MWresearch

Well thanks for that then.

Well I can't say that for sure because when studying hyperbolic functions, their relation to the cos(x) graph was never mentioned, so I don't know if its purposely not showing me that imaginary component of if imaginary components sometimes stop existing when you take the inverse a function.

I don't know, I just want to represent an indefinite series of radians as some function of arccos x. If there's some way to do that with archosh(x) then that's fine.

For arccos(x), outside of its domain on [-1,1], it turns into i*arccosh(x). But, if you took the inverse of accros(x) (no h), those parabola-like structures caused by i*cosh(x) disappear when that function is converted into the function cos(x). I want to know why that I*cosh(x) component suddenly disappears when I flip arccos(x) over the x axis to find the inverse, or if they are still there but just not being displayed?

Say I have f(x) and then I have Σ(f(x))dx (I'm using Σ in place of an integral since there's no character for an integral that fits on one line), is there some formula relating Σ(f(x))dx to f^-1(x)? Sort of like the opposite of the derivative rule for inverse functions?

Yeah still not working for what I'm trying to do, maybe there's no way.

In reference to logically interpereting the law, I want to know if its possible to use checks wherein the amount specified of the check can be substituted with a mathematical expression like e^(pi*I)+e^(ln(12+sin(pi/2))).

I don't know what you mean exactly, but it could lead to the right idea. If you increase by a smaller and smaller number that never increases past pi, then arccos should grow indefinitely large, but it does so logarithmically according to the hyperbolic trig function so it's still not cyclical. the only thing I could think that works is arccos(cos(pi*x))*pi or something like that, but I never see anything like that anywhere.

Yeah...that's more or less what I said within the first post except the exact domain of arcos(x) is [0, pi].

I'm running into a problem where I want to represent a series of radians as A*arccos(Bx+C)+D, like for instance pi/2, 3*pi/4, pi...18*pi/12.. or pi/2, pi, 3pi/2, 2pi, pi/2, pi, 3pi/2....ect. But, because the domain of arccos is limited and then becomes some imaginary hyperbolic trig function outside of the domain 0<x<pi, I'm having trouble figuring out how to do that. Like normally, you could represent a series of cyclical numbers like 1,-1,1,-1 as cos(pi*x). I want to do the same thing with radians using arccos(x) where it just keeps repeating periodically as x grows arbitrarily large

I don't know if I can really explain it on a more fundamental level. In your study of math, I'm sure you've dealt with both discrete mathematics and analogue mathematics, you have to know what I'm talking about on some level. Think of statistics, where originally you had a discrete distribution for the distribution of averages, forming a bell-like shape that had a bit of roughness to it. Then, you found that as the sample size went towards infinity, it became a continuous curve and turned perfectly into the Gaussian bell curve, that's the kind of thing I'm talking about. (-1)^(n) typically only occurs in a discrete series or summation or infinite product. However, if you wanted to describe (-1)^n over all real numbers including decimals and irrational numbers, not just natural integers, you would instead use e^(i*pi*x), where x replaces n. Or, a classic case, the integral. Again with the bell curve, you use to describe the probability resulting from the frequency distribution in a distribution of averages as a discrete summation. Then, when you let the sample size tend towards in infinity and the probability density function became a probability function via a discrete summation, you switched to a continuous, analogue form of an infinite sum when working with continuous data, an integral, called the "error function." It's a very very very very very very clear distinction between discrete and analogue math, I know for a fact you know what I'm trying to say but I don't know the proper terminology for distinguishing between systems of math like that. I want to take that same concept of transforming this discrete n mod k from a discrete system to an analogue system, which I suspect would involve trigonometric functions not limited to Euler's identity. Basically, how do you define n mod k for non-integer values?

Ok, that function for the double factorial does output 5!! as 15, so it works! But there's still something I can't figure out: what is the analogue of "mod k"? For instance the analogue version of (-1)^n is e^(i*pi*x), since the imaginary part is 0 when x is an integer or something similar to x^cos(pi*x)

