10 ms·
Imaginary Numbers May Be Essential for Describing Reality
- Ambolia 6y agoImaginary numbers are not so scary if you just think of them as 2d vectors with a different multiplication operation added to it.
- injb 6y ago*complex numbers. But yes you're right. The biggest problem (at least for me) is the name. Because everyone recognizes "real" and "imaginary" as adjectives, they think that's how these words should be understood in the context of numbers. But in truth, they are _proper nouns_, i.e. labels, and nothing more. As far as the adjective "imaginary" is concerned, all numbers are imaginary. Really the sooner we stop calling them "real" and "imaginary" the better. They are terrible choices of name, coined by people who didn't really understand them, and didn't want you to understand them either.
- janderson3 6y agoAbsolutely agree. What's your preferred terminology? I like "Forward" and "Reverse" for positive and negative number and "Lateral" for the imaginary unit, since it is just perpendicular to the Forward and Reverse numbers.
- klmadfejno 6y agoGood and evil.
- deleted 6y ago[deleted]
- _jal 6y agoSpooky and strange?
- andrewprock 6y agoHuman and Werewolf.
- kortex 6y agoSpooky Werewolf Algebras on the Monster group. Coming to arxiv this October.
- Robotbeat 6y agoI just think of them as different axes on a 2D Cartesian plane. That’s basically how they work. EDIT: similarly, quaternions as a number in 4D, with that extra dimension being handy for avoiding the instability problems when you get near “gimbal lock” in 3D rotations.
- phkahler 6y agoBut the quaternions sort of make the real component special dont they?
- kortex 6y agoThat it more an effect of how we use them in computer geometry. Unit quaternions with ijk plus real map to an xyz tilt plus a rotor. But I believe you could pick any such mapping. On the other hand, dual quaternions do care which is the dual component, since multiplying by complex numbers looks like true rotation, while multiplying by a dual looks like rotation around a point at infinity, aka translation. So in that case maybe there is something special about the real component.
- injb 6y agoThat's pretty close to what Gauss said they should be called: "If ... +1, -1, and √-1 had been called direct, inverse and lateral units, instead of positive, negative, and imaginary (or impossible) units, such an obscurity would have been out of the question" (https://shitohichiumaya.blogspot.com/2016/10/gausss-quote-for-positive-negative-and.html https://shitohichiumaya.blogspot.com/2016/10/gausss-quote-fo...)
- yccs27 6y agoI also like the name "lateral unit", maybe call the field "planar numbers"?
- jonnycomputer 6y agoDo you have a suggestion for an alternative to "real number"?
- gpm 6y agoContinuous numbers?
- macksd 6y agoImaginary numbers are also continuous, are they not?
- gpm 6y agoSo they are, so are many other spaces. I'd avoid calling complex numbers "numbers" personally. In the context of plain english (and not currently in the context of mathematics) I think of the word "number" as saying that this is somehow a measure of a quantity (We have "a number" of brands of cereal in the store). Complex numbers are not a measure of a quantity. I'd probably replace the "numbers" part with "field". I don't have a good replacement for the "complex" part, but I think that should change also (because telling people things are complex is almost as bad as telling them that they are imaginary). If no one proposed something better and I was king of the universe I guess I'd just name them after whoever discovered them.
- injb 6y agoWould you consider -3 a number? It's certainly not a measure of quantity. A better way of thinking about it is that it's 2 things: a magnitude and a direction. Complex numbers just expand the direction into two dimensions. As long as you accept negative numbers, then you already accept the fact that there can be more than one number of a given size. The complex numbers just contain infinitely many numbers of a given size, instead of only 2.
- gpm 6y agoIn plain english, no I don't think I would particularly consider -3 a number. If I say someone gave me a number of horses I don't mean they stole my horse. I think it is the right call to include them in the mathematical definition though. Unlike complex numbers they are almost always useful (they are needed to turn the object into a field, which is needed if you want everyday operations like subtraction to always remain within the set), have a much more intuitive definition of multiplication, are a smaller generalization, etc.
- GoblinSlayer 6y agoIn quantum physics complex numbers are used to represent circular symmetry, so they are more like modulus and phase.
