How many “super imaginary” numbers are there?

Question 1

How many “super imaginary” numbers are there? Numbers like $i$ ? I always wanted to come up with a number like $i$ but it seemed like it was impossible, until I thought about the relation of $i$ and rotation, but what about hyperbolic rotation? Like we have a complex number
$z = a + bi$
can describe a matrix
$\begin{bmatrix} a & -b \\ b & a\end{bmatrix}$
You can “discover” $i$ by doing (which is used for another discovery)
$\begin{bmatrix} c & -d \\ d & c\end{bmatrix} \cdot \begin{pmatrix} a \\ b \end{pmatrix} = \begin{pmatrix} ac - bd \\ ad + bc \end{pmatrix}$
$(a + bi) \cdot (c + di) = ac + adi + bci + bdi^2$
From here on you can infer that $i^2 = -1$ .

So what if we do the same thing, but a different matrix?
$z = a + bh$
can describe a matrix
$\begin{bmatrix} a & b \\ b & a\end{bmatrix}$
and we can discover it the same way
$\begin{bmatrix} c & d \\ d & c\end{bmatrix} \cdot \begin{pmatrix} a \\ b \end{pmatrix} = \begin{pmatrix} ac + bd \\ ad + bc \end{pmatrix}$
$(a + bh) \cdot (c + dh) = ac + adh + bch + bdh^2$
From here we infer that $h^2 = 1$ .

Also
$e^x = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!} + \cdots$
$\begin{align} e^{xh} & = 1 + \frac{xh}{1!} + \frac{(xh)^2}{2!} + \frac{(xh)^3}{3!} + \frac{(xh)^4}{4!} + \frac{(xh)^5}{5!} + \cdots \\ & = 1 + \frac{xh}{1!} + \frac{x^2}{2!} + \frac{x^3h}{3!} + \frac{x^4}{4!} + \frac{x^5h}{5!} + \cdots \\ & = \cosh{x} + h \cdot \sinh{x} \end{align}$

How many more numbers like this are there? And does that mean that for each set of trigonometric functions there exists a number which can turn multiplication into a rotation using those trigonometric functions?

(Sorry if I got some things wrong)

Question 2

Your $h$ -based number system is called split-complex numbers, and what you called $h$ is in my experience usually called $j$ (although, as @user14972 notes, $h$ is sometimes used). A related system introduces an $\epsilon$ satisfying $\epsilon^2=0$ , and this gives dual numbers. Linear transformations guarantee these two systems and complex numbers are the only ways to extend $\mathbb{R}$ to a $2$ -dimensional commutative associative number system satisfying certain properties. However:

The Cayley-Dickson construction allows you to go from real numbers to complex numbers and thereafter double the dimension as often as you like by adding new square roots of $-1$ , taking you to quaternions, octonions, sedenions etc.;
Variants exist in which some new numbers square to $0$ or $1$ instead, e.g. you can have split quaternions and other confusingly named number systems;
If you really like, you can take any degree- $d$ polynomial $p\in\mathbb{R}[X]$ with $d\ge 2$ and create a number system of the degree- $<d$ polynomial functions of a non-real root of $p$ you've dreamed up, e.g. $\mathbb{C}$ arises from $p=X^2+1$ , whereas this video explores $p=X^3-1$ .

How many “super imaginary” numbers are there?

Answer

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