Split Complex Numbers

Start with the real numbers, and apply the Cayley-Dickson process with $\epsilon=-1$. The resulting algebra is known as the split complex numbers, denoted $\CC'$, and satisfies \begin{equation} \CC' = \RR \oplus \RR L \end{equation} where \begin{equation} L^2 = 1 \end{equation} rather than $-1$. What are the properties of such numbers?

A general element of $\CC'$ takes the form $a+bL$, with $a,b\in\RR$. Just like the ordinary complex numbers, the split complex numbers are both commutative and associative. The (squared) norm is given by \begin{equation} |a+bL|^2 = (a+bL)(a-bL) = a^2-b^2 \label{splitC} \end{equation} which is not positive-definite. In particular, $\CC'$ contains zero divisors, for instance \begin{equation} (1+L)(1-L) = 1-L^2 = 0 \end{equation} Furthermore \begin{equation} \left(\frac12(1\pm L)\right)^2 = \frac12(1\pm L) \end{equation} so that $\frac12(1\pm L)$ act as orthogonal projection operators, dividing $\CC'$ into two null subspaces.

Another curious property of $\CC'$ involves square roots. How many split-complex square roots of unity are there? Four! Not only do $\pm1$ square to $1$, but so do $\pm L$. More generally, from \begin{equation} (a+bL)^2 = (a^2+b^2)+2abL \end{equation} and \begin{equation} (a^2+b^2)\pm 2ab = (a\pm b)^2 \ge 0 \end{equation} we see that a split complex number can only be the square of another split complex number if its real part is at least as large as its imaginary part. In particular, $L$ itself cannot be the square of any split complex number!

You may recognize the inner product (\ref{splitC}) as that of special relativity in 2 dimensions, with the spacetime vector $(x,t)$ in 2-dimensional Minkowski space corresponding to the split complex number $x+tL$. The hyperbolic nature of the geometry of special relativity [The Geometry of Special Relativity] leads to \begin{equation} e^{L\beta} = \cosh(\beta) + L\sinh(\beta) \end{equation} which can also be checked by expanding $\exp(L\beta)$ as a power series. For this reason, the split complex numbers are also called hyperbolic numbers. Hyperbolic numbers not only represent the points in 2-dimensional Minkowski space, but can also be used to describe the Lorentz transformations between reference frames, which are nothing more than hyperbolic rotations.

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