AcademyComplex Form
Academy
Argument
Level 1 - Math I (Physics) topic page in Complex Form.
Argument of Complex Numbers
The argument of a complex number is the angle it makes with the positive real axis in the complex plane.
Definition
Argument
\[\arg(z) = \theta \quad \text{where} \quad z = re^{i\theta}\]
The argument can be expressed in radians or degrees.
Principal Argument
The principal argument \(\text{Arg}(z)\) is the unique argument in the interval \((-\pi, \pi]\):
Principal Argument
\[\text{Arg}(z) \in (-\pi, \pi]\]
For example:
- \(\text{Arg}(1 + i) = \frac{\pi}{4}\)
- \(\text{Arg}(1 - i) = -\frac{\pi}{4}\)
- \(\text{Arg}(-1) = \pi\)
Multiple Values
The general argument has infinitely many values differing by integer multiples of \(2\pi\):
General Argument
\[\arg(z) = \text{Arg}(z) + 2k\pi, \quad k \in \mathbb{Z}\]
For instance, \(\arg(1) = 2k\pi\) for any integer \(k\).
Angle Notation
The shorthand notation \(\text{cis}\theta\) represents \(\cos\theta + i\sin\theta\):
Cis Notation
\[\text{cis}\theta = \cos\theta + i\sin\theta = e^{i\theta}\]
This allows us to write polar form compactly as \(z = r\text{cis}\theta\).