Another way to extend the Complex Plane (Insertsomethingthatgetsyourattention)

[Frogeyedpeas]

Hey guys so I was thinking about how to extend the Complex Plane out to a third dimension and I started reading the whole tidbit about Quaternions and their mechanics when I realized that I want to propose a whole new question. Now please feel free to prove me wrong if you can answer it because I haven't found a whole lot.

Imagine a number J who satisfies the solution to the following equation

Log_b(J) = -b

FOR ALL B:

There is no complex number that satisfies that solution and I believe (as uneducated as I might be in this subject) that there is no Quaternion, Octonion or any type of standard Algebraic extension of the number line that satisfies this equation. If this number J can be proposed as the new number extension to the complex plane, then,

We get numbers being described in the form of:

a + bi + cj. Now keeping in mind the ability for numbers to cross:

a + bi + cj + dji is what this number can look like...

What's your take on it?

 

[Frogeyedpeas]

Any replies, opinions, haikus, limericks, epic poetries, hard rock songs, and qwerty-piano recitals would be cool

 

[coelho]

Ok... if by Log_b you mean log in the base b, then we have:
$$log_b(J)=-b$$
$$\frac{ln(J)}{ln(b)}=-b$$
$$ln(J)=-b ln(b)$$
$$J=e^{-b ln(b)}$$
$$J=(e^{ln(b)})^{-b}$$
$$J=b^{-b}$$
Which is, at worst, a complex number (if b were complex). If b were real, so is J.

 

[disregardthat]

If you want a number J to satisfy that equation, you are not extending the complex plane, you are reducing it by the equvalence relation $$b^{-b} = c^{-c}$$ for all complex b and c. If the function z^{-z} := e^{-z\log(z)} for some branch of the logarithm is surjective, you will have collapsed the complex plane to a single point.