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Section 2.8 The Split Complex Numbers

(This section was originally published in [1].)

Start with the real numbers, and apply the Cayley–Dickson process with \(\epsilon=-1\text{.}\) The resulting algebra is known as the split complex numbers, denoted \(\CC'\text{,}\) and satisfies

\begin{equation} \CC' = \RR \oplus \RR L\tag{2.8.1} \end{equation}

where

\begin{equation} L^2 = 1\tag{2.8.2} \end{equation}

rather than \(-1\text{.}\) What are the properties of such numbers?

A general element of \(\CC'\) takes the form \(a+bL\text{,}\) with \(a,b\in\RR\text{.}\) 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\tag{2.8.3} \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\tag{2.8.4} \end{equation}

Furthermore

\begin{equation} \left(\frac12(1\pm L)\right)^2 = \frac12(1\pm L)\tag{2.8.5} \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 \(1\text{,}\) but so do \(\pm L\text{.}\) More generally, from

\begin{equation} (a+bL)^2 = (a^2+b^2)+2abL\tag{2.8.6} \end{equation}

and

\begin{equation} (a^2+b^2)\pm 2ab = (a\pm b)^2 \ge 0\tag{2.8.7} \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 (2.8.3) 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 \(x+tL\text{.}\) The hyperbolic nature of the geometry of special relativity [2] leads to

\begin{equation} e^{L\beta} = \cosh(\beta) + L\sinh(\beta)\tag{2.8.8} \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.