When does the next complex split occur?

Question

So I was thinking about complex numbers and how they came about and someting interesting occured to me:

the formation of complex numbers occurs because there exists a function (namely $f(x)=x^2$) that maps reals to a smaller subset of reals and therefore the by setting the equation $x^2 = p$ in $\{x\in\Bbb R:x\not\in f(\Bbb R)\}$. We are forced to create a new mathemtical value, $i$ to handle the equation.

Couldn't this extension occur elsewhere? For example, say there is some function $f_{\Bbb C}$ that maps the complex numbers to a smaller subset of the complex numbers then the expression

$f_{\Bbb C}(x) = j$ in $\{z\in\Bbb C:z\not\in f_{\Bbb C}(\Bbb C)\}$

will require the creation of a new mathematical constant.

Is this possible? Has it ever happened? This differs greatly from quaternions and other tools which are considered extensions of the complex plane. The most basic idea that comes to mind is the absolute value function.

The equation: $|x| = c$ where $c$ is not a positive real number or $0$ but is a member of the complex plane.

You would get numbers $p$, such that $|p| = -1$, and $t$ such that $|t| = i$. From there the expression $p\cdot t$ would be such that $|p\cdot t| = -i$. Then an entire algebra from these can be constructed.

But the only reason I'm not satisfied with this answer is because Complex numbers were naturally developed from the basic hyperoperators (addition -- multiplication -- powers (produces complex numbers) --> tetration ---> pentation etc...) Is there some hyperoperator down the line that will produce the next complex number?

Answer 1

First of all we have the fundamental theorem of algebra which says that the the roots of polynomial functions with complex coefficients all lie in the field of complex numbers.

You are intimating that integers are required to solve

$x+a=b$ with $a>b$ are natural numbers.

Fractions are required to solve

$ax=b$ where $a,\,b\in\mathbb{N}$ with $a\neq 0$.

Real numbers are required to solve

$x^2=a$ with $a\in\mathbb{N}$

etc.

My answer to your question is don't disregard the quaternions so easily: which could be considered to be required to solve an equation of the form:

$xy-yx=a$

where $a\neq 0$.

Answer 2

There are some things that come to mind when you ask this question. One example would be the Riemann Sphere - basically, you have the complex plane $\mathbb{C}$, and a very natural function like $f(z) = \frac{1}{z}$, and you have two basic problems:

1. $f(z)$ is never $0$.
2. $f(0)$ is undefined.

So you add the point $\infty$ to your plane and define the new plane's topology in a way that lets $f(z)$ be defined everywhere.

However, when you do that, you lose the beautiful algebraic properties of $\mathbb{C}$, as this is not even a field anymore. In fact, because $\mathbb{C}$ is an algebraically closed field, it has no finite extensions, meaning that if you want to extend $\mathbb{C}$ and still get a field, you will have to add infinitely many independent elements - and that wouldn't be "pretty", in much the same way that $\mathbb{R}$ is not a "pretty" extension of $\mathbb{Q}$ - it adds uncountable many constants, most of which don't have a nice representation.

Other functions have even more complicated and weirder Riemann surfaces that enable them to make more sense than on the standard complex plane, but all of these are of geometric nature and lack the beauty of the algebraic extension of $\mathbb{R}$ to $\mathbb{C}$

Answer 3

Picard's theorem states that any entire function on $ \mathbb{C}$ that omits more than 1 points is actually constant. i.e. either the range of the function is 1 point (function is constant), the range omits precisely one point (for example: $ x \rightarrow e^x$ is never $ 0$), or has as range all complex numbers (examples: all polynomials, $ \sin$ and $ \cos$, etc.)

so in your construction, we'd either end up with (generalisations) of quaternions, like Clifford algebras or higher order analogues, or we'd have to start with a non-holomorphic function (like you did with the absolute value function). Anyway, if you demand that it is a field, and contains the reals in some way, you'd end up with the reals, the complex numbers, the quaternions, or the octonions, due to Hurwitz's theorem, meaning that we either end up with something where we can't divide, or something already known.