What a Complex Problem

🔖 Background Information

Complex numbers have the form $a+bi$ where $a$ is called the real part of the number, $b$ is called the imaginary part of the number, and $i$ is the square root of -1. Another way of thinking of $i$ is $i^2=-1$. Some mathematical operations are defined on complex numbers as follows:

Addition:

$$(a+bi)+(c+di)=(a+c)+(b+d)i$$

Subtraction:

$$(a+bi)-(c+di)=(a-c)+(b-d)i$$

Multiplication:

$$(a+bi)(c+di)=(ac-bd)+(ad+bc)i$$

Division:

$$\frac{a+bi}{c+di}=\frac{(ac+bd)-(ad-bc)i}{c^2+d^2}$$

Negation:

$$-(a+bi)=-a-bi$$

Magnitude:

$$|a+bi|=\sqrt{a^2+b^2}$$

Conjugation:

$$\overline{a+bi}=a-bi$$

Real Part:

$$Re(a+bi)=a$$

Imaginary Part:

$$Im(a+bi)=b$$

This problem is available courtesy of Professor Jason James (James, 2017).

🎯 Problem Statement

Write a ComplexNumber class that implements operations on complex numbers. The class will include some extra methods that improve quality of life for developers (defined in the Acceptance Criteria).

✅ Acceptance Criteria

• You cannot use any existing complex number type or library - you must implement your own class from scratch.
• Write a class called ComplexNumber that takes in two floats as arguments (the real part and the imaginary part of the number).
• Implement addition, subtraction, negation, magnitude, conjugation, real, and imaginary methods on the ComplexNumber class as instance methods.
  • Each of these methods should take in no arguments or one argument of type ComplexNumber.
  • Each of these methods should have a return value of type ComplexNumber or float, as needed.
  • None of these methods should change the calling instance of ComplexNumber or any arguments.
• Implement a prettyPrint method on the ComplexNumber class that prints the complex number out to the console in the form “a + bi”.

📋 Dev Notes

• You need to think carefully about how you want to test this class. How can you implement unit tests on your custom class? Maybe, you could consider testing the real and imaginary part separately?
• You do NOT have to put your ComplexNumber class into another file using headers. However, there is an add-on for the portfolio that you can do for extra credit.

🖥️ Example Output

Within your C++ code, you might have some examples that look like this:

ComplexNumber numberOne = ComplexNumber(5.0, 7.0);
ComplexNumber numberTwo = ComplexNumber(2.5, 3.1);
 
ComplexNumber numberThree = numberOne.add(numberTwo);
cout << numberThree.prettyPrint(); // Returns 7.5 + 10.1i
 
ComplexNumber numberFour = numberOne.subtract(numberTwo);
cout << numberFour.prettyPrint(); // Returns 2.5 + 3.9i
 
float magnitude = numberOne.magnitude();
cout << magnitude; // Returns 8.602325267

📝 Thought Provoking Questions

1. Does it make sense to default any of the arguments in the constructor of ComplexNumber to a particular value? Why or why not?
2. What are the types of values that are returned from each method in ComplexNumber?
3. What happens if you try to initialize your ComplexNumber class with an integer instead of a float?
4. Does prettyPrint() print anything besides the number in “a + bi” form (even an endl)? Why or why not?

💼 Add-Ons For the Portfolio

(One Credit) Calculate the Phase

Add a method to your ComplexNumber class called phase which calculates the phase of the number. The phase of the number is given by:

$$\varphi =arctan\left(\frac{imaginary}{real}\right)$$

Be sure that the method is tested either by your driver function or assertions.

(Two Credits) Header File

Move your ComplexNumber class into a separate file from main() and create a corresponding header file. Then, #include the header file for your code in main(). This should not change the functionality of your code.

(Two Credits) Multiplication and Division

In the “Background Information” section, we defined multiplication and division of complex numbers. However, we did not implement them in the lab, originally.

Implement two methods on the ComplexNumber class for multiplication and division. Just like with the other operations:

• Each of these methods should take in one argument of type ComplexNumber.
• Each of these methods should have a return value of type ComplexNumber.
• Neither of these methods should change the calling instance of ComplexNumber or any arguments.

🔗 Useful Links

N/A

📘 Works Cited

James, J. (2017). What a Complex Problem! In Jason James’ Homepage. http://craie-programming.org/122/labs/compclass.html