Euler's formula: e^(itheta) = cos(theta) + isin(theta)

Polar form: z = re^(itheta) where r = |z| and theta = arg(z)

Why polar form is powerful:

• Multiplication: z1z2 = r1r2 * e^(i(theta1+theta2)) -- multiply moduli, add arguments
• Division: z1/z2 = (r1/r2) * e^(i(theta1-theta2))
• Powers: z^n = r^n * e^(intheta)
• Roots: z^(1/n) = r^(1/n) * e^(i*(theta+2k*pi)/n) for k = 0, 1, ..., n-1

Special values:

• e^(i*pi) = -1 (Euler's identity)
• e^(i*pi/2) = i
• e^(i2pi) = 1
• e^(ipi/3) = 1/2 + isqrt(3)/2

Conversion: a + ib = sqrt(a^2+b^2) * e^(i*arctan(b/a)) [with quadrant adjustment]