Once you have a good grasp of complex number it’s much easier to also understand a twin concept – Split Complex Number, we name is j, instead of square of i = -1, square of j is 1, but it’s not an one in ordinary sense but an imaginary number too.

Split complex numbers also satisfies

Everything is intuitive now when comparing to complex number, so exact as what |B||A| = |B| as A could be defined as on the unit circle, in split complex number system, we can define |A| as on the unit hyperbola and draw dot for B on another hyperbola curve:

So if we nudge A a little bit by times B that is

We get the shape of hyperbola

Following on Euler’s formula connecting exponential and tri formed complex number, here is the mirror version for connecting exponential and tri formed SPLIT complex numbers.

Split complex number can be expressed in matrix form as

and this identity look on both diagonals matrix can be further written as

The determinate of this matrix is

It’s clear now that introducing this concept of split complex number is greatly helpful in special relativity because we all know that anti-intuition distance of a^2 -b^2. So here are some preparation work:

Now computing the velocity and time in multiple inertial system becomes a super easy problem.