Imaginary Number Spaces

[AlexSamDen]

Introduction​

Hello everyone, this thread will be a continuation of this one. The reason is that I've encountered the case of Imaginary Number Space once again and while reading the aforementioned Q&A, I noticed that several important arguments weren't made, so I'm here to introduce them.

A basic thing to mention, so as to avoid misunderstandings

Imaginary numbers and complex numbers are essentially the same. While strictly speaking, the imaginary numbers are (b*i) and complex numbers are (a + b*i), imaginary numbers in social media, fiction and generally the Internet are widely referred to as (a + b*i), moreover their respective geometric interpretations that are usually used, while with different reasonings, have the same form, meaning that when we talk about feats of imaginary number space in fiction, it is sensible to treat them as complex number space.​

The justifications for the proposal​

1) Projecting the geometrical interpretation onto this case
The accepted geometrical interpretation of the set of basic complex numbers (that is a+bi) is a plane, as opposed to the set of real numbers being a line. That is due to the fact that the expression a+bi not only includes all the real numbers (while b = 0), but due to the fact that both a and b can take any real value and due to them being independent from each other, for every of the uncountably infinite values of b there are also uncountably infinite different values of a, essential making this the same as the standart (x;y) plane. This means that whenever you add an imaginary part to a number, it adds an additional imaginary axis to the geometrical interpretation due to a new separate variable being added. So going from "set of real numbers" to "set of complex numbers" is going from a 1D structure to a 2D one. Therefore, if we use this method further and take a regular space, which is a set of all 3D coordinates (x;y;z) and add an imaginary part to it, we would get a new axis for the space, making it a 4-dimensional comlex number space.
In short, when we add an imaginary part to a set, it's geometrical interpretation gets +1D, therefore imaginary number space is a regular space +1D which equals 3+1D = 4D

Of course, this is a theoretical approach, however this part just shows how it is sensible and highly likely to be the case we need to use. . There are heavier arguments in the second part

2) There actually is a defined imaginary number space in Maths and Physics. And it's 4D
Yes, you've read it correctly, it is an extension of imaginary numbers (and not just complex) and it is used rather widely. Some of our members that are well versed in 3D animation might know about it. It is the Quaternion space. It is a imaginary number space that consists of 1 real axis and 3 imaginary ones (I know that what I proposed in (1) is a bit different, but essentially it does not matter as they follow the same logic: either you add an Im axis to 3 Rez axises or a Rez axis to 3 Im ones). This space is often used for describing motion of 3D objects, hence the use in animation.

You may argue that going straight to them is a stretch, however this is the only term that is referred to as "Imaginary Number Space" in science. Moreover, it's been proven that there is no such thing as a working 3-dimensional imaginary number space:

In fact, Ferdinand Georg Frobenius later proved in 1877 that for a division algebra over the real numbers to be finite-dimensional and associative, it cannot be three-dimensional, and there are only three such division algebras: R, C (complex numbers) and H (quaternions) which have dimension 1, 2, and 4 respectively. From Quaternions, Wikipedia

This effectively means that the only possible meaning behind Imaginary Number Space in fiction is a 4D space (If there are no clear statements it isn't in this specific setting, that is)

The proposal itself​

If you hadn't guessed it already, my proposal is that we treat Imaginary Number Spaces as 4D structures. Of course I don't say we just throw 4D at whatever mention of this name. I have prepared three conditions for something to qualify for this:

1) It is explicitly called "Imaginary Number Space"
Seems rather logical, as it is what we're discussing here. I am aware that they can be not called such and yet effectively be one via mentions of imaginary axises, however to avoid confusion these verses can just straight up prove it via the direct statement of an additional axis, without bringing up the imaginary numbers in this meaning

2) It is in some way separated from regular space
Also reasonable, imaginary coordinates aren't called imaginary for no reason, they don't fit the normal coordinates, even ignoring the 3D and 4D structures somehow properly intersecting. It can be any kind of separation: from an invisible line to it being a separate realm entirely, not proposing anything harsh here, just a clear anti-feat

3) It is in some way shown to be superior to regular space

It is also not a strict condition, obviously 4D space is superior to a 3D one. Can essentially be shown anyhow, just at least somehow
—————————————————————————

You may have noticed that the conditions are rather lenient. The reason is that they aren't intended as strict rules, but rather a way to ensure that the structure we are scaling is not in a complete scaling vacuum, as there can be situations where INS just appears and disappears in a moment in the verse and I find scaling it in this situation rather unreliable. In this light, the meaning of the conditions is this: (1) is the defining of said structure, (2) is the absence of a clear anti-feat and with (3) we ensure that it occupies at least some presence in the narrative

What perks does having something accepted as such actually give? ​

There are basically only 2 default things it gives

1. If someone manipulates INS, they get 4D Spatial Manipulation
2. If it's some structure, it means that there is a 4D spatial structure in the cosmology. What it actually does is up to the verse

—————————————————————————
So that's everything I wanted to say. Thanks to everyone participating and have a good day

(Some references to the Wikipedia articles I studied before making this: Imaginary numbers; Complex numbers; Quaternions (EN) ; Quaternions (RUS) (personally think that while the English and Russian articles have the same set of information, the Russian one does a better job at properly explaining it). A separate thanks to my Math teacher)

 

[AlexSamDen]

Bump

 

[FinePoint]

4D in the context of this wiki is not just anything which can be described or visualized with four numbers.

Imaginary numbers are just a transitory extension of the real numbers, more akin to a rotation in the same set of axes than a new one.

