This Hypercomplex package provides the algebra of the quaternion, octonion and sedenion hypercomplex numbers.;

Feel free to utilize this package for academic, business, or personal use according to Maplesoft's license agreement. However, neither Maplesoft nor I are liable for any errors that may come from this package.  All I ask is that you give me credit.  Please do not reverse engineer this package; it is the ethical thing to do.

In the universe, we are aware that invisible entities exist that we want to learn.  Knowledge about these invisible entities add to our universal jigsaw puzzle.  Unfortunately, we currently only describe our universe with six properties:  Energies, Forces, Spaces, Times, Motions, and Matter. Any remaining properties remain inaccessible, and thus have eluded our metaphoric puzzle.   For the six currently known components, physicists have studied classical and modern physics of the universe at length. Einstein and Newton used mathematics (their seeing-eye dog) to help them understand the forces in our universe--in particular, gravity.  

Newton unified the heavens and the earth by showing that gravity is universal.  Einstein unified space and time by showing that gravity is the result of mass bending the local space-time continuum.    In addition to gravity, we know have quantum mechanical forces. Following Einstein and Newton, a mathematical understanding of these quantum mechanical forces would significantly increase our knowledge.  

The Cayley-Dickson hypercomplex numbers may be the mathematical 'seeing-eye dog' that leads to an understanding of quantum mechanical forces.  Already, The Pauli Spin Matrices and Maxwell equations have been defined in terms of quaternions.

There is something interesting about the rotational operator "-1".  Multiplying a real number with "-1" rotates the 1-D vector either left or right.  Multiplying by -1 an even number of times willl rotate the 1-D vector right and will rotate left for an odd number of multiplications. Complex numbers will change both the direction and magnitude of 1-D vectors using phasors.  

In addition, we can use vectors on a 2-D space of 1-D vectors through a stereographic projection from the Riemann sphere.    I indicated earlier that there was something interesting about the "-1" rotation.  We are also interested in derivations of "-1".  It is very `naïve` to represent the imaginary component as the square root of negative one.  In quaternions, the components of i, j, and k are not equal to each other, though squaring any one of them will result in "-1".  If one look at a double or dual complex numbers, we do not see these `naïve` representations, such as the square root of zero or the square root of one.  Nevertheless, students have been taught that the square root of negative one is a true exact representation of the imaginary component we called "i".  We cannot represent the exactly of the transcendental number pi, instead we use a symbol.  Therefore, we will do the same for imaginary components of hypercomplex numbers.  The only exception to this rule is their matrix representations.   

In another situation engineers explore quaternions to eliminate only gimbal lock (ex. satellites and space shuttles).  Graphic programmers also use quaternions for eliminating gimbal locks.  In abstract algebra, the quaternions are studied en passant only as a non-abelian group.  However, little credit is given to the fact that the Pauli Spin Matrices are actually derivations of quaternions.  Einstein studied tensors for his unification of General Relativity with Electro-Magnetism, but limited them to 360-degree rotation.  

However, in quantum mechanics, we need to represent intrinsic spin with a maximum of 720 degrees.  We cannot do this without the use of quaternions.  This also applies to the quaternionic 2-vector of the Dirac equations.  To resolve these difficuites, and upgrade the mathematical toolset of the scientific community, we pose the question:  how can physics and other sciences take advantage of the Cayley-Dickson Construction Hypercomplex Numbers"?

One of the major obstacles is simply "informing" the scientific community about this mathematical technique the apparent obstacle of the successive loss of mathematical properties as complexity is increased. This obstacle could be overcome by simply seeing that each lost property provides "power" in usage, instead of the opposite.  This is evident in a direct analogy when doing integration on Reals vs. ordinary complex numbers; ordinary complex numbers lose the properties of order, trichotomy, and self-conjugation. Despite this, there is more "power" in integration of the complex field than integrating over the Reals."  There are two reasons why quaternions were not embraced at the time of their discovery.    

1) Hamilton skills to promote the quaternions was not as good as Gibbs' and Heaviside's skills in promoting vectors analysis.

2) Gauss has proven that the fundamental theorem of Algebra only requires the use of ordinary complex numbers.    

Hamilton was simply ahead of his time.  Relativity was so new and different at this time that many scientists dismissed Einstein's theories.  Moreover, quantum mechanics was so strange to Einstein that he dismisses it with this quote, "I, at any rate, am convinced that He [God] does not throw dice."  In addition, the Cayley-Dickson Constructions Hypercomplex Numbers were so strange to many scientists that they dismissed them as mathematical novelties, as they did with fractals.   

I hypothesize that this largely unexplored branch of mathematics may be the tool needed for unification. We discovered general relativity via an old pure mathematical entity--Riemannian's geometry utilizing tensor analysis. We discovered String Theory via an old pure mathematical entity--Euler's beta function.  Similarly, I believe that we will discover future physics via an old pure mathematical entity--the Cayley-Dickson Construction Hypercomplex numbers.

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There are higher dimensional numbers besides complex numbers.  There are also hypercomplex numbers, such as, quaternions (4D), octonions (8D), sedenions (16D), pathions (32D), chingons (64D), routons (128D), and voudons (256D).  These names were coined by Robert P.C. de Marrais and Tony Smith.  It is an alternate naming system providing relief from the difficult Latin names, such as: trigintaduonions (32D), sexagintaquattuornions (64D), centumduodetrigintanions (128D), and ducentiquinquagintasexions (256D).

A Hamilton Quaternion is a hypercomplex number with one Real part (the scalar) and three imaginary parts (the vector).

This is an extension of the concept of numbers.  We have found that a Real number is a one-part number that can be represented on a number line, and a complex number is a two-part number that can be represented on a plane.  Extending that logic, we have also found that we can produce more numbers by doubling the parts.     

Quaternion --> a + b*i + c*j + d*k, where the coefficients a, b, c, d are real numbers

The hypercomplex number-quaternion-is a non-commutative division ring.  This is what we call a four-dimensional number.  Here is an example of a quaternion:  5 + 2i + 3j + 4k.  The first term is called the scalar term; it is simply a real number.  The other terms consisting of, i, j, k. are called the imaginary terms.  As a group, they are called the vector of the quaternion.  Vector algebra uses the same name vector as the quaternion number, but with a different meaning.  Although vector algebra is an offspring of quaternions, vectors are not numbers.  Starting with complex numbers, we lose the permanence of trichotomy.  The permanencies we lose with quaternion numbers are the trichotomy property and the commutative property under multiplication. Additionally the imaginary units are anti-commutative under multiplication.  Anti-commutative means the sign of the imaginary unit changes when we transpose the two operands under multiplication, e.g., i*j = -(j*i).  The imaginary elements, i, j, and k, give cyclic permutations with each other.

If we set the coefficients of the imaginary elements j and k to zero, the quaternion number becomes an ordinary complex number.  We can create all of the other algebraic numbers and the transcendental numbers from the quaternions simply by setting all the coefficients of the imaginary elements to zero.

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