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How the modern world arose from imaginary numbers

bradwood 5y ago
The best bit for me: ALL numbers are imaginary
polotics 5y ago
Mathematicians have always sucked at naming their variables, and constants, and types... This has given pundits of all ilk countless opportunity to debate...
nyc111 5y ago
> For some reason—whether a sense that there was some mistake, or someone copied something down wrong, or because it was so absurd—the manuscripts we have show that Heron ignored the minus sign and gave the answer as √63 instead. I wonder if the minus sign was in use in the time of Heron (First century AD). I couldn't tell from this Wikipedia page https://en.wikipedia.org/wiki/Plus_and_minus_signs https://en.wikipedia.org/wiki/Plus_and_minus_signs
dTal 5y ago
Another way to think of it is that all numbers are imaginary. Numbers aren't real. Platonism is wrong. Imaginary numbers aren't "out there" somewhere. The whole system of mathematics is an accumulated edifice of metaphors designed by human brains, for human brains, and there's no god "behind the curtain". It's just a tool of thought. It reflects the "reality" of the universe only insofar as we've looked at the universe, noticed patterns, and constructed metaphors around them. This is not a popular viewpoint! But it is the only scientifically supported one.
- hansbo 5y ago
  But even if you deep dive enough, there are discrete values, like in Quantum Mechanics. And as long as you have discrete values, you have integers, no? So integers do not seem only like human models, they seem to me as something innate in the universe.
  - dTal 5y ago
    "Discrete values" are also a human metaphor. You say there are two apples on your desk? I say there is a fuzzy quantum mess of probability distribution functions on your desk. "Two apples" is in your mind.
    - shusaku 5y ago
      I’m baffled that you’re invoking QUANTum mechanics to ascertain that discrete values don’t exist. At any rate, nominalism has a rich history, so I doubt these hacker news comments will solve the issue…
    - nh23423fefe 5y ago
      and spin?
  - igorkraw 5y ago
    are atoms discrete? we used to think so. we might never get a better model than quantum physics and it might still be wrong and fail to explain things. So there is a human idea of "discrete element" that we used to apply to everything - and as we look closer, it always breaks down. that doesn't mean it's a useless abstraction, but it is am abstraction - a tool for thought, a map, not the territory
  - helen___keller 5y ago
    Quantum mechanics is a mathematical description of the behavior of the universe, so wouldn’t invoking this to prove mathematical objects exist be begging the question? Not to say I agree with GP, but I don’t think it will be so easy to prove GP wrong either
- mannykannot 5y ago
  > But it is the only scientifically supported one. Really? What is the empirical evidence for it?
- throwaway17_17 5y ago
  I agree that this view it is not popular, but I also do not think that supporters often articulate their view/support well. I am a hard materialist and the amount of platonic-leaning discourse around the fundamentals of mathematics confuses me. I do not know how so many people (typical those outside philosophy and mathematical foundations) just assume a platonic style view.
  - dTal 5y ago
    I am currently reading "Where Mathematics Comes From" by George Lakoff and Rafael E. Núñez - the same Lakoff who authored the seminal "Metaphors We Live By", so I have a lot of time for him. At first it seems like they're just going to explore the pedagogical psychology of mathematics - how interesting! But then right at the end of the preface they hit you with "and by the way this is all there is to it, mathematical Platonism is a lie", which struck me immediately as straying out of their lane. But it seems their investigation into the titular question overwhelmingly led them to this conclusion. The argument is pretty simple - if there is a "platonic mathematics", we cannot have any direct experience of it. All mathematical thought, like all thought in general, is metaphorical. The predictive power of mathematics in the real world is unsurprising because we throw away the metaphors that don't work well. I do not like this conclusion. Mathematics has always been something of a religion for me. But I can find no flaw with the argument. From a scientific perspective, mathematics bottoms out at "what goes on in human noggins".
    - imbnwa 5y ago
      >The argument is pretty simple - if there is a "platonic mathematics", we cannot have any direct experience of it. Aside, but this is also Aristotle's exact argument against Platonism in general, though when he makes it in the Nichomachean Ethics he is specifically talking about ethical Good (if the definition/actual taking place of the Good lies in some other plane, we can't participate in it so no one is or can be good), but the idea is the same even when he's talking about what a soul is in De Anima. Aristotle doesn't believe in 'souls' in the way we think of them as religio-spiritual entities that exceed the capacity of the body; a 'soul' for Aristotle is the body but in a way that radically challenges the idea of a body as mere shell or vessel - soul is what any form of life repeats doing, as a body, in order to continue being itself. It should be noted that a lot of time at Aristotle's Academy was spent in Zoology, studying animals and their anatomy.
- mjburgess 5y ago
  This view isnt "scientifically supported" because science is neutral on (indeed, even presupposes) the existence of abstract objects. No one believes abstracta have a physical location -- they lack physical properties. The claim "2 + 2 = 4" is true -- and clearly not true invirute of anything anyone thinks... if we kill that person (/people), it is no less true. Indeed, if numbers don't exist (for example), do we suppose that we can't communicate issues of quantity with other species (, & possible alien life, etc. etc.) ? (If we can, what shared things are we talking about when we quantify?) It seems deeply implausible to say that our use of number is circumstantially psychological -- any description of reality is going to be indispensably quantitative --- quantity is what we are talkng about. We are not talking about ourselves.
  - echopurity 5y ago
  - jhedwards 5y ago
    I've thought about this problem quite a bit and, while my initial position was the same as above (math is not "real" per se) I had to concede that integers are real, because quantity is self-evidently real. If you have four oranges, the quantity "four" is right there. If you take away one of those oranges you know that the result cannot be split evenly without a remaining orange because of the properties of odd numbers. If you cut the remaining orange in half then you get a rational number, but is that self-evidently real? The halves of the orange are only "halves" because we consider them in relation to their origin, which we consider to be "one" orange. So rational numbers necessarily involve the human action of relating some quantity to a reference quantity, therefore they are a higher-level abstraction built on top of the fundamental physical property of quantity. In the end I decided that math is based on a foundation of quantity (and maybe "space" as well?) and everything else was a derived abstraction. I am very curious if anyone else has a good argument for other parts of math being fundamental.