Well, perhaps you are not understanding the point I am trying to make then. I would say we know with certainty, that, if we follow the pattern of a regular factorial, that we at least can know that 9*6*3 is a correct representation of a triple factorial. However, we don't actually know all in any way shape or form that 8!!! is 8*5*2, because it doesn't actually "end" on a 3. The way I see it, a factorial isn't "stop before the expression gets to 0", its "stop on value of k." From what I can see, the later is quantifiable, where you "stop on what k value you're using" which has already been put into terms of a generalized equation and works for apparent numbers evenly divisible by "k." I don't however, see any mathematical interpolation for "stop wherever the hell it happens to get closest to but not less than or equation to 0" under a single generalized equation, with no special "mod k" condition. In short, if it is just "the definition," then that interpolation was an assumption, which, means it could be the completely wrong way that mutifactorials are actually suppose to be interpolated and non-divisible values with respect to the whole of mathematics. If you don't stop on the value of "k," then it should be the same as not stopping at 1 in a regular factorial, which, would be equivalent to either an incomplete gamma function or some weird fractional argument for the gamma function. Suppose there exists continuous extension for all multifactorials with no special rules like "mod k" and "x>1", just one single formula and this generalized formula works flawlessly for all multifactorials divisible by their respective k values and for non-divisible values it generates an output close to but not exactly the equal to the number generated by the previously assumed classical interpolation. Perhaps for instance, it could turn out that this formula yields 8!!!=72.4463983759257389579 instead of 80, and something like 9!!=947.34735345 instead of 945, but, this equation would still generate 9!!!=162 since 162 is evenly divisible by 3. If that were the case, how would I know which set of standards is actually correct? Since the previous definition of a multifactorial is nothing but an arbitrary assumption, would it be correct to disregard or alter that definition if such an equation existed as described? How would I know which one is right? I mean saying a mathematician is wrong to assume something is a bold statement so if I put the time into investigating this and find something out that challenges that definition, I want to make sure that first, its actually logical to challenge or alter that classical definition or that doing so is acceptable by the mathematical community , and second that its not going to be illogically thrown away just because I don't happen to be a famous, wealthy, dead European male. This area to me is especially grey because, even if I do find such an equation, how would I know that it cannot be secretly altered in order to generate the same output as the classical definition? You know what I really wish is if there was a more scientific method for the exploration and proof for mathematics, kind of like induction. A method where, if if an equation works perfectly for a certain set of known values, then it must work for all values. Like a parabola, if a certain parabola works perfectly to describe 3 points, then shouldn't it be correct to assume it works perfectly for all points on its curve, since only 1 parabola can perfectly fit those 3 points? With only 2 points, I could see a problem, because there is more than one parabola that can fit any given two points, but, not 3. Every time I think about that though, I think about that old issue with prime numbers. I forgot who it was, but someone investigating prime numbers generated some formula that was suppose to describe the frequency or the value of all prime numbers, and it worked perfectly up until like the 43rd prime and then the formula stopped working for what seems to me like no apparent reason. But perhaps that mathematician just made a specific mistake that can be avoided or that circumstance is special.

Overall there doesn't seem to be a lot on multifactorials. Generalized formulas for multifactorials seem to only work for numbers evenly divisible by k, where k is the type of factorial "like double, triple, quadruple ect). But, my concearn is that the non-evenly-divided numbers are treated completely arbitrarily and not even conjectured, just arbitrary selected. For instance, 7!! (double factorial) would count down to 7*5*3*1, whereas 8!! would count as 8*6*4*2. See the difference? 8!! ends on 2, like its suppose to, and its evenly divisible by 2. 7!! however ends on 1, like its not suppose to because its not a mono-factorial. But, is it really "not suppose to" or is there some huge proof I'm missing that I've never see nor heard of before that proves those numbers are suppose to not end on k on the multiplication chain? If not, I would take it upon myself to redefine multifactorials at non-evenly-divisible values.

Counting a set of all reals does, but counting the specific integers 0 and 1 means nearly the opposite; it means specifically picking out the numbers 0 and 1 out of all numbers that exist.

Well , let's say there's a bowling ball dropping in 3D space on some planet that follows the parabola -x^2+3. If a person was standing next to that bowling ball, where would they look in physical reality to see the solution -x^2+3=4? The solution is of course imaginary, but no where can a person physically see the solution +/- i.

