Intractus Research: Research Blog 
The
Nature of Complex Numbers of Any Dimension


I
have been studying quantum computing lately and have been struggling
to understand the nature of complex numbers. To help myself
conceptualize this abstract number type, I have envisioned the
possibility of a more general class of number which I would call
multidimensional complex numbers. I have come to discover this is not
a novel idea at all, but was proposed as early as 1844 by the Irish
mathematician William Hamilton. Here is a short discussion on the
nature of complex numbers of any dimension. Hopefully this will help
you get a better grasp of this abstract concept from number theory. Imagine trying to explain the set of negative integers to a 4 yearold child. To check if she understands the concept of the positive integers, you could ask her how many sheep are in a pasture out the window (assuming you are in sheep country). She might say "2" for example, if she sees 2 sheep, 2 being a positive integer number. Those sheep are realworld animals we can all see as emergent phenomena in the reality of our current universe, nothing imaginary about them. This is a very classical, common sense observation of a realworld quantity. Now talk with an 8 yearold, his teachers are showing him a number line to help envision the real numbers as a 1dimensional sequence. You realize that this line can be extended in 2 directions, right and left, corresponding to positive and negative amounts. However, we can never directly observe negative 3 sheep in a field. Somehow 3 is a number, but it is less than 1, which is the smallest number of sheep we can directly observe in our reality. The concept of 3 sheep is an abstraction of mathematics, whose discovery helps us perform calculations such as: if I currently have 10 sheep, I owe Mary 5 sheep, and John owes me 4 sheep, how many sheep will I have after we settle our debts at the end of the month? If we represent the balance with respect to Mary as 5 sheep then if I add up all my balances I will see that I will end up with 9 sheep at month's end ( 10+(5)+4=9 ). Using this abstract concept of a negative integer allows us to do a calculation that predicts a potential future state. This state does not currently exist, (currently I have 10 sheep) and may never exist (maybe John declares bankruptcy due to some unforeseen event and defaults on his debt to me). But it has a possibility of becoming our realworld state in the future if all goes according to plan. And the mathematics of negative numbers allows us to calculate exactly how many sheep we would have in that event. You can think of a negative number as a 1dimensional complex number. Taking a positive integer, we can perform a 180 degree rotation on the number line by multiplying by (1) . This negation operation (multiplication by a negative number) is useful to represent simple hypothetical operations where realworld, fungible, amounts may entirely cancel each other such as how debt cancels income in finance or how positive and negative electric charges can cancel in physics. A negative number can be considered a special 1D case of a complex number, where only rotations of 180 degrees are allowed, keeping calculated outcomes entirely on the real number line. The traditional complex number involves a real and an imaginary component, often stated as a+bi, where "a" is called the real component and "b" is called the imaginary component. This can be considered a 2dimensional number. If you study operations on traditional complex numbers you will find that most of these operations can be modeled as involving rotations and translations in a 2D plane, where one axis is the real number line and the other orthogonal axis the imaginary number line whose basis vector is called "i", a number whose magnitude is 1 but is rotated 90 degrees off of the real number line, giving it a real component of 0. This 2D number scheme is useful in Algebra to factor polynomials that do not have real roots, and in representing operations in quantum computing, for example, where part of what is happening during a computation is apparent in the observer's universe and other histories are not observable in this universe but can be useful in calculating possible future realworld states such as those involving the interference of photons. Are there practical uses for a 3dimensional complex number including a third basis vector, call it "j", orthogonal to the real number basis "1", and also orthogonal to the imaginary number basis "i" ? Operations on this type of number could be analyzed as rotations within the 1i plane and then within the ij plane, basically moving the vector within a sphere or volume vs a circle or plane such as for conventional complex numbers. Why stop at 3 dimensions? Are there 4, 5, 6...n dimensional complex numbers? Can we find realworld descriptions that can be most succinctly and accurately formulated using this number type, such as possibly descriptions of highly entangled quantum systems, or other multiagent evolving systems? This reference on hypercomplex (> 2 dimensional) numbers, http://mathworld.wolfram.com/HypercomplexNumber.html , suggests the answer is yes. This is a good source to read more on the subject, in a more precise mathematical format. You can also read the original paper (http://www.emis.ams.org/classics/Hamilton/OnQuat.pdf) by Hamilton proposing what he calls the quaternion numbers, a set of numbers very similar to the generic higher dimensional comlpex numbers defined in the mathworld reference on hypercomplex numbers. Send a Comment 

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