10.3 Complex Numbers in Polar Form

Complex numbers in polar form offer a powerful way to visualize and manipulate numbers on the complex plane. By using magnitude and angle, we can easily perform operations like multiplication and division, which can be tricky in rectangular form.

Polar form shines when dealing with powers and roots of complex numbers. It allows us to use De Moivre's Theorem for exponents and find multiple solutions for roots, making it a versatile tool for solving complex equations.

Complex Numbers in Polar Form

Polar form of complex numbers

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• Polar form representation $r(\cos\theta + i\sin\theta)$ expresses complex numbers using magnitude and angle
• Modulus $r$ measures distance from origin to point in complex plane (3 units)
• Argument $\theta$ indicates angle from positive x-axis to point (45°)
• Relates to rectangular form $a + bi = r(\cos\theta + i\sin\theta)$ connecting Cartesian and polar coordinates
• Cis notation $r \cdot cis(\theta)$ provides compact way to write polar form where $cis(\theta) = \cos\theta + i\sin\theta$

Rectangular vs polar form conversion

• Rectangular to polar conversion calculates modulus $r = \sqrt{a^2 + b^2}$ and argument $\theta = \tan^{-1}(\frac{b}{a})$
• Polar to rectangular conversion finds real part $a = r\cos\theta$ and imaginary part $b = r\sin\theta$
• Quadrant considerations require adjusting argument for quadrants II, III, and IV (add π or 2π)
• Special cases include pure real numbers with $\theta = 0$ or $\pi$ (3+0i) and pure imaginary numbers with $\theta = \frac{\pi}{2}$ or $\frac{3\pi}{2}$ (0+2i)

Multiplication and division in polar form

• Multiplication in polar form multiplies moduli and adds arguments $r_1r_2[\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2)]$
• Division in polar form divides moduli and subtracts arguments $\frac{r_1}{r_2}[\cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2)]$
• Simplifies complex arithmetic compared to rectangular form especially for repeated multiplications or divisions

Powers and roots using polar form

• De Moivre's Theorem for powers raises modulus to power and multiplies argument by power $r^n(\cos(n\theta) + i\sin(n\theta))$
• nth roots of complex numbers use formula $\sqrt[n]{r}(\cos(\frac{\theta + 2\pi k}{n}) + i\sin(\frac{\theta + 2\pi k}{n}))$ with $k = 0, 1, 2, ..., n-1$
• Applies to solving complex equations and finding multiple solutions in trigonometric equations (cubic roots of unity)