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

 

[lekh2003]

I haven't said that this is an incorrect view, except if it is reduced to $\mathbb{R}^2$. I think, it just shouldn't be the first view. But, hey, let's consider it as vector space over the rationals.

I see your argument here, but I feel like its sometimes useful to view the complex numbers instead as endowing a convenient multiplicative structure on #\mathbb{R}^2#. Specifically a multiplicative structure which only really coincidentally happens to be possible because of the complex numbers, quaternions and octonions.

But separately, I find viewing it this way neat because it removes some of the fantastical feeling of viewing complex numbers. It shows that the natural properties of holomorphic functions lead into those for harmonic functions in general for $\mathbb{R}^n$