
Boost Users : 
Subject: Re: [Boostusers] [rational] proposed gcd and lcm for rational
From: Júlio Hoffimann (julio.hoffimann_at_[hidden])
Date: 20110722 07:05:49
Hi Thomas, hi all,
Interesting approach. :) I'm just saying that these concepts are not what
most people think as GCD/LCM. If Boost.Rational developers wants to add this
feature to the library, they should make clear what is happening in
Boost.Rational documentation. ;)
Best Regards,
Júlio.
2011/7/21 Thomas Taylor <thomas.taylor_at_[hidden]>
> Topposting gets me confused :S so I reordered this a bit
>
> >> On Jul 21, 2011, at 7:44 AM, Júlio Hoffimann wrote:
> >>
> >> Hi Thomas,
> >>
> >> Thank you for the intention, but as far as i know, GCD and LCM makes no
> >> sense in the rational field. Any rational number is divisible by all
> >> rationals. For instance, given two rationals b, r in Q, r != 0, you can
> >> write:
> >>
> >> b = (b * r^1) * r
> >>
> >> and then b is divisible by r.
>
> > 2011/7/21 Christopher Henrich <chenrich_at_[hidden]>
> >
> >> GCD and LCM can be defined for rationals, if one takes "divisible" to
> >> mean "divisible with an integer quotient" and, similarly, "multiple" to
> >> mean "multiple by an integer." Then, for example, GCD(1/2, 1/3) = 1/6
> and
> >> LCM(1/2, 1/3) = 1.
>
> > Hi Christopher,
> >
> > This is the first time i heard about this interpretation. For me, still
> > makes no sense, but if people thinks is useful, ok.
>
> Basically I used the approach desribed by Christopher. (see also
> http://en.wikipedia.org/wiki/Least_common_multiple#Rational unfortunately
> it
> does not cite anything for this paragraph (there even is a similar approach
> for modulo calculation
>
> http://de.wikipedia.org/wiki/Division_mit_Rest#Verallgemeinerung:_Reelle_Zahlen
>  sorry German only))
>
> Especially a lcm makes sense in my opinion, as any two numbers have a lcm;
> the question only is what base set [Zahlenraum] one uses. (the posted
> implementation gives rational lcms (base set = R), but it could be modified
> to only give integer lcms (base set = Z)).
>
> The gcd is certainly trickier as it appears to me to be less intuitive.
> However the best way to describe what it does is probably what I am using
> it
> for: Consider a 2d vector v=(a,b). Now we don't care about the length v
> but only for the direction, which is defined by the factor between a and b.
> Lets further assume that both a and b are rational. Multiplying both a and
> b
> with gcd(a,b) [c=a*gcd(a,b) and d=b*gcd(a,b)] will give v'=(c,d), where
> both
> c and d are integers and furthermore the smallest integers that represent
> the direction (or factor between a and b).
>
> HTH,
> Thomas
>
> >>
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>
>
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