When learning about higher dimensional complex numbers, one of the things you must first understand that the complex plane as known for centuries simply does not live in three dimensional space. If you look at it from a historical perspective people always tried to start with the two dimensional complex plane and tried to expand that into three dimensions.
It does not work that way; for example you can make 21-dimensional numbers by using 3-dimensional and 7-dimensional numbers. The dimension nicely breaks down via the prime number theorem (every natural number can be uniquely written as the factors of prime numbers).
Since 2 is not a divisor of 3, it is impossible to find the complex plane in a three dimensional world…
Now years ago I proved that for the complex multiplication the number i does not exist, yet now I started this new website why not give the same proof for the circular multiplication?
Click on the picture below to read that exiting proof… 😉
(On my browser I first have to click on the picture and after that enlarge it to get the readable stuff.)
Picture size is 550 pixels wide by 1650 pixels high, I tried to write it in such a way that advanced high school folks could understand it.