The mere fact that multiplying two imaginary numbers together gives you a real number proves that they are not truly a separate axis by default.

That is to say there's no basis to say a number with imaginary components is superior to one without them.

 

[AlexSamDen]

4D in the context of this wiki is not just anything which can be described or visualized with four numbers.

4 orthogonal axises can be indeed described with 4 numbers and they do qualify for 4D, so don't really see your point

Imaginary numbers are just a transitory extension of the real numbers, more akin to a rotation in the same axis than a new one.

That is simply not true. Not only does motion of something require some additional means to be overviewing it, but imaginary numbers also describe motions that are orthogonal to the axis of real numbers and in no way it is within it

The mere fact that multiplying two imaginary numbers together gives you a real number proves that they are not truly a separate axis.

With (X;Y) plane you also have the case when multiplying 2 numbers gives you a number, and yet they still perfectly function as 2 axises

That is to say there's no basis to say a number with imaginary components is superior to one without them.

Never I said that, I said that adding a new imaginary variable to an equation makes so that it has uncountably infinitely more solutions

 

[FinePoint]

4 orthogonal axises can be indeed described with 4 numbers and they do qualify for 4D, so don't really see your point

Rectangle-square. 4D spaces can be described with 4 numbers. Not all things described with 4 numbers are 4D.

That is simply not true. Not only does motion of something require some additional means to be overviewing it, but imaginary numbers also describe motions that are orthogonal to the axis of real numbers and in no way it is within it

Motion of the entire graph would require another axis, but motion of specific portions of it does not.
When you have an imaginary root it doesn't turn a 2D graph 3D, it just creates some weird movement.
The structure here is still 2D, it just has parts which require imaginary numbers to predict.

Never I said that, I said that adding a new imaginary variable to an equation makes so that it has uncountably infinitely more solutions

A normal 2D graph already has uncountably infinite places a point could exist. Introducing imaginary numbers into calculating specific ones doesn't turn it into a higher infinity. There is no cardinality increase.

 

[AlexSamDen]

Rectangle-square. 4D spaces can be described with 4 numbers. Not all things described with 4 numbers are 4D.

Rectangle and square are not 4 orthogonal lines, two pairs of them are parallel. The geometrical interpretation of the imaginary number addition are officially accepted to be orthogonal to each other

Motion of the entire graph would require another axis, but motion of specific portions of it does not.
When you have an imaginary root it doesn't turn a 2D graph 3D, it just creates some weird movement.
The structure here is still 2D, it just has parts which require imaginary numbers to predict.

A normal 2D graph already has uncountably infinite places a point could exist. Introducing imaginary numbers into calculating specific ones doesn't turn it into a higher infinity. There is no cardinality increase.

I think you misunderstood what I meant here. We don't replace the already existing part of the equation with an imaginary one, we add an imaginary one. (It's not ax + by = c →ax + bi = c, but rather ax + by = c → ax + by + di = c)

Moreover, you seen to be completely skipping the Quaternion part

 

[FinePoint]

Rectangle and square are not 4 orthogonal lines, two pairs of them are parallel. The geometrical interpretation of the imaginary number addition are officially accepted to be orthogonal to each other

I was using it as a reference to the fact that a square is always a rectangle, but a rectangle is not always a square.

That is to say: X having quality Y does not mean that everything with quality Y is X.

I think you misunderstood what I meant here. We don't replace the already existing part of the equation with an imaginary one, we add an imaginary one. (It's not ax + by = c →ax + bi = c, but rather ax + by = c → ax + by + di = c)

You can add as many variables as you want, imaginary or not.
It doesn't suddenly create a spatial axis, it just makes the shape of the graph more complicated.

Moreover, you seen to be completely skipping the Quaternion part

An algebraic dimension and a spatial dimension aren't the same thing.
A 3D system can have arbitrarily as many parameters in its calculation as you want without becoming a 4D object.

Ultimately, whether or not your new axis has imaginary components doesn't change anything about its overall size in a spatial sense.

So if all you're really saying is that a new axis having imaginary numbers doesn't disqualify it as a new spatial dimension, that I'd agree with.
What matters most is independence and geometry.

 

[AlexSamDen]

I was using it as a reference to the fact that a square is always a rectangle, but a rectangle is not always a square.

That is to say: X having quality Y does not mean that everything with quality Y is X.

That is true, however it is accepted that imaginary axises are indeed separate and orthogonal due to the specifics of their workings, so I really don't understand why we're discussing one of the basises of its interpretations

You can add as many variables as you want, imaginary or not.
It doesn't suddenly create a spatial axis, it just makes the shape of the graph more complicated.

Any variables? Yes. Independent ones? No

An algebraic dimension and a spatial dimension aren't the same thing.
A 3D system can have arbitrarily as many parameters in its calculation as you want without becoming a 4D object.

Ultimately, whether or not your new axis has imaginary components doesn't change anything about its overall size in a spatial sense.

So if all you're really saying is that a new axis having imaginary numbers doesn't disqualify it as a new spatial dimension, that I'd agree with.
What matters most is independence and geometry.

And to all of this I'll reply with mostly quotes from linked Wikipedia pages

Quaternions form a 4-dimensional vector space over the real numbers, with {1,i,j,k}

as a basis, by the component-wise addition

A quaternion is an expression of the form

a+bi+cj+dk,

where a, b, c, d, are real numbers, and i, j, k, are symbols that can be interpreted as unit-vectors pointing along the three spatial axes.

