structure Rational:NumberSig =
struct
type num = int * int

fun gcd(0,b) = b
  | gcd(a,b) = gcd(b mod a, a);

fun red(a,b) = 
    let val res = gcd(a,b)
    in (a div res, b div res) end;

fun (an, ad) ++ (bn, bd) = red(an*bd + ad*bn, ad*bd);
fun (an, ad) -- (bn, bd) = red(an*bd - ad*bn, ad*bd);
fun (an, ad) ** (bn, bd) = red(an*bn,         ad*bd);
fun (an, ad) // (bn, bd) = red(an*bd,         ad*bn);

fun ~~(n, d) : num = (~n ,d) 

fun ((an,ad):num) == ((bn,bd):num) = (an*bd = ad*bn);

end