What is $\tilde{\Bbb{C}}$

Question

$\tilde{\Bbb{C}}$ was defined in the following manner

$\tilde{\Bbb{C}} = \Bbb{R} \cdot 1 + \Bbb{R} \cdot e$

with $1 \cdot 1 = 1, 1 \cdot e = e \cdot 1, e \cdot e = 1$

Could you elaborate more on this please? Or simply what is its name?

Answer 1

There are three ways of extending multiplication to the plane which are useful in elementary geometry. One writes a planar vector formally as $a+be$ for some new element $e$. The arithmetic is then determined by knowledge of $e^2$. The best known is the case of the complex numbers ($e^2=-1$) (related to eulidean geometry). The case $e^2=0$ leads to the so-called dual numbers. Finally, $e^2=1$ is appropriate for hyperbolic geometry. A suitable reference is Yaglom "Complex numbers in geometry". Interesting exercise: compute the exponential function $\exp(a+b e)$ using the power series expansion of the exponential function in the other two cases.

Answer 2

A way to think about this set is $\mathbb{R}[\sqrt{1}]$ (in contrast to $\mathbb{C}=\mathbb{R}[\sqrt{-1}]$. $e$ here is a formal symbol whose square is 1, just like $i$ is a formal symbol whose square is $-1$. A little more formally is that this is isomorphic to $\mathbb{R}[x]/(x^2-1)$, where $(x^2-1)$ is the ideal generated by $x^2-1$.