Animated Solution for Mathematics - Complex Numbers: If $arg(z)<0$, then $arg(-z)-arg(z)=$

Imagine you are standing at the origin of the complex plane. You have a vector $z$ pointing somewhere into the fourth quadrant.

We are given the condition that $arg(z)<0$. In the language of complex numbers, a negative argument indicates a clockwise rotation from the positive real axis.

Let us define this angle as $\theta$. Thus, we have $arg(z)=\theta$, where $-\pi <\theta <0$. This vector is anchored firmly in the lower half of the complex plane.

Now, consider the complex number $-z$. Multiplying a complex number by $-1$ is geometrically equivalent to a reflection through the origin.

This reflection corresponds to a rotation of exactly $\pi$ radians, or $180^{\circ }$. If $z$ is at an angle $\theta$, then $-z$ must be at an angle $\theta +\pi$.

Therefore, the transformation is defined by:

We must ensure that our new argument, $\theta +\pi$, remains within the principal argument range, defined as $(-\pi ,\pi ]$.

Given the constraint $-\pi <\theta <0$, we add $\pi$ to every part of the inequality:

This simplifies to:

Since this result is strictly between $0$ and $\pi$, it sits comfortably within the principal range. Because the value is valid, no adjustment by $2\pi$ is required.

We now compute the difference $arg(-z)-arg(z)$. Substituting our established values:

The $\theta$ terms cancel out perfectly. This leaves us with the final result:

It does not matter where $z$ is located within the fourth quadrant; the difference between the arguments is always constant. This demonstrates the elegance of complex geometry, where the variables vanish to reveal a fundamental truth.