singular imaginary infinity

[AntonioLao]

Since by common standard mathematical acceptance, infinity is not a number whether rational or irrational, therefore, if it is not real then it must be imaginary. In fact, it is a pure imaginary number and cannot be defined as a complex number or any of those hypercomplex numbers (e. g. quaternions, vectors, tensors, spinors, twistors, etc.). Fortunately, its virtual reality can be demonstrated using two given rational functions ( a and b) such that their product is equal to their difference: ab = a - bor separately given by a = b/(1-b) and b = a/(1+a), that are expressed as functions of each other.

Consequently, if it is now assumed that a =b and definitely exclusively specifying their domains as the set of rational numbers then equivalently the next logically expressible equation is always true: a²=b²/(1-b²) and rearranged both sides give a²(1-b²)/b²=1 or a²/b²-a²=1. Furthermore, a²=a²/b²-1 whose inverses give their reciprocal equation as 1/a²=1/(a²/b²-1). By multiplying both sides by -1 the equivalent equation 1/a²=-1/(1-a²/b²) is also true. Taking the square root of both sides give 1/a=i/Ö(1-a²/b²). This clearly indicates that the reciprocal of a is equal to the quotient of unit imaginary dividend over a divisor that is topologically conformal to the g factor of special relativity where b is now the constant speed of light and ais the time rate of change of a given arbitrary distance. It’s reciprocal 1/a could then be defined as the distance rate of change of time. If a is exactly zero, implying absolute rest then 1/a is simply infinity which in physics could be used to imply infinite mass or infinite energy analogous to the big bang singularity. Nonetheless, the right-hand side is simply imaginary unity proving that ¥ = i. On the other hand, if as assumed that a =b or exactly where and when an object linear speed is equal to the speed of light (c) then 1/c is equal to a nonsensical singular imaginary infinity.
Obviously, real infinity versus imaginary infinity cannot be distinguished by these mathematical demonstrations. It can be said that there is no difference between real and imaginary infinity. In other words, if a =b = 0 then 0x0=0-0=0.

[Cubola Zaruka]

The term 'imaginary number' is a misnomer. Infinity is neither a real nor an imaginary number - it is a set with an infinite number of members.

1/a=t/((1-((a^2)/(b^2))))^1/2
If a=0, t=infinity, which means that a stationary object will take forever to move to somewhere else (unless that somewhere else moves towards it). The right hand side does not seem to equal unitary infinity as claimed.

[SteveA]

Since by common standard mathematical acceptance, infinity is not a number whether rational or irrational, therefore, if it is not real then it must be imaginary. In fact, it is a pure imaginary number and cannot be defined as a complex number or any of those hypercomplex numbers (e. g. quaternions, vectors, tensors, spinors, twistors, etc.). Fortunately, its virtual reality can be demonstrated using two given rational functions ( a and b) such that their product is equal to their difference: ab = a - bor separately given by a = b/(1-b) and b = a/(1+a), that are expressed as functions of each other.

Consequently, if it is now assumed that a =b and definitely exclusively specifying their domains as the set of rational numbers then equivalently the next logically expressible equation is always true: a²=b²/(1-b²) and rearranged both sides give a²(1-b²)/b²=1 or a²/b²-a²=1. Furthermore, a²=a²/b²-1 whose inverses give their reciprocal equation as 1/a²=1/(a²/b²-1). By multiplying both sides by -1 the equivalent equation 1/a²=-1/(1-a²/b²) is also true. Taking the square root of both sides give 1/a=i/Ö(1-a²/b²). This clearly indicates that the reciprocal of a is equal to the quotient of unit imaginary dividend over a divisor that is topologically conformal to the g factor of special relativity where b is now the constant speed of light and ais the time rate of change of a given arbitrary distance. It’s reciprocal 1/a could then be defined as the distance rate of change of time. If a is exactly zero, implying absolute rest then 1/a is simply infinity which in physics could be used to imply infinite mass or infinite energy analogous to the big bang singularity. Nonetheless, the right-hand side is simply imaginary unity proving that ¥ = i. On the other hand, if as assumed that a =b or exactly where and when an object linear speed is equal to the speed of light (c) then 1/c is equal to a nonsensical singular imaginary infinity.
Obviously, real infinity versus imaginary infinity cannot be distinguished by these mathematical demonstrations. It can be said that there is no difference between real and imaginary infinity. In other words, if a =b = 0 then 0x0=0-0=0.

I didn't read through everything in great detail, but your thoughts here agree with some other things I've noticed.

If we had two dimensions that were perfectly orthogonal, they don't actually end up interacting within a space and the scaling between dimensions in terms of size is arbitrary as they don't share a common metric/ruler/unit.

If we look at how a knowledge of space is constructed, it's via a learning over time hence one dimension of experience in time is processed to construct/derive/recognize knowledge regarding other properties. One dimension should be the largest and it constructs the others as smaller infinities, just like we could take any group of irrational numbers and multiplex their terms/digits etc. into a single irrational number that contains all of them.

If there is more than one infinite term involved in a solution, then the relative magnitude of each of these (which is more of a function of their forms of growth over time relative to each other, so they're all processes of construction) compared to each other must be known otherwise the results can't be determined, but if you can define all the unknowns/dynamics/infinities etc. as a function of a single (potentially unlimited) process, then they're no longer unknowns. The largest physical dimension would appear to necessarily be described in terms of physical memories over time because there wouldn't appear to be anything larger than that to utilize in determining physical properties unless there's simply some innate timeless knowledge that people possess that needs no physical confirmation of (in which case, that might be considered to be an even larger infinite quantity as it has no finite window of events in time by which its bounded, so at least relative to finite quantities of time, it would appear larger ... I'm not so certain on that last part, but that's just an idea).