    - dagw 5y ago
      integers are real, because quantity is self-evidently real. But the there are more integers than there are quantifiable 'things'[1]. Are integers that are a lot larger than, say the size of the power set of all fundamental particles in the universe still "self-evidently real". [1] Assuming a finite universe (or a finite number of finite universes) and a few other things.
- andreareina 5y ago
  Science has nothing to do with it, rather it's a question of philosophy and what we define as being real.
  - dTal 5y ago
    Science is "what we have evidence for". Is there evidence for some abstract mathematics that we didn't invent?
- dandanua 5y ago
  > there's no god "behind the curtain" have you looked there already? :)
- tasha0663 5y ago
  You can boil down pretty much everything to "an accumulated edifice of metaphors designed by human brains, for human brains". In the game Hearts, if you take most of the spades you lose. However, if you manage to take all the spades you win, and they call it "shooting the moon". In a similar fashion, when you reject everything as an unreal system of metaphors, Platonism "shoots the moon" by having us reexamine what we thought we meant by "real" in the first place.
- voldacar 5y ago
  This is one particular point of view, and it's extremely sloppy reasoning to say that it's "scientifically supported", given that mathematics is not a form of science. There are plenty of very smart people, not just mathematicians but also physicists & scientists who are mathematical platonists.
  - dTal 5y ago
    There are plenty of smart people - scientists even - who believe in all kinds of deities. Mathematics is indeed not a form of science. But the existence and shape of mathematics is an observable phenomenon, and so metamathematics - the study of what it is and where it comes from - can be studied scientifically. How do you know mathematics exists? Well, there's a textbook right there. Who wrote the textbook and why? A human, expressing metaphors inside their heads. How did those metaphors get inside that human's head? Ah, well, that's the interesting bit - the answer of course transpires to be "a combination of innate ideas imprinted by genetic evolution by natural selection, and sociology". And you don't have to stop there, you can explore in glorious detail exactly where each idea comes from, what innate monkey-ish tendency is being deployed, how exactly ideas like "infinity" fit in a mind designed for finding fruit and chasing things. We can similarly bring all manner of religious beliefs under the anthropological knife. It's not a pretty process though, to the people who believe in them.
    - carapace 5y ago
      > metaphors inside their heads. You're begging the question presupposing the non-Platonic viewpoint here. How do we know that metaphors are "inside the head"?
    - voldacar 5y ago
      You are just assuming the point you're trying to prove > There are plenty of smart people - scientists even - who believe in all kinds of deities. Okay? This is supposed to make me feel - how exactly? I'm not inherently disdainful towards theism or theists, but if I were, I guess your remark would make me like science less, or something? > We can similarly bring all manner of religious beliefs under the anthropological knife I'm not really sure we can, actually. At least not in some kind of non-contentious, "objective" sense. I don't really trust individual humans to give an accurate account of why they believe their beliefs, but I trust "anthropology" and "sociology" even less. My distrust for this on an individual scale comes from the fact that many beliefs & memes exist for purposes of social signalling, group identification, etc, and it might not actually be in your interest to know exactly why you believe what you do. But these auxiliary functions of beliefs, such as signalling etc, seem to me to scale up as you introduce groups and larger-scale activities such as "anthropology" and "sociology". Without some feedback loop keeping them honest, why would I expect anthropologists or sociologists to tell me a true story about why someone believes what they do, any more than that person or anyone else? In aerospace engineering, the feedback loop is that if your design is bad, your jet engine won't work. As a result, I generally trust aerospace engineers about jet engines. But what is there to stop sociologists, anthropologists, etc from just settling on some bullshit that agrees with their preconceived beliefs or flatters their group status and promoting it forever? But back to math. The history of mathematical ideas is complicated and interesting, but it isn't really that relevant to the question of whether the things those ideas are about are "real", which is equivalent to asking whether mathematical platonism is true or not. The question of platonism comes down to the definition of words like "real" and "exist". It is very easy to equivocate using these words, which is why most discussions about mathematical platonism are so low quality. I think the overall question isn't that meaningful so I'm not really a platonist or an anti-platonist. In most parts of human life, when I say "x exists", I mean that I can reach out and touch x, that it has a mass, temperature, surface texture, etc. In math, when I say "x exists" I just mean that I can talk about x without creating any logical contradictions. The square root of -1 may not exist in the same sense as my laptop here, but it exists in the sense that I can do things with it, such as add, multiply, raise to powers, etc, without reaching a contradiction in my formal system. So the whole "out there" thing doesn't really matter. There doesn't need to be an "out there" in order for me to meaningfully say that the square root of -1 exists. I think that a lot of philosophy is like this too, when you mentally zoom in really closely on a problem, it often reduces to some kind of equivocation or inconsistent language usage. Btw I don't really consider anthropology or sociology to be real intellectual disciplines, and I'm pretty on the fence about psychology and economics. I realize that is an unpopular opinion but I've thought about it a lot and I'm pretty certain that it's correct. Aerospace engineering is real because it attaches to some fundamental reality, namely that of the spinning fan blades, the combusting fuel, etc. If you get your engineering wrong, the fan blades won't spin. Likewise, math is attached to systems of axioms. When your do your math wrong, you get a contradiction. Sociology and anthropology don't attach to anything, they're like a closed loop, like theology. If you get your anthropology wrong, nothing really happens.
- canjobear 5y ago
