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