Considering you purposely ignored my multiple efforts to describe in detail what the words "interesting" and unique represent as well as not only admitting that they would be harder to interpret for the less socially inclined but that I am planning on changing it, I have much more grounds to think you are 14. Besides, now that I think about it, the project and purpose of the list is rather scientific, especially considering I was brought on just to compute statistics and other data. The project involves creating planets in a space environment and my job is to calculate as many relevant statistics as I can for different planets, orbital radius, acceleration due to gravity at the surface, volume, orbital tilt, likely temperature at its distance from its parent star, orbital period, distance from parent star, average velocity, etc. But moreover, as someone scientific, I am suppose to keep track of specific properties of commonly known substances as to determine what unique physical phenomena would occur as a result of a planet being composed in large of a particular type of substance or that has a specific inherent phenomena so that it can be graphically created if it is interesting enough. For gaseous wolds containing hydrogen, simply refer to Jupiter with its storms and identify its slushy surface under enough compression which eventually turns into metallic hydrogen, hence why metallic hydrogen is on the list. Of course, the planets also get creative. Since obsidian is made from cooled lava flows, a planet with a stable crust of almost pure obsidian would be said to result from a planet in its primordial state after cooling with its primal lava surface being composed almost entirely of silicon dioxide. A planet made of foam at least in its crust, however improbable that seems, would still have specific properties. It would have properties such as a very low density, very deep but localized surface deformations due to asteroid impacts, cavernous, much more surface area than average Earth ground which would allow for very large scale chemical reactions to take place quickly if the foam was to be made out of a particularly reactive substance, the surface may periodically undergo massive collapse in different crustal shelves due to the cavernous nature from the large amounts of erosive or corrosive fluid on its surface to penetrate deep within the crust in numerous locations. A world somehow composed if actinides, which, is improbable though I suppose not impossible if there happens to be multiple supernovas in one location wherein an excessive amount of heavy elements coincidentally gathered. It would have properties such as patches of crust undergoing both supercritical and sub-critical chain reactions, producing large glows almost visible from large distances in space. It's fate would ultimately be to shrink. A radioactive planet with most of its composition being actinides would shrink, assuming its gravity was not powerful enough to trap all the decay particles, and the fundamental composition of the planet would change over time and turn into different substances like technitium from the fission reactions, helium, xenon and samarium from decay and so on. But, actinides are also extremely dense and heavy, so the density of the planet would be considerably larger than Earth which in tun would create enormous amounts of compression which regularly creates massive tectonic events that makes the crust unstable, as predicted to occur with very large rocky planets. So, this list does actually have scientific value to me, in that I want to know, historically and presently, what are common substances and phenomena that occur so that I can compute statistics and properties of things that average people can relate to or understand. If I was a classical physicist I would use this list to see what things I should compute coefficients of friction for and probably sell the data to some company that makes commercial products. If I was a chemist I would want to know what substances are common so that when I'm finished making my new and improved sulfuric acid ultra kill-anything cleaner, I'd know what byproducts a person could easily encounter, and I could go on to make up more scenarios.

I was not asking anything akin to where imaginary numbers fit in mathematics, I was asking where they fit in physical reality.

Except for the small part where you purposely ignored the fact that I set parameters to for the list to prevent every single compound from being mentioned. In addition to that, not only would a list containing every known compound be possible to create, it would be one of the most helpful lists ever created in science which is probably why there are multiple books that already exist with exactly that information, a list of every known element and compound known. Not coincidentally, it's called "Chemistry."

And?

So does every single infinitesimal real value have a corresponding imaginary axis? How would that work in 3D space where you run out of ways to represent orthogonal dimensions?

I'm arguing in favor of the existence of numbers, not against. Or at least, I did not make an effort to say they do not exist, I merely wanted to know where imaginary numbers fit into existence from a mathematician's point of view.

Which does not ignore the other parameters I set. Neutronium is well within the realm of science, as is glass, wood, water, rock, dark matter and so on. You only focus on one parameter at a time, that leads to issues. The term "interesting" as a term I was hoping you'd understand more sociably, which again is not mutually exclusive with science. But as I said, I will have to revise the explanation of the list with the next edit.

I focus on accumulating as many ideas as possible with the goal that someone will use the list to pick specific items from for purposes of deciding what substances to test, commercial applications (which can include science) or artistic applications, or, if it so happens, allowing people to learn about what others know as common. I do not openly ask for people to debate whether items should be on the list. They may do so, but I do not specifically ask them to, I decide based on if I think they fit the parameters. I welcome a variety of ideas, including unusual substances like bile and dark matter I combine ideas with categorization, such as combining "obsidian" and "fulgurite" into a single "glass" category.

According to that logic if I add 0 and 1 = 1 then I add an infinite amount of numbers. But, the sum of all numbers between 0 and 1 is infinity, you are therefore saying infinity=1. There are infinite numbers between 0 and 1, but that doesn't mean we are distinguishing all of them in an infinite set just because we focus on the integers 0 and 1. The act of counting itself as I am sure you know is much more complicated, you are actually doing the opposite of what you describe, specifically seeking out certain numbers of a general infinite set, not necessarily creating another infinite set.