- ko27 6y ago> They are terrible choices of name, coined by people who didn't really understand them, and didn't want you to understand them either. Sorry to be blunt, but your comment is horrible. There is far more going on here than you realize. Real numbers have an isomorphism with our reality. We thought we didn't need imaginary numbers to describe reality. They were inteded to be used more as a mathematical tool. Another good example is euclidean vs non-euclidean geometry. They are both incredibly useful, but the former one was though to be "real" and the latter "imaginary". But it turns, physics is okay with us living in a non-euclidean universe (https://en.wikipedia.org/wiki/Shape_of_the_universe https://en.wikipedia.org/wiki/Shape_of_the_universe).
- fooker 6y ago> Real numbers have an isomorphism with our reality. You can probably get a nobel prize by proving this. :)
- bikenaga 6y agoMathematicians have largely abandoned the term "imaginary number" and use "complex number" instead. I never used the term "imaginary number" when I was teaching (except to point out that it wasn't very good terminology). But the old term still survives in some areas. For instance, the real and imaginary parts of a complex number: re (3 + 4 i) = 3 and im (3 + 4 i) = 4.
- denkquer 6y agoImaginary numbers are 1d. complex are 2d.
- darig 6y agoImagination itself isn't scary if you just think you live in a turtle's dream, flying through space. Granted, my math degree is 20 years old, but I'm pretty sure you're not talking about imaginary numbers. They couldn't exist in 2d.
- rimiform 6y agoThat's great, but my intuition fails me when I try to consider what it would mean for the answer to some question to be a complex number. It's even worse for quaternions, because I can't even use the spatial analogue there.
- munchbunny 6y agoI think about it this way: the complex-ness of the number is really just a representational trick. What it means, intuitively, depends on what the question is really asking and why the behaviors of math on complex numbers is well-suited to representing the answer, and that reason might change based on the context.
- srean 6y agoThere is more to it. Vectors typically don't have multiplication defined. Complex numbers form a field and that is a major extra feature. Multiplication and division I'd defined on them. If this property is not needed one can make do with 2d vectors instead. There are 4d and 8d analogues as well. Hamilton started an ambitious program to base mathematics of nature on the basis of the 4d ones -- quarternions
- gspr 6y ago> Imaginary numbers are not so scary if you just think of them as 2d vectors with a different multiplication operation added to it. Perhaps, but that view makes them seem arbitrary. It makes it seem as if you could just do the same thing in 3d, 4d, etc. and they'd all be equally impacful and meaningful.
- leereeves 6y agoYou can (almost). You can do the same thing again to get quaternions, and again to get octonions, and so on. https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_constru... That's different from extending 1d to 2d, 3d, etc as vectors.
- gspr 6y agoI know. But these are not as fundamentally important as the complex numbers.
- AnimalMuppet 6y ago"And so on"? I thought octonions are where it stops? At least, anything past the octonions is not able to be a normed division algebra, if I understand correctly.
- leereeves 6y agoIt doesn't stop, but as you said, the next step (sedenions), and all steps after that have non-trivial zero divisors and aren't division algebras.
- gspr 6y agoExactly. This is what I was hinting at when critical at people who say complex numbers are "just 2d reals with fancy multiplication". It's true, of course, but it makes it seem as if the 2 in "2d" is unimportant/arbitrary.
- 6y ago
- SavantIdiot 6y agoWhere the analogy to a 2D vector starts to diverge is raising something to a complex power. There's no meaning in raising 3 to the [x,y], but Euler clearly defined it for 3^(xi+y).
- pfortuny 6y agoThe thing (which is subtle) is that the dimensionality is 1. Which is, as far as I have read the intro to the preprint cited in the Quanta article, what the authors are mistaken. Algebraically, as vector spaces: C is the change of base of R when tensored by C which is not the same as R^2 as R-vector space tensored by C (not at all the same).
- elil17 6y agoAnyone whose ever studied vibrations (e.g. spring mass dampers) already knew this. This is a cool example but it’s not really any more “essential” than macro-level phenomena.
- Armisael16 6y agoHalf the article is explicitly talking about how this fundamentally is different than the phenomena you’re talking about. Imaginary numbers are merely convenient in the cases you’re describing; you can get the same results without using them. The paper this article is about is claiming that in QM this is fundamentally untrue, and models that don’t use complex numbers cannot match those that do.
