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

[a−bba]

You can “discover” i by doing (which is used for another discovery)

[c−ddc]⋅(ab)=(ac−bdad+bc)

(a+bi)⋅(c+di)=ac+adi+bci+bdi2

From here on you can infer that i2=−1.So what if we do the same thing, but a different matrix?

z=a+bh

can describe a matrix

[abba]

and we can discover it the same way

[cddc]⋅(ab)=(ac+bdad+bc)

(a+bh)⋅(c+dh)=ac+adh+bch+bdh2

From here we infer that h2=1.Also

ex=1+x1!+x22!+x33!+x44!+x55!+⋯

exh=1+xh1!+(xh)22!+(xh)33!+(xh)44!+(xh)55!+⋯=1+xh1!+x22!+x3h3!+x44!+x5h5!+⋯=coshx+h⋅sinhxHow 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)

**Answer**

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 ϵ satisfying ϵ2=0, and this gives dual numbers. Linear transformations guarantee these two systems and complex numbers are the only ways to extend 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∈R[X] with d≥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. C arises from p=X2+1, whereas this video explores p=X3−1.

**Attribution***Source : Link , Question Author : EEVV , Answer Author : J.G.*