Módulo e Fase de Complexo

Modulus and phase: the polar form of a complex number

Take z = a + bi. Its modulus, or magnitude, is |z| = √(a² + b²), which is just how far the point sits from the origin in the Argand plane. The argument, or phase, is arg(z) = atan2(b, a): the signed angle measured from the positive real axis. Using atan2 rather than a plain arctangent is what lets it cover all four quadrants correctly. Put the two together and you have the polar form z = r(cos θ + i sin θ) = r·e^(iθ), where r = |z| and θ = arg(z). That exponential version comes from Euler's formula (1748). To see it in action, take z = 1 + i: you get |z| = √2 and arg(z) = π/4, which means z = √2·e^(iπ/4).

Applications: signals, AC circuits and synthesis

Polar form turns up wherever amplitude and phase need to travel together. In signal processing each sinusoid becomes a phasor carrying both. The FFT leans on powers of the roots of unity e^(2πik/N). In AC electronics the impedance folds resistance and reactance into a single modulus-and-phase pair, while digital filters get read off pole/zero diagrams and FM synthesizers work by modulating the phase of a complex oscillator.

FAQ

Radians or degrees? Mathematicians tend to work in radians, while engineers usually reach for degrees. To switch, use deg = rad·180/π.

What is the principal argument? It is the one value of arg(z) that falls in (−π, π], which is exactly what atan2 hands back.

What happens at z = 0? The modulus |z| = 0, and the argument simply isn't defined, since any angle would do.

Why is polar form useful for products? Because multiplication collapses to r₁r₂·e^(i(θ₁+θ₂)). You multiply the moduli and add the arguments.