14.7. Another function on Complex numbers

Another operation we might want is multiplication. Unlike addition, multiplication is easy if the numbers are in polar coordinates and hard if they are in Cartesian coordinates (well, a little harder, anyway).

In polar coordinates, we can just multiply the magnitudes and add the angles. As usual, we can use the accessor functions without worrying about the representation of the objects.

Complex mult (Complex& a, Complex& b)
  double mag = a.getMag() * b.getMag();
  double theta = a.getTheta() + b.getTheta();
  Complex product;
  product.setPolar (mag, theta);
  return product;

A small problem we encounter here is that we have no constructor that accepts polar coordinates. It would be nice to write one, but remember that we can only overload a function (even a constructor) if the different versions take different parameters. In this case, we would like a second constructor that also takes two doubles, and we can’t have that.

An alternative it to provide an accessor function that sets the instance variables. In order to do that properly, though, we have to make sure that when mag and theta are set, we also set the polar flag. At the same time, we have to make sure that the cartesian flag is unset. That’s because if we change the polar coordinates, the cartesian coordinates are no longer valid.

void Complex::setPolar (double m, double t)
  mag = m;  theta = t;
  cartesian = false;  polar = true;

As an exercise, write the corresponding function named setCartesian.

To test the mult function, we can try something like:

Complex c1 (2.0, 3.0);
Complex c2 (3.0, 4.0);

Complex product = mult (c1, c2);

The output of this program is

-6 + 17i

The active code below uses the mult and setPolar functions. Feel free to modify the code and experiment around!

There is a lot of conversion going on in this program behind the scenes. When we call mult, both arguments get converted to polar coordinates. The result is also in polar format, so when we invoke printCartesian it has to get converted back. Really, it’s amazing that we get the right answer!

Now let’s try implementing the setCartesian function. Write your implementation in the commented area of the active code below. Read the comments in main to test out your code! If you get stuck, you can reveal the extra problem at the end for help.

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