Views On Complex Numbers

[Paul Colby]

Is there a way to define complex conjunction without the mapping $\mathbb{C}\rightarrow \mathbb{R}^2$? The involution, $\ast$, always seemed the gateway to $\mathbb{R}^2$.

 

[FactChecker]

If $z=r e^{i\theta}$, then $\bar z = r e^{-i\theta}$. There are other, more general, ways to reflect a vector through a general vector, plane, or other multi-dimensional subspace.

 

[Agent Smith]

In the bottom case, congruence doesn't depend on orientation, but by a combination of relations between sizes of sides, angles. I'm not sure the ancient Greeks who laid out such notions were even aware of general notions of orientation, orientability.

Methinks they were aware. Congruence and similarity are based on corresponding angles, sides. Gracias