[HN Gopher] Imaginary numbers are real
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Imaginary numbers are real
Author : Hooke
Score : 32 points
Date : 2022-07-23 17:22 UTC (5 hours ago)
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| unnouinceput wrote:
| Quote: "...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 wrote:
| Hah! 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 wrote:
| Of 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 wrote:
| Excellent. 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 wrote:
| Only if you ignore the implicit +- infront of sqrt
| riskneutral wrote:
| We don't even know if real numbers are "real."
| labster wrote:
| That's why I only use natural numbers. You can trust values
| from organic sources, like the number 5 and good old 23.
| joe__f wrote:
| Yeah, all this modern stuff is too rational for me
| mayoff wrote:
| Die ganzen Zahlen hat der liebe Gott gemacht, alles andere
| ist Menschenwerk.
|
| -- Leopold Kronecker
|
| ("God made the integers; all else is the work of man.")
| chowells wrote:
| Actually, 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.
| riskneutral wrote:
| > we know that the subset of R that can be used directly has
| measure zero
|
| The cardinality of the natural number set is the same as the
| cardinality of the rational number set. So, in some sense,
| you are saying that we know that the natural counting numbers
| are "real," which is a self-evident truth. In another sense,
| you are saying that fractions of an inch/cm/etc are "real"
| which is another self-evident truth.
|
| The question is whether uncountable infinity somehow exits in
| nature (in the case of real numbers, it is a question about
| the infinitesimally small scale). To say that it's a "polite
| fiction to make some proofs works" is too strong a statement,
| we do not know the answer to that question. At the same time,
| our measurement instrument will always be discrete and
| bounded, so the question is seemingly beyond science itself.
| chowells wrote:
| I'm not making a statement about the real world other than
| "it's impossible to communicate infinite information." I
| suppose that's a bit of a leap of faith, but I'm
| comfortable with it. Everything else I mean comes from
| math, with no connection to the real world.
| ChrisLomont wrote:
| We don't know that - some people posit that but it's far from
| being provable.
|
| All physically measured numbers have uncertainty, meaning the
| value obtained is not an actual number, but is a range,
| perhaps with some associated probability spread.
|
| This is not measure zero.
|
| We are also very capable "directly" using various things that
| may be continuums, such as energy, or time, or velocities, or
| many other physical quantities.
| bobbylarrybobby wrote:
| All the numbers we can use are computable (how can you use
| a number if you can't actually talk about it?) and there
| are only countably many of those.
| ChrisLomont wrote:
| >All the numbers we can use are computable
|
| That's not true :)
|
| A nice example is Chaitin's constant [1], which I can use
| in proof and books and define and on and on....
|
| And it's explicitly and most definitely NOT computable :)
|
| There are lots of numbers in lots of areas of
| mathematics, even symbolic mathematics that are not
| computable in the sense you want them to be computable.
| Chaitin's constant is the tip of a very big iceberg.
|
| You're using circular logic by claiming the only numbers
| I can are are the computable ones then claiming all
| numbers I can use are computable. That's not true. It's a
| circular argument.
|
| [1] https://en.wikipedia.org/wiki/Chaitin%27s_constant
| chowells wrote:
| No, I mean something simpler. The subset of R that can be
| identified is countable, because language is countable. No
| matter what system you devise for describing numbers, it
| will be countable. And that means that it cannot describe
| approximately 100% of the numbers in R. They can't be
| uniquely described; they can't be used in computations.
| They're phantoms, at best. You know they're out there, but
| they will never be usable the way numbers you can actually
| describe are.
|
| The only thing they give you is the ability to declare R to
| be Cauchy complete in some proofs. They're a polite
| fiction.
| ChrisLomont wrote:
| Yeah, I'm well aware of these style arguments ala Gregory
| Chaitin.....
|
| >The subset of R that can be identified is countable,
| because language is countable
|
| Define "identified" in mathematically precise terms
| without using circular reasoning - and there is the flaw
| in this line of claims. You will find such definitions
| miss common uses of real numbers in the same way no
| finite set of axioms catch all true statements about
| integers. You will maybe get a nice consistent math
| subset of the reals with measure zero, but it will not
| cover all the cases you want it to. Thanks Godel
|
| For example, using your "proof" of "because language is
| countable" would imply there can be no infinity, yet we
| use it all the time. Time, for example, may be a physical
| continuum, so a finite interval of time contains
| infinitely many time steps, and if your math cannot even
| represent physical reality then it's a pretty weak
| system.
