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.

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## Peter

/ September 19, 2016Hi

I use online complex number calculator –

https://www.hackmath.net/en/calculator/complex-number?input=1%2B3i&submit=Calculate

for double checking school’s examples and tasks….