  Reality is that which, when you stop believing in it, doesn’t go away. -Philip K Dick Even if you stop “believing in” math, your proofs are still either correct or incorrect.
  - dagw 5y ago
    The proofs are only correct in as far as you believe in the Axioms of mathematics that those proofs are built on. Stop "believing in" the axioms and the proofs are no longer true or even meaningful.
    - canjobear 5y ago
      But it is still the case that IF the axioms hold, THEN the proofs are either valid or not.
      - marcosdumay 5y ago
        So it's conditional reality. Now make that fit within your usual realistic philosophy...
        canjobear 5y ago
        Easy. There exist contingency relationships between axioms and theorems provable from those axioms.
        marcosdumay 5y ago
        Oh, yeas, your proof is quite real. And it's completely meaningless, being all about imaginary things. It can only have any meaning if you adhere to some scientific model. At this point you've deviated so much from the OP discussion that you could as well talk about angels dancing in pinheads. Any quantification of them is as real as your proof.
- tim333 5y ago
  >all numbers are imaginary.... mathematics is an accumulated edifice of metaphors designed by human brains Or you can say they exist but in a different way to physical reality. I mean pi probably still was 3.14159... before humans evolved so it's not our fault really. Personally I think maths not only exists but physical reality is a subset. I mean why else is there something rather than nothing? Scientifically it's the only hypothesis that works for that really.
- nathias 5y ago
  No it isn't scientifically supported because science does not include ontology and episemology as its domain. I'm not a platonist, but the reasons for not being platonists are philosophical.
- garmain 5y ago
  1) I might be missing something, but as far as I can tell "scientific support" in this context seems ill defined. To the extent that it seems rather meaningless term (in this context) 2) While the vast majority of the discourse on the interpretation of mathematics oscillates (fruitlessly) between Platonism and Nominalism, I tend towards a more Aristotelian view. See for example - Franklin: An Aristotelian Realist Philosophy of Mathematics - Keith Hossack: Knowledge and the Philosophy of Number. Note that these two authors do not converge on exactly the same interpretation. Apropos: North (2021): Physics, Structure, and Reality explores the relationship between "mathematical structure" reality and theoretical physics.
dandanua 5y ago
Life is complex – it has real and imaginary parts.
gumby 5y ago
“Imaginary” was an unfortunate choice of word. Positive integers and rationals feel “legit”* to most people, even (or especially) the non-mathematically trained. But “imaginary” numbers are no more or less legit than, say, transcendental numbers. But in middle school you learn this confusing thing and maths starts to fall off the rails. * Had to find a synonym for “real” here.
- jdrc 5y ago
  Yeah we should stop using the word imaginary, and just call them iNumbers (sorry Apple)
  - Qem 5y ago
    I like to call them "double-barreled numbers". I never liked the term "imaginary", because that gives mystical connotations and even hamper understanding, in my opinion.
- intrepidhero 5y ago
  That's what I've finally settled on in my mind. Since we use imaginary numbers to model and measure real phenomenon, they're not imaginary in the sense that I would usually use that word. There's got to be a better name... Transcendental complement maybe.
- starwind 5y ago
  when I did my math degree in college the profs and grad students called them "complex numbers." I think they know there's a problem so they want to get the students who would go on to teach used to not calling them "imaginary"
  - deleted 5y ago
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  - bee_rider 5y ago
    That doesn't quite get at it, though -- a complex number has an imaginary and real part. I guess we could call them "complex numbers with a zero real part," but 1) that's quite a mouthful i) it implies that the other part is something other than real -1) it seems weird to describe something sort of... subtractively like this
hgomersall 5y ago
There is a persuasive school of thought that argues exactly the opposite, that imaginary numbers are not real http://geometry.mrao.cam.ac.uk/1993/01/imaginary-numbers-are-not-real-the-geometric-algebra-of-spacetime/ http://geometry.mrao.cam.ac.uk/1993/01/imaginary-numbers-are... .
- topaz0 5y ago
  I'm familiar with (and partial to) the geometric algebra literature, but I think it's a bit disingenuous/clickbaity to say that it implies imaginary numbers aren't real. I think of it as meaning that the algebra of the imaginary numbers arises as a slice through a more useful, bigger algebra.
  - hgomersall 5y ago
    Yes, I think it's rather tongue in cheek. It's probably not defined as clickbait since it was written in '93!
    - topaz0 5y ago
      Fair enough!
- wrycoder 5y ago
  You can make some progress by thinking of the imaginary numbers as being a two dimensional vector space over the reals, equipped with a very peculiar multiplication operator that is a combination of the usual stretch with a rotation. Addition is the usual addition of components, but the multiplication (using polar coordinates) looks like C = ABexp(T1+T2), where the T are the phases of A and B. To multiply two positive reals, since their phases are zero, it’s only a stretch: AB To multiply two negative reals, since their phases are pi, it’s ABexp(pi+pi). But exp(2pi) is unity, so we have AB again, a positive number. To find the square root of a positive number D, find a positive real (zero phase) that, when multiplied by itself, gives D. AA, for example. So, real number arithmetic is a subset of complex number arithmetic, with zero phase. And therefore zero rotation. Now, find the square root of a negative number, say -4. To do this, we have to step off the real axis for the first time: out into the Argand plane! -4 is represented as 4exp(pi) OK? So we are looking for a number out here on the Argand plane that when multiplied by itself following our rule gives 4exp(pi). And that number is 2exp(pi/2), which is two units out on the “imaginary” axis of our Argand plane. 2exp(pi/2)2exp(pi/2) equals 4exp(pi/2+pi/2) equals 4exp(pi) QED
viktorcode 5y ago
I highly recommend video by Sabine Hossenfelder [1] on this topic. She's explaining the topic in layman terms, starting from what does it mean to exist (or be real) in physical sense. Complex numbers, as useful as they are, just an abstraction, a tool which can be replaced with different forms of calculations. That makes the question of "reality" rather complex (pardon the pun). [1] https://www.youtube.com/watch?v=ALc8CBYOfkw https://www.youtube.com/watch?v=ALc8CBYOfkw
- Psyladine 5y ago
  For those interested I also highly recommend Veritasium's video[1] as an excellent explainer as well, especially regarding the roots of imaginary/complex numbers in geometry. [1] https://www.youtube.com/watch?v=cUzklzVXJwo https://www.youtube.com/watch?v=cUzklzVXJwo
