Imaginary Number Spaces

[AlexSamDen]

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

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.

You yourself said that adding an imaginary axis doubles the total amount and referenced that multiple times afterwards

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.

And now you say that if you give a 2D plane an imaginary part, it's 3D. Unless I misunderstood your arguments (in which case I would like you to explain clearer), adding an imaginary axis to a plane results in a 4D space

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

We've kinda been discussing here when it might become relevant even without a direct mention of additional axises

 

[Agnaa]

You yourself said that adding an imaginary axis and referenced that multiple times afterwards

And now you say that if you give a 2D plane an imaginary part, it's 3D. Unless I misunderstood your arguments (in which case I would like you to explain clearer), adding an imaginary axis to a plane results in a 4D space

ughghhhhhbhh

You can add one complex axis.

You can change one axis into being complex.

You can give every extant real axis a complex component.

You can add 436 complex axes.

You can add two complex axes and remove three real axes.

All of these do different things.

 

[AlexSamDen]

ughghhhhhbhh

You can add one complex axis.

You can change one axis into being complex.

You can give every extant real axis a complex component.

You can add 436 complex axes.

You can add two complex axes and remove three real axes.

All of these do different things.

I could try to continue quoting you contradicting some of this, but I actually found a better argument

So in the light of this discussion I've actually been wondering, why didn't I see any structures with imaginary axises that has >1 real axises. And now I know why, they just don't work at all.
So let's take a space made of 3 axises: x, y and i. Now let's take a point (A, for example) in this space with coordinates (x1;y1;z1) with each of the numbers corresponding to 1 axis. Now, this is a space with an imaginary axis, so naturally it's gonna follow the basic principle of all complex numbers: for every vector going from 0 to any point in it, there corresponds a complex number. Then let's actually take a look at the vector 0A and the number that corresponds to it. The number is x1 + y1 + z1*i = (x1+y1) + z1*i. And exactly here arises the problem. The problem is that this number doesn't actually correspond to a 3-dimensional vector. It corresponds to a 2D one, which shouldn't happen as we took a point in 3 dimensions. And that's why it doesn't work, it just breaks the whole thing that complex numbers even work on and is therefore not possible.
Therefore there can't be more than 1 real axis

 

[Agnaa]

I don't understand why that wouldn't correspond to a 3-D vector.

The important part of imaginary numbers is that if you square them you end up with a real number, so it would simply have to be associated with a particular real axis, so we can determine how such operations apply to it.

 

[AlexSamDen]

I don't understand why that wouldn't correspond to a 3-D vector.

Because this corresponds to just a+bi, which in turn corresponds to a 2D vector. In that expression x1 and y1 are just numbers and therefore not independent and due to that, they can be simplified to just one variable

The important part of imaginary numbers is that if you square them you end up with a real number, so it would simply have to be associated with a particular real axis, so we can determine how such operations apply to it.

The only part of imaginary numbers for which squaring is important is the definition of the imaginary unit. The all other important parts of them correspond to their behavior in coordinate systems for rotation. And when we add the real axis to form a plane, we transition to complex numbers, for which the most important characteristic is them defining a vector and their behavior in algebra with their operations having a certain result within the coordinates. Such as, for example, if you multiply 2 complex numbers (a+bi) and (c+di) you get a complex number that corresponds to a vector √(a²+b²) times longer than the vector corresponding to the second number and also rotated counterclockwise by an angle a, for which (cos a) = a/(√(a²+b²))

 

[Agnaa]

Because this corresponds to just a+bi, which in turn corresponds to a 2D vector. In that expression x1 and y1 are just numbers and therefore not independent and due to that, they can be simplified to just one variable

What? They are independent, they're completely different axes that would have different values.

They're as perpendicular as real numbers and imaginary numbers are, normally.

The only part of imaginary numbers for which squaring is important is the definition of the imaginary unit. The all other important parts of them correspond to their behavior in coordinate systems for rotation.

Those come directly from how raising them to an exponent works, or rather, from how that can be interpreted.

 

[AlexSamDen]

What? They are independent, they're completely different axes that would have different values.

They're as perpendicular as real numbers and imaginary numbers are, normally.

The axises? Yes they are independent. The already defined point's coordinates in a separate expression? Not. In that expression they are just like the parameters, which while being able to take different values, are not independent, as ultimately only one of these possible values is actually present in the expression

Those come directly from how raising them to an exponent works, or rather, from how that can be interpreted.

