On p. 24 "Clifford Algebra and Spinors" (2.2 Double Ring of $\:{}^2\mathbb{R}$ of $\mathbb{R}$) by Petri Lounesto the author mentions Study numbers as an equivalent alternative to complex numbers.
The Study numbers are pair of two real numbers (a,b) $\in \mathcal{\mathbb{R}}^2$ such as $$a+jb, \quad j^2=1 ,\quad j \neq 1$$
He introduces Study conjugate as $(a+jb)^-=a-jb$, then they can be written in hyperbolic polar form e t c.
It seems that the use of such numbers can be as powerful as the use of complex numbers then.
My questions are
- Why Study numbers are not widely known and used?
- Can someone point me to the reference to the Study numbers? I can not find it either in Wikipedia, neither in literature.
PS The mathematician is Eduard Study. He approached the dual quaternions with the dual numbers.
I haven't heard of these being called the Study numbers; the seemingly more common name is split-complex numbers. Equivalently, it is the Clifford algebra $\operatorname{Cl}_{1,0}(\mathbb R)$ of a one-dimensional real vector space with a positive-definite quadratic form.
As a real algebra, these are isomorphic to a direct product $\mathbb R \times \mathbb R$, thus it is hardly an interesting research subject (anything about it follows from properties of $\mathbb R$ and direct products of rings).