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Imaginary numbers are real
- superb-owl 4y agoI really wish we'd just call them "rotation numbers" or something. "Imaginary" was always a ridiculous name.
- kzrdude 4y agoYes, I hope the consensus eventually moves away completely from "imaginary". Complex numbers is already ok, so it doesn't need to change, only the terminology for that i axis. So what can we contribute, do you know any authors that have already found some new and better terminology and promoted it?
- jonsen 4y agoElectrical engineers don't use i, because i is used for current. They use j instead. I've heard them called jolly numbers.
- HPsquared 4y agoNegative numbers are imaginary enough already, really.
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- alanbernstein 4y agoI kind of like "geometric numbers".
- tut-urut-utut 4y ago
- alangibson 4y agoPointlessly obnoxious comment of the day right there
- macksd 4y agoPretty sure by saying this about a set of numbers that exist and that are distinct from the Real set, you're actually making a point opposite from the one you think you are making.
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- pvg 4y agoIt's no more or less ridiculous than 'negative', 'real' or 'odd'.
- Chinjut 4y ago"That this subject [imaginary numbers] has hitherto been surrounded by mysterious obscurity, is to be attributed largely to an ill adapted notation. If, for example, +1, -1, and the square root of -1 had been called direct, inverse and lateral units, instead of positive, negative and imaginary (or even impossible), such an obscurity would have been out of the question." — Gauss
- FredPret 4y agoDirect, inverse, and lateral are so much more productive than the labels we usually use. Can't believe I only ran into it now.
- tasty_freeze 4y agoHow about "Cartesian numbers"? But my hunch is that somewhere someone has already taken that name for something else.
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- idoubtit 4y agoYou should know the basics of the subject before stating that everyone is wrong and that the current standard is "ridiculous". You apparently confused "complex number" and "imaginary part". The latter made sense in the historical context, which was finding real solutions of quadratic equations (with real coefficients, of course). And please keep in mind that complex numbers are not rotations, and they do not map well to them. For instance, which rotations would be represented by the complex numbers "3", "4", "1-2i" and "8i"? You can map {plane rotations} to { r, |r| = 1 } using z ↦ r*z, but that's a circle, not a 2D space. To summarize, complex numbers can be thought of plane vectors, with an obvious geometric way to add them, and a non-obvious way to multiply them. The set of "rotations" is too vague and loosely related to complex numbers, e.g. 3D rotations are often represented by quaternions, which are more "complex" than complex numbers.
- jacobolus 4y ago> keep in mind that complex numbers are not rotations Complex numbers are (isomorphic to) "amplitwists": similarity transformations between plane vectors. If you want a pure rotation, you need a unit-magnitude complex number. The complex number 3 represents scaling by 3. The complex number 4 represents scaling by 4. The complex number 1 – 2i represents a scaling by √5 combined with a rotation clockwise by arctan(2). The complex number 8i represents a scaling by 8 combined with a quarter-turn anticlockwise rotation. > complex numbers can be thought of plane vectors No, (physics-style displacement) vectors and complex numbers are distinct structures and should not be conflated. Complex numbers are best thought of as ratios of planar vectors. A complex number z = v/u is a quantity which turns the vector u into the vector v, or written out, zu = (v / u)u = v(u \ u) = v. (Concatenation here represents the geometric product, a.k.a. Clifford product.) Mixing up vectors with ratios of vectors is a recipe for confusion. > non-obvious way to multiply them Multiplication of complex numbers is perfectly “obvious” once you understand that complex numbers scale and rotate planar vectors and i is a unit-magnitude bivector. > 3D rotations are often represented by quaternions, which are more "complex" than complex numbers. Analogous to complex numbers, quaternions are the even sub-algebra of the geometric algebra of 3-dimensional Euclidean space. Used to represent rotations, they are objects R which you can sandwich-multiply by a Euclidean vector v = RuR* to get another Euclidean vector, where * here means the geometric algebra “reverse” operation. Those of unit magnitude are elements of the spin group Spin(3). For more, see http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf
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- otabdeveloper4 4y agoI'm pretty sure they're always called "complex numbers" when you start using them. "Imaginary numbers" is something from a highschool curriculum.
