This question discusses the YES answer. We don't need the √-1.
The complex numbers, using rational numbers (i.e., the Gauss set G) or mathematical real-numbers (the set R), are artificial. Can they be avoided?
Math cannot be in ones head, as explains [1].
To realize the YES answer, one must advance over current knowledge, and may sound strange. But, every path in a complex space must begin and end in a rational number -- anything that can be measured, or produced, must be a rational number. Complex numbers are not needed, physically, as a number. But, in algebra, they are useful.
The YES answer can improve the efficiency in using numbers in calculations, although it is less advantageous in algebra calculations, like in the well-known Gauss identity.
For example, in the FFT [2], there is no need to compute complex functions, or trigonometric functions.
This may lead to further improvement in computation time over the FFT, already providing orders of magnitude improvement in computation time over FT with mathematical real-numbers. Both the FT and the FFT are revealed to be equivalent -- see [2].
I detail this in [3] for comments. Maybe one can build a faster FFT (or, FFFT)?
The answer may also consider further advances into quantum computing?
[1] Article Algorithms for Quantum Computation: The Derivatives of Disco...
[2] Preprint FT = FFT
[2] Preprint The quantum set Q*
Fermat's Last Theorem (FLT) was likely already proven by his "enemy" Descartes, in the cartesian construction, but Fermat just did not want to say it. Instead, there is no confusion (see photo), and it is not just a Diophantine equation.
The cartesian construction allows one to go from 1D to 2D to 3D, to 4D, ... to nD.
All is in beautiful order, though it might not look like it.
Physically, as this question says, every path in a complex space must begin and end in a rational number -- anything that can be measured, or produced, must be a rational number.
Therefore, complex numbers are not physically needed, they are artificial and only a mathematical model.
We explore this observation in mathematics, in this new question at RG.
We are introducing a "new" model, that we presented years ago in an earlier question at RG, allowing us to better understand why "a negative times a negative is a positive". There is a well-known movie where that phrase is memorized although not understandable, in a high-school setting, for calculus, cited.
We recognize that complex numbers are useful there, but they both mathematically and physically are not necessary in numerical terms. However, in algebraic terms, they are.
For example, one can split zero using complex numbers, as (0) = (+1 -1) = (1 - √-1)(1 + √-1), where √-1 is the square-root of (-1), also called ¡.
See also our well-read earlier preprint, on complex numbers, and comments, at
https://www.researchgate.net/publication/331025041/
Responding to a puzzled private question, by not using complex numbers, we also do not use waves or calculate trigonometric functions.
The "wave picture" is gone. And it is much faster to calculate, without the added structure. It is artificial to postulate waves and complex numbers. We can do it also in algebra, not just in numbers.
This will become clearer in the preprint, when done.
Mathematics, as seen in differential and integral calculus [1] using points in the set Q, leads one to consider that reality is quantum.
The points are, everywhere:
Discrete: digital, to use a 21st century term, being separated from each other by a non-zero amount;
Rigorous: showing absolute accuracy with width 0; and
Isolated: surrounded by ``nothingness'', where even the word ``nothingness'' may be too much.
Each point is surrounded by a naturally open neighborhood, showing discontinuity, where all hence isolated points can show ``jumps'', enabling the three aforementioned quantum properties (3QP).
As a tool for physics and engineering, complex analysis provides both a geometric view and an analytically apparently rigorous justification [2-4], even though the real and imaginary parts use mathematical real-numbers (MRN) impacted by [1]. May complex numbers be also impacted, as a consequence?
1] Ed Gerck, “Algorithms for Quantum Computation: Derivatives of Discontinuous Functions.” Mathematics 2023, 11, 68.
https://doi.org/10.3390/math11010068, 2023.
[2] Stewart, I. and Tall, D., “Complex Analysis.” Cambridge University Press, 1983.
[3] Ahlfors, L., “Complex Analysis.” McGraw-Hill, Inc., 1979.
[4] Arfken, G. B., “Mathematical Methods For Physicists.” Elsevier Academic Press, 2005.
Fig.(1): Directions of change in B^3, from [1], attached.
This work begins by extending accordingly the structure of Boolean algebra significantly [1], with Boolean differential calculus (BDC), which is first seen as analogous to differential and integral calculus [2] using the set Q, and analogous to calculus [3-4] using the set R, notably studying the changes in functions with respect to their variables, in the well-known value representation of polynomials.