1970-01-01 5y ago
I've been calling them invisible numbers most of my life. It makes much more sense.
adityapatadia 5y ago
This is one video which shows history of imaginary numbers and their usefulness in current times. It's from Veritasium: https://www.youtube.com/watch?v=cUzklzVXJwo https://www.youtube.com/watch?v=cUzklzVXJwo
dboreham 5y ago
This is only mentioned in passing in the article, but the penny dropped for me when I read somewhere else that you should first think about negative numbers. They seem pretty real, but they're not. There's no negative quantity in nature. You can't have -5 rocks. Negative numbers are just the solution to equations of the form x + n = 0. Now after you let that sink in, imaginary numbers don't seem so weird after all. Just another kind of number defined as the solution to some equation.
- mettamage 5y ago
  I disagree. > You can't have -5 rocks You can have 5 rocks destroyed in the future. That to me, is what -5 physically means: a guarantee that the object associated with the number will be eliminated out of existence in the future and decrement the negative number by 1. Some people call it debt, but to me, emotionally that word feels too financial. So I prefer “a guarantee to be eliminated out of existence in the future “, or something like that. Conversely, 5 cows means: 5 cows currently in existence. 5 cows - 5 cows means: I see 5 cows and now they don’t exist any more and there is nothing. I wish my math skills were better, I am optimistic that I’d find a similar thing for imaginary numbers and maybe even complex numbers. With that said, I do get where you’re coming from and I find it a compelling perspective as well. It’s simply that I feel the perspective I described as well.
  - easywood 5y ago
    >> You can have 5 rocks destroyed in the future You have 5 rocks now, and zero rocks in the future. At no point are there negative rock-shaped holes in the fabric of space. The parent's comment still stands, he is talking about reality, not our mathematical interpretation of it.
  - marcus_cemes 5y ago
    > You can have 5 rocks destroyed in the future. Exactly, you use it to store some information that has no real quantity but may be converted to a real quantity in the future through some other process. I'm a polytechnic university student, we use imaginary numbers extensively in all sorts of places, especially whenever there is any oscillatory behaviour, such as an electrical signal or a light wave. A complex number is just a two-dimensional vector with real/imaginary components, whcih provides an amplitude and a phase (angle). An oscillating sinusoidal signal/wave may appear to be zero and completely static if you freeze time at the right moment, but as time progresses, it will continue oscillating, like a swing in a park. In a way, the magnitude represents the built up "momentum" of the system, whilst the real quantity is the immediate physical value at any given point in time (given by the phase). The amplitude is always the same at any given moment, even when the swing is vertical, it has momentum which will help it reach its maximum height. Personally, I still think they are just "invented", but I think the vast majority of engineers much prefer them to the alternative, manipulating trigonometric functions (every engineer's nightmare). They're a neat way to represent the exchange of potential and mechanical/electrical energy with a single value and some simplified mathematics (this is an engineer's, not a mathematician's, point of view). Like negative numbers, we could have chosen to have two positive quantities, balance and debt, instead we find use in merging these definitions, whether negative values make sense or not. We have become used to to negative numbers representing the "inverse" action, which makes sense when representing a phyiscal quantity such as velocity.
rthomas6 5y ago
What finally made imaginary numbers intuitively make sense to me is realizing that the number i just represents a 90 degree rotation on the complex plane. That's why i*i = -1 (it's rotating 180 degrees), why imaginary numbers are orthogonal to real numbers, why e^ix = cos(x) + i*sin(x) makes sense, everything. When you are dealing with 2 dimensions, complex numbers are a kind of hack for representing both dimensions without any kind of vectors or pairs or anything, just numbers.
- starwind 5y ago
  I majored in math and took a course on waves through the physics department. Everyone struggled with why imaginary numbers would show up and I was just like "we took the square root of this negative number, and the reals aren't closed when we take square roots, therefore imaginary." I missed out on a lot of physical intuition and had basically no concept of what the imaginary numbers actually meant
- ThrustVectoring 5y ago
  More relevant and less hacky, IMO, is that the real numbers aren't algebraically closed - there are polynomials with real coefficients that you cannot factor into getting real zeros. This is a big problem when you have something like a spring-mass-damper system. The springiness, damping, and mass of the system generate coefficients of a degree-two polynomial, and the zeros of that polynomial correspond to how the system moves over time.
- Epa095 5y ago
  With this intuition, why is it i*i=-1 and not i+i (90+90=180 after all)?
  - rthomas6 5y ago
    Because with complex numbers you are implicitly defining a point on the complex plane. So how on the Cartesian plane you can have 2x + 3y, and that defines the point (2,3), you can have 2 + 3i on the complex plane defining the same point. If you want to rotate the Cartesian coordinate -90 degrees, x' = x*cos(π) - y*sin(π) = -y y' = y*cos(π) + x*sin(π) = x 2x + 3y => -3x + 2y So check it out: (2 + 3i)*i = 2i - 3 It's the same as rotation!
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nuncanada 5y ago
Mathematicians have been constantly been giving bad names do their concepts... Imaginary Numbers is one of the atrocious ones, Imaginary Numbers are part of the fabric of our reality... Meanwhile Real Number were axiomatized in a way that creates completely unrealistic numbers, "ghost-like" numbers, numbers that can never be given a name... The first course in Analysis is ridden with examples of Continuous Functions that are not continuous in any sense humans would consider ever it...
- openfuture 5y ago
  Completely and utterly disagree, it may have been the case in the 1800s or whatever but the naming sense of modern mathematics is pretty on point and actually incredibly insightful. Consider the definition of a "connected" topological space and tell me it doesn't tell you something about society.
Agamus 5y ago
I'm not an expert here, but truly interested to hear responses to this question. To say that 1+1=2 is "true", does that not require a corollary in "reality" to something fundamental that can be called a "one" object? I believe this is called mathematical constructivism. Imagine, hypothetically, that we cannot identify something that is physically fundamental and individual. My question is whether any mathematics in that scenario could be considered "true" without such constructivism, in other words, without a physical correspondence to an unquestionably, physically fundamental "one" object.