- RicoElectrico 4y agoAt this point "imaginary numbers are not real" sounds like a strawman.
- xyzal 4y agoSabine Hossenfelder did a nice video on the topic. https://www.youtube.com/watch?v=ALc8CBYOfkw https://www.youtube.com/watch?v=ALc8CBYOfkw
- riskneutral 4y agoWe don't even know if real numbers are "real."
- labster 4y agoThat’s why I only use natural numbers. You can trust values from organic sources, like the number 5 and good old 23.
- chowells 4y agoActually, we know that the subset of R that can be used directly has measure zero. Everything else is a polite fiction to make some proofs work.
- avindroth 4y agoAny source? Curious as to what is meant by "used directly"
- joe__f 4y agoProbably just that, all physically measurable quantities are rational, and the rational numbers have measure zero in the reals
- ChrisLomont 4y ago>all physically measurable quantities are rational That's not even clear, unless you also assume you can measure exactly a unit length, which is not physically possible. Assuming you can measure something as a perfect rational value implies infinite precision, which is not possible. All physically measurable quantities have uncertainty is what I think you mean, but that doesn't say anything about possible cardinalities.
- alangibson 4y agoNumerical names are like names for musical notes: they make no sense anymore but it's not worth the effort to change them.
- cal85 4y agoWhat names? Aren’t musical notes called A, B, C etc?
- messe 4y agoMaybe they're referring to crotchets, (semi) quavers, minims, (semi)breves?
- foobarbecue 4y agoApparently those names were indeed worth the effort to change, since we did that here in the USA.
- sseagull 4y agoMaybe they meant scale degrees: Tonic, supertonic, mediant, subdominant, dominant, submediant, leading tone. Or modes: ionian, dorian, phyrgian, lydian, mixolydian, aeolian, locrian.
- hyperhopper 4y agoDo re mi fa so la ti do?
- pygy_ 4y ago"Ti"? Nonsense. Everyone knows it’s "si"! Edit, also B is also sometimes notated as H. In that case B means what is usually Bb. Edit 2: also, I just discovered that Eb was sometimes called S. What a mess :-)
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- LgWoodenBadger 4y agoCan another imaginary number do the same thing for division by zero, or am I getting the cause/effect backwards?
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- vletal 4y agoYeah. Great mind teaser. I can imagine an imaginary number (-/+) "inf" for divisions by number approaching zero from left/right, yet the algebra would not be possible to define properly because you can approach zero with a different rate. The undefined cases, where the left/right limits are not equal coulf get imaginary number "shrug" because it would be even less useful. Of is anyone able to define a useful algebra for these? I'm really curious.
- jacobolus 4y agoYou can add a non-zero number which squares to zero to your number system [edit: but this doesn't let you divide by zero]. This results in the "dual numbers" and is practically useful for "automatic differentiation", where we represent quantities in the form x + x'ε, and have the rule that ε² = 0. https://en.wikipedia.org/wiki/Dual_number https://en.wikipedia.org/wiki/Dual_number https://en.wikipedia.org/wiki/Automatic_differentiation#Automatic_differentiation_using_dual_numbers https://en.wikipedia.org/wiki/Automatic_differentiation#Auto... Quantities which square to zero are also implicit in spacetime. A "lightlike" vector which has equal displacements in space and time between two spacetime “events” (e.g. the displacement between two points along the path of a photon) has a squared length of 0, compared to "timelike" vectors with negative squared length and "spacelike" vectors with positive squared length. (conventions about signs vary from source to source) https://en.wikipedia.org/wiki/Spacetime#Spacetime_interval https://en.wikipedia.org/wiki/Spacetime#Spacetime_interval * * * In other contexts it makes sense to define 1/0 to be the quantity ∞. There are two relevant models here, with different practical applications. One model where we add a single number ∞ which is also equal to –∞, and makes the number line "wrap around" into a circle. Another model where we add two separate numbers +∞ and –∞ at the two ends of the number line. https://en.wikipedia.org/wiki/Projectively_extended_real_line https://en.wikipedia.org/wiki/Projectively_extended_real_lin... https://en.wikipedia.org/wiki/Extended_real_number_line https://en.wikipedia.org/wiki/Extended_real_number_line
- exabrial 4y agoIsn't i just a rotation about the axis of origin?
