Re: [HM] Further Fruitfulness of Fractions

Richard Guy (rkg@cpsc.ucalgary.ca)
Wed, 16 Sep 1998 09:47:10 -0600

Let me give a stupid and irresponsible reply -- no doubt John Conway
will give a sensible one. As Tom Lehrer said, the important thing
is to understand what you are doing, rather than to get the right
answer. I shall never be a mathematician, in the eyes of most,
because I don't think that logic and mathematics have anything to
do with one another. The basis for all manipulation of fractions
is equivalence thereof, known (at least many years ago in Britain)
as the Golden Rule. Here's the 'logic', if that's the right word
for the example quoted below.

You want either two-thirds of one quarter or one quarter of two-thirds
You 'can't do' two-thirds of one (quarter) or one quarter of two (thirds),
but you can do two-thirds of three (twelfths) and one quarter of eight
(twelfths) [or even one quarter of four (sixths), if you were more
economical], arriving at the respective answers two twelfths, two
twelfths and one sixth. In ancient Britain, and I suspect in many
other places where they used to (does anyone still?) teach addition
and subtraction of fractions, the Golden Rule seemed simple and
sensible (logical?) and it's nice to know that you can use it for
multiplication as well. It may be stretching it a bit to use it
for division, since 'compound fractions' are involved, and their
'logic' may not be clear, but for as long as I can remember, I've
advocated its use, even before the occasion, 50 years ago, when I
was inveighing against 'turn it upside and multiply' in a methods
class, and asked 'what on earth do you turn upside down before using
it?' and a bright student brought the house down with 'egg-timer'.

And thank you for your 'brilliant' remark. R.

On Wed, 16 Sep 1998, Samuel S. Kutler wrote:

> Friends:
>
> 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