|
| >You know they're out there, but they will never be
| usable the way numbers you can actually describe are
|
| Plenty of formal computational systems are capable of
| using the same set of numbers I can describe.
|
| You're conflating being able to list every number in use
| one at a time, and operating on sets of numbers, like
| computation has done almost since day 1 using interval
| arithmetic.
|
| >they can't be used in computations
|
| Interval arithmetic, formal proof system - both capable
| of using all the numbers as sets that I can use - well
| beyond (Lebesgue) measure zero.
|
| And to be pedantic, even countable numbers have infinite
| measure for certain measures :)
| avindroth wrote:
| Any source? Curious as to what is meant by "used directly"
| joe__f wrote:
| Probably just that, all physically measurable quantities
| are rational, and the rational numbers have measure zero in
| the reals
| ChrisLomont wrote:
| >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.
| bobbylarrybobby wrote:
| I'm assuming they mean that for a number to be used, it
| must be computable.
| chowells wrote:
| Source: |R| > |N| = |any string of symbols in any notation
| that has existed in the past, exists now, or will exist in
| the future|
| exabrial wrote:
| Isn't i just a rotation about the axis of origin?
| unsafecast wrote:
| It 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.
| superb-owl wrote:
| I really wish we'd just call them "rotation numbers" or
| something. "Imaginary" was always a ridiculous name.
| pvg wrote:
| It's no more or less ridiculous than 'negative', 'real' or
| 'odd'.
| kzrdude wrote:
| Yes, 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?
| HPsquared wrote:
| Negative numbers are imaginary enough already, really.
| [deleted]
| Chinjut wrote:
| "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 wrote:
| Direct, inverse, and lateral are so much more productive than
| the labels we usually use. Can't believe I only ran into it
| now.
| idoubtit wrote:
| You 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 wrote:
| > _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 - 2
| _i_ represents a scaling by [?]5 combined with a rotation
| clockwise by arctan(2). The complex number 8 _i_ 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
| tasty_freeze wrote:
| How about "Cartesian numbers"? But my hunch is that somewhere
| someone has already taken that name for something else.
| [deleted]
| alanbernstein wrote:
| I kind of like "geometric numbers".
| [deleted]
| tut-urut-utut wrote:
| alangibson wrote:
| Pointlessly obnoxious comment of the day right there
| macksd wrote:
| Pretty 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.
| [deleted]
| alangibson wrote:
| Numerical names are like names for musical notes: they make no
| sense anymore but it's not worth the effort to change them.
| cal85 wrote:
| What names? Aren't musical notes called A, B, C etc?
| hyperhopper wrote:
| Do re mi fa so la ti do?
| pygy_ wrote:
| "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 :-)
| [deleted]
| messe wrote:
| Maybe they're referring to crotchets, (semi) quavers, minims,
| (semi)breves?
| foobarbecue wrote:
| Apparently those names were indeed worth the effort to
| change, since we did that here in the USA.
| sseagull wrote:
| Maybe they meant scale degrees: Tonic, supertonic, mediant,
| subdominant, dominant, submediant, leading tone.
|
| Or modes: ionian, dorian, phyrgian, lydian, mixolydian,
| aeolian, locrian.
| pygy_ wrote:
| Adding to the other replies, there's an off by one error in
| the interval names.
|
| The unison is an interval of zero notes, the second an
| interval of one note, etc...
| RicoElectrico wrote:
| At this point "imaginary numbers are not real" sounds like a
| strawman.
| LgWoodenBadger wrote:
| Can another imaginary number do the same thing for division by
| zero, or am I getting the cause/effect backwards?
| jacobolus wrote:
| You 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'e, and have the rule that e2 = 0.
|
| https://en.wikipedia.org/wiki/Dual_number
|
| 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
|
| * * *
|
| 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_lin...