- pfortuny 6y agoThey do match. The thing is that the description is much more complex, nothing else. What I see from the preprint (notice that the authors have not discussed it with Quanta, because it is under peer-review) and bothers me is that they "add a real qubit" (...) which should be done very carefully because the two real qubits are not independent (they are linked by the hamiltonian). I have not read the whole preprint, though.
- Enginerrrd 6y agoTo a certain extent, you could probably say the same about the time term in the flat spacetime metric in relativity. There might be ways around it, but I'd wager that anything you come up with can be massaged a bit to make it look like time is an imaginary dimension of space where it's contribution to distance along the spacetime manifold is negative when squared.
- CyberRabbi 6y agoWas going to comment this. Complex numbers are a natural consequence of the basic math axioms and the fact that they are two dimensional (and not 3,4,etc.) is not arbitrary.
- ihunter2839 6y agoNot a physicist, so maybe I took the wrong gist away from the article, but - it seems like what they are saying is that systems that are generally modeled by complex equations have always had real valued equivalents that were known. So it's not that the complex representation isn't a useful mathematical tool, but that it was an alternate representation that was easier to work with. Can you represent a spring mass dampening with all real valued quantities, even if it's a real piece to work with?
- Xophmeister 6y agoOne could argue that all numbers are an abstraction; they don't "exist" in any physical form, even integers. Therefore, why is it such a surprise that complex numbers are sometimes also needed to model reality?
- deleted 6y ago[deleted]
- j-pb 6y agoInterestingly enough it would be "especially" integers that don't exist physically in mosts scales of reality if physics is to have any mathematical coherence.
- zeroonetwothree 6y agoEh, because of quantisation you could argue only integers exist ;)
- eigenket 6y agoQuantised is not synonymous with discretised. The fact that (for example) energy in a simple harmonic oscillator comes in discrete packets doesn't mean that the kinetic energy of a free particle is. Some quantities in quantum mechanics are as continuous as their classical analogues.
- GoblinSlayer 6y agoThen you learn about Zeeman effect.
- jMyles 6y agoThe phrase "mosts [sic] scales of reality" is doing an incredible amount of heavy lifting here, though, since the most reliable mathematics describing three of what appear to be the four fundamental forces of nature are indeed quantized. It's more accurate to say that only integers appear to be relevant in describing the nature of the nuclear forces and electromagnetism/electroweak force. And it seems that, at least for the moment, we're headed in a direction of quantization of gravity and a Planck unit of time as well.
- lizknope 6y agoI thought we already knew this. Euler's Identity is used in electrical engineering.
- pfortuny 6y agoYes but they are claiming that the link between the real and imaginary parts of the solutions to the Schrodinger Equation are "more linked" than just by the hamiltonian. Have looked over the introduction of the preprint and have some issues with it but am not an expert.
- leereeves 6y agoThat can be reformulated using only real numbers, similar to some math the article mentions for quantum mechanics: > Any property of complex numbers can be captured by combinations of real numbers plus new rules to keep them in line, opening up the mathematical possibility of an all-real version of quantum mechanics.
- Ma8ee 6y agoBut that is only for convenience. You can do everything with real trigonometry, just that it’s a bit more cumbersome. The imaginary numbers don’t correspond to anything in physical reality in electrical engineering.
- dnautics 6y agoNonsense. Just use a [[1 0] [0 -1]] matrix and it's kin and do away with "essential" imaginary numbers.
- eigenket 6y agoThe paper is describing a test of a particular alternative to quantum mechanics called "real quantum mechanics", its rather more specific than you are suggesting https://arxiv.org/abs/2101.10873 https://arxiv.org/abs/2101.10873 I know a few of the people on that preprint, and I'm pretty certain they all know you can embed the complex numbers in the real 2x2 matrices ;) As an aside I'm pretty sure the matrix you need is [[0,-1],[1,0]], which squares to -I, unless you don't want to use matrix multiplication in your construction.
- dnautics 6y agoas I was writing that matrix out from memory I was thinking there was something funny... Thank you for correcting me.
