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

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

    fun (a, m) ** (a', m'): complex = (a + a', m * m')
    and (a, m) // (a', m'): complex = (a - a', m / m')
    and ~~(a, m) : complex = (a, ~m)

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

    val pi = 4.0 * arctan 1.0;

    fun descartes {real=r, imag=i} =    
        let val argument =
                if r = 0.0 
                     then if i < 0.0 then ~pi/2.0
                                     else  pi/2.0
                     else arctan(i/r)
            val modulus = sqrt(r*r + i*i)
         in
            (argument, modulus)
        end
    and argand {modulus, argument} = (argument, modulus)

    and realpart (a, m) = m * cos a
    and imagpart (a, m) = m * sin a

    and modulus  (a, m) = m
    and argument (a, m) = a 

    fun x ++ y = let val r = realpart x + realpart y
                     and i = imagpart x + imagpart y
                  in
                     descartes{real = r, imag = i}
                 end
    fun x -- y = x ++ (~~ y)

end;