structure Complex : ComplexSig =
struct
    type complex = real * real;

    fun r X (a,b) : complex = (r*a, r*b)

    fun (a,b) ++ (c,d): complex = (a+c, b+d)
    and (a,b) -- (c,d): complex = (a-c, b-d)
    and (a,b) ** (c,d): complex = (a*c - b*d, b*c + a*d)
    and (a,b) // (c,d): complex = 
        let val factor = 1.0/(c*c + d*d)
         in
            factor X (a,b) ** (c,~d)
        end
    
    and ~~(a,b) : complex = (~a,~b)

    and (a,b) == (c,d) = (a=c) andalso (b=d)

    val pi = 4.0 * arctan 1.0;

    fun descartes{real, imaginary} = (real,imaginary)
    and argand {modulus, argument} = modulus X (cos argument, sin argument)

    and realpart (real, _) = real
    and imagpart (_, imag) = imag

    and modulus (r, i) = sqrt(r*r + i*i)
    and argument(r, i) =  
        if r = 0.0 
             then if i < 0.0 then ~pi/2.0
                             else  pi/2.0
             else arctan(i/r)
end;