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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.

Anixx
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Eddward
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    I suppose $j\neq -1$ either – Matías Ures Dec 20 '20 at 10:02
  • I guess so too. – Eddward Dec 20 '20 at 10:04
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    I'm curious—the complex numbers were invented to solve equations such as $x^2+1=0$. How do study numbers solve this problem—what study number, when multiplied by itself, makes $-1$? – Joe Dec 20 '20 at 10:05
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    Study numbers are well known, but their ring is simply isomorphic to $\Bbb R\times R$ via $(a+bj)\mapsto(\frac{a+b}2,,\frac{a-b}2$ and thus algebraically it doesn't bring any more information. – Berci Dec 20 '20 at 10:08
  • I mean this wouldn't be a field, in fact $(a+ja)(a-ja)=0$ so neither $a+ja$ nor (a-ja) is invertible – Matías Ures Dec 20 '20 at 10:12

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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).

lisyarus
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