How many “super imaginary” numbers are there?

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
[abba]
You can “discover” i by doing (which is used for another discovery)
[cddc](ab)=(acbdad+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+hsinhx

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)

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 pR[X] with d2 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=X31.

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

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