Quaternion are not only algebraically 4D, but its interpretation is also that of a 4D space. Unit vectors are base vectors that each stem from the 0 of the structure and along their respective axis, representing the base value, similarly to how 1 is on a regular axis. There are 3 spatial axises in addition to the already present real axis, making it a 4D spatial structure

 

[Astral_Trinity439]

I'll have to agree in that we should treat Imaginary Number Dimensions/Space as basically additional axis, but I'm not sure if we should treat them as just 1 additional axis or more (3, to be specific), totalling 2-dimensional and 4-dimensional complex planes respectively.

But I think I'll learn towards treating it as only +1-dimensional (totalling an X (real + 1) dimensional complex plane.

This is mainly because Quaternions are more so a specific extended branch of complex numbers, not the "general" definition.

In 1843, William Rowan Hamilton extended the idea of an axis of imaginary numbers in the plane to a four-dimensional space of quaternion imaginaries in which three of the dimensions are analogous to the imaginary numbers in the complex field.

In general, imaginary numbers (and by extension the imaginary axis) is just one axis orthogonal to some amount of real axis/axes present.

Geometrically, imaginary numbers are found on the vertical axis of the complex number plane, which allows them to be presented perpendicular to the real axis. One way of viewing imaginary numbers is to consider a standard number line positively increasing in magnitude to the right and negatively increasing in magnitude to the left. At 0 on the x-axis, a y-axis can be drawn with "positive" direction going up; "positive" imaginary numbers then increase in magnitude upwards, and "negative" imaginary numbers increase in magnitude downwards. This vertical axis is often called the "imaginary axis"[11] and is denoted
iR,{\displaystyle i\mathbb {R} ,} I,{\displaystyle \mathbb {I} ,} or ℑ.[12]

And by extension the complex number plane) are more so 2-Dimensional rather than 4-Dimensional, unless we SPECIFICALLY go to the field of 4-dimensional ones (Quaternions).

In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal x-axis, called the real axis, is formed by the real numbers, and the vertical y-axis, called the imaginary axis, is formed by the imaginary numbers.

While seldom used explicitly, the geometric view of the complex numbers is implicitly based on its structure of a Euclidean vector space of dimension 2, where the inner product of complex numbers w and z is given by ℜ(wz¯); then for a complex number z its absolute value |z| coincides with its Euclidean norm, and its argument arg(z) with the angle turning from 1 to z.

Other meanings of "complex plane"
The preceding sections of this article deal with the complex plane in terms of a geometric representation of the complex numbers. Although this usage of the term "complex plane" has a long and mathematically rich history, it is by no means the only mathematical concept that can be characterized as "the complex plane". There are at least three additional possibilities.

• Two-dimensional complex vector space, a "complex plane" in the sense that it is a two-dimensional vector space whose coordinates are complex numbers. See also: Complex affine space § Two dimensions.
• (1 + 1)-dimensional Minkowski space, also known as the split-complex plane, is a "complex plane" in the sense that the algebraic split-complex numbers can be separated into two real components that are easily associated with the point (x, y) in the Cartesian plane.
• The set of dual numbers over the reals can also be placed into one-to-one correspondence with the points (x, y) of the Cartesian plane, and represent another example of a "complex plane".

However, I could agree to treating Imaginary Number Space as possibly 4-dimensional IFF one or more among the following holds true:

• The verse directly references Quaternions, in which case we have absolutely no reason to assume INS is not 4-Dimensional in context.
• The verse shows that said Imaginary Number Space is capable of containing more than 1-Dimensional structures/things AND those things, when they're isolated inside the INS, hold no existence/extension in regular 3-Dimensional Space.

Since the OP proves how INS (or complex plane) can either be 2-dimensional or 4-dimensional, we can skip the 3-dimensional lowball/midball entirely.

Additionally, we should only treat INS as scaleable at all in terms of cosmology if:

1. It is infinite, or at least has no feats of being finite.
2. It is either a part of the Universe but isolated from the 3-regular dimensions or it contains Universes, serving as the space separating them. An example of how to prove the first type is to show that said INS is affected by the flow of Time in one way or another, or at least doesn't have it's own Temporal Axis and is still over arched by the continum's own Temporal Axis.

Note: Wikipedia pages being referred to here are Imaginary Numbers and Complex Plane.

Snip

Thoughts on the above?

 

[Voidnether]

Yeah, no, it's only 4-D if the real number contains the space-time continuum as elaborated here with how Honkai's imaginary space works firsthand where it's technically just outside the real space that's 4-D. I don't really know any specific example, other than this

 

[Voidnether]

I'll have to agree in that we should treat Imaginary Number Dimensions/Space as basically additional axis, but I'm not sure if we should treat them as just 1 additional axis or more (3, to be specific), totalling 2-dimensional and 4-dimensional complex planes respectively.

But I think I'll learn towards treating it as only +1-dimensional (totalling an X (real + 1) dimensional complex plane.

This is mainly because Quaternions are more so a specific extended branch of complex numbers, not the "general" definition.

In general, imaginary numbers (and by extension the imaginary axis) is just one axis orthogonal to some amount of real axis/axes present.

And by extension the complex number plane) are more so 2-Dimensional rather than 4-Dimensional, unless we SPECIFICALLY go to the field of 4-dimensional ones (Quaternions).




However, I could agree to treating Imaginary Number Space as possibly 4-dimensional IFF one or more among the following holds true:

• The verse directly references Quaternions, in which case we have absolutely no reason to assume INS is not 4-Dimensional in context.
• The verse shows that said Imaginary Number Space is capable of containing more than 1-Dimensional structures/things AND those things, when they're isolated inside the INS, hold no existence/extension in regular 3-Dimensional Space.

