[HN Gopher] How imaginary numbers were invented [video]
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How imaginary numbers were invented [video]
Author : peter_d_sherman
Score : 138 points
Date : 2021-11-11 07:01 UTC (1 days ago)
(HTM) web link (www.youtube.com)
(TXT) w3m dump (www.youtube.com)
| dexwiz wrote:
| Real numbers are for straight number lines, and imaginary numbers
| are for circular number lines. That is about all there is to it.
|
| It's really a shame we introduce imaginary numbers as quirky work
| around for negative square roots. I would say that is more of a
| side effect. Multiplying positive numbers always results in
| positive numbers, but due to the circular nature of imaginary
| numbers you can get positive or negative results.
|
| Complex roots are like saying this equation has no solution in a
| traditional Cartesian/Euclidean world, but does have solutions in
| a rotational one.
|
| I think we if introduced imaginary numbers as rotations
| initially, the seemingly magical things like Euler's Identify and
| Formula would look much more trivial.
|
| EDIT: I would also say the lie that we live in a 3+1D world (X,
| Y, Z + Time), helps confuse the mater. We probably live in a 6D +
| 1 world: X, Y, Z, Pitch, Yaw, Roll, and Time. Just as you can
| conceive of a flatland 2D universe without a Z axis, you could
| conceive of a 3D world without rotational orientation.
| mmcdermott wrote:
| I really like this explanation. The negative square roots
| explanation never sat right with me in high school and I think
| this touches on why.
| dls2016 wrote:
| > introduced imaginary numbers as rotations initially
|
| How would you do this without developing the exponential first?
| adamnemecek wrote:
| I think that i is not interesting because i^2 = -1 but because i
| * conjugate(i) = 1.
| jonsen wrote:
| It's the same thing i * i = -1 <=> -i * i = 1
| adamnemecek wrote:
| I know but the semantics are different.
| jonsen wrote:
| In what way does multiplying an equation with -1 change its
| semantics?
| nabla9 wrote:
| That was one of the most interesting 20 min history lessons I
| have ever watched.
| VHRanger wrote:
| Veritasium always makes great stuff
| satori99 wrote:
| I wish I was taught this history when I was taught the math.
|
| My schooling provided next to no historical context when
| learning mathematics. My math teachers just taught the math,
| and my history teachers were probably largely unaware of math
| history.
| ralusek wrote:
| > Imaginary numbers exist on a dimension perpendicular to the
| real number line.
|
| Correct me if I'm wrong, but I don't think that imaginary numbers
| exist in a dimension with any sort of fixed spatial relation to
| real numbers. Isn't the fact that we chose to graph the imaginary
| axis perpendicular to the real axis on the complex plane simply
| arbitrary? Or maybe better described as "a functional way to
| visually plot the negative root component of a number."
|
| My understanding is that visualizing the imaginary portion of
| complex numbers on a spatial plane with perpendicular axes also
| just happened to get a convenient 90 degree rotation by
| multiplying by i, and rotations like this just happen to show up
| all over nature. For example, the wave function they mention in
| the video, where they say it has an "imaginary component," seems
| like complete bullshit to me. You could write a program with real
| numbers in order to graph that wave...in fact I guarantee that's
| how a technical artist visualized it. In fact, it's the same with
| the Mandelbrot set, which supposedly "exists on the complex
| plane." No...it doesn't. It just happens that if you describe the
| arbitrary rotation operation that is performed in order to define
| whether or not a coordinate falls within the Mandelbrot set or
| not, the rotation from the way we've chosen to graph the complex
| plane means the equation can be written very concisely like this.
| AnotherGoodName wrote:
| Every independent variable should be graphed perpendicular. X
| and Y when graphed are perpendicular. If they are not graphed
| perpendicular it indicates that a change in X position must
| change Y (or vice versa) whereas being perpendicular makes it
| clear that they are independent and you can change one without
| any thought that you're changing the value of the other.
|
| Complex numbers have 2 independent variables. The real and the
| imaginary parts. You can change one part without being forced
| to change the other. So you graph them perpendicular.
|
| It's not really a convention, it's a rule that each independent
| variable is perpendicular (or else they aren't independent!).
| Complex numbers have 2 independent variables.
| ralusek wrote:
| Well yes, if we want to plot any sort of independent values,
| we have 3 spatial axes we are familiar with and will use them
| to graph accordingly. But I'm not going to argue that
| "housing prices therefore exist on an axes perpendicular to
| square footage." I just plotted the relationship that way.
| AnotherGoodName wrote:
| I think focusing on the perpendicular doesn't make much
| sense since of course we graph different variables
| perpendicular.
|
| The real point i think you're making is that complex
| numbers are just a notational shorthand for
| multidimensional vectors. And yeah. That's true.
|
| https://xkcd.com/2028/
| cool_dude85 wrote:
| There are valid reasons to plot variables that aren't
| perpendicular. Maybe you want a plot of the radius of a
| cylinder and its volume, for example. Then a change in
| the x axis changes the value of the y axis and the axes
| wouldn't be drawn perpendicular.
