(* F# Tutorial Exercises Addendum.

   Given the definition of Numbers:
*)

type Number =
  | Integer of int
  | Rational of int*int
  | Real of float
  | Complex of float*float;;

// a. Define a function inverse that returns the inverse of the number as
// Some(inverse) if the inverse exists, or None if there's no inverse.
// Integer(0) and Rational(0,1), for example, have no inverse as the
// denominator cannot be zero. A Rational(a,b) has inverse Rational(b,a)
// if a is not zero.  A Real(r) has inverse Real(1.0/r) if r is not
// 0.0.  A complex number a+bi has an inverse if not both a and b are
// zeros, in which case the inverse is
// let d = a*a+b*b in Some(Complex(a/d, b/d));

//   I'll get you started:

let inverse num =
  match num with
    | Integer(x) when (x<>0) -> Some(Rational(1,x))
    ...
    _ -> None  // default case


// b. Assuming you've written the multiply function from the previous exercise,
//    write a function divide that divides a by b by multiplying a with the
//    inverse of b, if it exists.  This function should also return an
//    Option<Number>, because the inverse may not exist (no divide by zero).

// Hint: try to write this function using Option.map.  If you can't do
// that, you can resort to pattern matching.