Probabilities, Euler's Identity and Super-complex Numbers

Today we are going to examine how super-complex numbers fit in with Euler's identity and probability amplitudes. If you are not familiar with super-complex numbers, read my post entitled: "Introducing Super Complex Numbers."

Euler's identity is as follows:

It has a cosine and one isine or imaginary number. It can be used to model a right triangle:

Or a propagating wave:

Here is the super-complex number version of Euler's identity:

The exponent has i sub-n and theta sub-n. This tells us we will take the cosine of angle theta, and the isine-thetas from 1 to n. In the example below, n = 3.

The super-complex Euler identity is useful if you are working with one cosine and multiple triangles that share the same cosine. Take the exponent i sub-n, theta sub-n; it shows precisely how many isines/triangles are involved. The super-complex Euler identity is a real time saver. Instead of writing a big long string of trig functions we can write one simple exponent.

Since it can represent multiple triangles, it can be used to represent a wing design for a stealth aircraft:

OK, so that wasn't really a stealth aircraft--just more triangles, but you get the idea. Below are multiple waves propagating through space that are modeled by the super-complex Euler ID:

We know that if we multiply Euler's identity with its complex conjugate we get 1. We can also take the inner products of cosine and isine to get probabilities, but notice we are stuck with just two probabilities:

Using the super-complex Euler ID, is it possible to have as many probabilities as we like--and they all add up to one? To answer this question we need to derive the super-complex Euler ID. Start with the normal Euler ID, then split up the isin(theta) into smaller parts. Each part shall have a coefficient of epsilon:

If we examine a unit circle diagram and some trig relations, we discover that each epsilon sub-j isin(theta) equals isin(theta sub-j). We make a substitution:

If we equate i with each i sub-j and i sub-k, we assume the product of any two indexed i's is -1. Unfortunately this will give us non-zero cross product terms.

Let's check to see if the cross product of two arbitrary indexed i's really do give us -1 and not 0. Take the sum of two arbitrary terms and multiply it by its super-complex conjugate. That will give us a real number we label b^2. Like the isine of Euler's ID, we split b into smaller parts. Call those parts c and d.

The last equation above confirms the product of any two indexed i's equals -1. The great thing about Euler's ID is the square of its absolute value yields the same result as a dot product. We want the super-complex Euler's ID to behave the same way--no non-zero cross products! So here's what we do:

At the second equation above, we assume there are no non-zero cross product terms on the right side; however, without the extra cross-product terms, the left side is greater than the right side. We cure this by increasing the value of the isines. We then throw in unit vectors (ej). We now have a multiplication that behaves like an inner or dot product. We make some further refinements below:

Note how each indexed i is converted to an indexed mu, which behaves like a unit vector, giving the result we want:

Each squared sine can represent a probability and the total is 1.

As you can see, the super-complex version of Euler's identity has a great deal more flexibility than the ordinary Euler's identity. It allows us to model complex systems and probability amplitudes with just one simple exponent expression.