Since the OP proves how INS (or complex plane) can either be 2-dimensional or 4-dimensional, we can skip the 3-dimensional lowball/midball entirely.

Additionally, we should only treat INS as scaleable at all in terms of cosmology if:

1. It is infinite, or at least has no feats of being finite.
2. It is either a part of the Universe but isolated from the 3-regular dimensions or it contains Universes, serving as the space separating them. An example of how to prove the first type is to show that said INS is affected by the flow of Time in one way or another, or at least doesn't have it's own Temporal Axis and is still over arched by the continum's own Temporal Axis.

Note: Wikipedia pages being referred to here are Imaginary Numbers and Complex Plane.

Thoughts on the above?

You might be mistaking with something else, imaginary axis doesn't necessarily grant a higher axis dimension when we know characters who are baseline Immeasurable already operates within the confines of imaginary time which works differently than the real time yet it doesn't require a second temporal dimension

 

[Astral_Trinity439]

You might be mistaking with something else, imaginary axis doesn't necessarily grant a higher axis dimension

My point is that imaginary number axes would, by default, be separate from the real dimensional axes, regardless of whether the real axes amount to only 1, 2, 3, or any higher number. Because imaginary number axis simply cannot be represented in real plane coordinates, a separate axis is needed to represent them.

when we know characters who are baseline Immeasurable already operates within the confines of imaginary time which works differently than the real time yet it doesn't require a second temporal dimension

That's simply because the verses in question never call out that form of "time" as separate, so we have no reason to treat it as such. Or we can either just assume said temporal Axis is not significant.

It's similar to how we treat the 5D space in which 4D universes are embedded as "exists by default" but don't treat it as significant enough to warrant ANY rating by default unless the verse calls out the Hyperspace with certain context added.

Yeah, no, it's only 4-D if the real number contains the space-time continuum as elaborated here with how Honkai's imaginary space works firsthand where it's technically just outside the real space that's 4-D. I don't really know any specific example, other than this

While I do agree with you on that we shouldn't treat Imaginary space as 4D by default with no further context given, absolutely not, but I think "containing a 4D continuum" is a very specific example.

The OP had already shown how complex plane can either be 2D (1-D imaginary part) or 4D (3D imaginary part).

Also, I think the example you gave lies more so beyond 4D since, in essence, a space containing/separating 4D space time continuums is by default 5D (the question we ask the verse about it is not "is it 4D or 5D" but "is it significant or not").

Which brings me to the requirements I suggested in my post:

It is infinite, or at least has no feats of being finite

The reason why I think "if it has no feats of being finite" as also valid is because imaginary numbers have the same cardinality as real numbers

 

[Voidnether]

My point is that imaginary number axes would, by default, be separate from the real dimensional axes, regardless of whether the real axes amount to only 1, 2, 3, or any higher number. Because imaginary number axis simply cannot be represented in real plane coordinates, a separate axis is needed to represent them.

That's simply because the verses in question never call out that form of "time" as separate, so we have no reason to treat it as such. Or we can either just assume said temporal Axis is not significant.

It's similar to how we treat the 5D space in which 4D universes are embedded as "exists by default" but don't treat it as significant enough to warrant ANY rating by default unless the verse calls out the Hyperspace with certain context added.

While I do agree with you on that we shouldn't treat Imaginary space as 4D by default with no further context given, absolutely not, but I think "containing a 4D continuum" is a very specific example.

The OP had already shown how complex plane can either be 2D (1-D imaginary part) or 4D (3D imaginary part).

Also, I think the example you gave lies more so beyond 4D since, in essence, a space containing/separating 4D space time continuums is by default 5D (the question we ask the verse about it is not "is it 4D or 5D" but "is it significant or not").

Which brings me to the requirements I suggested in my post:

The reason why I think "if it has no feats of being finite" as also valid is because imaginary numbers have the same cardinality as real numbers

I think this would only work if it's like something equivalent of aleph numbers or so, like if the imaginary number space is 5-D then if we use the aleph numbers analogy for example: It would mean that the imaginary number space here is equivalent to an aleph 1 worth of universe and that's like 5-D given the irrational numbers and all that

 

[Astral_Trinity439]

I think this would only work if it's like something equivalent of aleph numbers or so, like if the imaginary number space is 5-D then if we use the aleph numbers analogy for example: It would mean that the imaginary number space here is equivalent to an aleph 1 worth of universe and that's like 5-D given the irrational numbers and all that

By aleph numbers, you mean Aleph-1 (cardinality of R), right?

If yes, then indeed, but imaginary numbers by default have a cardinality equal to R (that is, Aleph 1), so an imaginary number space (which represents the axis that imaginary numbers exist on) is by default equal to a real normal spatial dimension that is significant (unless suggested otherwise).

Thus, the application isn't limited to only Imaginary Number spaces containing universes, but also to Universes that have an imaginary number space as part of itself.

So for example, if a Universe has 3 regular dimensions as well as an imaginary number space overarched by a temporal dimension, then the Dimensionality of the Universe would overall be either 5D (3 real + 1 imaginary = complex plane then overarched by Time) or 7D (3 real + 3 imaginary = complex plane then overarched by Time) depending on if we go with general definition or extended/Quaternion definition, or so I understand.

 

[AlexSamDen]

By aleph numbers, you mean Aleph-1 (cardinality of R), right?