| zsmi wrote:
| This is why it's fun to know the history of math.
|
| The complex plane was introduced by Caper Wessel in a
| paper that was published in 1799 so it would've existed
| before vector notation.
|
| My guess is by the time vectors got popular the complex
| notation, and theorems that people had proved which used
| complex notation, had already stuck. But I'm only a hack
| math historian so I can definitely be wrong here.
|
| I think it's important to keep in mind that math and
| science, much like the code base that I am trying my
| hardest to avoid, is evolved.
| rhdunn wrote:
| Dimension in this sense means degree of freedom, not a spatial
| dimension. That is, a number on the "imaginary dimension"
| cannot be represented on the "real dimension". This is similar
| to how a length does not represent a width or height.
|
| When visualizing the real and imaginary dimensions, it is
| convenient to represent them using x/y axes. However, some
| visualizations of complex functions use colour/hue as a way of
| representing the real or imaginary part of the result.
|
| Dimensional analysis is used in physics, etc. when manipulating
| fundamental units (time, length, luminosity, etc.). That is,
| m/s^2 has dimensions length=1 and time=-2.
| ralusek wrote:
| Right, my point is that length and time don't exist
| perpendicular to each other simply because there might be
| some utility in visualizing them that way. The complex plane
| seems to be generally taught as if imaginary numbers in some
| capacity exist on this perpendicular axis. Even Veritasium,
| who is supposed to be teaching deconstruction/first
| principles approaches to these concepts, is saying this.
| rhdunn wrote:
| The spatial unit vectors for a 3 spatial dimension [x,y,z]
| vector are i^=[1,0,0], j^=[0,1,0], and k^=[0,0,1]. These
| are all said to be orthogonal to each other, which is a
| generalization of perpendicularity [1] to non-spatial
| vector spaces.
|
| As a lay person who is not familiar with the mathematics
| would recognize the term perpendicular, using that in the
| video is fine.
|
| [1] https://en.wikipedia.org/wiki/Orthogonality
| pjbk wrote:
| That is correct. In the Clifford formulation that is equivalent
| (and in general a superset) of complex numbers you can choose
| the dimension or behavior to be whatever you want it to be in
| the resulting algebra. You can even assign it a particular
| metric or nullify it. The 'square to -1' that implies
| quadrature is just a specialization in that case. Many
| algebraic things of complex numbers like the exponential map
| still work or have general counterparts.
| zuminator wrote:
| I disagree, it's not simply arbitrary. Euler's formula
| ex=cos(x)+i[?]sin(x) exists outside of any physical
| representation of the complex plane. And the fact that we
| already graphed trigonometic functions lent itself naturally to
| a corresponding graph of complex functions. As sibling comments
| have stated, any graphical visual representation of completely
| independent variables is most efficiently represented with
| perpendicular axes, so the mapping is inevitable.
| VHRanger wrote:
| I'd say mathematical concepts are "discovered" rather than
| "invented". Even something like complex numbers.
| Grustaf wrote:
| I'd say _especially_ complex numbers. Few things seem more
| natural once you get used to them and see how they make
| everything fall into place.
| chas wrote:
| If you are not familiar, this is part of a very long-running
| discussion in the philosophy of mathematics:
| https://plato.stanford.edu/entries/philosophy-mathematics/#F...
| (For context on the intro to that section, Platonism is,
| roughly speaking, the idea that mathematical objects truly
| exist and mathematicians are discovering them)
| mxwsn wrote:
| I wonder if one can place mathematical concepts on a spectrum
| from discovery to invention? To me, the pythagorean theorem
| feels much more like a discovery of a "hidden" eternal truth
| that was once beyond our grasp. But to me, complex numbers seem
| more like a notational choice, more akin to an invention. To
| put it another way, I would expect aliens to have the
| pythagorean theorem, but perhaps different notation for complex
| numbers, like representing them with matrices and computing
| over them with linear algebra. Such a representation would
| reduce the distinction of the "imaginary axis" which is really
| not that special.
|
| I may be naive; I haven't worked or thought with complex
| numbers since completing my basic education.
| gizmo686 wrote:
| Definitions are invented, theorems are discovered.
|
| The Pythagorean theorem is a discovery about Euclidean
| geometry. But in order for that discovery to be meaningful,
| one must first invent Euclidean geometry, or at least
| something sufficiently similar to Euclidean geometry.
| count wrote:
| Pure math concepts like complex numbers are not naturally
| existing, just waiting for us to find them, they're human-
| defined tools to describe things. Like new words, they're
| invented. They wouldn't be there without us, as they are, for
| the most part, artifacts of our cognition.
| jonbronson wrote:
| That's self-evidently untrue. The properties that make a
| circle would be true regardless of whether a human ever set
| eyes on a perfect circle. Us identifying those properties is
| an act of discovery via research. Codifying those
| mathematical truths into a written notation is the only
| component of the process that could really be called
| invention.
| UncleMeat wrote:
| But we still chose the labels. We chose what the elevate to
| the level of a mathematical object. Heck, even the idea of
| "what is true" is not universal in mathematics.