And they're just going off definition. Which pretty much every number system does. This still means that particularly squaring is important only for the definition

 

[AlexSamDen]

Another thing I've found is that we can just throw the original real axis out of window and only use complex numbers themselves for this. It's called complex coordinate space

 

[Agnaa]

The axises? Yes they are independent. The already defined point's coordinates in a separate expression? Not. In that expression they are just like the parameters, which while being able to take different values, are not independent, as ultimately only one of these possible values is actually present in the expression

How are they not independent? One could be 10^10^10, and the other could be 2 or 0. One does not restrict the value of the other.

 

[AlexSamDen]

How are they not independent? One could be 10^10^10, and the other could be 2 or 0. One does not restrict the value of the other.

They don't restrict eachother, however there are restrictions on them, and that is the fact that the point is already chosen. While we have a huge amount of points we can choose from, once we choose one to represent the general, their values don't change. Yes we don't know them, but they are numbers and not variables. The parameter analogy is really important here, as in the expression 5x² +3x+ a = 0, a can take a lot of different values. Does it actually make it independent? No, because while the choice is great, a is still a single number and not a variable, in the expression it only takes 1 value at a time

 

[Agnaa]

They don't restrict eachother, however there are restrictions on them, and that is the fact that the point is already chosen. While we have a huge amount of points we can choose from, once we choose one to represent the general, their values don't change. Yes we don't know them, but they are numbers and not variables. The parameter analogy is really important here, as in the expression 5x² +3x+ a = 0, a can take a lot of different values. Does it actually make it independent? No, because while the choice is great, a is still a single number and not a variable, in the expression it only takes 1 value at a time

????????

This applies equally to when the two variables are a real number and a complex number. Or when they are four real numbers.

So you are not actually making a point; this post is still incorrect. You did not substantiate it.

 

[AlexSamDen]

????????

This applies equally to when the two variables are a real number and a complex number.

They are separated from each other by the imaginary unit

Or when they are four real numbers.

And that's the important part, you just cannot make an expression like that when you only have real numbers, as going from coordinates to a number representing a vector is only possible in complex numbers, as it relies on its nature to make such a thing feasible. This expression simply doesn't exist for 4 real numbers

 

[Agnaa]

They are separated from each other by the imaginary unit

And that's the important part, you just cannot make an expression like that when you only have real numbers, as going from coordinates to a number representing a vector is only possible in complex numbers, as it relies on its nature to make such a thing feasible. This expression simply doesn't exist for 4 real numbers

?????

They don't restrict eachother, however there are restrictions on them, and that is the fact that the point is already chosen. While we have a huge amount of points we can choose from, once we choose one to represent the general, their values don't change. Yes we don't know them, but they are numbers and not variables. The parameter analogy is really important here, as in the expression 5x² +3x+ a = 0, a can take a lot of different values. Does it actually make it independent? No, because while the choice is great, a is still a single number and not a variable, in the expression it only takes 1 value at a time

ONCE YOU CHOOSE A POINT

YOU HAVE CHOSEN A POINT

YOU CAN DO THIS WITH ONLY REAL NUMBERS

WHAT THE **** ARE YOU SAYING!?!?!?!?!?!?!

 

[AlexSamDen]

?????

ONCE YOU CHOOSE A POINT

YOU HAVE CHOSEN A POINT

YOU CAN DO THIS WITH ONLY REAL NUMBERS

WHAT THE **** ARE YOU SAYING!?!?!?!?!?!?!

The whole transition from a point A(x1; y1; z1) to a number C = x1 + y1 + z1*i is only possible if you have an imaginary axis. The contradiction only pops up in C, so when you have only real numbers, you indeed can choose a point and the coordinates are fixed, but there is no transition to anywhere from there and you don't get a contradiction. The cornerstone of the contradiction is the presence of both the imaginary axis and >1 real ones and it happens only in C, but you can't get C at all if you only have real numbers

 

[Deadguy999]

Let’s say I don’t understand 90% of things said here

 

[Agnaa]

The whole transition from a point A(x1; y1; z1) to a number C = x1 + y1 + z1*i is only possible if you have an imaginary axis. The contradiction only pops up in C, so when you have only real numbers, you indeed can choose a point and the coordinates are fixed, but there is no transition to anywhere from there and you don't get a contradiction. The cornerstone of the contradiction is the presence of both the imaginary axis and >1 real ones and it happens only in C, but you can't get C at all if you only have real numbers

I wouldn't call that a contradiction, but just trying to do an improper operation.

Like, think about the thing you're ultimately arguing here.