- pfortuny 6y agoI have read the introduction (only cursorily). What bothers me is that they may be conflating C as C-vector space, which is R (as R-vector space) tensor C (complex dim 1). with R^2 as R-vector space, tensored by C (complex dim 2). Because when they "add a real qubit", they seem to be adding a degree of freedom which you do not have when you have just one C-qubit. But -I repeat- I have only looked over the intro.
- GoblinSlayer 6y agoHuh? But it's literally mathematical definition that a complex number is a pair of real numbers. Spinors are more interesting.
- spekcular 6y agoThe necessity of complex numbers in quantum mechanics was also discussed a while ago in a blog post by Scott Aaronson: https://www.scottaaronson.com/blog/?p=4021 https://www.scottaaronson.com/blog/?p=4021. At the very least, he's able to show the problem with QM described using quaternions: superluminal signalling!
- deleted 6y ago[deleted]
- godelzilla 6y agoThis article is epistemological garbage. There's nothing more "strange" about "complex numbers" than any other mathematical construction. It's no surprise that they help "describe reality" because they're a relatively basic and fundamental concept of mathematics that was constructed in large part to describe physical observations. I can describe the rotation of my dinner plate using complex multiplication, but that doesn't make it magic or mysterious.
- cableclasper 6y agoRoger Penrose devoted more than half of his doorstopper of a book "Road to Reality" to complex numbers in which he uses Riemann surfaces to visualize the complex-number fabric of Reality™: https://openlibrary.org/works/OL3474173W/The_Road_to_Reality?edition=roadtorealitycom00penr_0 https://openlibrary.org/works/OL3474173W/The_Road_to_Reality... I was interested in plotting these Riemann surfaces and wrote a little article about it sometime ago: https://honeybee.freeddns.org/Visualizing%2520roots%252C%2520logs%252C%2520and%2520branches.html https://honeybee.freeddns.org/Visualizing%2520roots%252C%252...
- SavantIdiot 6y agoI would like to know why Schrodinger was so upset by the presence of i in his equation. It isn't clear to me why all the hate for i? It's not like it makes the math more complex: we're talking about quantum mechanics math!
- pygy_ 6y agoThere was initial backlash against complex numbers in the math community. i was deemed "imaginary" IIRC by its detractors, for the same reason the primeval atom was dubbed the "Big Bang": mockery, to accelerate its dismissal. This bad reputation may have carried up to the 1920’s...
- frongpik 6y agoImho, the presence of i in the schrodinger equation has a simple origin. The equation describes evolution of a wave and waves can be conveniently described with amplitude and phase components bundled together. With such description a wave can be moved or shifted by multiplying it by a complex number. Otoh, if the same wave is described by real numbers only, each number meaning the displacement from the null state, then shifting the wave pattern becomes a non trivial operation.
- mbrodersen 6y agoInformation doesn’t exist without physics. So you could argue that math does exist in physical form (neutrons, bits on a hard disk, sound waves, printed paper etc.)
- foxes 6y agoThe issue is not the "imaginary numbers not real etc!!" fluff. Maths is a tool to model reality and perhaps imaginary/real is a bad name. The interesting question why is C the correct choice of mathematical field to do "quantum mechanics" over -- what are the physical reasons this is needed. You do wonder what exactly is meant by "doing quantum mechanics". I assume the start of the story is going from functions over C, quantising and getting some Hilbert space of operators/ getting wavefunctions. Naively I would think that "quantum mechanics over the reals" might be less interesting somehow. Consider the difference between doing complex analysis and real analysis. C loses total ordering as a field, but then gains additional nice properties over R. Quaternions however are non commutative which is a bit strange. Complex differentiable functions are more restrictive than functions on RxR (need to satisfy a stronger property). At some level you would think as models of reality you might need one structure over the other.
- alpineidyll3 6y agoComplex numbers encode a specific and essential algebra, which is everywhere in nature. So do quaternions, and so on ad infinitum. Elementary field theories are often simply named by algebraic spaces they span. Plenty of algebras not thought of as numbers also map onto physical processes usefully. None of this is shocking or new.
- siproprio 6y ago> No instrument has ever returned a reading with an i. It's called a phase meter. You can buy it on amazon. An oscilloscope also works.