newRPL: The complexity of complex mode

[Claudio L.]

(08-28-2016 12:21 AM)Vtile Wrote:  It seems I still get definition that says that ATan(inf/inf)=Pi/4 which is not defined (ie. in wolfram alpha), yet \( \mathit{z}=\infty*e^{\frac{i\pi}{4}}\) seems to be legit.

This is what I have found most formal definition for complex number.
\(\mathbb{C}=\left \{ \mathit{z} |\mathit{z=a+bi,\:\: a,b\in \mathbb{R}} ,\:\: i^2=-1 \right \} \)
also from a few sources that when Im(z)=0 then the result is real number.
Also it is said that infinity is part of the real number continuum and
calculation rule for complex number is given that
z1*z2=(a1,b1)(a2,b2)=(a1a2-b1b2 , a1b2+b1a2) then
\(\mathit{z}_{\infty }=\infty(a,b)=(\infty,0)(a,b)=(\infty*a-0*b \; ,\; \infty*b+0*a)=(\infty*a \; ,\; \infty*b)=(\infty \; ,\; \infty) = \left |\infty \right |\angle \frac{\pi }{2}=\left |\infty \right |e^{\frac{\pi}{2} i}\)

The above is precisely why I don't think rectangular representation is "lossless". You lost a lot of information the moment you distributed that infinity into each component of the complex (that operation is invalid, infinity doesn't distribute). If instead of Infinity you had X, the result would be the complex (X*a,X*b).
The limit when X tends to infinity would keep the direction of (a,b) as X goes from 0 to infinity, hence the correct result is a vector of infinite length and the same direction of (a,b). This is easy to represent in polar coordinates, hard to do in rectangular.
In polar coordinates, you can do the complex multiplication without applying the distributed property that doesn't work with infinity:
\(\infty==>\infty*e^{i 0}\)
\(Z ==> (a,b) ==> r * e^{i \theta}\)

\(\infty*Z = \infty*r * e^{i(0+\theta)} = \infty * e^{i \theta}\)

This is why it's best to store a directed infinity in polar form.