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[3]{-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.

Leave a comment

4 Comments

  1. Peter

     /  September 19, 2016

    Hi

    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….

    Reply
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