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 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