The argument of a complex number is the angle its vector makes with the positive real axis.
If:
\[
z=a+bi
\]
then \(z\) can be plotted as the point \((a,b)\) on the complex plane. The argument tells us the direction
of this point from the origin.
1. Modulus and argument
The modulus is the length of the vector:
\[
|z|=\sqrt{a^2+b^2}
\]
The argument is usually computed with:
\[
\theta=\operatorname{atan2}(b,a)
\]
The function \(\operatorname{atan2}\) is important because it uses the signs of both \(a\) and \(b\).
This places the angle in the correct quadrant.
2. Principal argument
The most common single value of the argument is the principal argument:
\[
-\pi\lt\operatorname{Arg}(z)\le\pi
\]
This is written with a capital \(A\):
\[
\operatorname{Arg}(z)
\]
For example, if:
\[
z=-1+i
\]
then the point lies in Quadrant II, so:
\[
\operatorname{Arg}(-1+i)=\frac{3\pi}{4}
\]
3. Multi-valued argument
Angles repeat every \(2\pi\). Therefore, if \(\theta\) is one argument of \(z\), then all arguments are:
\[
\arg(z)=\theta+2\pi k,
\qquad k\in\mathbb{Z}
\]
For example, if the principal argument is \(3\pi/4\), then other valid arguments include:
\[
\frac{3\pi}{4}+2\pi,
\qquad
\frac{3\pi}{4}-2\pi,
\qquad
\frac{3\pi}{4}+4\pi
\]
4. Positive argument range
Sometimes it is useful to choose arguments in the range:
\[
0\le\theta\lt2\pi
\]
In this convention, angles are always nonnegative. For example:
\[
\arg(-i)=\frac{3\pi}{2}
\]
instead of \(-\pi/2\).
5. Custom branch ranges
A general branch range can be written as:
\[
\alpha\lt\theta\le\alpha+2\pi
\]
The ray at angle \(\alpha\) is the branch cut. Crossing this ray causes the chosen argument value
to jump by \(2\pi\).
6. Branch cuts
A branch cut is needed whenever we want a single continuous argument function.
For the principal argument, the branch cut is usually the negative real axis:
\[
\theta=\pi
\quad\text{or}\quad
\theta=-\pi
\]
Points on a branch cut require careful interpretation because the selected branch is discontinuous there.
7. Formula summary
| Concept |
Formula |
Meaning |
| Complex number |
\(z=a+bi\) |
Point \((a,b)\) on the complex plane |
| Modulus |
\(|z|=\sqrt{a^2+b^2}\) |
Distance from origin |
| Argument |
\(\theta=\operatorname{atan2}(b,a)\) |
Direction angle |
| Principal argument |
\(-\pi\lt\operatorname{Arg}(z)\le\pi\) |
Standard single value |
| All arguments |
\(\arg(z)=\theta+2\pi k\) |
Multi-valued angle family |
| Custom branch |
\(\alpha\lt\theta\le\alpha+2\pi\) |
User-selected argument range |
8. Common mistakes
- Do not use ordinary \(\arctan(b/a)\) without checking the quadrant.
- The argument of \(0\) is undefined because the zero vector has no direction.
- The principal argument is only one value; the full argument is multi-valued.
- Angles that differ by \(2\pi\) represent the same direction.
- Changing the branch range changes the selected argument, not the complex number itself.
- Points on the branch cut may cause a jump in the selected argument.