- unsafecast 4y agoIt is! Calling them 'imaginary' seems like a misnomer, they're just a different _kind_ of number. It becomes much clearer when you plot it in 2D space. The number line is just one dimension of the possibly infinite dimensions you can plot a number in. We're just labelling numbers wrong.
- game-of-throws 4y agoI had to laugh at the title. Imaginary numbers may be real, but they're also outside the set of real numbers ℝ.
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- unnouinceput 4y agoQuote: "...or millions of times colder than the insides of your fridge." I smiled. Eh, modern journalism, just a few order of magnitudes wrong here but who's counting anymore, yes?
- prof-dr-ir 4y agoHah! So if, as per the article, "imaginary numbers are square roots of negative numbers" then I am happy to report that: 1 = sqrt(1) = sqrt((-1) * (-1)) = sqrt(-1) * sqrt(-1) = sqrt(-1)^2 = -1 and all of mathematics collapses into a giant pile of logical nothingness. Ergo, imaginary numbers and mathematics cannot both be real.
- JadeNB 4y agoOf course it's a joke, but the problem with that argument isn't that imaginary numbers don't exist, but that the rule sqrt(a * b) = sqrt(a) * sqrt(b) doesn't hold for them. Or, rather, it does hold, but (as we should for real numbers, too!) we must regard sqrt(a) as being not a single number but a set of square roots of a (something we avoid doing in the real case by preferring non-negative numbers—a convenience not available for the complex numbers). With this understanding, the rule sqrt(a * b) = sqrt(a) * sqrt(b) holds, and your instantiation of it (with a = b = −1) shows that 1 and −1 are both equally valid square roots of 1, which is certainly true.
- prof-dr-ir 4y agoExcellent. Indeed, as you say already for the real numbers one could have chosen sqrt(9) = -3: after all, (-3)^2 = 9 just as much as 3^2 = 9. Taking into account this inherent ambiguity in the definition of the square root function becomes crucial when extending it to the negative numbers, as my example shows. So, long story short: although not everyone agrees, personally I prefer to define the imaginary numbers by means of an entity i that obeys i^2 = -1, and avoid talking about "square roots of negative numbers" as the (otherwise fine) article does.
- trinovantes 4y agoOnly if you ignore the implicit ± infront of sqrt
- amadeuspagel 4y agoThere's a nice video series by that title[1]. [1]: https://www.youtube.com/watch?v=T647CGsuOVU&list=PLiaHhY2iBX9g6KIvZ_703G3KJXapKkNaF https://www.youtube.com/watch?v=T647CGsuOVU&list=PLiaHhY2iBX...
- nitwit005 4y agoYou can always express some calculation that uses imaginary numbers in a way that avoids them. It might make things vastly more complex to so so, but it's possible. People used to do all sorts of math without negative numbers or zero.
- ndr 4y agoHow would you do sqrt(-1) in that case?
- nitwit005 4y agoThat's not something calculable. The symbol i is just a sort of placeholder for it.
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- XorNot 4y agoImaginary numbers don't answer that though: they assume it has an answer, and do operations anyway.
- audunw 4y agoOne of the ways to avoid imaginary numbers is to use matrices instead. You wouldn't exactly solve sqrt(-1). You'd solve the sqrt of the negative of the 2x2 identity matrix (-I) You can't solve sqrt(-1) without a second dimension. So even with imaginary numbers you kind of have an implicit -1+0i when solving sqrt(-1). Complex numbers is just a way of working with two numbers at once, just like 2 dimensional vectors and matrices. There nothing special to it, other than some syntactic shorthands.
