# Calculators and Complex Numbers: Index

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 $\ln(-5)$ and $\sqrt{-8}$. See the video below.

Part 2: The trigonometric form of complex numbers.

Part 3: Proving the theorem $\left[ r_1 (\cos \theta_1 + i \sin \theta_1) \right] \cdot \left[ r_2 (\cos \theta_2 + i \sin \theta_2) \right] = r_1 r_2 (\cos [\theta_1+\theta_2] + i \sin [\theta_1+\theta_2])$

Part 4: Proving the theorem $\displaystyle \frac{ r_1 (\cos \theta_1 + i \sin \theta_1) }{ r_2 (\cos \theta_2 + i \sin \theta_2) } = \displaystyle \frac{r_1}{r_2} (\cos [\theta_1-\theta_2] + i \sin [\theta_1-\theta_2])$

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 $z^q$ for rational numbers $q$.

Part 12: The Laws of Exponents for complex bases but rational exponents.

Part 13: Defining $e^z$ for complex numbers $z$

Part 14: Informal justification of the formula $e^z e^w = e^{z+w}$.

Part 15: Simplification of $e^{i \theta}$.

Part 16: Remembering DeMoivre’s Theorem using the notation $e^{i \theta}$.

Part 17: Formal proof of the formula $e^z e^w = e^{z+w}$.

Part 18: Practical computation of $e^z$ for complex $z$.

Part 19: Solving equations of the form $e^z = w$, where $z$ and $w$ may be complex.

Part 20: Defining $\log z$ for complex $z$.

Part 21: The Laws of Logarithms for complex numbers.

Part 22: Defining $z^w$ for complex $z$ and $w$.

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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