BDC allows various aspects of dynamical systems theory [4] such as: finite automata theory [5]; Petri net theory in parallel computing [6]; and supervisory control theory with applications to robotics [7], which areas are also impacted by these discoveries.
Using a first isomorphism (detailed forthwith), this work explores the immediate neighborhood of the binary digits 0 and 1, in all possible directions evaluated in the set Q, including Boolean values using the binary set B^n, continued into the rational set Q.
But the reassessment of Boolean algebra is not enough for improving computation. If the “wave picture” can also be gone, there would be no “wave-particle'' conundrum anymore [2,8], and many “problems” could be solved using the measurable set Q, not the immeasurable sets R and C [2,9]. This needs a reassessment also of complex theory [10-12].
We recognize that complex numbers seem useful in algebra, but they both mathematically and physically are not necessary in numerical terms, and cannot be calculated. However, use has been the justification for their use, since the celebrated “Ars Magna” of Girolamo Cardano in 1545 [13].
Physically, every path in a complex space must begin and end in the set Q -- anything that can be measured, or produced, must be a rational number [2]. Therefore, one can suspect, in terms of Nature (Nature is considered a preexisting and undefinable reality, more than 13.8 billion years old), where complex numbers are not physically needed.
Complex numbers are considered artificial and only a mathematical model --- but that can be useful in design and hand calculations (i.e., algebra, Euler identity), however with other shortcomings in numerical calculations (e.g., by computer), as is our interest.
(This segment of future publication is provided to foster cooperation, question are welcome, also by private message)
REFERENCES
[1] B. Steinbach and C. Posthoff, “Derivative operations of boolean functions.” Synthesis Lectures on Digital Circuits & Systems. Springer Cham., 2017. Including Fig.(1), attached.
[2] Ed Gerck, “Algorithms for Quantum Computation: Derivatives of Discontinuous Functions.” Mathematics 2023, 11, 68. https://doi.org/10.3390/math11010068 , 2023.
[3] R. Courant, “Differential and Integral Calculus, Volume 1; first published in Germany in 1930 as Vorlesungen ̈uber Differential - und Integralrechnung.” Ishi Press, New York, 2011.
[4] Tom M. Apostol, “Calculus, volume I.” John Wiley and Sons, India, 2005.
[5] Thatcher, J.W., Wright, J.B, “Generalized finite automata theory with an application to a decision problem of second-order logic.” Math. Systems Theory 2, 57–81. https://doi.org/10.1007/BF01691346 , 1968.
[6] J. B. Dennis, Petri Nets, pp. 1525–1530. Boston, MA: Springer US, 2011.
[7] Lopes, Y.K., Trenkwalder, S.M., Leal, A.B. et al, “Supervisory control theory applied to swarm robotics.” Swarm Intell 10, 65–97. https://doi.org/10.1007/s11721-016-0119-0 , 2016.
[8] Nicolas Gisin, “Indeterminism in Physics, Classical Chaos and Bohmian Mechanics: Are Real Numbers Really Real?” Springer Nature, Erkenn86, 1469–1481. https://doi.org/10.1007/s10670-019-00165-8 , 2021.
[9] Yuri Igorevich Ozhigov, “Constructive Physics (Physics Research and Technology).” Nova Science Pub Inc; UK ed., ISBN 1612095534, 2011.
[10] Stewart, I. and Tall, D., “Complex Analysis.” Cambridge University Press, 1983.
[11] Ahlfors, L., “Complex Analysis.” McGraw-Hill, Inc., 1979.
[12] Arfken, G. B., “Mathematical Methods For Physicists.” Elsevier Aca-demic Press, 2005.
[13] Ars Magna (The Great Art), pp. 1–26. New York, NY: Springer New York, 2007.
GK: The list can increase, and open new vistas.
Currently, Mathematics has a number of fictions stemming from the 17/18 centuries, reformed already in the 21st century. And clear the procedures for computers in Computer Science.
For example: infinitesimals, hyper-reals, continuity, limits, Cauchy epsilon-deltas, Cauchy accumulation points, Lie numbers, mathematical real-numbers, irrational numbers, complex numbers, etc.