| Intuitionist and classical logic have different ideas of
| what. it means.
| b3kart wrote:
| No human has ever set eyes on a perfect circle, because it
| (most likely) doesn't exist in nature. As such, I would
| argue that a perfect circle is a concept (or a _model_ , if
| you will) that we've _invented_ to make it easier for us to
| deal with an imperfect real world. I would not call
| identifying properties of such a model an act of
| _discovery_ : one can come up with any model, no matter how
| far from reality, and use some set of axioms to identify
| its properties, but none of it will make said model real or
| fundamental in any way. The best we can hope for is that
| the model will be _useful_ for making predictions about the
| real world.
| moffkalast wrote:
| Well it really depends on how far you push the definition.
| There are inherent properties that can be discovered, but
| the way they're calculated and described is purely
| arbitrary. You need to get the same result of an area of a
| circle in the end, but how you get there is invented. Far
| more feasible and evident for complex stuff than the basics
| of course.
|
| As shown in the video, the depressed quadratic was
| basically solved by 3 people in 3 different ways, with
| today's description and definition being different from
| that too.
| jhedwards wrote:
| It's not self-evidently untrue, this is an incredibly
| complex philosophical question with no easy "right answer".
| There is also a spectrum of positions about this.
|
| The Stanford Encyclopedia of Philosophy is a good resource
| for reading about this topic:
| https://plato.stanford.edu/entries/platonism-
| mathematics/#Ob...
|
| Edit: another good link:
| https://plato.stanford.edu/entries/nominalism-mathematics/
| pohl wrote:
| If that were the case, would you expect another civilization
| on some rock in different galaxy to arrive at entirely
| different concepts, or ones isomorphic to our own?
| anchpop wrote:
| I'd expect other civilizations to have also invented
| rockets, microwave ovens, radio communications, and more.
| Does another civilization arriving at the same thing as us
| mean it isn't an invention?
| robotsteve2 wrote:
| Their observations of nature and their ability to predict
| stuff should be consistent with ours.
|
| They might not use the same mathematical tools or the same
| physical models, but they should make the same predictions.
| That is, we might not be able to understand their theory of
| gravity, but whatever their theory is, it has to be able to
| make predictions about orbits, black holes, etc.
|
| I don't think we could assume much about the mathematical
| concepts they use beyond that.
| delecti wrote:
| That's a pretty hard question to answer, and I think
| impossible to answer definitively (at least unless we met a
| civilization that did arrive at different concepts). It's
| kinda like the allegory of the cave; it's hard to envision
| another way of looking at the world, but that doesn't
| necessarily mean it's impossible for there to be one.
| JackFr wrote:
| It's a good question to think about.
|
| A constant refrain of mine is that our brains have
| convinced us that they're universal understanders, but we
| really don't know that to be true.
|
| Imagine the difficulty of dealing with aliens who have a
| different mapping of the physical universe, different
| than our mathematics, which is both true but literally
| and physically incomprehensible to us.
| zsmi wrote:
| I really enjoy looking into the history of mathematics and
| physics. I think it gives one a much better appreciation of why
| things are defined the way they are, and also the limitations of
| those definitions.
|
| There is a really great book on the history of imaginary numbers.
| The history mostly focuses on how i was used to help solve
| algebra problems, so definitely one should be comfortable with
| high school algebra to get something from the text, but I don't
| think one needs much more math than that for the first half of
| the book. The second half gets more into how various use cases
| developed, in those chapters basic college level calculus would
| be a major plus. I read it more than 10 years ago though so no
| promises. :)
|
| An Imaginary Tale: The Story of [?]-1 Paul J. Nahin
|
| https://press.princeton.edu/books/paperback/9780691169248/an...
| gregfjohnson wrote:
| A Danish cartographer named Caspar Wessel came up with an early
| formal treatment of complex numbers, in his work "On the
| Analytical Representation of Direction" (wikipedia has a nice
| article about him). It was published in an obscure forum, and
| predates subsequent rediscovery of complex numbers by others. His
| formulation is IMHO beautiful, intuitive, and compelling. He did
| it in terms of directions on a map, replacing the "sign" of a
| conventional real number with a "direction" or "compass heading".
| So, one might say, "the nearest Starbucks is two blocks east and
| one block north". He was simply using what became known as the
| polar form of complex numbers. One can follow intuition and
| define reasonable notions of addition and multiplication by real
| values. But what of multiplying two "Directions"? Wessel derived
| what multiplication must mean, and went further in deriving a
| large number of identities involving his newly discovered
| "directional numbers".
|
| If you pick a specific important pair of directional numbers, the
| multiplicative identity (call it "1") and a number 90 degrees
| away from it (call it "i"), it is convenient to represent any
| directional number as a the sum of scalar multiples of these two
| numbers. Then, one considers the simple formula "(i + 1)(i - 1) =
| i^2 - 1". A straightforward geometrical argument demonstrates
| that i^2 must be equal to -1. ("Show HN":
| gregfjohnson.com/complex)
| cool_dude85 wrote:
| I've never liked the framing of "imaginary" numbers as "not
| reflecting reality" or somehow being less "real" or something.
