Our answer is YES. Irrationals, since the ancient Greeks, have had a "murky" reputation. We cannot measure physically any irrational, as one would require infinite precision, and time. One would soon exhaust all the atoms in the universe, and still not be able to count one irrational.
The set of all irrationals does not even have a name, because there seems to be no test that could indicate if a member belongs to the set or not. All we seem to know is it is not a rational number -- but what is it?
The situation is clarified in our book Quickest Calculus, available at lowest price in paper, for class use. See https://www.amazon.com/dp/B0BHMPMMTY/
There, Instead of going into complicated values of elliptic curves, and infinite irrationals, algebra allows us to talk about "x".
No approximating rational numbers need to be used, nor Hurwitz Theorem.
Thus, one can "tame" irrationals by algebra, with 0 (zero) error. For example, we know the value of pi. It is 2×arcsin(1) exactly, and we can calculate it using Hurwitz Theorem, approximately.
GENERALIZATION: Any irrational number is some function f(x), where x belongs to the sets Z, or Q -- well-defined, isolated, and surrounded by a region of "nothingness". The set of all such numbers we call "E", for Exact. It is an infinite set.
What is your qualified opinion?
MS: Microprocessors do not use mathematical real-numbers. All operations are done using addition and natural numbers. Coprocessors also do not calculate in mathematical real-numbers, only on natural numbers.
If x×x=2, then x=√2 -- well-defined and exact. And √2 = 2/√2, which can expanded as a continued fraction. We solved in algebra the irrational number √2.
Expressing pi in different language or sqrt(2) as xx=2 does not change anything.
It is exact referal even though you cannot numerically calclate.
JW: Numerically you can calculate, using "x". There is no mystery, no imprecision. All irrational numbers are accessible that way, pi is 2×arcsin(1), absolutely accurate -- numerically and in algebra. And i^i is also exact, it is equal to e^(-pi/2), and both are 0.20787957635076...
You can also numerically approximate, using Hurwitz Theorem as 207879/10000000. As well as desired. But only rationals are measurable, and can be produced as a fraction.
JW and all: For 2 days, starting today, Thursday, October 20, 2022, 12:00 AM PDT, the electronic version of Quickest Calculus will be at $0.00 on Amazon. The URL in the US is https://www.amazon.com/dp/B0BJ7JM38X
One doesn't need a Kindle device. The file will self-install for free in any computer or cellphone.
JW: The speed of light in vacuum is 1. No units are needed.
But, in American elementary school, the teachers hammer in the need to join a scale with a number. Nothing could be more subversive to a free thought.
Different numbers can mean the same value, and different values can have different units (e.g., meter, inch) as scales.
Units have NOTHING to do with values, and NOTHING to do with a value 's representation. The three are independent. There is no "unit theory" of numbers.
We can see the same in semiotics. Symbol, meaning, and referent are independent. This was explained very-well by Frege, and there is no need to repeat.
Another example. There is also no magnetic field (B), and no electric field (E), it depends on the movements relative to the electron. If you are stationary to the electron, there is no magnetic field. Again, thinking otherwise is an elementary school misconception.
The equations have no magnetic source. But magnetism can be accessed using algebra. And we can use it in the production of electric energy and in motion engines.
Three coils, spatially separated by 120 degrees, and fed electric fields separated by 120 degrees in phase, produce a single, rotating, magnetic field. The source of that magnetic field is found using algebra, with no moving parts. This is the principle why AC motors work, with no commutation (and sparks) on the rotor collectors.
The rotor collectors just feed a constant magnetic field in the rotor, which is rotated by the so-called stator. Mechanical motion of the rotor results.
One can also reverse. The rotor can have three coils fed a triphasic current rotated electrically as a transformer, or rotated mechanically as a generator with a DC coil, inducing a high current on the stator coils, for efficient energy triphasic generation/transformation. With no magnetic source, but calculated using algebra.
The number e -pi is unknown to be irrational, or transcendental. A transcendental number is a (possibly complex) number that is not the root of any integer polynomial, meaning that it is not an algebraic number of any degree. Every transcendental number must also be irrational, since a rational number is, by definition, an algebraic number of degree one. We conclude that neither number is transcendental.
The number pi is equal to 2×arcsin(1). This expression is well-defined and the function maps univocally 1, a member of the set N, thus it is has 3 quantum properties [c.f. "Quickest Calculus "] and, therefore, is exact.
The number e is equal to: 1/(((-1)^(√-1))^(1/(pi))), where pi is exact (as above) and 1 is exact. Thus, it has 3 quantum properties and is exact.
