How to decide the order of points and the sign of an argument in the complex plane

Key concept: When you work with arguments of complex numbers, the direction you trace around the origin changes the sign and the total angle. That is why the order of the points is not arbitrary.

Step by step

1) Why the order is usually written as $z_1 \to z_2 \to z_3 \to z_4$

Because the argument is tied to a directed angle. Once you choose a path around the complex plane, you are no longer just naming points; you are fixing a direction of rotation.

• If the points are taken in counterclockwise order, the angle change is positive.
• If they are taken in clockwise order, the angle change is negative.

So the order is chosen to match the intended orientation of the polygon or arc, not because the other orders are impossible.

2) Why a different order can give inscribed angles

If you reorder the points, you may end up measuring a different geometric angle — sometimes an inscribed angle rather than the intended directed angle in the complex plane. That changes the interpretation.

In many complex-number arguments, the goal is to track the change in direction along a specific traversal. Reordering the vertices changes that traversal.

3) About the sign: could it be $-\pi$?

Yes, it can be. If the orientation shown is clockwise, then the accumulated change in argument is negative. For example, a half-turn clockwise gives $-\pi$, while a half-turn counterclockwise gives $\pi$.

So the key question is:

• What direction are you moving around the origin?
• Are you taking principal values or cumulative changes?

Those two things are often mixed up.

4) Do we “switch the vectors around”?

Not exactly. You do not change the vectors just to force a preferred sign. Instead, you should:

1. Identify the actual directed segment or rotation.
2. Measure the argument change in that direction.
3. If needed, rewrite the angle using an equivalent principal value, but only after the geometry is handled correctly.

5) A practical rule

If you are finding an argument from several points, ask:

• Is the path traced in the positive direction or negative direction?
• Are the points listed in traversal order?
• Is the result supposed to be a principal argument in $(-\pi,\pi]$ or a total accumulated angle?

That usually clears up the sign confusion fast.

Pitfall alert

A common mistake is to treat arguments like ordinary undirected angles. In complex numbers, the orientation matters. Another frequent error is forcing everything into a principal value too early; that can hide whether the actual turn was $\pi$ or $-\pi$.

Try different conditions

If the same points are listed in the opposite order, the argument changes sign. If the figure is traversed counterclockwise instead of clockwise, the result becomes the positive counterpart. For a full turn, you may also need to add or subtract $2\pi$ to match the chosen branch of the argument.

Further reading

principal argument, directed angle, complex plane orientation