Each item above has reasons NOT to exist in the real world, here represented (as possible) by physics, biology, anthropology, and even humanities. Nothing is from a causeless no-source, a happenstance, everything is (somehow) stimulated, consequential.
This discards a probability model in quantum mechanics, things can be uncertain to an observer, but are not inexact to ta participant.
A simple 3-body problem cannot be solved by an observer, e.g., but the Solar system has an exact orbit for the least of its million bodies!
Perfection exists, we just don't yet understand it, nor have tools to describe it.
We don't say the last word, but the first -- mathematics needs to change, then physics, biology, engineering, computing, and so on.
Ed
C and rational numbers have nothing to do with each other. In any case your correct question may be rational versus real.
Quite a different direction to this question may be seen in my RG discussion
Alternative Fundamental Theorem of Algebra.
No one thinks of actually replacing C, just suggesting the possibility of using an algebra containing nilpotent elements, alternatively.
Further my understanding of an FFT is that of an integral containing
exp(ikx).(THE Fourier transformation) It is true you can make it discrete,but in principle contains the Complex math.
Ed
Im sorry that you are not really open to any discussion,There is certain freedom implicit in any discussion.
To sustain your point, if any, make a reasoned understandable presentation.
Using complex number goes against the "Principle of Closure", detailed in my latest publications.
Furthermore, it makes it "natural" to calculate trigonometric functions on the Fourier Transform (FT), which is both costly in time and imprecise, i.e., not needed. This does NOT invalidate Fourier Transform spectroscopy and spectrometers, as some not so careful people might think -- but makes them better, as more precise and faster.
The consequence is that FT ~ FFT, as explained in
https://www.researchgate.net/publication/367117742/
Responding to a private message, YES. The "Principle of Closure" is detailed in
https://www.mdpi.com/2227-7390/11/1/68
and applies there to mixing irrational numbers with rationals -- they do not close operations in the 4 four arithmetic operations.
That can be extended easily to complex numbers, that uses mathematical real-real-numbers -- they are already recognized NOT to close for all 4 arithmetic operations.
However, complex numbers can be used (for the nostalgic) in algebra using the Gauss set, with care. Albeit everything non-numerical that can be done (even with the Gauss set) in complex numbers, can be done also with rational numbers. So, in spite of the beauty of Euler's identity, it is better and faster to NOT use complex numbers in calculations.
This question should be used in its context, or there is no suitability of purpose.
We are to discuss the YES response to this question. People need to know how limiting it becomes to use complex numbers. We no longer need to calculate time-intensive trigonometric functions.
Ed
There is no violation of closure in the algebra of the complex numbers.
The addition, multiplication , even division of Complex numbers is still
a Complex number. That is why they talk about the Field of Complex numbers.
Please be more carefull....
Yea, throw Fourier Transform spectroscopy and spectrometers away?
No longer need anyone to come along and measure your land?
No longer care about balance of forces on a bridge?
To repeat above, in responding to a private message, complex numbers can be used (for the nostalgic) in algebra using the Gauss set, with care.
This does NOT use mathematical real-numbers, as the set C uses. This is a consequence of
https://www.mdpi.com/2227-7390/11/1/68
Albeit everything non-numerical that can be done (even with the Gauss set) in complex numbers, can be done also with rational numbers.
A private message questions that one could not make it "natural" to calculate trigonometric functions, and still use the Fourier Transform (FT).
But ... this does NOT invalidate the FT, FT spectroscopy, and spectrometers, or Shor's quantum algorithm, as some not so careful people might think -- but makes them better, as more precise and faster.
The consequence is that FT ~ FFT (where ~ means equivalence), as explained in
https://www.researchgate.net/publication/367117742/
Quantum Computation is born!
Not my private messages, mine are public.
You seem to be peddling for what pases as idiocy with C question,
wheras any serious research you do seems elswhere, in quantum computation.
In this way we understand zilch. Your only path to redemption is your personal direct and simple explanation about what you are really up to,
if you are capable.
Readers dont usually eat glas, they just want to understand.
Many people write to me, and my mailbox is very accepting.
Some simple people assume that their simple questions/answers are, somehow, special, worthy, or unique. They even may want to claim ... innovation! Some chutzpah!
But, we live in a fluid, and many of our thoughts are not even original, or our own. Ego interferes. The solution is to let the noise sink under its own weight, and keep the attention on the signal.