The subtraction e - pi is exact, and has 3 quantum properties [c.f. "Quickest Calculus "]. It can be approximated as a rational number, following Hurwitz Theorem.
For example, with 5 digits is -42331/100000.
Actually, the error is 0 with 5 digits, in modular arithmetic, because only rational numbers up to 5 digits can be physically produced [c.f. "Quickest Calculus "].
In discussion, we hope to unlock secrets of other numbers that are considered transcendental, or irrational, such as 𝜋 and √2, and allow them to be represented approximately as desired in the set of rational numbers, using the well-known Hurwitz theorem.
The only numbers one can physically measure are rational numbers. We can use modular arithmetic exactly in complete calculations, as a field, allowing "click-mathematics", like Lego. This is well known to be used in the Advanced Encryption Standard (AES) with 𝐺𝐹 (2^8), protecting numbers, text, voice images, or any file.
In addition to irrational numbers, that can be absolutely accurate in algebraic calculations, we expect to reach closed form solutions in quantum computing, making actual computing follow integer modular arithmetic.
Thus, trivially done and with no numeric computation.
So, quantum superposition can work in algebra, as it already does today.
The solution for all the unit roots of i, is a set of equations -- all simultaneously valid, even though the roots span the complex circumference.
I dont know what you mean by solved, I would use the word described.
ie.
xx=2
describes the irrational x, unique if x>0. Perhaps one could eventually find an equation in algebra to describe (sqrt(2)+sqrt(3))
Ive decided agaainst arguing with you about units. For me the word each is simply
a descriptive word in English and not any kind of number.
Yes, I know that xx+1=0 will be solved by x=i
How about (Z-x)(Z-x)=0 does that give just Z=x?
No, it gives Z=x+ey where ee=0. That is a new number for you
Ring of dual numbers.
However cannot imagine such stuff aid computing yet.
Look at my RG page for more completness.
JW: No one is questioning "No, it gives Z=x+ey, where ee=0". I know you had difficulties to imagine such stuff aiding computing. You do not have to imagine anymore. Free PDF book is at Book FREE Newest version -- Quickest Calculus: Self-Study & Class...
Mathematics HAS changed. Discontinuous functions can now be differentiated, and that changes a lot. For example, the Schrödinger equation. The Einstein field equation. The Navier-Stokes equations. Etc.
You can compute with these numbers using 2 by 2 matrices,
as i may be
0 1
-1 0
instead of a single number. Then i and e have transposes.
However what I see in that book does not match what I expected,
goes off in different direction. An algebraic rather than numerical calculus
would hinge on
f(x+eh) = f(x) +eh Df(x)
easiest shown by truncating the Taylor expansion. Entails the Lagrange
rather than limit definition of derivative.
Even though polynomial equations may define numbers as sqrt(2), sqrt(3),...these
are quite different from those numbers as i,e and the unit of hiperbolic tt=1,
labeled perhaps hiperreal,
and different also from generated by equations as exp(ix)=-1 (trascendental )to define pi.
Im not sure if they can all be labeled algebraic numbers.
So what kind of number are you dealing with? Only real, irrational and not trascendent? Or hiperreal? Or real trascendent?
A further distinction would be field or ring number.
You see this field is not as simple as just displaying an equation or so.
By looking into the differentiation of discontinuous functions, we change math and physics with new solutions.
We expand the validity of the Schrödinger equation, GR and QM meet, and quantum computing is done algebraically, not just circuit-based or by an annealing metaphor.
Ha, ha, you wont even use R?(because you can not yet stomach irrationals)
(Throw away all calculus and physics books)
You what with Schrodinger?
Not to mention not standing up for other hiperreal, once you indeed adopted i?
JW: One can digest irrationals in the book, by noting its 3 discrete (quantum) properties -- and nothing uncertain is left. No hyper-reals were adopted either.
Take your time -- math and physics HAVE changed. I published the Schrödinger conclusion way back in 1982, starting "click mathematics". See
Article Solution of the Schrödinger equation for bound states in closed form
The "black box" of quantum is more like Bohr imagined, less like the mythology of the Copenhagen interpretation.
Id rather not mix irrational numbers with Schodinger.
Your paper seems legit, but the art of finding bound states is far from new.
I admit no hiperreal other than i seems to be in applied math yet.
Ill just point out that Z is R+iR , if you dont stomach R, you dont stomach Z either and you dont mess around with complex functions in quantum mechanics. My guess is you wrote that QM article in Brazil, before you got into strange number issues, perhaps as strange as mine.