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- drdec 4y agoIt's funny that the article uses quantum mechanics to argue that complex numbers are "real", since I would argue that quantum mechanics shows that irrational numbers aren't "real". Quantum mechanics belies the idea of space bring a continuum and so you cannot actually physically manifest numbers like sqrt(2) and pi, because Euclidean right triangles and circles do not actually exist.
- rcxdude 4y agoSpace is quite continuous in quantum mechanics, it's only the interactions between particles which are quantised (in fact the governing equations of quantum mechanics are almost all continuous. It's only when you try to solve them that quantisation appears, in a similar way that a guitar string is continous but only certain shaped standing waves can exist on it).
- edgyquant 4y ago> mechanics shows that irrational numbers aren't "real". Yep the ancient philosophical math debate about what real even means beyond some points on a grid.
- siraben 4y agoOne of my favorite places where imaginary numbers pop up is in solving recurrence relations[0]. For instance, the Fibonacci numbers are an example of a recurrence relation and you can obtain the closed form solution by solving the characteristic polynomial. However, you can also get a closed form solution involving imaginary numbers when the roots of the polynomial are imaginary, even though the sequence only contains real numbers! From that you can express the result completely with real numbers if you use the Euler transformation of complex numbers. [0] https://math.unm.edu/~jvassil/Recurrence%20Relations.pdf https://math.unm.edu/~jvassil/Recurrence%20Relations.pdf
- transfire 4y agoDubious. Any Computer Scientist can tell you that any Turing complete system can emulate any other such system. Perhaps the Quantum aspect of reality makes the difference, but I deeply suspect it’s the same.
- trainfromkansas 4y agoBefore I studied math, I always slightly resented imaginary numbers as being "math wankery" and just defined because mathematicians had a compulsion to generalize and define new nonsense because they could, and not because it made any sense to. On the way to changing my mind, I learned that the Fundamental Theorem of Algebra only works for complex numbers (and not "real" numbers), the beauty and simplicity of rotations in the complex plane, but maybe most convincing to me was a history lesson about quaternions. Quaternions are an extension of the complex numbers, but they're not typically taught in higher math education these days, which contradicted my resentment that mathematicians were just obsessed with getting more and more abstract and general for the sake of it. Of course, they were in vogue in the 19th century (Maxwell's equations were originally written down using them), but mathematicians soon realized they just weren't as useful or as "nice" philosophically as complex numbers and just about anything you can accomplish with quaternions were better accomplished with vectors of complex numbers. That story played a big part in persuading me that there really is something special about complex numbers -- that maybe they're the most "natural" or "real" numbers of them all.
- dontlaugh 4y agoQuaternions are actually often used in games to represent rotation, largely because they can’t gimbal lock.
- blooalien 4y agoBugger me… I just commented on that very thing, and then read a few more comments and find you beat me to the punch by several days. LOL
- DavidSJ 4y agoAlso, note the origin story of the complex numbers: Contrary to legend, they weren't discovered out of a desire by mathematicians to have roots to all quadratics such as x^2 + 1 = 0. It's perfectly sensible for an equation like that to just lack a solution: this just means the standard parabola never drops below zero. Analogously, if we calculate a rocket's payload mass to Low Earth Orbit and the answer is negative, we don't feel a need to find some deep meaning behind negative mass: we just say the rocket can't get to orbit at all. Simple. It's cubics for which complex numbers were introduced. Cubics (with real coefficients), unlike quadratics, always have real roots, since one arm goes to +∞ and the other to -∞, so it has to cross the x axis somewhere in between. But when the cubic formula was finally discovered, it had this strange property that you frequently had to take square roots of negative numbers, then add those weird square roots to "regular" numbers, and if you just shut up and calculated, the weird parts would always cancel out and you'd get a "regular" number that solved the original equation. That is, you had to pass through the complex numbers in order to find the real solutions.
- TeeMassive 4y agoAll measures are real numbers.