The complex numbers have been followed without deep questioning, and their justification, in medieval Italy, was to solve the equation x^2+1=0.
But, they lead one astray from the actual solution!
This was revealed by the FFT, that can now be entirely calculated with rational numbers, and without any trigonometric functions, as mentioned above.
Then, we conclude that FT ~ FFT, as
Preprint FT = FFT
explains. For the savvy, a letter cn b a wrd! This is the shortest work I could write, but it took me more time ...
There are no mathematical real-numbers either. Thus, the set C is deprecated and the set G is pretty superfluous. The sets R and C cannot be calculated (Gisin,Ozhigov, Gerck) with probability 1.
Mathematics gets simpler, and correct, accessible to natural processes, and accessing them in quantum mechanics.
Then, the Shor quantum algorithm of 1994, without any practical result since then, only theoretical results, opens the calculations of the FFT, to the components of numbers.
This reveals the prime numbers as "lumps", resonances in the number line, much in the same way that a physical chord has physical resonances. Easier to calculate, with such a physical model.
I dont care how successful or not you may be, all I say is your RG scipts are really bad, and you dont answer anything convincing, when things you say seem formally wrong.
"There are no mathematical real numbers"?
You think conventional math has been replaced by QM??
What do you mean by a set not being calculated???
Actually the more history something gets in math, the more solid it gets.
Of course, you probably merely argue that any real number on a computer is actually a rational nummber. We have heard that one before.
Ed
I happen to agree with some of you math criticism, but when you say C has no
closure, for example, that is going too far.
I, with you do not like infinitesimals , nor the limit def for derivatives.
An idea of mine also maintains that rings have more potential than thought, versus fields.
But math itself evolves very slowly, and you cannot change it by throwing out
questions that may sound scandalous to the old conservative gaudians. It only does so gradually, the older part always remains in reserve. I know from throwing out a few suggestions myself.
When I mention rings there is a howl about non invertible elements, even if it is only a line in the plane. Usually new results are added without questioning the previous, otherwise someone always jumps. They jump anyway very often.
I really dont understand your involvement with Gaussian intergers for example, Z+iZ. That is more for pure number theory
Math has always been in ones head.
Why should we change that? Prefer machines instead?
Try a matrix
0 -1
1 0
isomorphic to the properties of i
Or
x -y
y x
isomorphic to a C number x+iy
Further, Complex nummbers can be implemented on any reasonably advanced programming language.
Ex fortran, Z=CMPLX(X,Y)
declaration
COMPLEX Z
This represents the one-sided, effective, end of trolls in RG answers: deny them a public, free, citable audience. Deny abuse masquerading as virtue.
No more, and more, ..., "relativistic mass", "discussion" of SR, fake "non- acceptance" of a macroscopic quantum reality, acceptance of "continuity", defense of "mathematical real-numbers", and other loss of time in RG,under the name of free speech. To yell "FIRE" in a theater is a crime under US law. Evolution is upon us!
Some comments above (some now deleted) 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.
See also the questions:
https://www.researchgate.net/post/Can_one_end_trolling_in_RG_answers and
https://www.researchgate.net/post/Can_questions_become_a_research_tool_in_RG
If the idea is to get rid of i, thats easy. You just formulate the bracket method (x,y) explaining the rules to multiply and add. No i
Now , if the idea is to get rid of Complex numbers theselvs, then good luck.
A ton of mathematicians indicating the importance of what a Field means will
fall on your back, and you will never recover.
JAS: Yes.
Time is an indication that, using variational analysis and looking for the path with least time, that ALL points must be rational numbers -- so, there is no space for mathematical complex numbers.
Thus, mathematical complex numbers are like mathematical real-numbers: they are not measurable. See also Gisin and Ozhigov. Their references and other reasons are given in https://www.mdpi.com/2227-7390/11/1/68 In summary, get rational!
Any mention that math itself evolves very slowly, or that one cannot change it by throwing out questions that may sound scandalous to the old conservative guard ... forgets that quantum mechanics exists, discontinuity exists, jumps exist ... changes exist. Nature is not gradual.
A ton of mathematicians indicating the importance of what mathematical complex numbers are, are barking up the wrong tree and the sooner they stop the better.
But no one falls on their back, they will all recover. The new vistas is their own reward. Old traps now have signs saying "not this way!".