I have about 40 legit physics articles.
What I can do is use e in liu of i, with i=e-e(T)
e(T) the transpose of e, using also e(T)e(T) =0 , ee=0
and e e(T)+ e(T) e =1
From xx=-1 there are really two i: i or -i, or the matrix with its transpose.
Dito with e.
The black box is reserved for Black Body radiation that Planck studied.
In popular terms it can be used for most anything.
JW: And in 1983 we published the study "Scaling Laws for Rydberg Atoms in Magnetic Fields" in the Physical Review Letters 50(5):324-327, using the same method as in the 1982 paper -- and still stands.
Now, we published the book "Quickest Calculus" revealing in a free PDF the "secret sauce" -- use the set Q, abandon R and C. New results in math and physics, not only in QM, will follow.
Discontinuous functions can be differentiated, and we present a theory. It goes against Courant and Apostol.
Dear Ed,
You wrote:
"Discontinuous functions can be differentiated, and we present a theory."
OK, just to catch your main idea, can you please find a derivative for the function f(x)=x XOR c defined on integers? Here the constant c is an integer, x runs over integers, XOR is bit-by-bit 'exclusive OR" operation on integers represented by their base-2 expansions.
Thanks.
A discontinuity in electron density would just give rise to diffusion, so in spite of obeying Schrodinger, I dont know how realistic this may be. Boundary conditions routinely can give a first derivative discontinuity.
Problems with magnetic field would lead to complex wave functions.
I dont see any advantage to only using Q. Surely you could get the same result if you restore Q to R and or C?
It might be a good take in math, but I would doubt in Physics.
JW: The future is open to us, as we see that a new unbounded class of solutions exist -- they were always there, just hidden.
Who would think that the magnetic field has the same unit as the electric field, and both are part of a better description? SR revealed that to us, and it helps research -- with new vistas under this light.
Mathematical real-numbers, and complex numbers that are also mathematically continuous, are not found in nature. Why invent them? This will lead to contradictions.
It is worthwhile to see this only in mathematics, to avoid the prejudices of physics. Physics is not reality, it is a story about reality, was the opinion of Bohr. QM is not reality, it is a fitting story.
Mathematics is closer to reality, surprisingly. We just have to base it on nature, not on how it defines nature.
In the theory of special functions unbounded ones often exist, but are discarded for physical reasons , so watch out.
When you change frames in SR you encounter corrections in both fields
Judge for yourself if they have the same units by writting force for each field
electric
F=qE
Magnetic
F=q v cross B
I dont think they have. For example for E it would be Newtons/coulomb
Work it out for B
Mathematics can be close or far from reality. Depends how you use it, what you use it for.
Your perhas correct that at a really microscopic level things are jumpy and not continuous. However no one is really interested in the detail of all the jumpy stuff, so they revert to continuous as approximation. Hence R and C, as if it were Classical Physics.
There is no real expectation of finding a coincidence between physical jumpiness and the mathematical one of using Q
Anyhow, that is my defense of "conventional physics"
JW: Maybe you are confusing the conversations. I did not say unbounded in value, I meant unbounded in number-- that is, infinitely many solutions.
Mathematics cannot be continuous "as an approximation". It is discrete.
In physics, E and B have the same units in SI CGS. You have to add c. It is well-known.
Normally my understanding is that for continuous events you are implicitly using real numbers to avoid gaps in the real number axis. This may be ideal
rather than practical.
In science carreers it is usual to give an explanation about measurement,
and approximation and models. You have your model for reality, not reality itself. So after such explanation you have to say the hell with the distinction say between rational or irrational numbers, or philosophical distinction beween continuous and discrete. It depends on scale and the problem, in part by your choice of description.
In classical physics it would be strange to have discretness in the orbit of some object. You would have problems to define velocity.
Thus a histogram can gradually look like a continuous distribution as you gather more data.
Units for E are newtons/coulomb
Units for B are weber/square meter or Tesla
you think they are the same?
In cgs (not mks) you do add a c to the Lorence force formula.
Well, even if they had the same units, it is clear that there is a large
difference in physical influence, say B involves speed or current of charge and not so E.
So it has more sense to use different units.
JW: Nature or reality cannot be defined by mathematics or any science, because we do not KNOW their limits or parts.
In classical physics or mathematics it is strange to mandate what does not exist: continuity. Discretness must be allowed in the orbits of objects, and values of numbers.