- mkl 4y agoIt's the other way around: real numbers are imaginary. All numbers are imaginary. Numbers are very useful concepts, and can be used in various ways to describe parts of reality. These descriptions can stop making sense if inappropriate kinds of numbers are used (e.g. -5 makes perfect sense as a displacement, but not as a count, and 5i makes perfect sense as part of a description of simple harmonic motion, but not as a displacement). The use of the word "imaginary" to describe parts of complex numbers is quite unfortunate [1], because it misleads people into thinking there's something fake or weird about them, leading to confusion and entire articles trying to convince people they're valid. They're a totally normal, expected, and widely used feature of number systems as we understand them today. The name they've got comes from how surprising and foreign they seemed to the people who first encountered them. Irrational numbers were similarly surprising and foreign to the people who first encountered them. Today we're perfectly comfortable working with irrational numbers like π, but the original reaction lives on in other meanings of "rational" and "irrational": based reasoning, and not based on reasoning, respectively. "Imaginary" seems like a tougher hurdle, and while the article says "imaginary numbers" many times, mathematicians and physicists don't really. We talk about complex numbers, and describe complex numbers with 0 real part as "pure imaginary". "Imaginary" and "real" are just labels for the axes of the complex plane, and neither is literal in this context. [1] As is the word "complex", which most commonly now means "complicated", but is used here with another meaning: made up of multiple parts.
- wruza 4y agothe original reaction lives on in other meanings of "rational" and "irrational": based reasoning, and not based on reasoning, respectively Doesn’t it simply mean “cannot be written as a ratio”? Edit: the parent is correct, they were called absurd numbers originally.
- mkl 4y agoI wasn't too clear in that sentence, but I mean the definitions about thinking. Colloquially: "rational" = "clear-headed", "irrational" = "crazy".
- TMWNN 4y ago
- ahmadmijot 4y agoThere were some suggestions to change 'imaginary numbers' to 'lateral numbers' but it is hard to do when the name stuck. Naming thing is hard.
- ubj 4y agoOne of my favorite examples of the reality of "imaginary" numbers is Euler's identity: e^{i \pi} + 1 = 0 Or even better, raising i to the ith power: i^i = e^{-\pi / 2} That looks pretty real to me.
- foxes 4y agoApart from the waffle the actual news [0] was that some people claim that they need complex quantum mechanics, quantum mechanics over C instead of just R. For some reason our universe might require complex structure. [0] https://physics.aps.org/articles/v15/7 https://physics.aps.org/articles/v15/7
- ardit33 4y agoEveryone needs a shift of perspective on this: Imaginary numbers are functions... not numbers! Functions are real if you can reproduce their output, even though you don't know the inward working of them.
- khana 4y ago
- amai 4y agoSee also https://news.ycombinator.com/item?id=29643921 https://news.ycombinator.com/item?id=29643921
- amai 4y agoUnfortunately this article repeats the same misconceptions about imaginary/complex numbers and their role in physics as most popular articles do. Imaginary numbers were invented too early, that is why the usual explanations following the historical development are so mysterious. The notion of Sqrt(-1) does not help to explain what they actually are. In fact imaginary "numbers" are not numbers. They cannot be ordered by size. Imaginary "numbers" are also not vectors in the geometrical sense. This is because you can multiply two geometrical vectors and get a scalar as result (scalar product). But if you multiply two imaginary "numbers" you get another imaginary "number", so this is different to a geometrical vector. In fact imaginary "numbers" are transformations or operators, because they multiply like operators/matrices and they act on geometrical vectors like rotation matrices. Group theory makes all that very clear: Imaginary "numbers" are just the elements of the group SO(2), the group of rotation operations in 2D (multiplied with a scaling factor). Regarding quantum mechanics what the scientists showed is, that a theory which does break SO(2) rotation symmetry (so called real quantum mechanics) does not describe nature. That should surprise no one as the SO(2) symmetry (or equivalently called U(1) symmetry (https://en.wikipedia.org/wiki/Circle_group#Isomorphisms https://en.wikipedia.org/wiki/Circle_group#Isomorphisms)) is necessary to describe conservation of the electric charge. (https://en.wikipedia.org/wiki/Charge_(physics)#Abstract_definition https://en.wikipedia.org/wiki/Charge_(physics)#Abstract_defi...)