Saying that imaginary number spaces are Tier 2 because:

1. Only complex spaces of power 2^n can have their co-ordinates represented by a single number.
2. We'll ignore the n=0 and n=1 cases because they're smaller than ordinary reality.
3. That leaves 2^2 as the smallest space, which should be Tier 2.

But we don't actually need spaces to have their co-ordinates represented by a single number for them to exist as realities. Our reality is proof of that, since we describe it with multiple real number axes.

So why does that constraint matter at all?

 

[AlexSamDen]

But we don't actually need spaces to have their co-ordinates represented by a single number for them to exist as realities. Our reality is proof of that, since we describe it with multiple real number axes.

So why does that constraint matter at all?

Not a single number, but by an expression, a function of sort.
And you're just wrong that we can't express our reality as one, as this function exists Ax + By + Cz + Dt + E = 0. It defines a single space in a 4D structure. So there is a function expressing our reality. Every (at least the ones following the Cartesian system) number system can be expressed using a single function

 

[Agnaa]

Not a single number, but by an expression, a function of sort.

Then such an expression would be about as easy to write if you made one of the axes imaginary.

And you're just wrong that we can't express our reality as one, as this function exists Ax + By + Cz + Dt + E = 0. It defines a single space in a 4D structure. So there is a function expressing our reality. Every (at least the ones following the Cartesian system) number system can be expressed using a single function

That's not what I said.

---

Look, at this point you've just been spouting nonsense without any tether to consistency or relevance to our system for dozens of posts now.

I'm going to stop responding. If any staff member is convinced by your points, they should let me know, and I'll explain why it's wrong.

 

[Antvasima]

If this thread is based on nonsense, should we close it now?

 

[Agnaa]

I'm of the kind persuasion that likes leaving those sorts of threads up.

 

[AlexSamDen]

Then such an expression would be about as easy to write if you made one of the axes imaginary.

Except this system will contradict one of the main properties of complex numbers

That's not what I said.

The I would like you to explain what else you meant when you said that you can't define our reality in a single expression

---

Look, at this point you've just been spouting nonsense without any tether to consistency or relevance to our system for dozens of posts now.

I'm going to stop responding. If any staff member is convinced by your points, they should let me know, and I'll explain why it's wrong.

I prefer to leave the threads with a two-sided conclusion, so I will respond anyways

1) There isn't any relevance to our system, because we have NO system for such a situation and that is the reason why this whole thread was created
2) I will describe a bit how it looks from my side:
I argue that there is no 3D complex space and in support of my stance provide several actual scientific arguments, such as the Frobenious and Hurwitz theorems, the Complex Coordinate Space (which DT actually suggested using) and Quaternions, all of which reject the idea of a 3D complex space in tandem with the main properties of complex numbers

You, on the other hand, provided no actual scientific terms or examples of it actually existing in science to back up your stance, with the main argument in your latest posts being the theoretical existence of a system made of 2 real axises and 1 imaginary, which not only doesn't have any analogs in actual mathematics, but also doesn't keep the main property of complex numbers, that is defining a vector.

That how it looks from my side, once I skimmed through the discussion once more. Feel free to correct the points made

 

[FinePoint]

You, on the other hand, provided no actual scientific terms or examples of it actually existing in science to back up your stance,

99% of what we do here has no actual scientific example.

 

[AlexSamDen]

99% of what we do here has no actual scientific example.

However most of our tiering system does, and this topic falls more into this category

 

[FinePoint]

However most of our tiering system does, and this topic falls more into this category

I'd argue nothing past 3-A has an 'actual scientific example'.

Maybe some mathematical basis, but even then applied theoretically.

 

[AlexSamDen]

I'd argue nothing past 3-A has an 'actual scientific example'.

Maybe some mathematical basis, but even then applied theoretically.

There are still clear definitions, laws and properties of those, so I'd consider them scientific enough. We can't even normally describe Black Holes without tapping into additional maths.

