. Write an essay (suggested length of 3–5 pages) in which you do the following:
1. Explain how the complex number system is an extension of the real number system.
2. Describe the individual parts of a complex number.
3. Explain how complex numbers combine under the following operations:
• Use one supporting example for each operation.
• Include both algebraic and graphical interpretations in your responses.
Note: The graphical interpretation should demonstrate how to add and divide complex numbers solely using the graph of each complex number (not based upon the algebraic computation).
4. De Moivre’s theorem states, “If z = r(cos u + i sin u), then zn = rn(cos nu + i sin nu).”
• Verify de Moivre’s theorem for n = 2.
a. Provide a correct proof that includes written justification for each step.