Simon Peyton-Jones <simonpj@microsoft.com> writes
[..] If someone could write a sentence or two to explain why gcd 0 0 = 0, (ideally, brief ones I can put in the report by way of explanation), I think that might help those of us who have not followed the details of the discussion.
Here it is. gcd n m :: Integer = if n == 0 && m == 0 then 0 else greatest integer that divides both n and m It is set so according to classic definition (by Pythago, Euclid ...) GCD x y = such g that divides both x and y and is a multiple of any g' that divides both x and y. In particular, GCD 0 0 :: Integer = 0 and can be nothing else. Because 0 divides 0 and divides nothing else, everything divides 0 (z*0 = 0). Other comments ------------------------------------------------- * People say, D.Knuth writes gcd 0 0 = 0 in his book. And he is a known mathematically reliable person. * Example. gcd 12 18 :: Integer = 6 or -6, because 6 divides 12 and 18, and any other such number divides 6. * The Haskell report says "greatest integer" for domain = Integer, and thus, chooses the sign `+' for the result. This choice is not a mistake and helps to write shorter programs. * All the above agrees with the modern generic theory of ideals (started in |XX by Kummer, Gauss, Lagrange) for any commutative domain R having the properties of (a /= 0, b /= 0 ==> a*b /= 0), existence of unique factorization to primes. According to it gcd x y = least by inclusion ideal of kind (g) that includes the ideal (x, y), where Ideal (a,b..c) =def= {x*a + y*b +..+ z*c | x,y..z <- R} - a subset in R. `includes` (as set) is a partial order on ideals, and it is true that (1) a divides b <==> (a) includes (b), (2) (a) includes (a). Specializing this definition to Integer, we obtain the source formula. ----------------- Serge Mechveliani mechvel@botik.ru