| It's just an ordered pair of numbers with a convenient extension
| of multiplication that preserves nice properties of the reals.
|
| I think it's because of the name. If you called them double
| numbers or paired numbers nobody would say that.
| 1970-01-01 wrote:
| In college, our professor said "We should be saying i is for
| invisible numbers, because you won't see them unless you know
| about them. But they're as real as death and taxes."
| dls2016 wrote:
| > If you called them double numbers or paired numbers nobody
| would say that.
|
| I'm not sure that's true.
|
| Just think of solving x^2+k=0. It's clear that for k<0 you get
| two solutions and k=0 you get one solution. But when k>0 the
| graph doesn't touch the x-axis... so why should I expect a
| "double number" or "paired number" to be the solution?
|
| I'm teaching college algebra right now and introduce 'i'
| algebraically as a solution to x^2+1=0... but then we talk
| about graphing quadratics and there's no simple connection
| between the geometry/graph and the algebra.
|
| Even if I had the time to talk about the geometry of
| multiplication and such, it's still a big leap to the graph of
| z^2+1 and its roots in the complex plane.
|
| And it's this leap which, IMO, makes them seem "not real".
| lordnacho wrote:
| > I'm teaching college algebra right now and introduce 'i'
| algebraically as a solution to x^2+1=0... but then we talk
| about graphing quadratics and there's no simple connection
| between the geometry/graph and the algebra.
|
| Take an equation like x^2 -2x + c
|
| When C is some large negative number, the roots are symmetric
| about x = 1. As you increase C, roots are real until C = 1,
| basically the two roots meet in the middle.
|
| When you increase it beyond 1, the roots become complex
| numbers, but they stay symmetric, they "lift off" from being
| real into being complex, but still symmetric (conjugates)
| where 1 is always the real part but the imaginary parts
| become sqrt(C - 1). You can conveniently imagine the complex
| plane on top of the real XY plane and the roots go orthonogal
| to the direction they were going when they were real, from
| the point where they met.
|
| That's kind of how I visualize it for quadratics. For higher
| orders you cut a complex circle into n equal pieces. Haven't
| quite figured it out in my head yet.
| dls2016 wrote:
| My point is that we step off the real line to talk about
| the points where the function vanishes... but what about
| the neighborhood of the roots where we're plugging complex
| numbers into the function and _not_ getting a real number
| as output?
|
| There's a big conceptual leap here.
| lordnacho wrote:
| Oh yeah, Veritasium has some great visualizations of that
| too. I cant remember what the video was called but it was
| about fractals and the Newton Raphson method, and he had
| these input/output complex planes left and right to
| illustrate it.
| dls2016 wrote:
| Sure we can locally visualize a conformal mapping. But in
| the historical development of complex numbers, or
| pedagogically in a college algebra class... this would be
| putting the cart before the horse.
|
| Again why would anyone posit that a _pair_ of numbers
| suddenly appears when trying to solve x^2+k=0 as k goes
| from negative to positive?
| sigstoat wrote:
| > I think it's because of the name.
|
| indeed. the name is terrible; causes a lot of folks
| consternation, or tricks them into thinking strangely about the
| complex numbers.
|
| anyone happen to know of any languages where they're not
| referred to as "imaginary", or anything that implies they're
| less "real"?
| mathnmusic wrote:
| Gauss complained about the name too. He suggested using
| "Direct, inverse and lateral numbers" instead of "positive,
| negative and imaginary numbers".
|
| Imaginary numbers are not pairs. Complex numbers are.
| Continuing Gauss, I'd rename "complex numbers" as "planar
| numbers".
| ogogmad wrote:
| Dual numbers and double numbers are also planar.
|
| Don't know how relevant this is, but I've been thinking about
| a better naming scheme for hypercomplex number systems. I
| came to it after seeing a paper about the "dual-complex
| numbers", which are _not_ a straightforward complexification
| of the dual numbers as one might expect. Hopefully, the
| scheme should be pronounceable, and without the possibility
| of confusing it for something else. This town needs law! I 'm
| thinking of asking for suggestions for what it should be
| exactly.
| agumonkey wrote:
| wow, `naming is hard` is way older than I thought
| Natsu wrote:
| I kinda think of them as 'orthogonal' numbers.
| carlmr wrote:
| That I think is the best alternative name I've heard yet.
| Since you describe two orthogonal dimensions.
|
| Also vector numbers could be somewhat of a useful name,
| since they behave like a 2D vector (or even higher
| dimensional vectors for e.g. quaternions)
| ogogmad wrote:
| "Vector numbers" for something purely 2D seems dodgy to
| me. There are lots of unital algebras, even in 2 and 4
| dimensions, and none of them should hog the name "vector
| numbers" as they all have equal entitlement to the name.
| pcrh wrote:
| Agreed. It's a similar semantic problem to the use of the term
| "significant" in statistics. P<0.05 as being "significant",
| while P>0.05 as "non-significant" has a technical meaning that
| doesn't equate to the common use of the term "significant".
