Complex Numbers – Polar Form, Operations, De Moivre & Roots

Complex Plane – Orientation & Plotting

• Complex numbers are written $a + b i$ where:

  • $a$ = real part (plotted on horizontal axis)

  • $b i$ = imaginary part (plotted on vertical axis)

• Quadrant summary (using the mnemonic "cos-first, sin-second”):

  • QI a>0,\;b>0 ⇒ \cos>0,\;\sin>0

  • QII a<0,\;b>0 ⇒ \cos<0,\;\sin>0

  • QIII a<0,\;b<0 ⇒ \cos<0,\;\sin<0

  • QIV a>0,\;b<0 ⇒ \cos>0,\;\sin<0

• Purely real points lie on the horizontal axis (angle $0$ or $\pi$). Purely imaginary points lie on the vertical axis (angle $\pm\tfrac{\pi}{2}$).

Rectangular ↔ Polar Conversions

• Polar form: $z = r(\cos\theta + i\,\sin\theta)$

  • $r = |z| = \sqrt{a^{2}+b^{2}}$ (modulus)

  • $\theta = \arg z$ (principal argument, measured in degrees or radians)

• Converting polar → rectangular:

  1. Evaluate $\cos\theta$ and $\sin\theta$.

  2. Multiply both by $r$.

• Example (from lecture)

  • Given $2(\cos30^{\circ} + i\,\sin30^{\circ})$

  • $\cos30^{\circ}=\tfrac{\sqrt3}{2},\; \sin30^{\circ}=\tfrac12$

  • $x = 2\cdot\tfrac{\sqrt3}{2}=\sqrt3$, $y = 2\cdot\tfrac12 = 1$

  • Rectangular answer: $\sqrt3 + i$.

• Example with $r=10$, $\theta = 270^{\circ}$ → $10(\cos270^{\circ}+i\sin270^{\circ}) = 10(0- i) = -10i$.

Products & Quotients in Polar Form (Section 5.3)

• Product rule (very “friendly”): $z<em>1 z</em>2 = r<em>1 r</em>2\bigl(\cos(\theta<em>1+\theta</em>2)+i\,\sin(\theta<em>1+\theta</em>2)\bigr)$

  • Multiply moduli, add arguments.

  • Order does NOT matter.

• Quotient rule (be careful with order!) $\frac{z<em>1}{z</em>2}=\frac{r<em>1}{r</em>2}\Bigl(\cos(\theta<em>1-\theta</em>2)+i\,\sin(\theta<em>1-\theta</em>2)\Bigr)$

  • Divide moduli, subtract arguments (top – bottom).

  • Adjust any negative or >$360^{\circ}$ angles by adding/subtracting $360^{\circ}$ or $2\pi$ to land back in one revolution.

• Worked example (textbook Ex. 1): $z=3(\cos20^{\circ}+i\sin20^{\circ}),\;w=15(\cos140^{\circ}+i\sin140^{\circ})$

  • Product: $45\bigl(\cos160^{\circ}+i\sin160^{\circ}\bigr)$

  • Quotient: $\tfrac15\bigl(\cos240^{\circ}+i\sin240^{\circ}\bigr)$ because $20-140=-120 \rightarrow 240^{\circ}$.

De Moivre’s Theorem – Powers

• Statement: $[r(\cos\theta+i\sin\theta)]^{n}=r^{n}\bigl(\cos(n\theta)+i\,\sin(n\theta)\bigr)$ for any integer $n$.

• Steps

  1. Raise modulus: $r^{n}$.

  2. Multiply angle: $n\theta$.

  3. If required, convert final polar result to rectangular by evaluating trig values.

• Classroom examples

  • $\bigl[2(\cos20^{\circ}+i\sin20^{\circ})\bigr]^{3}=8(\cos60^{\circ}+i\sin60^{\circ})=8\left(\tfrac12 + i\,\tfrac{\sqrt3}{2}\right)=4+4i\sqrt3$

  • $\bigl[6(\cos60^{\circ}+i\sin60^{\circ})\bigr]^{5}=7776(\cos300^{\circ}+i\sin300^{\circ})=3888-3888i\sqrt3$

  • Example mixing rectangular start:

