I’m using the Twelve Days of Christmas (and perhaps a few extra days besides) to do something that I should have done a long time ago: collect past series of posts into a single, easy-to-reference post. The following posts formed my series on how the trigonometric form of complex numbers, DeMoivre’s Theorem, and extending the definitions of exponentiation and logarithm to complex numbers.
Part 1: Introduction: using a calculator to find surprising answers for and . See the video below.
Part 2: The trigonometric form of complex numbers.
Part 3: Proving the theorem
Part 4: Proving the theorem
Part 5: Application: numerical example of De Moivre’s Theorem.
Part 6: Proof of De Moivre’s Theorem for nonnegative exponents.
Part 7: Proof of De Moivre’s Theorem for negative exponents.
Part 8: Finding the three cube roots of -27 without De Moivre’s Theorem.
Part 9: Finding the three cube roots of -27 with De Moivre’s Theorem.
Part 10: Pedagogical thoughts on De Moivre’s Theorem.
Part 11: Defining for rational numbers .
Part 12: The Laws of Exponents for complex bases but rational exponents.
Part 13: Defining for complex numbers
Part 14: Informal justification of the formula .
Part 15: Simplification of .
Part 16: Remembering DeMoivre’s Theorem using the notation .
Part 17: Formal proof of the formula .
Part 18: Practical computation of for complex .
Part 19: Solving equations of the form , where and may be complex.
Part 20: Defining for complex .
Part 21: The Laws of Logarithms for complex numbers.
Part 22: Defining for complex and .
Part 23: The Laws of Exponents for complex bases and exponents.
Part 24: The Laws of Exponents for complex bases and exponents.