Complex number arithmetic: addition, multiplication, polar/rectangular conversion and modulus/argument.
A complex number calculator handles arithmetic on numbers of the form a + bi (where i² = −1), including addition, multiplication, division, modulus, argument, conjugate, and conversion between rectangular and polar form.
Complex numbers are essential in electrical engineering (AC circuit analysis), signal processing (Fourier analysis), quantum mechanics (wavefunctions), and computer graphics (rotations). Despite the "imaginary" label, complex numbers describe very real physical phenomena.
z = r(\cos\theta + i\sin\theta) = r e^{i\theta}
Inputs: z = 3 + 4i
Real numbers cannot describe rotational/oscillatory phenomena compactly. Complex numbers naturally encode magnitude AND phase in a single object — useful for AC voltages, mechanical vibrations, and wave functions. Many "real-valued" problems are easier when temporarily extended into the complex plane and projected back at the end.
Multiplying by a complex number rotates and scales: multiplying by e^(iθ) is a pure rotation by θ radians; multiplying by a positive real r is a pure scaling by factor r. Multiplying by a general complex z = re^(iθ) does both at once. This is why complex numbers are the natural way to describe 2D rotations.
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