If yes, then indeed, but imaginary numbers by default have a cardinality equal to R (that is, Aleph 1), so an imaginary number space (which represents the axis that imaginary numbers exist on) is by default equal to a real normal spatial dimension that is significant (unless suggested otherwise).

Thus, the application isn't limited to only Imaginary Number spaces containing universes, but also to Universes that have an imaginary number space as part of itself.

So for example, if a Universe has 3 regular dimensions as well as an imaginary number space overarched by a temporal dimension, then the Dimensionality of the Universe would overall be either 5D or 7D (depending on if we go with general definition or extended/Quaternion definition), or so I understand.

Quaternion space is by itself 4D, not +3D, so in this case it's 5D

 

[Astral_Trinity439]

Quaternion space is by itself 4D, not +3D, so in this case it's 5D

One of the axis is real tho, no?
I'd assume we equate the real part to any amount of real dimensions that exist in-verse and treat the imaginary number part alone as 3 axes, in total making a complex plane of X+3 dimensional where X = real Spatial dimensions.

 

[AlexSamDen]

One of the axis is real tho, no?
I'd assume we equate the real part to any amount of real dimensions that exist in-verse and treat the imaginary number part alone as 3 axes, in total making a complex plane of X+3 dimensional where X = real Spatial dimensions.

Afaik the real part is always a line, Quaternions just kinda "grow axises" in the "other" direction, thought I can be corrected

 

[Astral_Trinity439]

Afaik the real part is always a line, Quaternions just kinda "grow axises" in the "other" direction, thought I can be corrected

Hmm, but we'll have to make it apply to actual instances in fiction too, no?
And in that we'll have to equate the real part to the real dimensions in context.

 

[AlexSamDen]

Hmm, but we'll have to make it apply to actual instances in fiction too, no?
And in that we'll have to equate the real part to the real dimensions in context.

For as I know, in the OP I proposed treating INS as separate from regular spatial axises, as if we add them, then INS would have to encompass said axises and therefore the reality

 

[Astral_Trinity439]

For as I know, in the OP I proposed treating INS as separate from regular spatial axises, as if we add them, then INS would have to encompass said axises and therefore the reality

Per what you said in the OP, you treat Imaginary Number space to complex plane itself

But there's a slight difference. INS itself is basically what the name says, imaginary number space, it only refers to the imaginary part. When we talk about the whole of the 4 axes, we mean the complex plane as there is 1 real axis besides the 3 regular axes.

Otherwise what you said would be equal to saying we're assuming all universes with an imaginary space inside them have an additional real dimension, not just additional imaginary number ones, which is false since "real" dimension just refers to the "regular" dimensions. In real space that'll be 3, while in geometric space that's only 1 most of the times (due to only talking about real axis lines, not real planes).

 

[AlexSamDen]

Per what you said in the OP, you treat Imaginary Number space to complex plane itself

But there's a slight difference. INS itself is basically what the name says, imaginary number space, it only refers to the imaginary part. When we talk about the whole of the 4 axes, we mean the complex plane as there is 1 real axis besides the 3 regular axes.

Otherwise what you said would be equal to saying we're assuming all universes with an imaginary space inside them have an additional real dimension, not just additional imaginary number ones, which is false since "real" dimension just refers to the "regular" dimensions. In real space that'll be 3, while in geometric space that's only 1 most of the times (due to only talking about real axis lines, not real planes).

As I've said, I can stand corrected. Was making this, so that the baseline is 4D, what is beyond that wasn't that thought through

 

[Lycoris4812]

Math.

 

[Antvasima]

@Agnaa

What do you think about this?

 

[DontTalkDT]

C^3 is nice C-Vector space. Why would we default to a division algebra over the real numbers?

 

[AlexSamDen]

C^3 is nice C-Vector space. Why would we default to a division algebra over the real numbers?

It seems my knowledge on the terminology is a bit lacking. Can you link or explain what you are referring to?

 

[DontTalkDT]

A vector space is composed of a set of vectors (V) and a set of scalars (F). The set of scalars have to be a field. The vectors and scalars together have to fulfill certain axioms on how they behave regarding addition and scalar multiplication. If they do, V is called a F-vector space.
C^3 denotes the set of all vextors of the form (x, y, z) for which x, y and z each are a complex numbers.
C is supposed to denote the field of complex numbers (in lack of special characters)
C^3 is a C vector space.
As in, you can multiply (x,y,z) with some complex number c in the way you would expect of a vector space. See wikipedia for details.

The result is a complex number (vector) space.

 

[Agnaa]

Introduction​

Hello everyone, this thread will be a continuation of this one. The reason is that I've encountered the case of Imaginary Number Space once again and while reading the aforementioned Q&A, I noticed that several important arguments weren't made, so I'm here to introduce them.