  1. Convert $\sqrt3 - i$ to polar ⇒ $r=2,\;\theta=300^{\circ}$.

  2. Raise to 6th power ⇒ $2^{6}(\cos(1800^{\circ})+i\sin(1800^{\circ})) =64(\cos\pi+i\sin\pi)=-64$.

De Moivre’s Theorem – n-th Roots

• Formula for all $n$ roots ($k=0,1,\dots,n-1$):
  $<br>\sqrt[n]{z}=\sqrt[n]{r}\;\Bigl(\cos\tfrac{\theta+360^{\circ}k}{n}+ i\,\sin\tfrac{\theta+360^{\circ}k}{n}\Bigr)<br>$

• Geometric facts

  • Exactly $n$ roots, equally spaced by $\Delta \theta = \tfrac{360^{\circ}}{n}$.

  • All roots lie on a circle of radius $\sqrt[n]{r}$.

• 5th-root example (lecture):

  1. Original number: $-\tfrac12 + i\tfrac{\sqrt3}{2}=2\bigl(\cos120^{\circ}+i\sin120^{\circ}\bigr)$.

  2. 5th-root modulus: $\sqrt[5]{2}$.

  3. Base angle: $\tfrac{120^{\circ}}{5}=24^{\circ}$.

  4. Add $72^{\circ}$ (because $360/5$) repeatedly:

  • $24^{\circ},<br>96^{\circ},<br>168^{\circ},<br>240^{\circ},<br>312^{\circ}$.

  1. Full root list: $\sqrt[5]{2}\Bigl(\cos24^{\circ}+i\sin24^{\circ}\Bigr),\;\dots$ (five total).

• 4th-root example with an existing square-root modulus:

  • $z=-1-i$ ⇒ $r=\sqrt2,\;\theta=225^{\circ}$.

  • Careful with the combined index: current radical already has index 2, new root index 4 ⇒ overall $\sqrt[8]{2}$.

  • First root angle $=\tfrac{225^{\circ}}{4}=56.25^{\circ}$, spacing $=90^{\circ}$.

  • Subsequent angles: $146.25^{\circ},236.25^{\circ},326.25^{\circ}$.

Handling Indices Inside Existing Radicals

• If modulus already appears as $\sqrt[m]{r}$ and you take an $n$-th root, total index becomes $m\,n$.

• Example shown: $\sqrt{2}$ (index $2$) then 4th root ⇒ overall $\sqrt[8]{2}$.

Common Pitfalls & Instructor Warnings

• Product: easy; quotient: angle order matters (top – bottom).

• Always reduce final angles to the 0^{\circ}!\le!\theta<360^{\circ} (or 0\le\theta<2\pi) range.

• In root problems NEVER forget to divide the original principal angle by $n$ before adding multiples of $\tfrac{360^{\circ}}{n}$.

• Keep $i$ outside square-roots; writing $i\sqrt{\cdot}$ inside the radical is incorrect.

• When converting rectangular → polar, match signs to pick the correct quadrant.

Real-World / Course Logistics Mentioned

• Quiz 40 answer key posted on D2L (file B-12).

• Test #3 scheduled for “tonight”.

• Final exam: Tuesday the 11th.

• Course evaluation earns up to 5 bonus points (submit after final for reliable credit).

• Instructor catch-phrase: “multiplication is friendly, division is tricky”; “our best friend Jesus wants to live in your heart.”

Quick Reference Sheet

• Modulus: $r=\sqrt{a^{2}+b^{2}}$

• Argument (deg): $\theta=\operatorname{atan2}(b,a)\;(\text{adjust to quadrant})$

• Polar → Rectangular: $x=r\cos\theta,\;y=r\sin\theta$

• Product: $r<em>1r</em>2,\;\theta<em>1+\theta</em>2$

• Quotient: $\tfrac{r<em>1}{r</em>2},\;\theta<em>1-\theta</em>2$

• Power: $r^{n},\;n\theta$

• Roots ($n$): $\sqrt[n]{r},\;\tfrac{\theta+360k}{n}\;(k=0\dots n-1)$

Practice, double-check your angle reductions, and you’ll avoid almost every arithmetic trap discussed in class!