phendrenad2 5y ago
Imaginary numbers are just a shorthand way of representing things in nature that relate to one another via the sine function, for example, charged particles in a magnetic field. You can bust out calculus to describe the motion, or you can use a convenient set of rules that represents the partially-solved equation.
Khaine 5y ago
Veritasium has a good video on how complex numbers were invented https://www.youtube.com/watch?v=cUzklzVXJwo https://www.youtube.com/watch?v=cUzklzVXJwo
abecedarius 5y ago
Details on Heron's missed sqrt(-63): https://nrich.maths.org/13402 https://nrich.maths.org/13402 Apparently there's not much to say about how that mistake came up, though.
gweinberg 5y ago
Quaternions still suck though. Octonions even more so.
analog31 5y ago
Here's what I wonder: If somebody is hung up on complex numbers, will getting them over that hump get them any closer to understanding how or why complex numbers are used in physics or engineering?
- dataflow 5y ago
  I think they'll understand it if you get them to solve a few Fourier series/transforms with sinusoids, and then with complex exponentials, and observe the difference in difficulty.
- ganzuul 5y ago
  The behavior of complex numbers under certain operations is seemingly the same mechanism as is seen in physical systems. The style of notation used in engineering usually denotes a periodic motion or a compound of many waves. Of interest for physics is that without the imaginary part, real numbers alone are a leaky abstraction. This structure extends to quaterions and octonions.
  - analog31 5y ago
    Indeed, I'm a physicist with a background in electronics, so I'm fluent in that stuff. The question is whether explaining complex numbers to a non physicist is really going to bring them any closer to physics.
    - ganzuul 5y ago
      Considering complex numbers lead to group theory and group theory may model atomic structure, this metaphysical correlation is at least worthy of note. It is almost as if we are living in a distortion of truth.
gregfjohnson 5y ago
A very helpful intuition is to consider a notion of "Compass numbers"; one might say, "the nearest coffee shop is East 2 blocks and North 3 blocks". One might say the the real numbers are restricted to just two directions. Extend them to allow the "sign" of a number to be any direction. It is intuitive to imagine what addition of "Compass numbers" might look like, and what multiplication by a positive scalar real number might look like. The question is, what should multiplication of two compass numbers neither of which is real be? One of the earliest developments of complex numbers was done by a Danish cartographer, Caspar Wessel. He wrote a lovely paper that was published in an obscure forum in 1797. He is now credited as the first person to understand the correspondence of complex numbers and vectors on a plane. It seems quite natural that a cartographer would be interested in "numbers" that can point in any direction. If you posit the existence of a multiplicative identity and (arbitrarily) label it "1", and then take a compass number 90 degrees away from it and (arbitrarily) label it "i", (and furthermore assume field axioms), the formula "(1 + i) * (1 - i)" forces the conclusion that i * i is the compass number pointing in the opposite direction of the multiplicative identity.
- gregfjohnson 5y ago
  To elaborate slightly; (1 + i) * (1 - i) = 1 - i^2. The left side is somewhere on the complex plane on a circle of radius 2 centered at the origin. The right side is somewhere on a circle of radius 1 centered at 1. The circles meet at one and only one place, namely 2. So 2 = 1 - i^2, or i^2 = -1.
vishnugupta 5y ago
> So let’s picture multiplication by –1 as half a rotation, anticlockwise, around a circle (in our case, the circle passes through 1 and –1). It’s actually a rotation by 180 degrees. Sorry but this just didn't work for me. My mental model is to use +/- as directional indicators. For me multiplication by -1 is same as multiplication by +1 but in the opposite direction. I would rather think of "i" as rotation by 90°. In this realm +/- stand for anti/clockwise. So +/- continue to stand for direction and "i" tells me if I need to rotate or not. It neatly lines up with different operations. As an example (+i) × (-i) == -i² == 1. In terms of movements you are rotating 90° anti-clockwise followed by 90° clockwise bringing you back where you started. On the other hand, Nautilus's article is bit too much of mental gymnastics for me to follow their reasoning. > So let’s picture multiplication by –1 as half a rotation..It’s actually a rotation by 180 degrees..What happens if we only do half of this rotation? It’s halfway to multiplying by –1, which you can think of as the same as multiplying by √–1...
- garmain 5y ago
  I'm curious. I seem to be missing something, because I feel like what you're saying tracks very much with the "alternative interpretation". That is, I don't see the difference (in my mind's eye) at all. Is there maybe a different way to explain what you perceive to be critically different?
  - vishnugupta 5y ago
    For me it is about expressing a concept as compactly as possible. In case of Nautilus they first ask us to imagine -1 as 180° and then proceed say half of it as 90° rotation and finally give it a name i. For some this may work fine as a way to understand. However it gets awkward to further explain. If 𝑖 stands for half of -1 (as they say) then does that mean 𝑖 == -(1/2)? If not then why not? and so on. Compare that with: 𝑖 is a unit of rotation which is 90°. That's it. Using this as a base it's easier to explain why 𝑖² = -1 (two units of anti-clockwise rotation) or why 𝑖³ = -i (three units of anti-clockwise rotation is same as one unit of clockwise rotation) and so on. More fundamentally, they ask us to imagine an operation on 1-d (-1) as happening through 2-d (180° rotation). It seems convoluted way to introduce 𝑖. I'd rather explain 𝑖 from the first principles and proceed to show its consequences such as 𝑖² = -1.
DarkCrusader2 5y ago
Can not recommend this series of videos from Welch Lab[1] enough on imaginary numbers. Deep dives into the history of how complex numbers were "invented" to enable mathematical representation which weren't possible by existing set of numbers. Also draws parallel to "invention" of negative numbers and zero to overcome the limitations of the then present number system (for eg. how to balance books without negative numbers). [1] https://youtube.com/playlist?list=PLiaHhY2iBX9g6KIvZ_703G3KJXapKkNaF https://youtube.com/playlist?list=PLiaHhY2iBX9g6KIvZ_703G3KJ...
j7ake 5y ago
I always imagine complex numbers by putting them on a Cartesian plane (2D linear space). Is there an extension of this for numbers that can be put on a 3D space?
- lordgrenville 5y ago
  That's my understanding of quaternions (but I might be way off).

