<div class="gmail_quote">2011/12/10 Stephen Bloch <span dir="ltr"><<a href="mailto:sbloch@adelphi.edu">sbloch@adelphi.edu</a>></span><br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
<div class="im"><br>
On Dec 9, 2011, at 3:31 PM, Daniel King wrote:<br>
<br>
> On Fri, Dec 9, 2011 at 15:27, Carl Eastlund <<a href="mailto:cce@ccs.neu.edu">cce@ccs.neu.edu</a>> wrote:<br>
>> What does "divides" even mean in Q? I think we need David to explain<br>
>> what his extension of GCD and LCM means here, in that "divisors" and<br>
>> "multiples" are fairly trivial things in Q.<br>
><br>
> I don't suppose to understand all the math on this page, but I think<br>
> it uses the same definition that dvh is using.<br>
><br>
> <a href="http://mathworld.wolfram.com/GreatestCommonDivisor.html" target="_blank">http://mathworld.wolfram.com/GreatestCommonDivisor.html</a><br>
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</div>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.<br>
</blockquote><div><br></div><div>If we for rational numbers x and y define "x divides y" to mean "y/x is an integer",</div><div>then I believe the definition</div><div> d is a gcd of x and y </div>
<div> <=> i) d divides a and y </div><div> ii) e divides x and y => d divides e</div><div>coincides with the MathWorld definition.</div><div> </div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
I would wonder: is this the ONLY "reasonable" function on rationals which, when restricted to integers, coincides with the usual definition of gcd?<br></blockquote><div><br></div><div>Not sure, but this seems relevant.</div>
<div><br></div><div><a href="http://trac.sagemath.org/sage_trac/ticket/10771">http://trac.sagemath.org/sage_trac/ticket/10771</a></div><div><br></div><div>-- </div></div>Jens Axel Søgaard<br><br><br>