A basic thing to mention, so as to avoid misunderstandings

Imaginary numbers and complex numbers are essentially the same. While strictly speaking, the imaginary numbers are (b*i) and complex numbers are (a + b*i), imaginary numbers in social media, fiction and generally the Internet are widely referred to as (a + b*i), moreover their respective geometric interpretations that are usually used, while with different reasonings, have the same form, meaning that when we talk about feats of imaginary number space in fiction, it is sensible to treat them as complex number space.​

The justifications for the proposal​

1) Projecting the geometrical interpretation onto this case
The accepted geometrical interpretation of the set of basic complex numbers (that is a+bi) is a plane, as opposed to the set of real numbers being a line. That is due to the fact that the expression a+bi not only includes all the real numbers (while b = 0), but due to the fact that both a and b can take any real value and due to them being independent from each other, for every of the uncountably infinite values of b there are also uncountably infinite different values of a, essential making this the same as the standart (x;y) plane. This means that whenever you add an imaginary part to a number, it adds an additional imaginary axis to the geometrical interpretation due to a new separate variable being added. So going from "set of real numbers" to "set of complex numbers" is going from a 1D structure to a 2D one. Therefore, if we use this method further and take a regular space, which is a set of all 3D coordinates (x;y;z) and add an imaginary part to it, we would get a new axis for the space, making it a 4-dimensional comlex number space.
In short, when we add an imaginary part to a set, it's geometrical interpretation gets +1D, therefore imaginary number space is a regular space +1D which equals 3+1D = 4D

Of course, this is a theoretical approach, however this part just shows how it is sensible and highly likely to be the case we need to use. . There are heavier arguments in the second part

Only happens if you add an axis instead of replacing it. And if you knew the series was talking about adding an axis, you could get 4-D anyway.

And, even then, I think your reasoning is wrong. Giving real numbers complex parts isn't a +1, it's a x2. There would be 6 axes if each real number line was given a complex component.

2) There actually is a defined imaginary number space in Maths and Physics. And it's 4D
Yes, you've read it correctly, it is an extension of imaginary numbers (and not just complex) and it is used rather widely. Some of our members that are well versed in 3D animation might know about it. It is the Quaternion space. It is a imaginary number space that consists of 1 real axis and 3 imaginary ones (I know that what I proposed in (1) is a bit different, but essentially it does not matter as they follow the same logic: either you add an Im axis to 3 Rez axises or a Rez axis to 3 Im ones). This space is often used for describing motion of 3D objects, hence the use in animation.

The only citations and quotes you've provided this don't describe that as "imaginary number space".

You may argue that going straight to them is a stretch, however this is the only term that is referred to as "Imaginary Number Space" in science. Moreover, it's been proven that there is no such thing as a working 3-dimensional imaginary number space:

This effectively means that the only possible meaning behind Imaginary Number Space in fiction is a 4D space (If there are no clear statements it isn't in this specific setting, that is)

I think this only happens if you don't allow any other real axes. Heck, I'd guess that in that case, you can only end up with spaces of pseudo-dimensionality 2^n.

The proposal itself​

If you hadn't guessed it already, my proposal is that we treat Imaginary Number Spaces as 4D structures. Of course I don't say we just throw 4D at whatever mention of this name. I have prepared three conditions for something to qualify for this:

1) It is explicitly called "Imaginary Number Space"
Seems rather logical, as it is what we're discussing here. I am aware that they can be not called such and yet effectively be one via mentions of imaginary axises, however to avoid confusion these verses can just straight up prove it via the direct statement of an additional axis, without bringing up the imaginary numbers in this meaning

2) It is in some way separated from regular space
Also reasonable, imaginary coordinates aren't called imaginary for no reason, they don't fit the normal coordinates, even ignoring the 3D and 4D structures somehow properly intersecting. It can be any kind of separation: from an invisible line to it being a separate realm entirely, not proposing anything harsh here, just a clear anti-feat

3) It is in some way shown to be superior to regular space

It is also not a strict condition, obviously 4D space is superior to a 3D one. Can essentially be shown anyhow, just at least somehow

This seems completely pointless as an overall standard.

This is just a way to give TYPE-MOON one extra dimension. Just argue it for that one series, instead of adding a bunch of bloat to our explanation pages for an incredibly niche concept.

---

I disagree.

 

[AlexSamDen]

Only happens if you add an axis instead of replacing it. And if you knew the series was talking about adding an axis, you could get 4-D anyway.

First: I am talking about adding one
Second: I'm making an analogy for how transition from a set of real numbers goes to a set of complex numbers and to how it goes from regular space coordinates to imaginary (complex) number space coordinates

And, even then, I think your reasoning is wrong. Giving real numbers complex parts isn't a +1, it's a x2. There would be 6 axes if each real number line was given a complex component.

I'm pretty sure you can add 1 at a time pretty fine, as that part isn't fully defined within the articles and given that just addig a bi to the equation doesn't actually x^2 the axises, but exactly +1

The only citations and quotes you've provided this don't describe that as "imaginary number space".

That's kinda the reason I provided links. There is a lot to consider in both of those and explaining everything would require too much copy-paste

Quaternions form a 4-dimensional vector space over the real numbers, with {1,i,j,k}

as a basis, by the component-wise addition

I think this only happens if you don't allow any other real axes. Heck, I'd guess that in that case, you can only end up with spaces of pseudo-dimensionality 2^n.

Both Frobenious and Hurwitz theorems talk about this. And if you allow other real axises, doesn't that by your own arguments make this space at least 4D when adding an imaginary one?

This seems completely pointless as an overall standard.

Don't know about you, but from time to time I see people asking in QnA about treatment of imaginary numbers and axises. The original idea even came from one such QnA by @Oblivion_Of_The_Endless , so it is not pointless

This is just a way to give TYPE-MOON one extra dimension. Just argue it for that one series, instead of adding a bunch of bloat to our explanation pages for an incredibly niche concept.