The best bit for me: ALL numbers are imaginary

Mathematicians have always sucked at naming their variables, and constants, and types... This has given pundits of all ilk countless opportunity to debate...

nyc111 5y ago

> For some reason—whether a sense that there was some mistake, or someone copied something down wrong, or because it was so absurd—the manuscripts we have show that Heron ignored the minus sign and gave the answer as √63 instead. I wonder if the minus sign was in use in the time of Heron (First century AD). I couldn't tell from this Wikipedia page https://en.wikipedia.org/wiki/Plus_and_minus_signs https://en.wikipedia.org/wiki/Plus_and_minus_signs

dTal 5y ago

Another way to think of it is that all numbers are imaginary. Numbers aren't real. Platonism is wrong. Imaginary numbers aren't "out there" somewhere. The whole system of mathematics is an accumulated edifice of metaphors designed by human brains, for human brains, and there's no god "behind the curtain". It's just a tool of thought. It reflects the "reality" of the universe only insofar as we've looked at the universe, noticed patterns, and constructed metaphors around them. This is not a popular viewpoint! But it is the only scientifically supported one.

hansbo 5y ago
But even if you deep dive enough, there are discrete values, like in Quantum Mechanics. And as long as you have discrete values, you have integers, no? So integers do not seem only like human models, they seem to me as something innate in the universe.
- dTal 5y ago
  "Discrete values" are also a human metaphor. You say there are two apples on your desk? I say there is a fuzzy quantum mess of probability distribution functions on your desk. "Two apples" is in your mind.
  - shusaku 5y ago
    I’m baffled that you’re invoking QUANTum mechanics to ascertain that discrete values don’t exist. At any rate, nominalism has a rich history, so I doubt these hacker news comments will solve the issue…
  - nh23423fefe 5y ago
    and spin?
- igorkraw 5y ago
  are atoms discrete? we used to think so. we might never get a better model than quantum physics and it might still be wrong and fail to explain things. So there is a human idea of "discrete element" that we used to apply to everything - and as we look closer, it always breaks down. that doesn't mean it's a useless abstraction, but it is am abstraction - a tool for thought, a map, not the territory
- helen___keller 5y ago
  Quantum mechanics is a mathematical description of the behavior of the universe, so wouldn’t invoking this to prove mathematical objects exist be begging the question? Not to say I agree with GP, but I don’t think it will be so easy to prove GP wrong either
mannykannot 5y ago
> But it is the only scientifically supported one. Really? What is the empirical evidence for it?
throwaway17_17 5y ago
I agree that this view it is not popular, but I also do not think that supporters often articulate their view/support well. I am a hard materialist and the amount of platonic-leaning discourse around the fundamentals of mathematics confuses me. I do not know how so many people (typical those outside philosophy and mathematical foundations) just assume a platonic style view.
- dTal 5y ago
  I am currently reading "Where Mathematics Comes From" by George Lakoff and Rafael E. Núñez - the same Lakoff who authored the seminal "Metaphors We Live By", so I have a lot of time for him. At first it seems like they're just going to explore the pedagogical psychology of mathematics - how interesting! But then right at the end of the preface they hit you with "and by the way this is all there is to it, mathematical Platonism is a lie", which struck me immediately as straying out of their lane. But it seems their investigation into the titular question overwhelmingly led them to this conclusion. The argument is pretty simple - if there is a "platonic mathematics", we cannot have any direct experience of it. All mathematical thought, like all thought in general, is metaphorical. The predictive power of mathematics in the real world is unsurprising because we throw away the metaphors that don't work well. I do not like this conclusion. Mathematics has always been something of a religion for me. But I can find no flaw with the argument. From a scientific perspective, mathematics bottoms out at "what goes on in human noggins".
  - imbnwa 5y ago
    >The argument is pretty simple - if there is a "platonic mathematics", we cannot have any direct experience of it. Aside, but this is also Aristotle's exact argument against Platonism in general, though when he makes it in the Nichomachean Ethics he is specifically talking about ethical Good (if the definition/actual taking place of the Good lies in some other plane, we can't participate in it so no one is or can be good), but the idea is the same even when he's talking about what a soul is in De Anima. Aristotle doesn't believe in 'souls' in the way we think of them as religio-spiritual entities that exceed the capacity of the body; a 'soul' for Aristotle is the body but in a way that radically challenges the idea of a body as mere shell or vessel - soul is what any form of life repeats doing, as a body, in order to continue being itself. It should be noted that a lot of time at Aristotle's Academy was spent in Zoology, studying animals and their anatomy.
mjburgess 5y ago
This view isnt "scientifically supported" because science is neutral on (indeed, even presupposes) the existence of abstract objects. No one believes abstracta have a physical location -- they lack physical properties. The claim "2 + 2 = 4" is true -- and clearly not true invirute of anything anyone thinks... if we kill that person (/people), it is no less true. Indeed, if numbers don't exist (for example), do we suppose that we can't communicate issues of quantity with other species (, & possible alien life, etc. etc.) ? (If we can, what shared things are we talking about when we quantify?) It seems deeply implausible to say that our use of number is circumstantially psychological -- any description of reality is going to be indispensably quantitative --- quantity is what we are talkng about. We are not talking about ourselves.
- echopurity 5y ago
- jhedwards 5y ago
  I've thought about this problem quite a bit and, while my initial position was the same as above (math is not "real" per se) I had to concede that integers are real, because quantity is self-evidently real. If you have four oranges, the quantity "four" is right there. If you take away one of those oranges you know that the result cannot be split evenly without a remaining orange because of the properties of odd numbers. If you cut the remaining orange in half then you get a rational number, but is that self-evidently real? The halves of the orange are only "halves" because we consider them in relation to their origin, which we consider to be "one" orange. So rational numbers necessarily involve the human action of relating some quantity to a reference quantity, therefore they are a higher-level abstraction built on top of the fundamental physical property of quantity. In the end I decided that math is based on a foundation of quantity (and maybe "space" as well?) and everything else was a derived abstraction. I am very curious if anyone else has a good argument for other parts of math being fundamental.
  - dagw 5y ago
    integers are real, because quantity is self-evidently real. But the there are more integers than there are quantifiable 'things'[1]. Are integers that are a lot larger than, say the size of the power set of all fundamental particles in the universe still "self-evidently real". [1] Assuming a finite universe (or a finite number of finite universes) and a few other things.
andreareina 5y ago
Science has nothing to do with it, rather it's a question of philosophy and what we define as being real.
- dTal 5y ago
  Science is "what we have evidence for". Is there evidence for some abstract mathematics that we didn't invent?
dandanua 5y ago
> there's no god "behind the curtain" have you looked there already? :)
tasha0663 5y ago
