, but I still wish that you had said
in it a little of what you wrote in the message below?
I repeat that, except for terminology (which is important) I totally agree
with your explanation.
I know that your book is a short book, but shouldn't you have said a little
of what you wrote below in THE BOOK OF NUMBERS?
Best wishes,
Sam Kutler
> On Wed, 16 Sep 1998, Samuel S. Kutler wrote:
>
> [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