1.5 De Moivre's Theorem

• De Moivre's Theorem is faster than multiplying out Cartesian forms for larger powers of complex numbers.
• Compare the efficiency of De Moivre's Theorem with Cartesian multiplication for finding powers of complex numbers.

  1️⃣ De Moivre's Theorem uses the formula $z^{n} =$$r^{n}(\cos n\theta + i \sin n\theta)$.

  2️⃣ Cartesian multiplication involves expanding $(a + bi) \times (c + di)$.

  3️⃣ De Moivre's Theorem is faster for larger powers.

  4️⃣ Cartesian multiplication is cumbersome and prone to errors.
• De Moivre's Theorem enhances both speed and simplicity in complex number calculations.
• What is the polar form of a complex number?

  $z =$$r(\cos \theta + i \sin \theta)$
• In polar form, $r$ represents the modulus
• The argument in polar form is the angle from the positive real axis.
• What is the modulus of the complex number $3 +$$4i$?


  5
• The argument of $3 +$$4i$ is approximately 0.927
• Match the components of polar form with their meaning:

  r (Modulus) ↔️ Distance from the origin

  θ (Argument) ↔️ Angle from the real axis
• Polar form uses distance and direction to represent complex numbers.
• What is the complex exponential function defined as for $z =$$x +$$iy$?


  e^{z} = e^{x}(\cos y + i \sin y)</latex>
• Euler's formula states that $e^{i \theta} =$$\cos \theta +$$i \sin \theta$, connecting complex exponentials to trigonometric functions
• What is the exponential form of 2(\cos \frac{\pi}{4} + i \sin \frac{\pi}{4})</latex>?

  $2e^{i \frac{\pi}{4}}$
• Match the complex forms with their representations:

  Cartesian ↔️ $x +$$iy$

  Polar ↔️ $r(\cos \theta + i \sin \theta)$

  Exponential ↔️ $re^{i \theta}$
• Euler's formula states that $e^{i \theta} =$$\cos \theta +$$i \sin \theta$.
• What does De Moivre's Theorem state for $z^{n}$?


  z^{n} = r^{n}(\cos n\theta + i \sin n\theta)</latex>
• In De Moivre's Theorem, $n$ represents the power
• If z = 2(\cos \frac{\pi}{4} + i \sin \frac{\pi}{4})</latex> and $n =$$3$, then $z^{3} =$$- 4\sqrt{2} +$$4i\sqrt{2}$.
• What is the exponential form of a complex number $r(\cos \theta + i \sin \theta)$?


  $re^{i \theta}$
• De Moivre's Theorem applies to all integers $n$.
• De Moivre's Theorem states that for $z =$$r(\cos \theta + i \sin \theta)$, then $z^{n} =$$r^{n}(\cos n\theta + i \sin n\theta)$ is the result
• What is the general form of a complex number in De Moivre's Theorem?

  z = r(\cos \theta + i \sin \theta)</latex>
• De Moivre's Theorem is used for finding roots of unity
• What is the purpose of De Moivre's Theorem?

  Simplifying complex number powers
• De Moivre's Theorem contrasts with Euler's formula because it raises complex numbers to powers.
• De Moivre's Theorem can be visualized geometrically on the Argand diagram.
• What geometrical transformation does the argument $\theta$ undergo when raising a complex number to the power $n$?


  Rotation around the origin
• The modulus $r$ in De Moivre's Theorem scales the distance from the origin on the Argand diagram.
• What does the argument $\theta$ represent in polar form?


  Angle from positive real axis
• The modulus $r$ in polar form represents the distance from the origin.
• The argument $\theta$ measures the angle from the positive real axis to the complex number.
• What is the complex exponential function defined for z = x + iy</latex>?

  $e^{z} =$$e^{x}(\cos y + i \sin y)$
• Euler's formula states that $e^{i \theta} =$$\cos \theta +$$i \sin \theta$, linking complex exponentials to trigonometric functions.
• The exponential form of a complex number is $re^{i \theta}$.
• What is the $n$-th power of a complex number $z =$$r(\cos \theta + i \sin \theta)$ according to De Moivre's Theorem?


  $z^{n} =$$r^{n}(\cos n\theta + i \sin n\theta)$
• What happens to the modulus when raising a complex number to the power $n$ on the Argand diagram?


  It scales by $r^{n}$
• The argument $\theta$ rotates the point around the origin by $n\theta$ radians when $z$ is raised to the power $n$.
• De Moivre's Theorem simplifies finding powers of complex numbers in polar form.
• What is the result of (\cos \frac{\pi}{6} + i \sin \frac{\pi}{6})^{4}</latex> using De Moivre's Theorem?

  $- \frac{1}{2} +$$i \frac{\sqrt{3}}{2}$
• The geometric interpretation of complex number powers simplifies calculations by visually representing transformations on the Argand