# How many “super imaginary” numbers are there?

How many “super imaginary” numbers are there? Numbers like $$ii$$? I always wanted to come up with a number like $$ii$$ but it seemed like it was impossible, until I thought about the relation of $$ii$$ and rotation, but what about hyperbolic rotation? Like we have a complex number
$$z=a+bi z = a + bi$$
can describe a matrix
$$[a−bba] \begin{bmatrix} a & -b \\ b & a\end{bmatrix}$$
You can “discover” $$ii$$ by doing (which is used for another discovery)
$$[c−ddc]⋅(ab)=(ac−bdad+bc) \begin{bmatrix} c & -d \\ d & c\end{bmatrix} \cdot \begin{pmatrix} a \\ b \end{pmatrix} = \begin{pmatrix} ac - bd \\ ad + bc \end{pmatrix}$$
$$(a+bi)⋅(c+di)=ac+adi+bci+bdi2 (a + bi) \cdot (c + di) = ac + adi + bci + bdi^2$$
From here on you can infer that $$i2=−1 i^2 = -1$$.

So what if we do the same thing, but a different matrix?
$$z=a+bh z = a + bh$$
can describe a matrix
$$[abba] \begin{bmatrix} a & b \\ b & a\end{bmatrix}$$
and we can discover it the same way
$$[cddc]⋅(ab)=(ac+bdad+bc) \begin{bmatrix} c & d \\ d & c\end{bmatrix} \cdot \begin{pmatrix} a \\ b \end{pmatrix} = \begin{pmatrix} ac + bd \\ ad + bc \end{pmatrix}$$
$$(a+bh)⋅(c+dh)=ac+adh+bch+bdh2 (a + bh) \cdot (c + dh) = ac + adh + bch + bdh^2$$
From here we infer that $$h2=1 h^2 = 1$$.

Also
$$ex=1+x1!+x22!+x33!+x44!+x55!+⋯ e^x = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!} + \cdots$$
exh=1+xh1!+(xh)22!+(xh)33!+(xh)44!+(xh)55!+⋯=1+xh1!+x22!+x3h3!+x44!+x5h5!+⋯=coshx+h⋅sinhx \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)

Your $$hh$$-based number system is called split-complex numbers, and what you called $$hh$$ is in my experience usually called $$jj$$ (although, as @user14972 notes, $$hh$$ is sometimes used). A related system introduces an $$ϵ\epsilon$$ satisfying $$ϵ2=0\epsilon^2=0$$, and this gives dual numbers. Linear transformations guarantee these two systems and complex numbers are the only ways to extend $$R\mathbb{R}$$ to a $$22$$-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 $$−1-1$$, taking you to quaternions, octonions, sedenions etc.;
• Variants exist in which some new numbers square to $$00$$ or $$11$$ instead, e.g. you can have split quaternions and other confusingly named number systems;
• If you really like, you can take any degree-$$dd$$ polynomial $$p∈R[X]p\in\mathbb{R}[X]$$ with $$d≥2d\ge 2$$ and create a number system of the degree-$$ polynomial functions of a non-real root of $$pp$$ you've dreamed up, e.g. $$C\mathbb{C}$$ arises from $$p=X2+1p=X^2+1$$, whereas this video explores $$p=X3−1p=X^3-1$$.