One has no problems to define velocity with the set Q, being the ratio of two rationals, the denominator is not 0 by construction. Gravity is not a force. The third law of Newton is not mandated by a contrarian demon. SR demands that E and B have the same units, and it is called gauss.
These are, in the 21st century, other models, with better predictions and less contradictions.
Ed
I just dont have the same style of taste or criteria in either physics or math.
than you. It is useless.
About knowledge of fields E;B Ive worked long enough with them in or out of labs to at least develop a good feeling on what each of them does.
One does know certain stuff.
But its good that finally you seem ready to dissociate math and science.
In normal conversation I will most ofter speak as a scientist than mathematician, dont try to spook me with irrational numbers please.
JW: Unless you finally accept SR, you won't accept that E and B are two aspects of the same thing. Studying SR will help make sense of EM fields, but then you will have to add QM to abandon all ideas of actions at a distance. You seem to be just 2 paradigm shifts away. You can bridge in time.
About irrationals, one can not add or subtract a rational and get a rational, so it doesn't make sense to use mathematical real-numbers-- addition/ subtraction cannot be counted in mathematical real-numbers.
Sorry Ed
I happen to know the implications of SR as far as em theory rather well.
When you change frames you can have a bit less of one or more of the other, for example. However that does not mean that they are "the same thing"
So far no need of QM, for that you have to apply your brain to phase shifting effects like the Aharonov Bohm experiment, or effects of the scalar gravitational potential on particles.
I wont even comment you spiel on numbers.
JW: You don't really use SR if it does not mean that E and B are the "same thing". The laser, diamagnetism, and other EM effects, depend on QM, and cannot be represented using Maxwell equations. This is not arguable using reason -- they are an experimental fact.
Irrationals can be used in algebra, but cannot be used numerically. Many evidences, as well. And, new results become possible.
JW: There are still many physicists who find it easier to think in terms of Maxwell's equations than in terms of the curl and divergence of a four-tensor.
But, as Feynman states, physicists should always be willing to keep several different formulations of the same physical laws in mind as there is no rule that tells us which form is the best when the necessity of generalizations arises due to more precise experiments or some other reason breaking the old laws.
But, the SR and QM are exceptions. There is also a clear need for the Lagrangian or Hamiltonian formulations of mechanics, because essentially they DO go beyond (contrary to weakpedia) to Newton's laws, and have on-ramps to SR and QM. It is also much easier to find the Schrödinger equation or Feynman's path integrals if you know Hamiltonian and Lagrangian mechanics.
JW: In algebra we can add and subtract two irrationals and get a rational, as in 1 = 3-2 = (√3 - √2).(√3 + √2). This can be valid for any x, in √x. An irrational raised to the power of an irrational, can also be a rational.
The "golden rule" is just: do not use irrationals, numerically. Algebra deals correctly with numerically infinite values, as √2 - √2 = 0. If one tries that numerically, not even 100 significant digits will give an absolutely correct result.
Ed
Yes I know
If you like Ill recommend you reading the adaptation of em to superconductivity, explained in a chapter of Reitz Milford where the London equation is explained. General theory is often challenged by special topics.
Just dont bug me with irrationals any longer please.
"More precisely, the set of computable numbers is countably-infinite,like the integers, while the set of real numbers is continuously-infinite, like the mathematical points on a line. Consequently,real numbers are uncomputable with probability one. Or,equivalently, the set of computable numbers has measure zero among the set of all real numbers." by Nicolas Gisin, in “Indeterminism in Physics, Classical Chaos and Bohmian Mechanics: Are Real Numbers Really Real?.” Springer Nature, Erkenn 86, 1469–1481. https://doi.org/10.1007/s10670-019-00165-8, 2021.
Irrational numbers follow the same physical considerations, which invalidates Cantor on MRN.
This is now a RG preprint, with more mathematical details on calculus by our book "Quickest Calculus", at
https://www.researchgate.net/publication/366124503/
In 1874 Cantor published the proof that the set of all algebraic numbers (including the set – of all rational numbers) is denumerable. His major achievement consists in having shown that the set R of all mathematical real numbers (MRN) is uncountable, i.e. that any bijection R to Z — is impossible.
Cantor's proof relies on the existence of MRN where continuity exists. QM shows in physics that continuity does not exist.
My preprint shows that continuity can exist mathematically, although it leads to non-physical results. Thus, MRN need to be abandoned in calculus.
Either way is logically correct, but only the non-existence case of MRN leads to solve physical results.
