The Journal of Online Mathematics and Its Applications, Volume 8 (2008)
The Most Marvelous Theorem in Mathematics, Dan Kalman

Geometric condition for zw to be a negative real

A particular case of complex multiplication is of interest in Marden's Theorem. The product of two complex numbers is a negative real number if and only if the angles of the two numbers add up to ±π. Geometrically, this is clear from the polar form rules on the previous page. The angles of two complex numbers add up to ±π if and only if the polar coordinates of their product has an angle of ±π. But that is exactly the condition under which a point in the plane lies on the negative x axis, representing a negative real number in the complex plane.