1) I don't argue about TYPE-MOON, I know pretty much nothing on that verse.
2) As I already said, there are verse that need some kind of norm for this. Moreover, I'm not particularly proposing adding a page, I'm proposing at least making a point of reference, a bad is to stand on

 

[AlexSamDen]

A vector space is composed of a set of vectors (V) and a set of scalars (F). The set of scalars have to be a field. The vectors and scalars together have to fulfill certain axioms on how they behave regarding addition and scalar multiplication. If they do, V is called a F-vector space.
C^3 denotes the set of all vextors of the form (x, y, z) for which x, y and z each are a complex numbers.
C is supposed to denote the field of complex numbers (in lack of special characters)
C^3 is a C vector space.
As in, you can multiply (x,y,z) with some complex number c in the way you would expect of a vector space. See wikipedia for details.

The result is a complex number (vector) space.

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

 

[Agnaa]

First: I am talking about adding one
Second: I'm making an analogy for how transition from a set of real numbers goes to a set of complex numbers and to how it goes from regular space coordinates to imaginary (complex) number space coordinates

Then as I said, if you're talking about adding one, then that would already qualify as 4-D.

An additional axis of space is 4-D (in some sense), regardless of whether it's called "imaginary" or not.

I'm pretty sure you can add 1 at a time pretty fine, as that part isn't fully defined within the articles and given that just addig a bi to the equation doesn't actually x^2 the axises, but exactly +1

I don't understand your argument here. It doesn't seem to engage with the argument I provided.

That's kinda the reason I provided links. There is a lot to consider in both of those and explaining everything would require too much copy-paste

The links you provided did not describe them as "imaginary number spaces", and this quote you provide here doesn't, either.

Both Frobenious and Hurwitz theorems talk about this. And if you allow other real axises, doesn't that by your own arguments make this space at least 4D when adding an imaginary one?

This doesn't engage with the important part of my point there.

Don't know about you, but from time to time I see people asking in QnA about treatment of imaginary numbers and axises. The original idea even came from one such QnA by @Oblivion_Of_The_Endless , so it is not pointless

Because those people are also curious about TYPE-MOON.

1) I don't argue about TYPE-MOON, I know pretty much nothing on that verse.

Where else does this come up? From a quick search, I can find no other mentions of "imaginary number spaces".

 

[PrimeHydra64]

Where else does this come up? From a quick search, I can find no other mentions of "imaginary number spaces".

I'm not going to go into the mathematic side but some japanese media has imaginary number spaces and with them having the same kanji "虚数空間":

Fate: https://typemoon.fandom.com/wiki/Imaginary_Number_Space

Tensura:

Imaginary Space

Imaginary Space ｢虚数空間, kyosū kūkan, lit. "Imaginary Numbers Space"｣ is a Chaos world[1] and a skill used by Rimuru Tempest. It's a super evolved version of Stomach and Isolation. It's a world with an infinitely sized space.[2] Everything inside is said to be mixed together in chaos.[3] It's also...

tensura.fandom.com

HoYoverse:

Imaginary Space - HoYoverse Universe Index

The Imaginary Space is a 'space' outside of the regular world structure. It is a mathematically consistent space that is to an extent independent from a world. It...

hoyodex.miraheze.org

Xenogear(though it has a different kanji for space):

Imaginary Space

Imaginary Space, also referred to as the Imaginary Number Domain or the Imaginary Realm, is the "spiritual" or "metaphysical" half of the Lower Domain where the...

www.xenoserieswiki.org

 

[Agnaa]

I can imagine it being worth a note somewhere, if we come up with a particularly noteworthy ruling on it, then.

 

[AlexSamDen]

Then as I said, if you're talking about adding one, then that would already qualify as 4-D.

An additional axis of space is 4-D (in some sense), regardless of whether it's called "imaginary" or not.

I'm trying to make a default case here, just like a set of complex numbers is uncountably infinitely bigger than the set of real numbers, the imaginary (complex) number space is also uncountably infinitely bigger than regular space

I don't understand your argument here. It doesn't seem to engage with the argument I provided.

Doesn't it? In that part you argued that adding imaginary parts squares the dimensionality, rather than addin 1 to it. To which I reply that going, for example, from ax + by + cz = d to ax + by + cz + fi = d, adds exactly one acis and not 3 or 6

The links you provided did not describe them as "imaginary number spaces", and this quote you provide here doesn't, either.

Quaternions (from Latin quaterni , four ) are a system of hypercomplex numbers that form a vector space of dimension four over the field of real numbers

Quaternions form a 4-dimensional vector space over the real numbers, with {1,i,j,k} as a basis

These two are basically textbook definitions of one. The Wikipedia article won't call it such more directly due to specifics of preference of calling it space over complex numbers and formulating the imaginary part as expressions, due to it defining the Quaternion space by itself with very little to stand on for the basis

This doesn't engage with the important part of my point there.

It does. You say that everything might go wrong if there are more real axises, while I'm saying that it can indeed go right

Because those people are also curious about TYPE-MOON.

Well, not like I know for definite about this one

Where else does this come up? From a quick search, I can find no other mentions of "imaginary number spaces".

The interest in how wiki treats them came from Tensura a couple months ago and due to me actively participating in the QnA forum, I came to notice a relative lot of questions about imaginary numbers, so decided to make that

 

[Agnaa]

I'm trying to make a default case here, just like a set of complex numbers is uncountably infinitely bigger than the set of real numbers, the imaginary (complex) number space is also uncountably infinitely bigger than regular space

No, it's just another way to create an axis.

Doesn't it? In that part you argued that adding imaginary parts squares the dimensionality, rather than addin 1 to it. To which I reply that going, for example, from ax + by + cz = d to ax + by + cz + fi = d, adds exactly one acis and not 3 or 6

Not squares the dimensionality, multiplies it by two.