| b0rsuk wrote:
| Imaginary numbers are just as imaginary as negative numbers. To
| disprove my claim, post a photo of -4 cats.
| pphysch wrote:
| The innovation of "imaginary" numbers is that we can concretely
| reason about multiple dimensions in a unified manner (unlike
| classical geometry, which relies on geometric primitives). `i` is
| not some magic, "imaginary" value, it is an _invented_ syntax
| that means something like "rotate by tau/4 radians" or
| "orthogonal to the default vector". It turns out that physical
| problems are much more accurately modeled in 2 dimensions than in
| 1 dimension, and even more so in higher dimensions.
|
| > Freeman Dyson: "Schrodinger put the square root of minus one
| into the equation, and suddenly it made sense ... the Schrodinger
| equation describes correctly everything we know about the
| behavior of atoms. It is the basis of all of chemistry and most
| of physics. And that square root of minus one means that nature
| works with complex numbers and not with real numbers."
|
| This quote is emblematic of the mysticism that some
| mathematicians and academics cannot resist using to advance their
| careers as public intellectuals. Reality is certainly not based
| solely in "real numbers" (one dimension), but nor is it based
| solely in "complex numbers" (two dimensions). The idea that
| mathematics is "the language of the universe" that can be
| precisely "discovered" by brilliant minds is a ridiculous notion
| that only serves the status of the mathematical elite.
| Mathematics is fundamentally about _designing_ models and
| abstractions that _help_ us reason about real phenomena with
| minimal cognitive resources. Everyone does it, and anyone can do
| it. Disclaimer: I have a degree in mathematics.
|
| "All models are wrong, but some are useful" [1]
|
| [1] - https://en.wikipedia.org/wiki/All_models_are_wrong
| leoc wrote:
| It is a bit odd that the video really talks up the idea that
| complex numbers sever algebra from geometry, then without pause
| goes straight into a geometric interpretation of complex
| numbers https://youtu.be/cUzklzVXJwo?t=1153 .
| [deleted]
| cmehdy wrote:
| I thought the discussion between "mathematics only models the
| world" vs. alternatives wasn't settled?
|
| (most recently I'm referring to Roger Penrose's views on the
| "epistemic argument against realism"[0], although I don't fully
| know where I stand myself)
|
| [0]
| https://www.wikiwand.com/en/Philosophy_of_mathematics#/Epist...
| ectopod wrote:
| > "rotate by tau radians"
|
| tau/4 radians?
| pphysch wrote:
| Thanks!
| agumonkey wrote:
| I've always found the geometric interpretation as easy as
| useful but felt I was not grasping the whole thing because it
| was too cute and easy.
| alphanumeric0 wrote:
| Yes, 'imaginary' numbers are invented syntax. I think imaginary
| numbers are just an additional tool tacked on to our existing
| tools in order to explore new areas. They are a patch written
| to cover some cases our previous tools failed to cover.
|
| > The idea that mathematics is "the language of the universe"
| that can be precisely "discovered" by brilliant minds is a
| ridiculous notion that only serves the status of the
| mathematical elite. Mathematics is fundamentally about
| designing models and abstractions that help us reason about
| real phenomena with minimal cognitive resources.
|
| Yes, creating abstractions and designing models are aspects of
| math, but I believe the modern definition of mathematics has
| expanded to include the study of abstract objects.
|
| Abstractions can describe concepts (concepts that exist in many
| different places) but are not the same as the concepts. I
| believe these concepts are discoverable and independent of any
| notations/models/abstractions we create. So in that sense, I
| believe mathematics can be discovered.
|
| e.g.
|
| exponential growth/decay (the spread of COVID in an
| unvaccinated population, bacterial growth, etc.) - invented or
| discovered?
|
| (https://en.wikipedia.org/wiki/Exponential_growth#Examples)
|
| fractals (pattern of rivers, trees, blood vessels, etc.) -
| invented or discovered?
|
| I can describe the details of these with tools like 'geometric
| progression', and 'the Mandelbrot set'. Those are tools aiding
| my understanding of these concepts, but the concepts themselves
| certainly seem like they were discovered.
| ogogmad wrote:
| Historically, the complex numbers were discovered by accident
| while solving cubic equations using the general formula. The
| general formula only works in general if you can take the
| square root of a negative number.
|
| One of the first attempts to provide a geometric meaning to the
| complex numbers was by John Wallis, and I haven't been able to
| make much sense of it. I suspect he didn't see it the way we
| do. Also, there's no indication that in spite of the work Euler
| did on the complex numbers, that he knew of their geometric
| meaning. The mystical sounding name "imaginary number" was
| coined by Descartes in the 1600s at least partly because he
| didn't have the modern view of them.
|
| Of course, teaching by explaining historical developments is
| not a common thing in mathematics, and the above facts
| illustrate why. But you have to be aware that things do start
| off being mysterious before they're fully understood.
|
| The mystery has been somewhat reawakened with the quaternions
| and octonions, and some other hypercomplex number systems. And
| mystery gets some people out of bed, so don't be too hard on
| it.
