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

From: J. Ian Johnson (ianj at ccs.neu.edu)
Date: Fri Dec 9 15:08:58 EST 2011

What? The greatest common divisor would definitely not divide 1, unless it truly were 1.
----- Original Message -----
From: "Jens Axel Søgaard" <jensaxel at soegaard.net>
To: "David Van Horn" <dvanhorn at ccs.neu.edu>
Cc: "Racket Dev List" <dev at racket-lang.org>
Sent: Friday, December 9, 2011 2:04:46 PM GMT -05:00 US/Canada Eastern
Subject: Re: [racket-dev] feature request: gcd, lcm for rationals

One definition of greatest common divisor in a ring R is: 
d is a greatest common divisor of x and y when: 
i) d divides both x and y 
ii) If e is a divisor of both x and y, then d divides e 

Now let's consider the ring Q. Since Q is a field, 1 divides all elements. 

This implies that 1 is a greatest common divisor of any non-zero x and y. 
( ad i) 1 is a divisor of both x and y 
ad ii) 1 is a divisor of e ) 

It is therefore not obvious that gcd should be extendend as you suggest. 
But maybe we can finde another name for the operation? 

/Jens Axel 

2011/12/7 David Van Horn < dvanhorn at ccs.neu.edu > 

It would be nice if gcd and lcm were extended to rational numbers, which seems in-line with Scheme's philosophy (but not standards) on numbers. 

(define (gcd-rational . rs) 
(/ (apply gcd (map numerator rs)) 
(apply lcm (map denominator rs)))) 

(define (lcm-rational . rs) 
(/ (abs (apply * rs)) 
(apply gcd-rational rs))) 

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