# [racket-dev] feature request: gcd, lcm for rationals

2011/12/10 Stephen Bloch <sbloch at adelphi.edu>
>*
*>* On Dec 9, 2011, at 3:31 PM, Daniel King wrote:
*>*
*>* > On Fri, Dec 9, 2011 at 15:27, Carl Eastlund <cce at ccs.neu.edu> wrote:
*>* >> What does "divides" even mean in Q? I think we need David to explain
*>* >> what his extension of GCD and LCM means here, in that "divisors" and
*>* >> "multiples" are fairly trivial things in Q.
*>* >
*>* > I don't suppose to understand all the math on this page, but I think
*>* > it uses the same definition that dvh is using.
*>* >
*>* > http://mathworld.wolfram.com/GreatestCommonDivisor.html
*>*
*>* Interesting: the Mathematica people have extended the gcd function from
*>* the integers to the rationals, not by applying the usual definition of gcd
*>* to Q (which would indeed be silly, as everything except 0 divides
*>* everything else), but by coming up with a different definition which, when
*>* restricted to integers, happens to coincide with the usual definition of
*>* gcd.
*>*
*
If we for rational numbers x and y define "x divides y" to mean "y/x is an
integer",
then I believe the definition
d is a gcd of x and y
<=> i) d divides a and y
ii) e divides x and y => d divides e
coincides with the MathWorld definition.
>* I would wonder: is this the ONLY "reasonable" function on rationals which,
*>* when restricted to integers, coincides with the usual definition of gcd?
*>*
*
Not sure, but this seems relevant.
http://trac.sagemath.org/sage_trac/ticket/10771
--
Jens Axel Søgaard
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