The way one learns in mathematics does not lead to solve physical results, only to mathematical results. One can't find all solutions to the Schrödinger equation, nor to the Navier-Stokes equations. The book by Apostol gives a good clue: abandon irrationals in calculus. One can consider them algebraically but not numerically -- they are uncomputable.
The preprint shows an unbounded class of valid solutions to the Schrödinger equation, using the derivative of discontinuous functions. Quantum computing needs that to calculate prime numbers, etc.
When I was at the MPQ, we were surrounded by "proofs" of the quantum nature of matter and radiation. No experiments were ever continuous.
The question on such experiments was not the units. We can define c=1 (no units) and all works.
So, the "proof" of the quantum nature is all around us, no matter how we measure it. In this reason, even monetary exchange is quantum, with the peppercorn as the least legal unit.
The irrationals and the mathematical real-numbers are, then, fictions. Algebra, however, can deal with unknow quantities, as if they are real. It can use "mathematical paint" with no atoms and achieve "continuity". I cite this in my recent preprint in RG.
Back to the real world, with "physical paint" that has atoms, modeled by integer numbers, modular arithmetic has no error. We use it in cryptography, and no one can use mathematical real-numbers or irrationals in cryptography -- no notion of "continuity" works in pure number theory, in cryptography.
The question one still has is about time. Is time quantum? As SR teaches, one has to consider spacetime. Is spacetime quantum for a comoving observer? This question seems to be still open, experimentally.
However, are we projecting our bias? Are we conditioned by irrationals and mathematical real-numbers to consider an infinite expression for 1/3? Any infinity is non-physical in value, even spacetime.
One has no experimental indication of continuity. The Brown Movement by Einstein and Perrin was, by itself, conclusive on the atomic nature of matter, and the atomic bomb was just a consequence.
A private opinion says, "by logic the first axioms namely the definitions for space and time should "connect" space and time to spacetime." And they do, by setting c=1.
But, for example, in those units, E=m. So, time and energy are equivalent in value, but not equal -- E is absolute, but m is relative.
This is explained by considering a number a 1:1 mapping between a symbol and a value -- a semiotic object.
Number is not a free invention of the human mind, as Dedekind said in 1888! The value of pi is, physically, a rational. That is all we can measure, and things can measure and produce. Others, such as AGC in RG, and Gichin in Switzerland, say that, too.
The mathematical value of pi does not consider atoms, it uses a "mathematical paint" that has "continuity" and no atoms. It is not real, and relegate irrationals to a mathematical fiction in the real world, with atoms.
We know that m changes with the proximity of masses. OTOH, E doesn't. Numbers must represent that, and can, as a semiotic object.
Confirming this question, support comes from various sources.
According to AGC, in an RG group, irrational numbers cannot be physically measured. According to Vladimir Sergeevich Anashin, in private communication, investigating the derivative of XOR, 2-adic numbers are not continuous. This is important to computer programs, that must use 2-adic numbers.
Mathematical real-numbers cannot be computed, according to Nicolas Gisin, so they would not be continuous either.
As my RG preprint shows, mathematical real-numbers is a poor fit in arithmetic and calculus, and should be restricted to algebra, as well as irrational numbers.
This is also put well, IMO, by P. Nahin, in the book "When Least is Best" cited in my RG preprint.
Thanks for the private messages. The most powerful mathematics is the one discovered, not the one invented. Solipsistic ideas have no resonance.
That is why physics can act with disruption, and mathematics does not need experimentation.
Prime numbers will evidence that, and irrationals are valid in algebra. Pi (+ kxpi) is equal to 2×arcsin(1) (+ kxpi), exactly.
Irrational numbers are uncomputable with probability one. In that sense, numerical, they do not belong to nature.
But algebra (this question) can deal with irrational numbers.
Algebra deals with unknowns and indeterminates, exactly.
There is no “personal” sense of algebra. It just is a combination of arithmetic operations.There is no “algebra in my sense” -- there is only one sense, the one mathematical sense that has made sense physically, for ages. I do not feel free to change it, and did not.
But we can reveal new facets of it. In that, we have already revealed several exact algebraic expressions for irrational numbers. Of course, the task is not even enumerable, but it is worth compiling, for the wary traveler. Any suggestions are welcome.
The comments above show how following the RG ToS is necessary for a scientific conversation.
But the legal responsibility of enforcing the RG ToS falls only on RG, per EU law.
Meawhile, it is suggested to read the question text, with the latest references. Also, private messaging is available.
irreducibility versus zeros being rational or irrational both lead to some way to solve equation .Normally , One of the two said options must lead to non-solvability of equation for when a method of general solution fails, the method of particular solution evolves.
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