You wouldn't go from ax/by/cz to ax/by/cz/fi, you'd go from ax/by/cz to ax/dxi/by/eyj/cz/fzk.

These two are basically textbook definitions of one. The Wikipedia article won't call it such more directly due to specifics of preference of calling it space over complex numbers and formulating the imaginary part as expressions, due to it defining the Quaternion space by itself with very little to stand on for the basis

Neither of these call it "imaginary number space".

They're talking about imaginary numbers and their use in vector spaces, but they're not calling them "imaginary number space"s. The latter is fictional sci-fi technobabble.

It does. You say that everything might go wrong if there are more real axises, while I'm saying that it can indeed go right

I'm saying that there can be real axes alongside them, resulting in other numbers of total axes.

 

[AlexSamDen]

No, it's just another way to create an axis.

Well, if we really go into it, with the way you describe addition of an Im axis, there is actually no way for a 3D space to be created like this

Not squares the dimensionality, multiplies it by two.

Sorry, thought that x2 was x^2 not *2. But anyways, the "3" remark was actually made in case you meant *2, so that's still right

You wouldn't go from ax/by/cz to ax/by/cz/fi, you'd go from ax/by/cz to ax/dxi/by/eyj/cz/fzk.

Doesn't it actually prove my point further about INS being higher than regular space?

Neither of these call it "imaginary number space".

They're talking about imaginary numbers and their use in vector spaces, but they're not calling them "imaginary number space"s. The latter is fictional sci-fi technobabble.

They depict a space made of imaginary numbers. The sci-fi technoblabber while it is a blabber (not with all verses though) still picks something to use as a basis. Right now we're further and further into agreeing that the 3D counterpart simply doesn't exist, so what science term did they use as a basis then?

I'm saying that there can be real axes alongside them, resulting in other numbers of total axes.

And again, returning to your arguments, the resulting number of axises if you throw new real ones is still ≥4

 

[Agnaa]

Well, if we really go into it, with the way you describe addition of an Im axis, there is actually no way for a 3D space to be created like this

???

Doesn't it actually prove my point further about INS being higher than regular space?

And again, returning to your arguments, the resulting number of axises if you throw new real ones is still ≥4

No, because you could construct such a place by replacing axes rather than adding new ones.

And if we know they're adding new ones, we can use that fact to provide appropriate tiers.

They depict a space made of imaginary numbers. The sci-fi technoblabber while it is a blabber (not with all verses though) still picks something to use as a basis. Right now we're further and further into agreeing that the 3D counterpart simply doesn't exist, so what science term did they use as a basis then?

That is largely irrelevant.

 

[AlexSamDen]

No, because you could construct such a place by replacing axes rather than adding new ones.

And if we know they're adding new ones, we can use that fact to provide appropriate tiers.

So how does replacing axises actually work different from adding them to a lower structure? What is the difference between replacing one of axises of a 3D structure with an imaginary one and adding an imaginary one to a 2D structure? You're still doing the same thing, especially when considering that to replace an axis you first remove it and then add a new one in its place. They are the same processes

That is largely irrelevant.

It is relevant if we actually want to determine how to work with them

 

[Agnaa]

So how does replacing axises actually work different from adding them to a lower structure? What is the difference between replacing one of axises of a 3D structure with an imaginary one and adding an imaginary one to a 2D structure? You're still doing the same thing, especially when considering that to replace an axis you first remove it and then add a new one in its place. They are the same processes

Replacing an axis in a space gives the same result as adding an axis to a lower space.

But you're saying that because they're called "imaginary number spaces" that this means an axis is added, when an axis could just as easily be changed. That is where the difference matters.

It is relevant if we actually want to determine how to work with them

If it ultimately has no meaning, and is not a proper concept in physics, I do not think we should be reverse-engineering some nonsense based on the words they used. We should not assume particular meanings for terms like "hyperdimension", even if we can theoretically trace its roots.

 

[AlexSamDen]

Replacing an axis in a space gives the same result as adding an axis to a lower space.

Exactly what I said

But you're saying that because they're called "imaginary number spaces" that this means an axis is added, when an axis could just as easily be changed. That is where the difference matters.

Well I did aim to draw an analogy, as to make a set of complex numbers, we add an axis and not change it, same being there, but anyways, if adding an imaginary axis to a 2D plane is the same as replacing one of 3D ones, then we will get a 4D space anyway as we will double the amount of axises for 2D

If it ultimately has no meaning, and is not a proper concept in physics, I do not think we should be reverse-engineering some nonsense based on the words they used. We should not assume particular meanings for terms like "hyperdimension", even if we can theoretically trace its roots.

Imaginary number space is a space made of complex or imaginary numbers and these do exist, so it is existing. A hyperspace in turn can easily mean a myriad of different things from some 1-B things, to just a big space if we take "hyper" as "very big"

 

[Agnaa]

Well I did aim to draw an analogy, as to make a set of complex numbers, we add an axis and not change it, same being there, but anyways, if adding an imaginary axis to a 2D plane is the same as replacing one of 3D ones, then we will get a 4D space anyway as we will double the amount of axises for 2D

?????????????????????

Look, I've already given the answer with proper reasoning (there's nothing meaningfully distinguishing an axis of complex numbers from any other real axis; so just knowing that some of them are imaginary does not matter).

It does not feel productive to continue with this.

Imaginary number space is a space made of complex or imaginary numbers and these do exist, so it is existing.

But doesn't inherently mean anything relevant for our profiles.