Introducing Super Complex Numbers

We know that i and -i are square roots of -1. They are part of the axis of imaginary numbers. Combine them with real numbers and you get complex numbers. Complex numbers are often used to model rotations, spins, oscillations, vibrations. They can even be used to model error margins. 55 +/- 3 can be expressed as 55 +/- 3i.

Complex numbers are useful any time you have a real-number value combined with a number that fluctuates. A good example is a wave. The real number tells you how long the wave is or how far it has moved along the x-axis. The imaginary number tells you the vertical measurement along the i axis. (See diagram below.)

Point A above is the sum of the real part and the imaginary part. Complex numbers work well in the above example, but suppose you want to model, say, an electric wave and a magnetic wave? For that you may want to use super complex numbers. Super complex numbers have an extra imaginary axis so you can model two waves for the price of one:

As you can see, i1 and i2 are both equivalent to i. They are both square roots of -1. Multiplications between them yield -1 or 1 in the same manner as plain old i. Operations of super complex numbers are similar to complex numbers.

Division with super complex numbers is tricky just as it is with complex numbers. You use super complex conjugates ( super complex numbers with the signs reversed) to get a real number solution:

The type of super complex number we've been working with so far is called a type-2 super complex number. It's type-2 because there are two imaginary axes. A complex number is a type-1 super complex number, since it only has one imaginary axis. Real numbers are type-0 for an obvious reason. All this implies we can have type-3 or even type-infinity super complex numbers.

Suppose we have multiple waves propagating through space? We can model the entire system with the following expression:

So far, all our waves have conveniently moved along the x-axis. What about the y and z axes? Or some combination of axes? Suppose we have waves moving along a vector? The imaginary axes would have to become imaginary vectors with the same angular relationship to the real-number vector as they had with the x-axis.

Below is a super complex vector expression:

But why stop at super complex vectors when we can have super complex tensors? The diagram below shows a type-n super complex tensor of rank-2. Below that is a general expression that can fit any super complex number or tensor.

Imagine being able to model a highly complex system filled with fixed values and variations. The weather perhaps? You could also model beams in a building as they vibrate during an earthquake. The beams could make up real-number tensors, and all the different vibrations could make up the imaginary-number tensors. These are but a couple of examples of what you can do with super complex numbers. Their application is only limited by your imagination.

For more information of this topic see "Probabilities, Euler's identity and Super-complex Numbers."