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 That’s because if we change the polar coordinates, the cartesian coordinates are no longer valid.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);
product.printCartesian();

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!

#include <iostream>
#include <cmath>
using namespace std;

class Complex
{
  double real, imag;
  double mag, theta;
  bool cartesian, polar;

public:
  Complex ();
  Complex (double r, double i);
  void calculateCartesian ();
  double getReal ();
  double getImag ();
  void calculatePolar ();
  double getMag ();
  double getTheta ();
  void printCartesian ();
  void printPolar ();
  void setPolar (double m, double t);
};

Complex add (Complex& a, Complex& b);
Complex subtract (Complex& a, Complex& b);
Complex mult (Complex& a, Complex& b);

int main() {
  Complex c1 (2.0, 3.0);
  Complex c2 (3.0, 4.0);
  Complex product = mult (c1, c2);
  product.printCartesian();
}
====
Complex::Complex () { cartesian = false;  polar = false; }

Complex::Complex (double r, double i) {
  real = r;  imag = i;
  cartesian = true;  polar = false;
}

void Complex::calculateCartesian () {
  real = mag * cos (theta);
  imag = mag * sin (theta);
  cartesian = true;
}

double Complex::getReal () {
  if (cartesian == false) calculateCartesian ();
  return real;
}

double Complex::getImag () {
  if (cartesian == false) calculateCartesian ();
  return imag;
}

void Complex::calculatePolar () {
  mag = sqrt(pow(real, 2) + pow(imag, 2));
  theta = atan(imag / real);
  polar = true;
}

double Complex::getMag () {
  if (polar == false) {
    calculatePolar ();
  }
  return mag;
}

double Complex::getTheta () {
  if (polar == false) {
    calculatePolar ();
  }
  return theta;
}

void Complex::printCartesian () {
  cout << getReal() << " + " << getImag() << "i" << endl;
}

void Complex::printPolar () {
  cout << getMag() << " e^ " << getTheta() << "i" << endl;
}

Complex add (Complex& a, Complex& b) {
  double real = a.getReal() + b.getReal();
  double imag = a.getImag() + b.getImag();
  Complex sum (real, imag);
  return sum;
}

Complex subtract (Complex& a, Complex& b) {
  double real = a.getReal() - b.getReal();
  double imag = a.getImag() - b.getImag();
  Complex diff (real, imag);
  return diff;
}

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

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;
}

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!

Q-2: What is the correct output of the code below?

int main() {
  Complex c1 (2.0, 3.0);
  Complex c2 (3.0, 4.0);
  Complex c3 (1.0, 0.0);
  Complex c4 (3.5, 2.5);
  Complex product = mult (c1, c2);
  Complex diff = subtract (c4, c3);
  Complex sum = add (product, diff);
  sum.printCartesian();
}

3.5 + 19.5i
Incorrect! Try using the active code above.
-3.5 + 19.5i
Correct!
-3.5 - 19.5i
Incorrect! Try using the active code above.
-3.5 + 19.5
Incorrect! Try using the active code above.

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.

#include <iostream>
#include <cmath>
using namespace std;

class Complex
{
  double real, imag;
  double mag, theta;
  bool cartesian, polar;

void Complex::setCartesian (double r, double i) {
  // ``setCartesian`` should set real and imag to
  // r and i respectively and set the cartesian flag.
  // Write your implementation here.
}

Complex add (Complex& a, Complex& b);
Complex subtract (Complex& a, Complex& b);
Complex mult (Complex& a, Complex& b);

int main() {
  Complex c1 (2.0, 3.0);
  Complex c2 (3.0, 4.0);
  Complex product = mult (c1, c2);
  product.printCartesian();
  // Should output 1.5 + 2.7i
  product.setCartesian(1.5, 2.7);
  product.printCartesian();
}
====
Complex::Complex () { cartesian = false;  polar = false; }