To illustrate that mathematical complex numbers do not exist, consider what has been called an example of "casus irreduciblis" in third-degree equations, of mathematical complex numbers "infesting" numbers in the real number line:
cubic-root(2-√(-4)) + cubic-root(2+√(-4))
where √ is the square-root. This expression seems not to be 3 numbers, and not in the real line, as they should all be.
However, using the set Q* we can calculate the 3 numbers in the real line, which are:
-2, 1+√3, 1-√3
You can check, using polar notation and trigonometry.
Thus, using mathematical complex numbers can be avoided in these (and other) cases, using the set Q*.
This has been added to the preprint above:
APPENDIX I
Another discussion may affect what is currently considered the iron-clad applicability of the well-known Galois theory.
Before Galois the concern was with generic solution formulas where the coefficients of the polynomial are parameters. Solvability and unsolvability of particular equations not in a parametric family was not really subject to a general theory until Galois placed it in the context of field extensions, Galois group actions and so on, using MCN.
In short, for the subsequent modern analysis of whether casus irreducibilis (see Section 2) can be circumvented or Galois theory is to be still considered valid, of course the first thing to require would be (but is not the case!) that there are no MCN solutions. No MCN "infesting" the number line of rational numbers, and masquerading as MCN. This condition will be pursued elsewhere.
For quantum computing, although not supported by some authors at Catech, and to this question:
Software can use only the set Q, while the hardware can use the set N for better results.
This is continued in:
https://www.researchgate.net/post/Can_quantum_mechanics_work_BETTER_without_imaginary_numbers/1
Some people think of using imaginary and complex numbers is in a dimension 90 degrees orthogonal to a baseline dimension.
But, one can do that better without imaginary numbers. They mix imaginary with real parts, even when using the set G.
Let's think of it: no computer uses imaginary numbers! And yet, all can calculate anything.
In connection to this question, a mathematical imaginary number presents a disturbing property: there is no way
to distinguish what should be different -- i from-i.
As we look to explain prime numbers, it is important to realize that a prime number is not a mysterious accident in set Z, which happens to use base 10.
A prime number is an objective value, valid for friends and foes.
This works because a number is a semiotic property.
The name can change, the referent can also change, but the sense stays constant: a prime number is a prime number.
In other words, a number is a 1:1 mapping between a name and a value.
Different pairs (name, value) have the same sense for any base.
As we can jokingly summarize, by this mathematical property, a prime number on Earth is a prime number in Betelgeuse. An objective property, even though numbers might be subjective.
The form z=a+ib is called the rectangular coordinate form of a complex number, that humans have fancied to exist for more than 500 years.
We are showing that is an illusion, see [1].
Quantum mechanics does not, contrary to popular belief, include anything imaginary. All results and probabilities are rational numbers, as we used and published (see ResearchGate) since 1978, see [1].
Everything that is measured or can be constructed is then a rational number, a member of the set Q.
This connects in a 1:1 mapping (isomorphism) to the set Z. From there, one can take out negative numbers and 0, and through an easy isomorphism, connect to the set N and to the set B^n, where B={0,1}.
We reach the domain of digital computers in B={0,1}. That is all a digital computer needs to process -- the set B={0,1}, addition, and encoding, see [1].
The number 0^n=0, and 1^n=1. There Is no need to calculate trigonometric functions, analysis (calculus), or other functions. Mathematics can end in middle-school. We can all follow computers!
REFERENCES
[1] Search online.
I have read your paper on the set Q*. I am probably missing something, but it what is the difference between the set Q* and the set Z? There seems to be a one-to-one correspondence in division and multiplication, e.g.,
r1 Exp(i phi1) * r2 Exp(i phi2) = r1*r2 Exp(i(phi1+phi2)) r1 * r2 = r1*r2 .
What about addition and subtraction? How do you define A1+A2 ?
SB: Thank you for your posting.
The difference between the set Q* and the set Q is that in Q* the same number x in Q and Q* can enjoy infinitely different rotations in Q*, while in Q only positive and negative values are allowed. There is no angle in Q, but two trivial ones.
The difference between the set Q* and the set Z is clear, as there is no denominator in Z, but the trivial 1.
Two numbers a/b and x/y are equal in Q* when their cross product ay=bx, which allows multiple full-circle rotations without any difference -- as long as in the same direction.