|
| > It turns out that physical problems are much more accurately
| modeled in 2 dimensions than in 1 dimension, and even more so
| in higher dimensions.
|
| Are you talking about matrices?
| pphysch wrote:
| > Are you talking about matrices?
|
| Yeah, matrices turn out to be a better general representation
| than adding more ambiguous symbols beyond `i`. Still not the
| "correct" one by any means, because e.g. they don't represent
| exotic (non-integer, etc.) dimensions well.
| [deleted]
| teleforce wrote:
| In mathematics in addition to real and complex numbers there
| are higher dimensional numbers namely hypercomplex numbers i.e
| quaternion and octanion numbers, and no other valid numbering
| systems beyond that. The latter hypercomplex numbers are better
| at accurately model reality compared to the former real and
| complex numbers inherent limitations. As an example,
| electromagnetic waves unlike sound waves has polarization
| component that can be easily modeled using quaternion numbers
| compared to just using imaginary numbers. Yet for centuries
| until now scholars have been reluctant to use quaternion
| numbers and insisted on conventional imaginary numbers
| technique for Maxwell's Equations electromagnetic solutions
| popularized by Heaviside rather than the quaternion techniques
| being used by Maxwell himself! History is repeating itself as
| our current scholars are very much reluctant to use
| hypercomplex numbers, not unlike our predecessors who were
| reluctant in using imaginary numbers although the advantages of
| the higher dimensional numbers are plain obvious.
| pjbk wrote:
| > in addition to real and complex numbers there are higher
| dimensional numbers namely hypercomplex numbers i.e
| quaternion and octanion numbers, and no other valid numbering
| systems beyond that.
|
| To be precise, no other multi-dimensional numbers beyond
| octonions (which lack associativity) that follow the behavior
| of _complex_ numbers. There are indeed an infinite amount of
| multivector numbers with different algebraic rules.
| BeetleB wrote:
| Nitpick: Anything in reality that can be modeled with complex
| numbers can be with real numbers. Complex numbers provide
| incredible _convenience_ , but that's all. There's no
| phenomenon that _requires_ complex numbers to explain -
| Schrodinger 's Equation included.
| echopurity wrote:
| If there is such a thing as "mathematical elites", they don't
| benefit from mystical nonsense but strongly oppose it.
|
| This nonsense thrives because mathematical education is
| completely divorced from actual mathematics and mathematicians.
| morokhovets wrote:
| I cannot agree with calling complex numbers just two
| dimensional.
|
| Function of a complex variable is very different from a
| function of two variables. You can say these are two different
| departments of mathematics.
|
| Real numbers are not algebraically complete, but extending it
| with 'i' makes it complete. Adding another dimension to go to
| 'two dimensions' does not do anything like this.
|
| Mathematicians are fascinated with complex numbers because it
| is THE extension of real numbers that completes them in very
| important sense but it comes with so many unexpected and
| fascinating properties.
|
| Quantum phase is not two-dimensional, it is complex and it
| amazes me much much more than two-dimensionality would.
| iamcreasy wrote:
| > I cannot agree with calling complex numbers just two
| dimensional.
|
| I once read, imaginary number is isomorphic to 2d vectors.
| Upitor wrote:
| Isomorphic as vector spaces, meaning that their additive
| structure is the same. But the complex numbers are usually
| not used as a vector space, but rather as an algebraic
| field, i.e. considering both their additive and
| multiplicative structure.
| marginalia_nu wrote:
| They are very much equivalent. You can express complex
| numbers with a 2x2 matrix of real numbers. R
| = 1 0 0 1 I = 0 1
| -1 0
|
| I and R form a basis that spans something that behaves like a
| complex plane.
|
| You have the expected identities
|
| R^2 = R
|
| IR = I
|
| I^2 = -R
|
| I^3 = -I
|
| I^4 = R
|
| Transposition is complex conjugate. You can put them into
| exponentials, e^Ix = cos(x) + I sin(x); everything works as
| you would expect.
| pphysch wrote:
| Two-dimensional does not necessarily mean two spatial
| dimensions. It is dimensionality, or orthogonality, in the
| most abstract sense. For all practical purposes, complex
| numbers represent a two-dimensional system.
|
| It is not a coincidence that they arose in 16th century Italy
| in the context of "completing the square" and related 2D
| methods/intuitions for solving equations.
| morokhovets wrote:
| > For all practical purposes, complex numbers represent a
| two-dimensional system
|
| Yet operations on complex numbers are not the same as
| operations on vectors on simple two-dimensional plane. This
| is my point.
| pphysch wrote:
| Complex numbers and (2D) vectors/matrices are different
| representations of multi-dimensional number systems. For
| each operation over one, you can find an analogous
| operation in the other.
|
| You can even find attempts to mix the two
| representations, like i+j+k vector syntax. But matrices
| generalize better to higher dimensions and are easier to
| parse.
| morokhovets wrote:
| I agree with you here, but I don't agree on downplaying
| complex numbers to be just a base vector and orthogonal.
