Animated Solution for Mathematics - Complex Numbers: $z$ and $w$ are two nonzero complex numbers such that $|z|=|w|$ and $Argz+Argw=\pi$ then $z$ equals

Imagine standing at the center of the Argand plane, looking out at a circle of radius $r$. This is where our complex numbers $z$ and $w$ reside.

The problem provides a beautiful starting point: $|z|=|w|=r$. This tells us that both $z$ and $w$ are dancing on the same circle, equidistant from the origin.

We are given the condition $Arg(z)+Arg(w)=\pi$. Let us set $Arg(w)=\theta$. This means $w$ is at an angle $\theta$ from the positive real axis. Consequently, $Arg(z)=\pi -\theta$.

Now, let us translate this into the language of algebra. We know that any complex number can be written in polar form as $r(\cos \varphi +i\sin \varphi )$.

For $w$, this is simply:

For $z$, we use our derived argument:

This is where the magic of trigonometry happens. We need to simplify $\cos (\pi -\theta )$ and $\sin (\pi -\theta )$. Using the supplementary angle identities, we know that $\cos (\pi -\theta )=-\cos \theta$ and $\sin (\pi -\theta )=\sin \theta$.

Substituting these back into our expression for $z$, we get:

Now, let us find the expression for the conjugate $\bar{w}$. The conjugate $\bar{w}$ is the reflection of $w$ across the real axis, which means its angle is $-\theta$.

Thus, we have:

Now, compare our expressions:

If we factor out a $-1$ from the expression for $z$, we get:

Look closely at the term inside the parentheses—it is exactly $\bar{w}$. Therefore, we arrive at the elegant conclusion:

This result is not just an algebraic manipulation; it is a geometric truth. Reflecting $w$ across the real axis gives $\bar{w}$, and then reflecting that across the origin gives $z$. You have just mastered the symmetry of complex numbers!