You can boil down pretty much everything to "an accumulated edifice of metaphors designed by human brains, for human brains". In the game Hearts, if you take most of the spades you lose. However, if you manage to take all the spades you win, and they call it "shooting the moon". In a similar fashion, when you reject everything as an unreal system of metaphors, Platonism "shoots the moon" by having us reexamine what we thought we meant by "real" in the first place.
voldacar 5y ago
This is one particular point of view, and it's extremely sloppy reasoning to say that it's "scientifically supported", given that mathematics is not a form of science. There are plenty of very smart people, not just mathematicians but also physicists & scientists who are mathematical platonists.
- dTal 5y ago
  There are plenty of smart people - scientists even - who believe in all kinds of deities. Mathematics is indeed not a form of science. But the existence and shape of mathematics is an observable phenomenon, and so metamathematics - the study of what it is and where it comes from - can be studied scientifically. How do you know mathematics exists? Well, there's a textbook right there. Who wrote the textbook and why? A human, expressing metaphors inside their heads. How did those metaphors get inside that human's head? Ah, well, that's the interesting bit - the answer of course transpires to be "a combination of innate ideas imprinted by genetic evolution by natural selection, and sociology". And you don't have to stop there, you can explore in glorious detail exactly where each idea comes from, what innate monkey-ish tendency is being deployed, how exactly ideas like "infinity" fit in a mind designed for finding fruit and chasing things. We can similarly bring all manner of religious beliefs under the anthropological knife. It's not a pretty process though, to the people who believe in them.
  - carapace 5y ago
    > metaphors inside their heads. You're begging the question presupposing the non-Platonic viewpoint here. How do we know that metaphors are "inside the head"?
  - voldacar 5y ago
    You are just assuming the point you're trying to prove > There are plenty of smart people - scientists even - who believe in all kinds of deities. Okay? This is supposed to make me feel - how exactly? I'm not inherently disdainful towards theism or theists, but if I were, I guess your remark would make me like science less, or something? > We can similarly bring all manner of religious beliefs under the anthropological knife I'm not really sure we can, actually. At least not in some kind of non-contentious, "objective" sense. I don't really trust individual humans to give an accurate account of why they believe their beliefs, but I trust "anthropology" and "sociology" even less. My distrust for this on an individual scale comes from the fact that many beliefs & memes exist for purposes of social signalling, group identification, etc, and it might not actually be in your interest to know exactly why you believe what you do. But these auxiliary functions of beliefs, such as signalling etc, seem to me to scale up as you introduce groups and larger-scale activities such as "anthropology" and "sociology". Without some feedback loop keeping them honest, why would I expect anthropologists or sociologists to tell me a true story about why someone believes what they do, any more than that person or anyone else? In aerospace engineering, the feedback loop is that if your design is bad, your jet engine won't work. As a result, I generally trust aerospace engineers about jet engines. But what is there to stop sociologists, anthropologists, etc from just settling on some bullshit that agrees with their preconceived beliefs or flatters their group status and promoting it forever? But back to math. The history of mathematical ideas is complicated and interesting, but it isn't really that relevant to the question of whether the things those ideas are about are "real", which is equivalent to asking whether mathematical platonism is true or not. The question of platonism comes down to the definition of words like "real" and "exist". It is very easy to equivocate using these words, which is why most discussions about mathematical platonism are so low quality. I think the overall question isn't that meaningful so I'm not really a platonist or an anti-platonist. In most parts of human life, when I say "x exists", I mean that I can reach out and touch x, that it has a mass, temperature, surface texture, etc. In math, when I say "x exists" I just mean that I can talk about x without creating any logical contradictions. The square root of -1 may not exist in the same sense as my laptop here, but it exists in the sense that I can do things with it, such as add, multiply, raise to powers, etc, without reaching a contradiction in my formal system. So the whole "out there" thing doesn't really matter. There doesn't need to be an "out there" in order for me to meaningfully say that the square root of -1 exists. I think that a lot of philosophy is like this too, when you mentally zoom in really closely on a problem, it often reduces to some kind of equivocation or inconsistent language usage. Btw I don't really consider anthropology or sociology to be real intellectual disciplines, and I'm pretty on the fence about psychology and economics. I realize that is an unpopular opinion but I've thought about it a lot and I'm pretty certain that it's correct. Aerospace engineering is real because it attaches to some fundamental reality, namely that of the spinning fan blades, the combusting fuel, etc. If you get your engineering wrong, the fan blades won't spin. Likewise, math is attached to systems of axioms. When your do your math wrong, you get a contradiction. Sociology and anthropology don't attach to anything, they're like a closed loop, like theology. If you get your anthropology wrong, nothing really happens.
canjobear 5y ago
Reality is that which, when you stop believing in it, doesn’t go away. -Philip K Dick Even if you stop “believing in” math, your proofs are still either correct or incorrect.
- dagw 5y ago
  The proofs are only correct in as far as you believe in the Axioms of mathematics that those proofs are built on. Stop "believing in" the axioms and the proofs are no longer true or even meaningful.
  - canjobear 5y ago
    But it is still the case that IF the axioms hold, THEN the proofs are either valid or not.
    - marcosdumay 5y ago
      So it's conditional reality. Now make that fit within your usual realistic philosophy...
      - canjobear 5y ago
        Easy. There exist contingency relationships between axioms and theorems provable from those axioms.
        marcosdumay 5y ago
        Oh, yeas, your proof is quite real. And it's completely meaningless, being all about imaginary things. It can only have any meaning if you adhere to some scientific model. At this point you've deviated so much from the OP discussion that you could as well talk about angels dancing in pinheads. Any quantification of them is as real as your proof.
tim333 5y ago
>all numbers are imaginary.... mathematics is an accumulated edifice of metaphors designed by human brains Or you can say they exist but in a different way to physical reality. I mean pi probably still was 3.14159... before humans evolved so it's not our fault really. Personally I think maths not only exists but physical reality is a subset. I mean why else is there something rather than nothing? Scientifically it's the only hypothesis that works for that really.
nathias 5y ago
No it isn't scientifically supported because science does not include ontology and episemology as its domain. I'm not a platonist, but the reasons for not being platonists are philosophical.
garmain 5y ago
1) I might be missing something, but as far as I can tell "scientific support" in this context seems ill defined. To the extent that it seems rather meaningless term (in this context) 2) While the vast majority of the discourse on the interpretation of mathematics oscillates (fruitlessly) between Platonism and Nominalism, I tend towards a more Aristotelian view. See for example - Franklin: An Aristotelian Realist Philosophy of Mathematics - Keith Hossack: Knowledge and the Philosophy of Number. Note that these two authors do not converge on exactly the same interpretation. Apropos: North (2021): Physics, Structure, and Reality explores the relationship between "mathematical structure" reality and theoretical physics.