[SteveA]

Consider Euler's Formula:

e^ix=cos(x)+i*sin(x)

For x=1, we have a radian rotation: e^i=cos(1)+i*sin(1) and this position can also be considered to be the product (cumulative angle of rotation) of a large number of smaller rotations.

We can construct a complex version of e like this:

e(complex)=lim(1+i/n)^n) as n->infinity

But if we rotate/exponentiate this vector over a full cycle of 2pi, do we actually come back to where we started?

If we compute the magnitude/length of this imaginary vector for n, we get:

|(a+ib)^c|=|a+ib|^c=((a^2)+(b^2))^(c/2)

For n=1, we have:

((1^2)+(1^2))^(1/2)=2^(1/2) ~= 1.414 > 1

n=2, ((1^2)+(1/2^2))^(2/2) = 5/4 =1.25 > 1
n=3, ((1^2)+(1/3^2))^(3/2) ~= 1.17 > 1
...
For any n, |(1+i/n)^n|!=1 because:

|(1+i/n)^n|=(1+1/n^2)^n!=1 unless there exists an n for which 1/n^2=0, because a^n!=1 if a!=1 and n is non-zero integer.

Though the magnitude/length of this radian unit shrinks for larger values of n and approaches 1, there is nothing requiring it to ever equal 1 (despite some claims to the contrary).

http://en.wikipedia.org/wiki/Limit_(mathematics)

Suppose ƒ(x) is a real-valued function and c is a real number. The expression:
means that ƒ(x) can be made to be as close to L as desired by making x sufficiently close to c. In that case, we say that "the limit of ƒ of x, as x approaches c, is L". Note that this statement can be true even if . Indeed, the function ƒ(x) need not even be defined at c.

Much of the time it's assumed that a limit represents a numeric identity and not simply a rational boundary to a convergence. Saying that a function approaches an irrational limit is even worse as the limit is not even a known quantity, yet these "numbers" are often treated as a single 1-D value on a number line despite the fact that they're only describable in specific ways as a pair of boundaries, that are implementation specific - we can find different constructions of pi that could be shown to never be able to equal each other, yet it's still generally assumed that pi refers to a single number, placeable on a 1-D number line.

Notice that being unable to compute a number between two irrational limits does not mean that they're equal. If we determine that there exist no integers between 2 and 3, but we can also show that there's a non-zero difference between 2 and 3, then we've simply shown these numbers are adjacent to each other (within a specific numbering system).

What we really have is simply a mechanism by which we can reduce the errors to being infinitesimally significant and ignorable for purposes of physical analogy - notice that someone showing that if we rotate a protractor on a piece of paper over a complete circle we begin at where we started is not actually proving this to infinitesimal precision and if someone instead began at some specific atom and then proved we could return to it, then they're using a finite value for n and only proving it true for finite polygons and not an ideal infinitely subdivisible perimeter (in which case they couldn't even leave the origin in a physically verifiable manner).

So let's say we, as is typical, assume that this limit of convergence represents an identity instead (such as the .999...=1 claims) and we proceed to apply it to computing a Fourier Transform or the Riemann Hypothesis.

As long as the divergences remain infinitesimal, we're doing good enough to satisfy 90% of the mathematicians out there, but consider something like this "identity" in trigonometry:

cos(2pi*x)^2+sin(2pi*x)^2=1

And we could similarly say that:

1=lim(cos(2pi*x)^2+sin(2pi*x)^2) as x->infinity

So let's show what's actually happening:

1=lim((lim((1+i/n)^n) as n->infinity)^(2pi*x)) as x->infinity

But this would generally be recognized, for good reason, as a non-standard representation. Yet when the same computation is rewritten as an (implicity precise) identity, few people seem to complain. It's ironic that the truth of this statement is generally seen as increased by holding it to a higher standard (an identity), whereas the truth of it should be greater when only held to weaker requirements (the limit/boundary of a converging process).

If we have no relationship between x and n in the above nested limits, then the result is not specifically 1 but indeterminant. If we specify x=n^2 then we do approach a finite limit, but it is not 1 and if we specify x<<n then we approach the expected limit of 1.

The fact that physical analogies are used and approximations in the form of limits of convergence, masks the ability for people to see alternate solutions to the same physically indistinguishable systems (traversing cycles of a circle over any finite number of times may leave each cycle only infinitesimally different than the previous one - at a minimum we have a difference in the time at which each cycle occured, but logically, unless it's actually composed of a polygon with a finite number of vertices/roots then there is no guarantee that you're ever repeating even a single cycle - and it's great how logic agrees with physical observations as well - there are neither perfect physical circles, nor does traversing them leave them unaltered - we have the quantum in mathematics/logic/thought as well).

Anyway, we're likely in for yet another rewrite of real numbers (and then likely the claims will once again be that this time they really did leave no logical stone unturned and it's truly "rigorous" ... I'm beginning to wonder what rigor means).

[AntonioLao]

For me, Euler formula gave a very strong implication of the intimate connection between the real and the imaginary. However, it does not explicitly or for that matter implicitly describe the directional property of a physical quantity. See other new thread on real matching imaginary.