|
| If we take a matrix representation of a complex number it
| is usually done as a 2x2 matrix of very specific
| structure. I completely agree that it is easier to work
| with. But looking at them this way misses very important
| place of them in the grand scheme of things.
|
| Complex numbers are actually what real numbers really ARE
| under the hood, we just aren't taught to think this way.
| 'i' is what real numbers miss to be completed. And you
| don't need 'j's, 'k's and others.
|
| If your point is that introducing 'i' above traditional
| real numbers syntax is ugly - I completely agree.
| pphysch wrote:
| > Complex numbers are actually what real numbers really
| ARE under the hood, we just aren't taught to think this
| way. 'i' is what real numbers miss to be completed. And
| you don't need 'j's, 'k's and others.
|
| This is an unnecessarily absolute statement. On what
| basis are you claiming that all number systems are
| _fundamentally_ two-dimensional, and not one-dimensional,
| three-dimensional, or some other dimension?
|
| I'm guessing that it is because you spent a lot of time
| working with mathematics in a 2D context, i.e. on paper
| or blackboard or screen.
| morokhovets wrote:
| > On what basis are you claiming that all number systems
| are fundamentally two-dimensional, and not one-
| dimensional, three-dimensional, or some other dimension?
|
| I never said anything like this. I was talking about
| complex numbers only.
|
| I suggest to stop here, we are talking about two
| different things. But, if anything, there is a comment in
| this thread by adrian_b which explains what I mean in
| more detail.
| ogogmad wrote:
| > It is not a coincidence that they arose in 16th century
| Italy in the context of "completing the square" and related
| 2D methods/intuitions for solving equations.
|
| That's... not how it happened
| pphysch wrote:
| According to the OP, it did. I'm sure that there were
| many independent inventions of higher dimensional number
| systems.
| ogogmad wrote:
| The complex numbers prefigured other "hypercomplex"
| number systems by several centuries. And the modern
| 2-dimensional view of them was only described close to
| the year 1800. Before that, they were purely algebraic.
|
| The video (I briefly skimmed it) shows that they were
| invented to solve a problem in algebra. Nobody thought of
| them as being 2D back then.
|
| I'm not a historian or anything, but your claim is
| textbook "whig history" -- and as far as I can understood
| you, I already proved you wrong in a previous comment.
| pphysch wrote:
| I am not sure where this aggressive condescension is
| coming from. You appear to be conflating the inherent
| multi-dimensionality of complex numbers with their
| geometric/2D spatial visualization, which came later. I'm
| pretty sure we are in agreement here.
| [deleted]
| morokhovets wrote:
| To give a simple example - Mandelbrot set is a direct
| consequence of amazing properties of complex numbers and has
| nothing to do with two dimension.
|
| Well, it looks great in two dimensions
| adrian_b wrote:
| The complex numbers are 2-dimensional, but their 2 dimensions
| are not the 2 dimensions of a normal 2-dimensional geometric
| plane, they are 2 other dimensions.
|
| The 2 dimensions of a geometric plane correspond to the 2
| orthogonal translations of the plane.
|
| The 2 dimensions of a complex number do not correspond to
| translations, but to scalings and rotations of the geometric
| plane.
|
| The multiplication of the complex numbers corresponds to the
| composition of scalings and plane rotations, which are
| invertible operations and that is why the set of complex
| numbers is a commutative field, unlike the set of points of a
| geometric plane, which does not have such an algebraic
| structure.
|
| The set of complex numbers can be viewed as a plane, but it
| must be kept in mind that this plane is a distinct entity
| from a geometric plane.
|
| (The Cartesian product of a geometric plane with a complex
| plane forms a geometric algebra with 4 dimensions.)
| ogogmad wrote:
| > The 2 dimensions of a complex number do not correspond to
| translations, but to scalings and rotations of the
| geometric plane.
|
| Similar story for other 2D number systems:
|
| For the dual numbers, they express scalings and "Galilean"
| rotations (i.e. shears).
|
| For the double numbers, they express scalings and
| "Minkowski" rotations (i.e. Lorentz boosts).
|
| Unfortunately, some of the nice theory of the complex
| numbers doesn't generalise easily to the dual numbers or
| double numbers. I'm thinking specifically of complex
| analysis which is very, very nice, and much nicer than real
| analysis. But I think these planar number systems have
| their own intriguing character: For instance, see "screw
| theory" and "automatic differentiation" for two distinctive
| applications of the dual numbers.
| morokhovets wrote:
| That's exactly what I was trying to convey, thanks.
| mrtedbear wrote:
| Veritasium is fantastic, I can't think of many current or past TV
| shows that are of a similar quality to his videos. A while back
| he did a video of his own startup story, which was very cool:
| https://www.youtube.com/watch?v=S1tFT4smd6E
| vishnugupta wrote:
| While I enjoyed the history part towards the end it got super
| hand wavy; especially as it went from describing "i" as a
| rotation to e^ix; Better Explained does a _fantastic_ job of
| explaining these from the first principles.
|
| https://betterexplained.com/articles/a-visual-intuitive-guid...