This follows a physical model observed in spacetime, and maps to gravity, quantum mechanics, electromagnetism, etc.
SB: We define A1+A2
by multivector addition.
Likewise, we define the other operations by corresponding multivector operations.
This means that the set Q* inherits all operations that have been already developed for multivectors, in a complete description -- unlike vector algebra, where vector multiplication is NOT closed, and there is no vector division. This is important, e.g., in electromagnetism.
SB:The Gibbs vector calculus is still studied today, e.g., in Multivariable Calculus, even though it is an outdated model of vectors, with no path forward in scientific terms.
For example, the cross-product of two vectors is not a vector, which is well-known.
This paper adds a more general objection in the quotient space of two vectors, otherwise well-defined in 2D and 4D, showing that the Gibbs description in 3D is deficient.
This motivates a study of a current model, such as in Geometric Algebra, by Clifford.
In Preprint The Ratio of Two Vectors As a Notion of Quotient in Gibbs Ve...
This is a summary.
Imaginary numbers are denied in quantum mechanics and Erwin Schrödinger said it first, in 1926.
We understand that numbers find a natural origin in set theory but common set theory does not explain physics and does not explain reality.
However, a special set people hint at could exist, that features phases with phase transitions. Could this represent gravity, electromagnetism, the laser, etc.?
We are proposing the set Q* [1], providing a solution to these physical questions in a natural setting with numbers, using the properties of Clifford multivectors. Critically, one can calculate the ratio of elements, which Gibbs vector calculus cannot do.
Further, and to answer this question, many proofs exist that rational numbers can be counted by natural numbers. The new set Q* satisfies this, opening new possibilities in the sets N and Z.
[1] Preprint The quantum set Q*
Avoiding the complexity after all we will come to simplicity of using natural numbers only to count sheeps...
We are correcting a 500 year error -- complex numbers do not exist and are not needed.
The sooner they are not used, the better. FFT=FT and can be faster. Galois theory needs to change. Polynomials have coefficients in the set Q. The quintic can be always solved in the set Q. The formula for third-degree polynomials works always in the set Q. The equation for second-degree polynomials change. Discontinuous functions can be differentiated and integrated. Mathematics evolves.
What a wonderful world is the Q set! No continuum problem, no Lebesgue measure, and pi=22/7 without any error!
Responding to a private message, complex numbers were already denied, when we denied mathematical real-numbers [1]. But, how about imaginary numbers, or the Gauss set, using rational numbers to resurrect complex numbers?
No, already denied by Erwin Schrödinger in 1926. And, we deny that today -- using computers, where R=Q and C does not exist
Soon, we will reveal how the Fourier Transform and the FFT can be a mapping from Q to Q -- and much faster --without passing through C, G, or R. The beginning of that is in the RG preprint "FT=FFT".
We are not saying the last word, but just the first.
If one gets a rational number (which I say happens in any physical measurement), one can get a natural number from it by a known isometric relationship, Q --> N.
One cannot get an irrational number by adding numbers in the set Q -- the set Q is closed for addition, +Q--> Q.
Any number x in the set Q, to add computability, obeys N --> B^n --> B ={0,1}. Computers can calculate anything measurable-- p-adic numbers, and the sets R, C, G, do not appear!
No physical measurement can be in sets R or C (which has two sets R).
No physical measurement can be in set G, either, because a physical path must begin and in the set Q -- and one men can always get two set Q numbers between any two set G numbers.
QED.
The reference of a sentence is its truth value, whereas its sense is the thought that it expresses. Frege justified the distinction in a number of ways.
One can see a number as pointing to its truth value (called a value) and expressing a thought (called its digit in the base used).
The digit is the sense, the meaning of the number.
The digit can change at will, subjectively, but the value stays objectively fixed.
Thus one can write F in hexadecimal, while the value is 15 in base 10. The number is this 1:1 mapping between a name ("F") and a value (15).
Or, in reverse, one can use 15 for the name, pointing to the value F.
Or, one can use 1111 in 64-bit signed binary for the name, pointing to the same value (15).
A number is seen as this flexible semiotic concept, very used in computers.
How this applies to avoid complex numbers? One can have that thought (as a name) but point to a value with an angle, a member of the new set Q*.
