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Quaternions And Transcomplex Numbers

Some ideas of mathematics are more intimidating than others. An example of this are the quaternions, a name that perhaps we may relate to electrons, protons or some other strange concepts of the weird quantum mechanics. However, quaternions are just another field of study of mathematics, just as the prime numbers, or the complex numbers. 

Quaternions

Quaternions are the creature of Sir William Rowan Hamilton (1805–1865), a man with a life background that can be compared to that of Johannes Kepler in their childhood and later discoveries.

Quaternions (from 'quadruples') were created by Hamilton in his way to extend the complex numbers system to four dimensions. (Some authors associate  this term with the old Latin word for a group of four soldiers; others, more realistic, associate this term with the Tetractys, the four row arrangement of the first 10 numbers in a pyramidal way.) . About the discovery of quaternions, quoting Crowe1:

Hamilton was prepared, perhaps to discover, more significantly to accept as legitimate, "four—dimensional" complex numbers (as quaternions), even if no geometrical justification were to be available.

This lack of geometrical justification that the author speaks is because a four–dimensional entity cannot be adequately represented in a three–dimensional axis system made of two real axes and one imaginary axis. The complex numbers are usually represented adding another axis to the Cartesian XY–plane. That new axis, the iZ–axis,  perpendicular to the plane is used to plot the imaginary numbers. In a three—dimensional system like this, the real functions can be plotted on the XY plane, and the complex numbers represented on the XiZ plane. Complex functions are plotted drawing two axes systems, one for the domain of the function, and the other to plot the range of the function. But no matter the configuration of the axes, there is no way to plot 4–axis entities into a 3–coordinates system.

Quaternions were also somewhat unintuitive because addition of two quaternions give another quaternion, etc, but the multiplication of two quaternions was not commutative, that is, for two quaternions Q and Q', QQ' does not in general equal Q'Q.  Recall that commutativity implies that the order of the factors does not influence or determine the final result of an operation.

In spite of this weird behavior, quaternions have found its way in limited areas of applications of geometrical operators, many of them useful for computer graphics. But in general, the quaternions is not an encompassing field as the complex numbers are. In that sense, the quaternions cannot be considered a true extension of the complex numbers to a fourth dimension.

Quaternions are numbers of the form Q = w + ix + jy+ kz, where w, x, y , z are real numbers, and i, j, and k are unit vector, along the x, y, and z axes respectively. 

Introduction to Transcomplex numbers

The transcomplex numbers are also four–entries entities, but they retain every and all the properties of the complex numbers. Transcomplexs are complex numbers of ordered pairs of the form T = (a, b) + i(c, d). To simplify the explanation, we can assume that a, b, c, and d are real numbers in different dimensions (four numbers in four dimensions). To clearly express the intention of each of the numbers, the 4-axis co-ordinate system is redefined as follows:

  • a is a number in the x-direction, whatever this direction is. That x-direction we it the X-axis
  • b is a number in another axis perpendicular the X-axis. This axis we call the Y~-axis to differentiate it from the X-axis because the x-axis is for real numbers only
  • c is a number in an axis orthogonal to both of the previous axis. This axis is for the imaginary numbers
  • d is a number in an axis orthogonal to all other three axes.

To obtain a consistent co-ordinate system that supports these four definitions a unitary axial system called the basis is defined such that:

  • 1 = (1, 0, 0, 0) = the unitary real number 1
  • 1~ = (0, 1, 0, 0) = the unitary image–real unitary number 1~
  • i = (0, 0, 1, 0) = the unitary imaginary number i
  • i~ = (0, 0, 0, 1) = the unitary image-imaginary number i~

Each one these units are called axial units. From this basis, and previously defined scalar products:

  • a = a(1, 0, 0, 0)
  • b = b(0, 1, 0, 0)
  • c = c(0, 0, 1, 0)
  • d = d(0, 0, 0, 1)

When expressed in their expanded form transcomplexs are of the form T = a + b~+ ic + i~d where a is a real number, b is an image–real, i is the imaginary unit and i~ is the image of the imaginary unit. This means that the transcomplex introduces two new types of numbers: the image–real and the image–imaginary

Some simple transcomplex numbers

Let T = (a, b, c, d), then the usual real numbers, the usual imaginary numbers and the usual complex numbers can be derived from the class of all transcomplex.

  • The real numbers. When b, c, and d are 0, then T = (a, 0, 0, 0) which is abbreviated as T = a. In this case we have that the set of all real numbers is a simple case of transcomplex numbers.
  • The imaginary numbers. When a, b, and d = 0, the T = (0, 0, c, 0) which is the case of the simple imaginary numbers. Thus, the set of all imaginary numbers is a subset of the transcomplex numbers.
  • The complex numbers. When b = 0 = d, then T = (a, 0, b, 0). This transcomplex has the image-real and the image-imaginary equal to 0. Thus it is a complex number which implies  that the set of all complex numbers is a subset of the transcomplex. Complex numbers are a simple case of transcomplexs.

Following the above schema, we can obtain 13 additional types of transcomplexs ranging from the simple origin (0, 0, 0, 0) to the full four-dimensional space (a, b, c, d) when none of the entries is 0.

Simple Operations with Transcomplex Numbers

The basic addition of two transcomplexs T = (a, b, c, d) and T' = (a', b', c', d') is intuitively defined as 

Formula for the addition of two transcomplex numbers.;

no problem with this simple and natural definition. 

The multiplication of T times T' defined as 

Formula for the multiplication of two transcomplex numbers.

Now, the multiplication operation deserves a few commentaries.

  • When b = b' = 0 and d = d' = 0 then T and T' become plain complex numbers and the multiplication then T * T' = (aa' - cc', 0, ac' +ca', 0) which is the usual definition of multiplication of complex numbers.

  • There are some instances where the result of a multiplication of transcomplex numbers is zero even when none of the factors is zero. One simple case of this when we multiply a pure real number times a pure image-real number. Let a be a real number, then a = (a, 0, 0, 0). Let b' be an image-real number, then b' = (0, b, 0, 0);  therefore a*b' = (a*0 - 0*0, 0*b' - 0*0, r*0 - 0*0, ) = (0, 0, 0, 0). There are eight possible cases. The Transcomplex Calculator is a simple tool you can download to do complex numbers multiplication.

That there exist the possibility that two numbers multiplied give a zero as result may sound crazy and insane. But note that if one of the factor T or T' is zero, the product is necessarily zero. So the usual properties of the real numbers and the complex numbers field are preserved.

Obviously, here is not the place to fully expose in full the theory. For that reason, the reader is invited to download the free book Foundations of Transcomplex Numbers where the transcomplexs foundations are developed beginning with the real numbers systems with strong emphasis in the properties of the ordered pairs. Complex numbers are developed from ordered pairs of real numbers, and T–numbers in turn, are developed from ordered pairs of complex numbers.

In the book is shown how the transcomplex numbers field retain all the properties of the complex numbers field; nothing is violated; in fact, the complex numbers becomes a subset of the T–numbers. 

But since the T–numbers are 4–entries numbers, that implies that to represent then adequately, we need a four–axes coordinate system. That system of coordinates is also developed as part of the Foundations with its basis of mutually orthogonal unit vectors.

E. Pérez Feb-17

The quaternions and the transcomplex numbers are tightly related to the concept of the fourth dimension. To learn more about why there can be many fourth dimensions, see the article Can there be different fourth dimensions?

References in print

[1] Crowe Michael J. (1967). A History of Vector Analysis: The Evolution of the Idea of a Vectorial System. Dover Publications, Inc. New York.

References on line