Imaginary Axis

[Furudo_Erika]

Do imaginary axes qualify for higher dimensions?

Like if you make a chart with a real axis and an imaginary one, you get a "complex plane", which.

Also, in mathematical science, complex numbers ("C") are considered just as "real" as the real numbers.

Though the set of complex numbers encompasses the sets of both the reals and imaginaries.

Also:
"Like other complex number variables, complex time is two-dimensional, comprising one real time dimension and one imaginary time dimension, changing time from a real number line into a complex plane. Introducing it into Minkowski spacetime allows a generalization of Kaluza–Klein theory"

 

[Furudo_Erika]

Bump

 

[Furudo_Erika]

And also, the set of complex numbers, "C", is regarded as equal to R^2; C = R^2

"William Rowan Hamilton introduced the approach to define the set C of complex numbers as the set R^2
of ordered pairs (a, b) of real numbers"

So then, would C qualify for a higher tier mathematically?; if R is Low 2-C, would C be Low 1-C relative to it?

 

[Furudo_Erika]

In a different perspective, it could seem that every complex number has an infinite number of "snapshots" of every iteration of a real or imaginary number.

In (a,b):
If "a" is the real part, and "b" is the imaginary, then:

(0,0), (0,1), (0,2), (0,3), (0,4), (0,5), (0,6), (0,7)...
(0,0), (1,0), (2,0), (3,0), (4,0), (5,0), (6,0), (7,0)...
(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (1,7), (1,8)...
(1,1), (2,1), (3,1), (4,1), (5,1), (6,1), (7,1), (8,1)...

A bit weird of an example, but this shows for every real or imaginary part, there's an infinite number of the other part.

Which is similar to an (x,y) plane.
Where for every "x" coordinate, there's an infinite number of possible points "y" could be at, and vice versa.

 

[Guardian_Doge]

Yes they qualify, Imaginary time as an example treats the real flow of time as a spatial dimension since it’s simply another vector (direction). Geometric dimensions are modeled off R the same way for all intents and purposes, so a 3 dimensional space, plus two dimensional time would strictly result in a Low 1-C structure.

 

[Furudo_Erika]

Ah, so what about hypercomplex numbers?
Such as quaternions (H), which have one real, and three imaginary axes.

Would it be R^4 then?

Also H = C^2
Which C = R^2 already

 

[Furudo_Erika]

Bump

 

[Guardian_Doge]

Apparently yes, quaternions are used to describe the mathematics of maxwells equations which describe the fifth dimension (fourth dimensional space) of electromagnetism in some brane cosmology so they would be R^4.