Intractus Research: Research Blog
Nature of Complex Numbers of Any Dimension
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
multi-dimensional 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 year-old 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 real-world 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 real-world quantity. Now talk with an 8 year-old, his teachers are showing him a number line to help envision the real numbers as a 1-dimensional 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 real-world 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 1-dimensional 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 real-world, 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 1-D 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 2-dimensional 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 2-D 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 2-D 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 real-world states such as those involving the interference of photons.
Are there practical uses for a 3-dimensional 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 1-i plane and then within the i-j 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 real-world descriptions that can be most succinctly and accurately formulated using this number type, such as possibly descriptions of highly entangled quantum systems, or other multi-agent evolving systems? This reference on hyper-complex (> 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 hyper-complex numbers.
this Web Site's Banner Images
of the images you see in the banners on the www.intractus.com web site
pages were generated by machine learning algorithms developed by
Intractus Research during training on various image processing, polyphysics design optimization and
financial modeling tasks. Our process often involves tapping
into the learning networks for
visual feedback during training both as a verification exercise to
ensure that the connectionist structure is learning and also because it
is curiously interesting and beautiful to directly see what these
learning algorithms are doing. These sorts of images are often called
"psychedelic" images as they visually represent some of the evolving
structure of an idea as it is being formed within a learning network,
combining real world 2D visual input with imaginary/internally
The images on the products_and_services page were captured during an object recognition task and represent spatial neural net activity as the algorithm learns to recognize where/if certain objects appear in a 2D image. These images bring to mind the well known emoji of an illuminating light bulb as representing an idea being created.
The images on the research_blog and contact_us pages were captured while an algorithm was learning to perform depth estimation on a 2D image. Some of the images on the research_blog page bear little resemblance to the original image, containing mostly imagined/internally generated structure that isn't really in the original picture, or is it? If these are psychedelic then we caught the algorithm while it was having a bizarre dream or bad trip, or actually just very early in the evolution of the concept. The images on the contact_us page show a more obviously productive evolution of an image depth estimation concept from very early random looking ideas that gradually refine into distinct objects with certain depth relations and clear size, shape and location.
The images on the about_us page are not psychedelic in the visual sense like the others. These images show the progression of a wavelet function generator that evolves to fit a complex curve from actual time series financial market data. You can clearly see the initial stair-step basis wavelet function in the first image. As the function parameters become more complex during training, the error between the learned wavelet function and the sampled data is reduced. By the final image the error is quite small. In fact, this particular function "overfits" the sampled data. Close inspection reveals that the generated function has modeled jumps that were present in the data between distinct sampling points. These periodic jumps do not reflect a hidden parameter of the underlying process creating the data stream, but rather are an artifact of sampling.
With years of experience coding and improving our learning algorithms, Intractus Research consultants are able to avoid many of the common pitfalls (such as overfitting and excessive CPU time) experienced when trying to create a machine learning based solution to a new problem. We have developed a suite of internally developed/proprietary tools and techniques (such as the visualization of psychedelic images, and optimal control of hyperparameters) to help our human consultants efficiently create working solutions to various types of intractable data-driven problems.