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The thing could not exist in real world; it is imaginary concept, expressed in a equations in a paper, that could run as a simulation on a PC. That could be interesting for people with some mathematical/physical/electrical background, and would require some knowledge in fields of linear algebra, EE, maybe circuit analysis..

To start, what is a quaternion? Well, in a simple words, its a number, that consist of 4 numbers: a "number" plus a imaginary 3D vector (that consists of the resting 3 numbers). There is a great video, interactively explaining "why and how"; with really great educational animations. What these 4D quaternions are able to represent - are a uniform scaling and rotations in 3D world.

As per electricity, wikipedia defines electric charge and current as well; Short video is also possible to look at, in order to get some clues. So in a real world, we have some wire, that transfers electrons, and we say that "charge flows", and the "rate of that flow" is a "current". So you see that this concept alredy became imaginary, but that's just a side note. "Amount of charge" and "rate of flow" are definetely some measurable quantities, that we represent as a real numbers. Then, on the papers, we've built an equation systems, that defines laws - Ohm's law and a Watt's law. The charge and current interactы with wires(resistors) and simple elements, like capacitors and inductors according to that laws:

Ohm's and Watt's laws

Now lets imagine another world, lets call it "quorld", where exists some analog to charge, called "quarge". The "quarge" related to quorld's matter, "quatter" in a same way, that a charge relates to a matter in our world. Same laws are applied, with one exception. A quarge quantity is being measured by quaternion, instead of a real number. Thinking further on, there exists some qurrent - "a rate of flow of a quarge", and people queople in a quorld has invented their elements - quapacitors, that stores quarge, so on..

And here comes a fact, rendering above chart useless. A feature of quaternions is that multiplication of two quaternions is noncommutative;

I, multiplied by R is not equal to R, multiplied by I

So, the order of multiplications does matters. It could be even helpful to define conjugate: a quaternion with inversed imaginary part.

So the conjugate of a product of two quaternions is the product of the conjugates in the reverse order:

(RI)* = I* R*

Also, there is another helpful property that they are associative:

(RI)I = R(II)

My ultimate goal is to write simulator, that can simulate quaternionic circuits, quircuits. And to check is there any mathematical contradictions, that proves non-functionality, or impossibility of existing such a system mathematically.

That involves a task of writing those 12 equations in a quaternionic form, with a correct multiplication order, as well as writing differential equations for a components: quapacitor, qinductor, qresistor(are they same component in that world?).

So if you have some math/physics/EE background and skills - you're welcome to assist me in those, that's primary goal of the question. I am a programmer, and for now I have web browser program with prototype, that is currently working with complex numbers. I would update with a working prototype, if question would be appreciated.

Also any thoughts on the topic are welcome, for example, how much dimensions such world could have? Is a mass "quass" also a quaternionic quantity in such a world? What are consequences of this, any thoughts, so on..

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closed as too broad by JBH, AlexP, Cyn, Alex2006, Measure of despare. May 2 at 15:23

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ I think you would have mote chances of getting a proper answer at either the Math or Computer Science stacks. $\endgroup$ – Renan May 2 at 0:15
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    $\begingroup$ @Renan I would ask there as well, but I think that I could also have some not quite math-related answers from people in here. To the end, It's an imaginated quorld, isn't 'it? :-) $\endgroup$ – xakepp35 May 2 at 0:18
  • $\begingroup$ You've got a point. $\endgroup$ – Renan May 2 at 0:24
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    $\begingroup$ How does Coulomb's law look like for this quadricharge? It cannot look like ours, because ours says that the force is proportional to the product of the charges. In our world, voltage is a difference of potential; but in the quadriworld, how do you define electric potential? Is the quadrielectric field still conservative? Unless you sort of the basics first, I just don't believe you when you say that "same laws are applied"; my first feeling is that the posited quadricharge does not behave like electric charge at all. @Renan: Isn't this a perfect example of a "high concept" question? $\endgroup$ – AlexP May 2 at 0:30
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    $\begingroup$ @xakepp35: You appear to not understand. Voltage is a difference of potential, and potential can be defined only if the electric field is conservative. But we don't know whether the quadrielectric field is conservative, because you didn't tell us how two quadricharges interact. So unless you figure out the basic properties first nobody can help you. The equations which you wrote in the question are not fundamental, they actually fall out of the general equations of electromagnetism. You cannot start with $U = RI$, you must start with the basic rules. $\endgroup$ – AlexP May 2 at 0:41
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My ultimate goal is to write simulator, that can simulate quaternionic circuits, quircuits. And to check is there any mathematical contradictions, that proves non-functionality, or impossibility of existing such a system mathematically.

If I understand your question correctly, you're asking whether it's possible to write all the equations listed in that chart if the variables are quaternions. The answer to that is yes-- kinda. The quaternions are an associative division algebra, which basically means that all the normal properties of the real numbers are there, with the glaring omission of commutativity. However, the lack of commutativity can really screw some stuff up. For instance:

  • Equations like $I = V/R$ become ambiguous-- do you mean $I = VR^{-1}$ or $V = R^{-1}V$? These are no longer the same thing, so you have to choose a convention and that convention would have to be motivated by some physical explanation.
  • "Polynomials" no longer necessarily have roots-- see this math stack exchange post. I put that in quotes because this actually depends on what you're counting as a polynomial in a non-commutative system such as the quaternions. This can be a problem because for (linear) electric circuits, you can usually find a polynomial that describes the system.