|
| https://betterexplained.com/articles/intuitive-understanding...
| travisgriggs wrote:
| I love Veritasium. Derek has done a number of videos that are
| awesome like this. The videos are great and educational, but he
| also manages to subtly weave in an almost spiritual subtext to
| them as well. For example, in this one, you learn about imaginary
| numbers in some cool ways, history, how awesome e^ix is, etc, but
| also this yen-yang balance between embracing not only what is
| real but taking a chance on what is imaginary.
|
| Some of my other favorites:
|
| - https://m.youtube.com/watch?v=rhgwIhB58PA
|
| - https://m.youtube.com/watch?v=HeQX2HjkcNo
|
| - https://m.youtube.com/watch?v=OxGsU8oIWjY
|
| - https://m.youtube.com/watch?v=3LopI4YeC4I
|
| - https://m.youtube.com/watch?v=pTn6Ewhb27k
|
| I think he's going to lose the $10,000 bet on the car though.
| lxe wrote:
| Some of popular educational/informational channels and podcasts
| make this patronizing assumption about the audience, watering
| everything down to make it "accessible". Veritasium doesn't do
| this, yet maintains huge popularity.
| kzrdude wrote:
| I didn't click through all (mobile, too slow) so don't know if
| it's in there, but one favourite i have is the one where he has
| an ice cube, a metal block and a plastic block and ask people
| on the street which block is colder.
|
| It teaches a lot about what temperature vs feels cold vs heat
| is.
| agumonkey wrote:
| That's my favorite video of him. This one ties history,
| education, simple geometry, advanced abstractions in an
| entertaining script. Pretty high bar.
| 6gvONxR4sf7o wrote:
| My favorite mental model for imaginary numbers is that they are
| placeholders or stepping stones. A speed of (30 + i*10) miles per
| hour is nonsense, but sometimes in the course of calculating a
| nice real 40 mph, you might take a detour through some steps
| including complex numbers. Having them handy sure will make the
| derivation easier.
|
| Same as negative numbers. I can't be negative six feet tall, but
| negative feet (as a magnitude) make the calculations way easier.
|
| You can always reformulate these derivations into derivations
| that don't use e.g. imaginary numbers, but there's no need to. If
| I were apointed Senior Exectutive Math Concept Namer For
| Humanity, I'd call them something like "algebraic placeholders"
| or "algebraic closure stepping stones" or something in that vein
| instead of "imaginary numbers."
| lxe wrote:
| I remember just playing with my good old TI-83 Plus Silver
| Edition, plugging a bunch of symbols together, like e, and i,
| etc... when I suddenly got the e^i*pi = -1.
|
| How in the world do these completely unrelated (to me, at the
| time, at least) constants ended up with -1? It was mind blowing.
| This made me actually interested in math.
| pjbk wrote:
| If you realize where e comes from, which is compounding and
| what we call an exponential process, being that "the current
| rate of change is proportional to the present value", then the
| exponential of a complex number is no different to the exponent
| of a real number. In the case of one-dimensional real numbers
| you assign the slope of the function to the current value. The
| complex case is exactly the same, but if you think instead in
| the two-dimensional Argand plane and complex algebra, the slope
| is the tangent to a circle proportional to the angle at that
| radius (which curve is the one that at every point the slope is
| equal to its complex value?).
|
| Therefore with the rules of complex arithmetic the tangent
| provides the "curling" effect in the curvature of a periodic
| circle, while in the real case you get the common compounding
| shape of the exponential curve. The relation to sine and cosine
| is just the projection into direct(real)/quadrature(imaginary)
| components in either fixed or intrinsic coordinates. Same when
| you expand the exponential into its complex power series.
|
| BTW, this is actually the essence of infinitesimal
| transformations in continuous groups and the exponential map,
| which generalizes this concept to other types of numbers or
| abstract objects (i.e. Lie group theory).
| faceyspacey wrote:
| I've been wondering about the connection between the "curling
| effect" and the exponential curve of e for a while. Do you
| have any links where I can learn more in depth?
| vishnugupta wrote:
| Thanks to the way math is taught in India I still have very
| little _understanding_ of so many of these concepts though I 'm
| very good at remembering formulae/steps, mechanically
| calculating them, and applying them to real-world problems.
|
| It wasn't until I stumbled upon Better Explained[1] I began to
| appreciate the need to understand math at a deeper/common-
| sensical level. Now I make it an effort to teach math to my son
| using examples, analogies so that he _gets_ concepts like
| positional numbering system, why we need it, its advantages
| etc.,
|
| https://betterexplained.com/articles/intuitive-understanding...
| timonoko wrote:
| Ouch. There is some logic and sanity in e^ix.
| mode80 wrote:
| you would like this:
| https://betterexplained.com/articles/intuitive-understanding...
| coremoff wrote:
| you might like this:
| https://www.youtube.com/watch?v=mvmuCPvRoWQ
| motohagiography wrote:
| If we still had math duels, my life would have taken a very
| different trajectory. We need these.
| echopurity wrote:
| This video is epistemological trash.
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