 

[Vietthai96]

Black Hole is real phenomenon, which do confront to real theory, of course we couldn't know everything about it, but that doesn't make it "less real"

Anyway, i don't understand the need to force a default dimensionality value on Imaginary Number Space where different fictions portray it differently, we tiering it based on what the verse show us. The only reason we could apply 4D to space-time with 3 spatial dimensions and 1 temporal dimension because it is a proven thing, but that doesn't stop fiction from having space-time with 38635 spatial dimensions and 4253 temporal dimensions, or even infinite dimensional

 

[AlexSamDen]

Black Hole is real phenomenon, which do confront to real theory, of course we couldn't know everything about it, but that doesn't make it "less real"

My point was exactly that Black holes are real, but still require purely theoretical mathematics to describe them

Anyway, i don't understand the need to force a default dimensionality value on Imaginary Number Space where different fictions portray it differently, we tiering it based on what the verse show us. The only reason we could apply 4D to space-time with 3 spatial dimensions and 1 temporal dimension because it is a proven thing, but that doesn't stop fiction from having space-time with 38635 spatial dimensions and 4253 temporal dimensions, or even infinite dimensional

You encounter INS in a fictional work. The author does not specify what kind of structure is this (doesn't directly call out axises, as, honestly, very few works can afford it without sounding strange). However, this INS has the characteristics I proposed in the OP. You know it should mean something, as there is no reason for these characteristics to exist just because, but you, again, don't have a straightforward statement. What do you do? You try to interpret what scientific structure this one is based on

I just think that if it has a clear scientific theory under it and appears in fiction from time to time, we should create a point of reference what this could actually be. Kinda like Absolute Infinity and L1A. It doesn't appear often and in even less situations it actually can mean L1A, but a standart for it is still there and it can allow a verse to not specify some certain things to qualify for the tier (I know that it's not the best example, but still, we do sometimes use some niche things to scale without direct feats)

 

[Vietthai96]

My point was exactly that Black holes are real, but still require purely theoretical mathematics to describe them

Dude what?, we use math to describe everything, or are you saying the table couldn't be described by math?, the point is Black Hole are real phenomenon, but INS isn't, imaginary number are purely mathematic and not real phenomenon, let alone INS

You encounter INS in a fictional work. The author does not specify what kind of structure is this (doesn't directly call out axises, as, honestly, very few works can afford it without sounding strange). However, this INS has the characteristics I proposed in the OP. You know it should mean something, as there is no reason for these characteristics to exist just because, but you, again, don't have a straightforward statement. What do you do? You try to interpret what scientific structure this one is based on

Not only INS, but also every other unspecified space will not be tiered, we don't give arbitrary giave statistics to anything that is not specified about their nature. We didn't give specific tiering to random mention of string theory, and some other dimensional theories except space-time due to it is proven theory, why should we have to give a default statistics to random mention of INS?

I just think that if it has a clear scientific theory under it and appears in fiction from time to time, we should create a point of reference what this could actually be. Kinda like Absolute Infinity and L1A. It doesn't appear often and in even less situations it actually can mean L1A, but a standart for it is still there and it can allow a verse to not specify some certain things to qualify for the tier (I know that it's not the best example, but still, we do sometimes use some niche things to scale without direct feats)

Absolute Infinity give nothing, unless the verse itself make it 1:1 with Cantor's ideas about Absolute Infinity, which is tier 0

 

[AlexSamDen]

Dude what?, we use math to describe everything, or are you saying the table couldn't be described by math?, the point is Black Hole are real phenomenon, but INS isn't, imaginary number are purely mathematic and not real phenomenon, let alone INS

I said purely theoretical a table doesn't need that for its description

why should we have to give a default statistics to random mention of INS?

Not really a default one, I still provided some conditions to even try to apply the theory

Absolute Infinity give nothing, unless the verse itself make it 1:1 with Cantor's ideas about Absolute Infinity, which is tier 0

Iirc one of Hoyoverse verses got L1A, for some time at least, due to this

 

[FriendOfTheTeaParty]

Iirc one of Hoyoverse verses got L1A, for some time at least, due to this

Hoyoverse is the reason why absolute infinity isn't low 1-A anymore

 

[FinePoint]

imaginary number are purely mathematic and not real phenomenon

Well, they may be needed to fully describe/predict a real phenomenon, like alternating current.

 

[FriendOfTheTeaParty]

Well, they may be needed to fully describe/predict a real phenomenon, like alternating current.

What if a verse acknowledges imaginary time as existing btw?

 

[Vietthai96]

What if a verse acknowledges imaginary time as existing btw?

Then we will see how the verse explain this concept, no? Imaginary time loop on itself iirc

 

[Vietthai96]

Not really a default one, I still provided some conditions to even try to apply the theory

Still depending on how the verse portray INS, some verse could make it infinite-dimensional, again, random mention of INS without going in dept of its nature mean nothing

Iirc one of Hoyoverse verses got L1A, for some time at least, due to this

Low 1-A Absolute Infinity was removed due to it being quantity of all quantity which make no sense at all, so now either it is tier 0 if it is 1:1 with Georg Cantor or nothing.

