Complex Numbers: De Moivre's Theorem & Polar Form

TAKEAWAYS:

• Complex numbers can be written in polar form $r(\cos \theta +i\sin \theta )$ using modulus $r$ and argument $\theta$.
• De Moivre’s Theorem lets you raise complex numbers to any power or find roots easily: $[r(\cos \theta +i\sin \theta ){]}^n=r^n(\cos n\theta +i\sin n\theta )$.
• nth roots of a complex number are equally spaced on a circle in the complex plane.
• Applications include solving polynomial equations and analyzing AC circuits in physics.
• NEB exam focus: Problems often ask for roots, powers, or conversions between forms.

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1. Complex Numbers: Basics

Complex numbers are numbers of the form $z=a+ib$, where $a$ and $b$ are real numbers, and $i=\sqrt{-1}$.

Key Parts of a Complex Number

• Real part (Re(z)): $a$
• Imaginary part (Im(z)): $b$
• Modulus (Magnitude): $|z|=\sqrt{a^2+b^2}$
• Argument (Angle): $\theta ={\tan }^{-1}\left(\frac{b}{a}\right)$ (measured in radians from the positive real axis).

Example 1: Find Modulus and Argument

Find the modulus and argument of $z=1+i$.

Solution:

• Modulus: $|z|=\sqrt{1^2+1^2}=\sqrt{2}$
• Argument: $\theta ={\tan }^{-1}\left(\frac{1}{1}\right)=\frac{\pi }{4}$ radians.

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2. Polar Form of Complex Numbers

A complex number $z=a+ib$ can be written in polar form as: $$z=r(\cos \theta +i\sin \theta )$$ where:

• $r=|z|=\sqrt{a^2+b^2}$
• $\theta ={\tan }^{-1}\left(\frac{b}{a}\right)$

Example 2: Convert to Polar Form

Convert $z=-1+i\sqrt{3}$ to polar form.

Solution:

• Modulus: $r=\sqrt{(-1{)}^2+(\sqrt{3}{)}^2}=\sqrt{1+3}=2$
• Argument: Since $a=-1$ and $b=\sqrt{3}$, $\theta ={\tan }^{-1}\left(\frac{\sqrt{3}}{-1}\right)=\frac{2\pi }{3}$ (second quadrant).
• Polar form: $z=2\left(\cos \frac{2\pi }{3}+i\sin \frac{2\pi }{3}\right)$.

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3. De Moivre’s Theorem

De Moivre’s Theorem states: $$[r(\cos \theta +i\sin \theta ){]}^n=r^n(\cos n\theta +i\sin n\theta )$$ This allows us to raise complex numbers to any power easily.

Example 3: Apply De Moivre’s Theorem

Find $(1+i{)}^4$ using De Moivre’s Theorem.

Solution:

1. Convert $1+i$ to polar form:
   • $r=\sqrt{1^2+1^2}=\sqrt{2}$
   • $\theta =\frac{\pi }{4}$
   • Polar form: $\sqrt{2}\left(\cos \frac{\pi }{4}+i\sin \frac{\pi }{4}\right)$
2. Apply De Moivre’s Theorem: $$(1+i{)}^4=(\sqrt{2}{)}^4\left(\cos \left(4\cdot \frac{\pi }{4}\right)+i\sin \left(4\cdot \frac{\pi }{4}\right)\right)=4(\cos \pi +i\sin \pi )=4(-1+i\cdot 0)=-4$$

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4. Roots of Complex Numbers

The nth roots of a complex number $z=r(\cos \theta +i\sin \theta )$ are given by: $$z_k=r^{1/n}\left[\cos \left(\frac{\theta +2k\pi }{n}\right)+i\sin \left(\frac{\theta +2k\pi }{n}\right)\right],k=0,1,2,\dots ,n-1$$ These roots are equally spaced on a circle in the complex plane.

Example 4: Find Cube Roots of $8$

Find all cube roots of $8$.

Solution:

1. Express $8$ in polar form:
   • $8=8(\cos 0+i\sin 0)$
2. Apply the roots formula for $n=3$: $$z_k=8^{1/3}\left[\cos \left(\frac{0+2k\pi }{3}\right)+i\sin \left(\frac{0+2k\pi }{3}\right)\right],k=0,1,2$$
3. Calculate each root:
   • For $k=0$: $z_0=2(\cos 0+i\sin 0)=2$
   • For $k=1$: $z_1=2\left(\cos \frac{2\pi }{3}+i\sin \frac{2\pi }{3}\right)=2\left(-\frac{1}{2}+i\frac{\sqrt{3}}{2}\right)=-1+i\sqrt{3}$
   • For $k=2$: $z_2=2\left(\cos \frac{4\pi }{3}+i\sin \frac{4\pi }{3}\right)=2\left(-\frac{1}{2}-i\frac{\sqrt{3}}{2}\right)=-1-i\sqrt{3}$

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5. Comparison: Rectangular vs. Polar Form

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6. Applications of De Moivre’s Theorem

1. Solving Polynomial Equations: Finding roots of $z^n=a+ib$.
2. AC Circuit Analysis: Representing impedances in polar form.
3. Signal Processing: Fourier transforms use complex exponentials.

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7. Common Mistakes to Avoid

• Forgetting the modulus: Always calculate $r$ before applying De Moivre’s Theorem.
• Incorrect quadrant for $\theta$: Use ${\tan }^{-1}$ carefully (check signs of $a$ and $b$).
• Missing roots: For $n$th roots, always include all $n$ values of $k$.

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8. NEB-Style Questions

Short Answer (2 marks)

1. Write $1-i$ in polar form.
2. Find $(1+i{)}^3$ using De Moivre’s Theorem.

Long Answer (5 marks)

3. Find all fourth roots of $-16$ and represent them graphically.
4. Solve $z^3=8i$ using De Moivre’s Theorem.

Exam Tip

• Always convert to polar form before applying De Moivre’s Theorem.
• Draw diagrams for roots—they are equally spaced on a circle.
• Check units: Ensure angles are in radians (NEB expects radians unless specified).

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Final Note: Practice converting between forms and solving roots/powers. NEB often tests both calculation and graphical representation of complex numbers!