Modulus and argument
|z| = √(a² + b²); arg(z) = atan2(b, a) in (−π, π].
Modulus: distance from origin. Argument: angle from positive real axis. Use atan2 (or arctan with quadrant adjustment) for arg.
Worked examples
z = 3 + 4i: |z| = 5, arg(z) = arctan(4/3) ≈ 0.927 rad.
z = −1 + i: |z| = √2, arg(z) = 3π/4.
arg(zw) = arg(z) + arg(w); |zw| = |z||w|.
Frequently asked questions
Argument range?
(−π, π] is the principal argument. Add multiples of 2π for general.
Why mod-arg form useful?
Multiplication and powers become easy: arguments add, moduli multiply.