dividing irrational infinity

[AntonioLao]

In a mathematical sense, dividing irrational numbers can be properly classified as constant uniform self-division. That is the idea where and when the outcome(s) of dividing does not alter the integrated identity of whatever being divided. In fact, infinite divisions tend to approach the true irrational identity. This mathematical truth can well be illustrated by the investigations of the theory of infinite series.

The theory classifies the mathematical distinctions among a variety of well develop and research series of constants, for examples: (1) arithmetic series, (2) geometric series, (3) power series, (4) reciprocal power series and many more named after their discoverers. One came to mind is the famous Taylor series applied widely in the physical sciences. However, in this context, only two of the well known irrational constants namely p and e will be mentioned here. The former is the ratio of the circumference to the diameter of any arbitrary circle. The latter is the base of natural logarithms. Without proof, p is equal to (4)-(4/3)+(4/5)-(4/7)+(4/9)-(4/11)+(4/13)-(4/15)+(4/17)-(4/19)+…to infinity. Clearly, this is considered as an alternating series (plus and minus addends) involving reciprocals of unity power of positive odd integers. On the other hand, e is equal to the infinite sum of reciprocals of whole numbers factorials: (1/0!)+(1/1!)+(1/2!)+(1/3!)+(1/4!)+(1/5!)+(1/6!)+(1/7!)+(1/8!)+…to infinity. For p and e to take on their respective true exact values, the divisions must be carried all the way to additive and subtractive terms at infinity.

[SteveA]

Thank you for your post. If you haven't guessed already, I enjoy this subject

I'm at work and can't post much now, but when you say "true exact values" do you mean that the upper and lower limits of these two series become equal to each other and if so, what is the definition of equality that you're using?

[AntonioLao]

Pi and e are two irrational constants used abundantly in advanced physics, yet both are derived from infinite sums of rationals. The open bounds for pi is 3 and 4 while its true value is closer to 3 after all the infinite sums. The lower bound of e is 2 while it approaches the integer value 3 but never actually take the exact value of 3. So between 2 and 4 we can find pi and e.

[SteveA]

Pi and e are two irrational constants used abundantly in advanced physics, yet both are derived from infinite sums of rationals. The open bounds for pi is 3 and 4 while its true value is closer to 3 after all the infinite sums. The lower bound of e is 2 while it approaches the integer value 3 but never actually take the exact value of 3. So between 2 and 4 we can find pi and e.

I'm going to play a bit ignorant here just to emphasize the potential problem.

From the above description, I could assume that if pi and e lie between 2 and 4 then pi may be equal to the rational number 31/10 and e may be equal to the rational number 5/2. Is there a way to show that I can't select a rational number that pi and e are equal to?

[AntonioLao]

I just realized in a world of whole number roundings and approximations, rounding to the nearest whole number both pi and e equal 3.

[SteveA]

I just realized in a world of whole number roundings and approximations, rounding to the nearest whole number both pi and e equal 3.

Actually, if we measured pi as the distance along the perimeter of a circular form embedded within a 2 dimensional space we could either come up with a precise value of 3 or 4 depending upon whether or not the dimensions were orthogonal or described a triangular lattice.

For example, if we computed the area of a circle using the percent values for which x^2+y^2<r^2, then the perimeter of this circle would have a value of 4 if the sampled points were uniformly and independently distributed in the x and y dimensions because the perimeter could always be described by orthogonal unit steps in one of 4 directions.

But even if we considered pi to be an infinite summation, the upper and lower bounds never equal each other:

Consider the upper and lower bound at the second term to be the pair {4,4-4/3} the distance between these is 1/3.

After adding the next term, our bounds on pi are {4-4/3+4/5,4-4/3} and the difference between these terms if 1/5.

So we have a decaying term in the form 1/(2n+1), but does this value ever equal the integer 0? No, it ends up equally an irrational 0 or infinitesimal and these aren't identities IMO. If we declare that 1/(2n+1) can be less than some e>0, such that:

0<=1/(2n+1)<e

Then does this similarly state that double this value is also less than e?

Is 2/(2n+1)<e, if 1/(2n+1)<e?

