newRPL: The complexity of complex mode

[Vtile]

(08-23-2016 07:43 PM)Claudio L. Wrote:  Some cases like (+∞,3) or (∞,∞) could be rejected as syntax error to enforce some discipline on the user and reduce the number of cases.

Hold on, this complex infinities goes partially beyond my knowledge (i don't understand the directed infinity), but (+∞,3) did take my attention.

PS. Now when I have spend too much time front of my computer rewriting this post, I think that the directed infinities are (lossy) polar representations.

It all boils down calculating the Theta with the some rules of the complex plane (or trigonometric sections of the circle).

This can be seen when we transform complex number \(Z=(\infty,3)=\infty + j3 \) to polar notation of
\(\left | Z \right | \angle \theta ^{\circ}\) = \(\sqrt{a^2+b^2}\angle ATan(\frac{b}{a})^{\circ}=\sqrt{a^2+b^2}\angle ACot(\frac{a}{b})^{\circ}\), with the values we get
\(\left | Z \right | = \sqrt{a^2+b^2} = \infty\) , when a (.. or b) approaches infinity
\(\theta = ArcTan(\frac{b}{a}) = ArcTan(\frac{0}{\infty}) = ArcCot(\frac{\infty}{0}) = 0\) => \(\infty \angle 0^{\circ}\) and because we were dealing with positive infinity of real part the theta is indeed 0 degrees (or should it be \(\theta = 2^{-\infty}\) hehe).

Now suddenly we can not return back since..
\( \begin{align*}
& a = Cos(\theta^{\circ})*\left | Z \right | = Cos(0^{\circ})*\left | \infty \right | = 1 *\infty =\infty\\
& b = Sin(\theta^{\circ})*\left | Z \right |= Sin(0^{\circ})*\left | \infty \right | = 0 *\infty = 0 (\neq 3)
\end{align*}\)

Quote:(Inf,3) = Same as real Infinity? the angle tends to ∡0° if you look at it in polar mode

It can not be, the imaginary part stays as 3j even though in calculations when we took angle of it \(\theta = ArcTan(\frac{b}{a}) = ArcTan(\frac{0}{-\infty}) = ArcCot(\frac{-\infty}{0}) = 0), still in rectangular form it doesn't vanish anywhere it just loose its purpose in practical calculation, since we can not use it for calculating an real angle of it since we do not have enough paper and the effects compared to the real component lost it importance.

So how I understand this "directed infinity" is that it is "partially continuous" and we should keep attention in which sector in im / re plane it stays. Then it can be expressed with the "nonlossy" rectangular (complex plane) form.


How I have understood the polar form is that it is "just a techical simplification" of complex system to avoid the polynomial calculation when adding and multiplying (with sliderule).

Partially related history snippet: The notation |Z|∡θ instead of |Z|*e^iθ did come to use somewhere at the first half of the 20th century (I have one book from 40s..50s era or so that mentions it as a new spreading notation in (atleast) electrical engineering).

Quote:(0,Inf) = Malformed, should be (Inf ∡90°)
(0,-Inf) = Malformed, should be (Inf ∡-90°)

Malformed why?

Quote:(3,Inf) = (Inf ∡90°)??

Comes from ArcTan.

Quote:(Inf,Inf) = Undirected infinity?

actually if a=inf and b=inf and inf=inf then it would be inf in angle of 45deg +n*90deg ???

Like said this subject is on the edge of my understanding / knowledge and partially over it.

PPS. what comes to the question if NewRPL is simple calculator the answer is NO, it is not a calculator, but a mathematical environment. Sorry it is first and foremost your project at this moment so it is your decision in the end.

PPPS. Interesting (new for me, I haven't studied it) and not entirely related is the spiral vector theory proposed by Sakae Yamamura (honored by IEEE atleast one time ), could be nice to understand it in the level that I could make a technically working library for it in the future for UserRPL & NewRPL. Let see starts to go too much to science for my interests.

edit. change \(\theta = 1^{-\infty}\) to \(\theta = 2^{-\infty}\), jumping lately to number theory concepts is vaporising my calculus and magnifying my careless errors.