Also, even before the removal, random mention of Absolute Infinity get you nothing, Honkai got Low 1-A because the verse have proof Absolute Infinity is the cosmology structure of the verse, specifically it is the Imaginary Tree. The same as random mention of some big cardinal like Woodin Cardinal or Mahlo Cardinal mean nothing, they must be the underlying framework, foundation of the cosmology to be tiered

 

[AlexSamDen]

Still depending on how the verse portray INS, some verse could make it infinite-dimensional, again, random mention of INS without going in dept of its nature mean nothing

If you didn't notice I was providing a minimum rating rather than a definite. I also mentioned in the OP that the verse's own description is above what was proposed. The proposal was only for the INS that meet the criteria in OP and don't have a verse providing its own definition

 

[PedjaTarzan]

Hoyoverse is the reason why absolute infinity isn't low 1-A anymore

Or the person behind the downgrade edited wikipedia source to make downgrade happen in the first place, and then link the scans sourced from unknown website without even linking it (every scan was from Gyazo) so how the **** is hoyoverse at fault
cant even use accepted standards in this age anymore
(i wont even mention how no staff thread was made for this specific scaling standard)

 

[Vietthai96]

If you didn't notice I was providing a minimum rating rather than a definite. I also mentioned in the OP that the verse's own description is above what was proposed. The proposal was only for the INS that meet the criteria in OP and don't have a verse providing its own definition

And you didn't read what i said, if the verse just randomly mention INS without explaining anything about its nature, then it will be granted nothing, in other word, there is no minimum

 

[AlexSamDen]

And you didn't read what i said, if the verse just randomly mention INS without explaining anything about its nature, then it will be granted nothing, in other word, there is no minimum

And if you read the criterias, the verse is obligated to provide some explanation on its nature, but by itself it doesn't really provide enough, but there are hints to the proposal being possible

 

[Vietthai96]

And if you read the criterias, the verse is obligated to provide some explanation on its nature, but by itself it doesn't really provide enough, but there are possible hints to the proposal being possible

That is case-by-case, which can be evaluatrded and solved in a CRT, and we do this all time, so i don't understand the purpose of this thread?, at one point you want to force a default statistics on it, iirc, 4 spatial dimensions?, then you went on debating with Agnaa about it add another axis to real number axes, which Agnaa said they could simply replace the real number axes. One comment above this one you said about applying a minimum to INS, and now you said the verse is obligated to provide some explanation, but if it is not enough, there are possibly hints for the proposal to be possible.

So......what exactly do you want?

 

[AlexSamDen]

That is case-by-case, which can be evaluatrded and solved in a CRT, and we do this all time, so i don't understand the purpose of this thread?, at one point you want to force a default statistics on it, iirc, 4 spatial dimensions?, then you went on debating with Agnaa about it add another axis to real number axes, which Agnaa said they could simply replace the real number axes. One comment above this one you said about applying a minimum to INS, and now you said the verse is obligated to provide some explanation, but if it is not enough, there are possibly hints for the proposal to be possible.

So......what exactly do you want?

I proposed 4D, as there are several theorems that there exist complex spaces only with dimensionality of 2^n or infinite, I chose to propose 4D as there are more arguments supporting it, as well as an already existing system defining it, but I didn't restrict higher options. Me and Agnaa were arguing, due to him thinking there might be a 3D complex space if we change the way it is created and I was explaining why this space would contradict itself.
The obligated explanation was a point from the very start, I don't support giving ratings in complete scaling vacuum. This whole thread will only apply if the criteria are met and the criteria themselves are the hints on the possibility that what was discussed in this thread can apply to this INS
If it still seems confusing what the possible hints mean, I will now explain the situation with INS in Tensura, as this was the verse that got me interested in the topic in the first place
So there is a Unique Skill "Infinite Prison" that traps the target in an infinite Imaginary Number Space. This skill was used to seal the literal manifestation of the Type 1 Concept of natural space, who, for context, rips holes in space just by not restraining his presence, and no matter what he did, he couldn't get free himself. This skill is also described as a strange skill that despite being a Unique Skill, its power reaches the Ultimate Level, with the Ultimate level in question is being able to effortlessly destroy entire large space-time continuums just by unleashing haki, so basically 4D.
These things are not enough to warrant a 4D rating for this INS, but the possibility is clear as day and so this thread can just get it to 4D if it passes. This thread is meant for situations like this