For what range of values m, would m/(2n+1)<e be true? Obviously the result is only that it's true for finite values of m, but if we took a Fourier transform and computed:

?=lim(sin(2pi*k)) as k->1,2,3...

The k approaches infinity, yet the precise of pi is only determined to within finite precision. (And we can find things like the Riemann Hypothesis being indeterminantly stated or the Navier-Stokes equations being ill-defined etc.)

Most mathematicians know what the equations are intended to describe, but the equations themselves are not in the form that's intended and various discontinuities in the assumptions regarding the mathematics of the system occurs with indeterminant results.

There are other ways we could define pi also, but we can similarly show that these different forms can in many cases not ever create an identical structure with infinitesimal precision, hence various definitions of pi can be describing different structures that are only accurate to within finite physically and geometrically verifiable precision and not precise logical identities. For example, if one structure creates a growth in the divisor that is relative prime to any other possible divisor constructed by another construction, then the two could never create identical objects.

Consider this question, if a>0 and b>0 can ab=0? So if we can show that a(0)>0 and b>0, then we can prove that all a(t)>0 by iterating the inequality a(t)=a(t-1)*b>0, if a(0)>0 and b>0. So the summation 1/2+1/4+1/8+...<1

1/2<1,
1/2+1/4<1,
1/2+1/4+...+1/2^n<1

Now we subtract 1/2^n from both sides (ok, that's a slight joke)

1/2+1/4+...<1-1/2^n<1

But the end result is still true. Just ignore the 1-1/2^n term.

Notice that instead:

1/2+1/2=1
1/2+1/4+1/4=1
1/2+1/4+1/8+1/8=1
1/2+1/4+...1/2^n+1/2^n=1
1/2+1/4+...1/2^n=1-1/2^n<1

And we can prove that 1/2^n>0 as n->infinity using the recursive identity that a>0, b>0 implies ab>0.

Anyway, there have been many attempts over the years to rectify this problem but the paradoxes and indeterminacies remain (I've gone through most every attempt to prove .999...=1 and they all have flaws and those flaws do not even need to remain infinitesimal if these assumed identities are reused in other computations that can reamplify the infinitesimal differences).

For example, with .999... we could write this as 1-10^-n and if we compute:

.368~=1/e=lim((1-10^n)^(10^n)) as n->infinity

Whereas if the term 1-10^n was actually identical to 1 as n->infinity, then we could substitute it with 1, but that results in a different solution:

1=lim(1^(10^n)) as n->infinity

Hence the 1/10^n term never equals 0 as n->infinity.

Cantor's Diagonal argument is misguided also and we can show that he can't even construct all rational numbers using his diagonal, much less irrational ones.

We can also similarly "prove" that every irrational is equal to a rational using the same argument that there exists a rational number within e of every irrational number.

There have been other rewrites of the real numbers and every claim is that, yes, it's finally acknowledged that the previous revision did contain problems and was not "rigorous", and then there's simply another layer of obfuscation layered over the problem.

Now despite all that, I won't claim there are plenty of uses for real numbers and there are differences between rational numbers and irrational numbers, but I think there's a lot being overlooked because of the common assumption that this area of mathematics is well understood and there's not much left to found in it (similar to claims made other areas of science - until later on it's realized that those insignificant digits end up being not so insignificant and infinitesimal mathematical differences are logical inequalities, and other areas of mathematics open up).

[AntonioLao]

the dimensions were orthogonal or described a triangular lattice.

In 3D (suppressing the time dimension) space-time would these be similar to the tessellations of cubes or tetrahedrons? I'm trying to associate the cubic topology to bosons and the tetrahedron topology to both quarks and leptons.

[SteveA]

In 3D (suppressing the time dimension) space-time would these be similar to the tessellations of cubes or tetrahedrons? I'm trying to associate the cubic topology to bosons and the tetrahedron topology to both quarks and leptons.

I think the cubic form is more of a mental construct using independent properties - for example, a year and a car model are treated as independent and if there could be, for example, 70 years over which cars were made and there have been 1000 models of cars then this gives a volume of 70,000 year models, but in reality this is not the actual space.

