Imaginary Number Spaces

[AlexSamDen]

So do I bump this or what?

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Replied to you but there wasn't an answer, so are you still here or not interested anymore?

I might call DT and Qawsedf here to have a final opinion on all arguments made, after that might ask to close it, or maybe we can make this a more general discussion of imaginary numbers, as it can help people who encounter them but don't have enough knowledge to ask their questions in a thread where there already are people with some knowledge (and maybe experience) in dealing with them

 

[FinePoint]

I might call DT and Qawsedf here to have a final opinion on all arguments made, after that might ask to close it, or maybe we can make this a more general discussion of imaginary numbers, as it can help people who encounter them but don't have enough knowledge to ask their questions in a thread where there already are people with some knowledge (and maybe experience) in dealing with them

Well they're not particularly relevant for our purposes but a little note somewhere of that fact and a link to a simple explanation wouldn't be the worst.

 

[DontTalkDT]

A complex-valued axis counts as one dimension in a C-vector space and as two in an R-vector space.
So the number depends on how many complex axis we have and which field we use for scalar multiplication. The latter will be a detail we basically never have, so... yeah.

There actually is a bit of a problem, and that is the fact that Quaternions are also a vector space, just 4D and the reason it isn't 3D is exactly the Frobenious theorem a summary of which I quoted in OP. The problem with assuming a 3D complex vector space is the fact that while additions work fine there, divisions and multiplications do not, effectively meaning that this space does not perform one of its key purposes and correlations

No, it does perform just fine. You don't usually try to divide vectors by each other (the operation has little meaning), while division and multiplication by a scalar work as usual.

Quaternions wouldn't usually be considered to even fall into the realm of complex numbers. Different stuff.

 

[AlexSamDen]

A complex-valued axis counts as one dimension in a C-vector space and as two in an R-vector space.
So the number depends on how many complex axis we have and which field we use for scalar multiplication. The latter will be a detail we basically never have, so... yeah.

Most sources refer to complex spaces as only with dimensionality divisible by 2, and some even go for 2^n, so kinda the reason I tried 4..

No, it does perform just fine. You don't usually try to divide vectors by each other (the operation has little meaning), while division and multiplication by a scalar work as usual.

While I agree that division isn't that big of a part, multiplication is. Such things as Moivre's formula set a concrete form how complex multiplication works and it has a good transition to geometry and generally regarded as a great thing

Quaternions wouldn't usually be considered to even fall into the realm of complex numbers. Different stuff.

Both yes and no from me here, Quaternions aren't complex numbers per se, but an extension to the already existing system on the same principles. So yeah they aren't complex numbers by definition, but they do behave like you would expect from such an extension

 

[Tempestdragon6]

Although I want to join this discussion I feel like I would just make a chaos

 

[AlexSamDen]

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Hello, Agnaa, I'm sorry for bothering you, since you already withdrew from the thread, but recently I've picked up a question related to this topic and I didn't know how else to ask. Especially considering that the nature of this thread compliments the question

So, we've been discussing Imaginary Number Space as a completely separate construct here, but I'm wondering about another situation:

There is an (at least) 3D construct/realm and the INS is stated to be a part of it. Given that INS requires an Imaginary axis (at least one) to be present and that axis is as Physical as a regular one and is also independent from/orthogonal to all Real axises, would this construct be considered a 4D spatial construct? (Given the significant size of the INS)

 

[Agnaa]

This seems like a question that's actually about a specific verse's CRT.

You should just discuss it there, instead of giving me 3% of the information, and then hoping to beat your opponents over the head with my answer.

 

[Ascka23]

Imaginary Numbers are simply the extra-dimensional extension of a number line.

C = R^2, if you're going to extend it to our 3-D universe then that would be C^3 = (R^3)^2 = R^6

By what you're proposing, it would be more logical for a 3-D equivalent of Imaginary Space to be 6-D instead of 4-D, however Imaginary Numbers is rarely ever applied to situations like this, even in fiction. Additionally, under many context, Imaginary Number spaces usually refers to complex planes in fields of mathematics which is just 2-D.