|
| https://en.wikipedia.org/wiki/Extended_real_number_line
| kragen wrote:
| Adding some kind of infinitesimals to the reals can be
| useful, as is done with hyperreal numbers, dual numbers, and
| other kinds of hypercomplex numbers, but it doesn't allow you
| to _divide_ by zero; there is no hyperreal or hypercomplex
| number _x_ that solves _x_ = 1 /0.
|
| As I understand it, if you add 1/0 = [?] you can't treat that
| [?] as a quantity in the usual algebraic ways. In particular
| you can't multiply [?] by 0 and get back some well-defined
| quantity such as 1; if you allow that, you quickly find that
| you can prove that all finite numbers are equal. The standard
| tricky example of this is in https://www.math.toronto.edu/mat
| hnet/falseProofs/first1eq2.h...:
|
| Let _a_ = _b_.
|
| Then _a_ 2 = _ab_ , _a_ 2 + _a_ 2 = _a_ 2 + _ab_ , 2 _a_ 2 =
| _a_ 2 + _ab_ , 2 _a_ 2 - 2 _ab_ = _a_ 2 + _ab_ - 2 _ab_ = _a_
| 2 - _ab_ = 2( _a_ 2 - _ab_ ) = 1( _a_ 2 - _ab_ ).
|
| If we then divide both of these last expressions by _a_ 2 -
| _ab_ we get 2 = 1. This is only invalid because _a_ 2 - _ab_
| = 0. Adding a different [?] _[?]_ for each value of _n_ /0,
| as lisper suggests, doesn't help.
|
| So, if you want to extend your number system with 1/0 = [?],
| you either need to throw out some of the standard laws that
| permit algebraic manipulations like that, or you end up with
| all finite quantities being equal.
| lisper wrote:
| The problem with any finite number of infinities is that you
| can't maintain the identity (a/b)*b = a. To make that work
| you need a different number to represent a/0 for every value
| of a, including all of your infinities, so you end up with an
| infinite hierarchy of infinities.
| jacobolus wrote:
| You already can't maintain the identity (a/b)b = a when you
| have a zero element, except by declaring "division by zero
| is illegal". Adding [?] is not fundamentally different, but
| just takes a slightly modified list of exceptions.
|
| In many contexts this is worth the hassle.
| lisper wrote:
| Well, yeah, but the _whole point_ of inventing a new kind
| of number to represent a /0 analogous to inventing a new
| kind of number to represent sqrt(-1) is to eliminate
| these you-are-not-allowed-to-do-that kind of exception.
| With sqrt(-1) it turns out to be straightforward. With
| a/0 it isn't, which is why sqrt(-1) is a thing in math
| and a/0 isn't.
| jacobolus wrote:
| For me, the whole point of modifying the number system is
| modeling different kinds of structures. Both complex
| numbers and projective numbers are interesting in their
| own right and also useful for making models of both
| physical reality and other kinds of mathematical
| patterns. Each system has its own internal logic with its
| own quirks.
| [deleted]
| wrnr wrote:
| No, but there are algebraic structures that allow for this,
| like the Riemann sphere. The proper way to talk about this is
| the concept of "zero dividers". For Z the only zero divider is
| zero, 0/0=0.
|
| The closest thing to what you describe is the the dual number
| (that together with the imaginary and hyperbolic numbers make
| the geometric numbers), which has zero dividers, and is defined
| as k^2=0 (where k is not in R).
|
| This is a very interesting number and it can help clean up a
| lot of problems when using complex numbers.
| yk wrote:
| No, or at least not in a field. From the field axioms you can
| directly proof that 0 _x=(a-a)_ x = ax - ax =0, and therefore 0
| doesn't have a well defined inverse. You can look at other
| algebraic structure, but those behave less like numbers.
| vletal wrote:
| Yeah. 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.
| game-of-throws wrote:
| I had to laugh at the title. Imaginary numbers may be real, but
| they're also outside the set of real numbers R.
| [deleted]
| [deleted]
| xyzal wrote:
| Sabine Hossenfelder did a nice video on the topic.
|
| https://www.youtube.com/watch?v=ALc8CBYOfkw
| amadeuspagel wrote:
| There's a nice video series by that title[1].
|
| [1]:
| https://www.youtube.com/watch?v=T647CGsuOVU&list=PLiaHhY2iBX...
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