From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: Q: related to GDC and maybe something new
Date: 10 Jan 1998 01:24:58 GMT

In article <980105050024.B2D800@news.nbnet.nb.ca>,
Jason Duplessis <gender@nbnet.nb.ca> wrote:

>I have found a new(??) way to reduce fractions. I'll show you the algorithm
>I use with an example as I'm not sure how to do it otherwise.

Well, "reducing fractions" simply means to replace  a/b  with (a/d)/(b/d),
where  d  is the greatest common factor of  a  and  b. What you've rediscovered
is the "Euclidean algorithm" for computing that greatest common factor.
(thats the last denominator -- 6 in your case -- which divides in without
remainder.) 

Alternatively, you could describe your procedure as the continued-fractions
approximations process. You could start with any real number, and continue
inverting and subtracting off integer parts until you're exhausted, then
reconstruct your fraction. You get a sequence of fractions -- always in lowest
terms -- which approximate your original number (in some sense optimally). 
If the origianl	 number is itself a fraction, the last approximant is the
original number, reduced as I said to lowest terms.
		
>initial fraction                                          reduced fraction
>  66/84                                                       = 11/14
>Here I reverse it
>  84/66 = 1+18/66                                    = 1+3/11 = 14/11
>            66/18 = 3+12/18                  = 3+2/3 = 11/3
>                      18/12 = 1+6/12 = 1+1/2 = 3/2
>                                12/6 = 2

>I want to know if you can replace the first part up to the "12/6 = 2"
>with an equation (not GCD) and also what the midway answer (2) represents 
>as a part of the 11/14 fraction.

I don't think it has any particularly deep meaning, except as part of
the whole continued-fractions expansion [0,1,3,1,2] of your number.

dave