So the space of orthogonal dimensions grows multiplicatively (I can't say exponentially because the "width" of each dimension can be different), whereas the entirity of time should grow linearly.

If we look at a motion within a space, we're always moving constrained by the space and always moving toward something - you can't move orthogonal to everything within a space, so if motion within the universe is always toward something, then we should have the point, line, triangle, tetrahedron etc. progression of volumes (the mental form of possibilities arising from assumed independent motion in different dimensions would be a hypercube growth). The volumes for that triangular/tetrahedral growth are described by polygonal numbers.

So we have potential mismatches in the volumes described these various types of spacial construction and this places restrictions on the possible volumes of spaces that can describe both these precisely (and then we can have a 1 to 1 mapping of points between these spaces).

If there is fundamentally no space, but instead there are objects and subjection motion between them, then I think the triangular form is a more appropriate form, but if we additionally consider that it's memory that determines where one is currently at, then we additionally have another form of spacial growth (there are still similarities to the triangular form), but it's factorial, permutational, or combinatorial growth and this describes more closely both statistical and objective perspectives but neither subjective motion within space, nor a space describing possible combinations of orthogonal properties.

Consider that if we have an object that is objectively seen, it can be described by various quantities of constituent components - for example, a car is comprised of various quantities of wheels, panes of window, metal and a steering wheel and headlights etc.

If two people see the "same" car, they're not actually seeing it identical in all respects but are simply seeing quantities of previously established common objects for communication (at the lowest level we have at least a single property shared by everything in the universe and that property encapsulates the concept of the universe, though this alone isn't sufficient for communication over time).

So anyway, if we have a space of observers seeing a common object, then the possible manners in which they can see it are described by the quantity of permutations of the elements of the object, if each element is fundamentally viewed serially (so this is similar to an observer seeing an object via. a memory of seeing the, for example, 3 elements of it, A, then B and then C and the observer is now at the "width 3" object that contains a single unit of A, B and C.

Notice that no two experiences should be precisely identical, otherwise they would have already been the same experience and not two experiences.

So every such object, as a composite is unique, even if it may be factored or subdivided in some manner to describe it as components.

This means that if an object is seen as described by 4 sub-objects or properties, that each of those subproperties must be unique as well and so if we described a viewed of each of these over time, someone might see ABCD to describe the composite object, but another might see BADC instead and this gives us 4!=4*3*2*1 possible observational points in which the composite object can be seen and the object encompasses 4 units of time, so it should be 4 planck units wide.

So we have a likely linear growth for a fundamental time, various polyagonal growths for motion within various dimensional space, multiplicative forms for mental spaces of possibilities and factorial forms describing the growth of possible observers in an objective space. (In some ways the factorial growth is just a special case of the multiplicative space of mental possibilities).

If there's a single quantity of space that these could all exist with 1 to 1 mappings between all of them, then we'd need to find solutions for these that are all equal and it would appear we may have to limit the length of memory for a motion, otherwise the space of possible motions continues to grow and can't be constrained within a fixed space, though there may be ways that allow a growth via. a selection of 1 of n possibilities over time, in order to have the cubic/orthogonal space of these possibilities construct a series of volumes that continually has a 1 to 1 mapping to the other spaces.

Anyway, just some ideas ...

As a simple example, 3! is a nice number for this as a single "state" could describe many different spaces from which it's being seen as 3! or an object composed of 3 elements being viewable from 6 direction in space with these 3 objects rotating or orbiting around objectively, yet subjectively being seen as variously reorder timelines in which these 3 things are encountered, and we could map it a 2-D polygonal surface as a triangle with 3 units length on each side or a space of 2 orthogonal properties, one binary and the other ternary (? 1 of 3 possible states).

4! could extend upon all these by allowing yet another dimension/element to exist and it could be seen by more observers.

There are correlations here with optimal packing of spheres or maximal reliability of communication within some dimensional space (it's "Leech Space" for 4 dimensions.

[AntonioLao]

I recall a theorem in plane geometry that 2 successive orthogonal reflection is equivalent to twice the vertical angles of rotation.