Complex numbers: z = a+bi = r(cosθ+i·sinθ) = re^{iθ}. Modulus r = √(a²+b²). Argument θ = atan2(b,a). Euler's formula: e^{iπ}+1=0 connects all fundamental constants. Multiplication in polar: r₁r₂∠(θ₁+θ₂).
How to Use
Enter a complex number:
- Rectangular: a and b (real/imag)
- Output: Polar form r∠θ
- Extras: Conjugate, powers
Euler's Form
z = re^{iθ}. This unifies trigonometric and exponential functions. de Moivre's theorem: zⁿ = rⁿe^{inθ}. nth roots: n equally-spaced points on circle of radius r^{1/n}.
Applications
AC circuit analysis (impedance), Fourier transforms, quantum mechanics (wave functions), and fractal geometry (Mandelbrot set) all use complex polar form.
Step-by-Step Instructions
- 1Enter real part a.
- 2Enter imaginary part b.
- 3View polar form.
- 4Check modulus and argument.
- 5Compute powers.