Quantum computing shows that there are no non-abelian anyons, no complex numbers, no mathematical real-numbers, no irrationals, no infinitesimals. They are 500-year illusions.
No phantasy, no hype. Just physics demonstrating mathematical expectations, or denying them.
Qunantum computing arrived without hype at the Max Planck Institute of Quantum Optics in 1982, but has being denied for 40 years ... yet, it works based on the quantum properties of integers. Anyone can use their mind also.
The Hindu mathematician Madhāva developed calculus 250 years before Newton and Leibniz. Things take time ...
Computers can calculate anything, and fast, only using the Boolean basic set B={0,1}. Some results take zero time: 0^n=0, 1^n=1.
Quantum computing opens an entirely new paradigm.
One can, finally, consciously, exactly, calculate everything.
Quantum computing comes from physics, and in physics one does not ask another human, the work of past humans, mathematics, or even God, one asks nature.
Quantum computing is just physics hearing nature and demonstrating mathematical expectations, or denying them, which opens new vistas -- promising rigorous results, fast.
Our universe lasted for 13.8 billion years, and never halted.
Computer Science did not get there yet ...
And maybe the Big-Bang was not the first. The universe may be too old to be counted in years. And never halted.
In the Vedas, the Hindu read that each kalpa (a day of Brahma) is divided into 14 manvantara periods.
Everyone can also understand quantum computing, in the times to come. No innuendo, nothing to defend or attack. Any conflict will die off?
Yes, one expects. Then, we will finally follow nature ... and be thankful for the realizations.
What a silly question . Attend a MIT computer memory design class ...students must add, subtract , multiply and divide in the electrical complex domain. Look the class up ...free online ..
Ed,
Electrical wiring use complex numbers ...guess you
are not interested in taking MIT computer memory
design classes .
Milo
MG: I did electrical engineering at ITA first, in order to better understand physics.
Electrical engineers use sqrt(-1) even though sqrt(-1) is a false number. The angle information does not need that error. One can use the new set Q*, just use the set Q, or calculate in binary as computers do exclusively. Electrical engineering needs to advance. MIT students are learning wrong things.
I am on vacation in the Sierra Nevada, California. I am near Columbia, CA.
Camping is well-organized here, and there are literally 1,000s of places to look for.
As rest is part of the work, I am working as well.
How can cryptography work post-quantum? It cannot use complex numbers, infinitesimals, irrationals, or p-adic numbers, because they are purely imagined -- they are non-physical.
For exanple, it cannot use the supposed insolvability of quintic and higher polynomials by radicals (Galois theorem), because quantum computing allows that to us by NOT using the set C.
Let's not fall into a false "Torricelli paradox" -- one needs to use physical facts, objective to friend or foe. Let us not imagine a reality -- let us observe it.
Avoiding collisions in hash-functions seems a good place to start. Collisions must happen, by dimension reduction. But they are hard to find.
This leads to a new ZSentry, using properties of a blockchain-- but without trace, so no adversary can find it in a safe time.
Not even with quantum computing and a "long enough" time.
Quantum computing provides a way to qualify post-quantum cryptography!
It must resist quantum computing ... and helps define "how long". it does not matter that everything becomes ... computable.
One can choose a billion years as "long enough", or 10^10 years ....
Been camping in our Gold Country many times ...usually in Butte, or Yuba County, nearer where the largest 57# nugget was found ...
MG: Yes. Being a camper is a great experience. One learns to use what is available, respect natural life, nature, and live silence!
One can mine the gold of truth in silence! A tree, for example, grows when pruned. By forcing to have less, we discover the inner gold, that never leave us, that never can be robbed, and that illuminates all we do.
Ed,
Another proof that irrational numbers exist as lines is
found in another MIT class, reverse engineering
Origami shapes . The class is a prerequisite to
computer design classes . Look it up, if you dare.
Milo
.
(x1,y1) +(x2,y2) =(x1+x2, y1+y2)
(x1,y1)(x2,y2) = (x1 x2-y1y2, x1 y2+x2 y1)
Some one sees i around?
However the QM answer is baloney, Complex does not mean containing i necesarily. If not use (0,1)
or that which solves xx+1=0, by field extension xx+(1,0)=(0,0)
Is that better.?
(0,1)(0,1)=-(1,0) according to above.
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