These problems aside, I think it's important to take a step back and consider why we describe physical systems the way we do. There seems to be some conception that math is some fundamental descriptor of the universe and that everything literally is numbers. To my mind at least, this is backwards-- reality just does what it's gonna do, and we humans try to come up with models that predict that reality in the most efficient way possible-- oftentimes, with the help of math.

To further explain my point, consider quantum mechanics. Oftentimes, I hear people trying to explain the use of complex numbers by saying that quantum mechanics is necessarily complex. And sure, the way we commonly see Schrodinger's equation written uses complex numbers-- but it's actually trivial to rewrite it as two separate equations using only real numbers. Complex numbers are simply a more convenient way of expressing it, which is the reason they're used. This isn't to say that real numbers are more fundamental than complex ones. At the end of the day all mathematical constructs are just as artificial as each other. We can express the real numbers as Dedekind cuts of equivalence classes of pairs of equivalence classes of pairs of natural numbers, and we can express the natural numbers as the minimal inductive subset of the real numbers containing 0-- what mathematical concept is more fundamental is just a matter of what you define as the starting point.

The point I'm trying to get at is that simply saying "what if electricity was defined by quaternions" is pretty much impossible to give a meaningful answer to unless you start with a physical motivation of how electricity works in your world and why we would want to describe it with quaternions in the first place. If we wanted to, we could describe electricity in our world with quaternions, since the complex numbers are equivalent to a certain subset of quaternions and we can (and do!) describe electricity with complex numbers already. But it would ultimately be useless and wouldn't really add anything. So, my advice would be to first think about what kinds of effects could "quelectricity" have in your world that would make quaternions a useful tool for describing it.

EDIT: Response to OP's Questions in Comments

Power quaternion obviously should be parallel to impedance R, so that P=R|I|^2 so it is just scaled with a norm(length) of a current quaternion, squared.. Do I have a logic good here? ... So probable power P is equal to V multiplied by I conjugate.. Not quite sure, but seems so. Both impedance and power then has same meanings, but impondance is more like "normalized" power consumption(specific, unit, per-element) and power is same thing, but upscaled according to actual current flow, or voltage applied..

If you're just looking for a set of consistent equations, and you want $P=R|I|^2$ to be true (as it is for complex power), then yes, $P=VI^*$ and $V=RI$ can be used to derive everything else. This is similar to the form used for complex power, only order now matters. We're still lacking any kind of physical motivation here though.

As for coordinates I see 3 approaches. Bold move would be try to tie it to regular 3d world(or 7d, or 15d). Coward mowe would be taking norm(length, magnitude) every time we couldn't plug something in. And strange move would be considering everything (coordinate x, time t) a quaternion. So now we haven't a standard geometry in a common sense(axes are not real numbers, but the very axis becomes 4d now) but all formulas should work without any modifications. That's also interesting. What do you think?

Here's where I'm kinda confused with what you're asking-- voltage is a scalar value, it doesn't really have coordinates. It can vary with coordinates in space and time, but it itself is just a single value (same with all the quantities on that chart). This is also a large part of why I spent so much time talking about physical motivation-- lacking that, any response to the question " what would happen if electricity was described by quaternions" is basically just going to be "the electricity would act like quaternions. That's fine and dandy, but its not really a worldbuilding question-- it's just a math one where all the variables you're dealing with are named after stuff in our world.

Interestingly enough, quaternions actually were originally used to describe electromagnetism, since the magnetic field and electric field have three components and cross products are built into the rules for quaternion multiplication. I think it might be interesting to you to read about this, because it gives an example of how the quaternion's properties can be used to model some systems. I can't really think of how they'd be used to model a potential or power-- but if you can come up with some reasonable solution, more power to you! But like I said before, without a physical reason why the electric potential in your world should be modeled by quaternions, your question is just asking how math with quaternions works, which is something well explored elsewhere on the internet.

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    $\begingroup$ This. Maths is a tool to model with, our models are just well informed guesses at what’s going on, and when modelling you try make the simplest model possible that explains your observations. If quaternions make the model more elegant for some reason then use them, otherwise don’t. $\endgroup$ – Joe Bloggs May 2 at 8:13
  • $\begingroup$ There are 2 variants of this world - both are like.. Mirrored copies, one is left-sided, another is right-sided. I am looking at I=R^-1 * V, where impedance inverse is multiplied by voltage, and sits to the left of it. That dictates correct order of another two: R=VI^-1 and V=RI $\endgroup$ – xakepp35 May 2 at 13:01
  • $\begingroup$ Power quaternion obviously should be parallel to impedance R, so that P=R|I|^2 so it is just scaled with a norm(length) of a current quaternion, squared.. Do I have a logic good here? $\endgroup$ – xakepp35 May 2 at 13:07
  • $\begingroup$ So probable power P is equal to V multiplied by I conjugate.. Not quite sure, but seems so. Both impedance and power then has same meanings, but impondance is more like "normalized" power consumption(specific, unit, per-element) and power is same thing, but upscaled according to actual current flow, or voltage applied.. $\endgroup$ – xakepp35 May 2 at 13:09
  • $\begingroup$ As for coordinates I see 3 approaches. Bold move would be try to tie it to regular 3d world(or 7d, or 15d). Coward mowe would be taking norm(length, magnitude) every time we couldn't plug something in. And strange move would be considering everything (coordinate x, time t) a quaternion. So now we haven't a standard geometry in a common sense(axes are not real numbers, but the very axis becomes 4d now) but all formulas should work without any modifications. That's also interesting. What do you think? $\endgroup$ – xakepp35 May 2 at 13:20

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