dandanua 5y ago

Life is complex – it has real and imaginary parts.

gumby 5y ago

“Imaginary” was an unfortunate choice of word. Positive integers and rationals feel “legit”* to most people, even (or especially) the non-mathematically trained. But “imaginary” numbers are no more or less legit than, say, transcendental numbers. But in middle school you learn this confusing thing and maths starts to fall off the rails. * Had to find a synonym for “real” here.

jdrc 5y ago
Yeah we should stop using the word imaginary, and just call them iNumbers (sorry Apple)
- Qem 5y ago
  I like to call them "double-barreled numbers". I never liked the term "imaginary", because that gives mystical connotations and even hamper understanding, in my opinion.
intrepidhero 5y ago
That's what I've finally settled on in my mind. Since we use imaginary numbers to model and measure real phenomenon, they're not imaginary in the sense that I would usually use that word. There's got to be a better name... Transcendental complement maybe.
starwind 5y ago
when I did my math degree in college the profs and grad students called them "complex numbers." I think they know there's a problem so they want to get the students who would go on to teach used to not calling them "imaginary"
- deleted 5y ago
  [deleted]
- bee_rider 5y ago
  That doesn't quite get at it, though -- a complex number has an imaginary and real part. I guess we could call them "complex numbers with a zero real part," but 1) that's quite a mouthful i) it implies that the other part is something other than real -1) it seems weird to describe something sort of... subtractively like this

hgomersall 5y ago

There is a persuasive school of thought that argues exactly the opposite, that imaginary numbers are not real http://geometry.mrao.cam.ac.uk/1993/01/imaginary-numbers-are-not-real-the-geometric-algebra-of-spacetime/ http://geometry.mrao.cam.ac.uk/1993/01/imaginary-numbers-are... .

topaz0 5y ago
I'm familiar with (and partial to) the geometric algebra literature, but I think it's a bit disingenuous/clickbaity to say that it implies imaginary numbers aren't real. I think of it as meaning that the algebra of the imaginary numbers arises as a slice through a more useful, bigger algebra.
- hgomersall 5y ago
  Yes, I think it's rather tongue in cheek. It's probably not defined as clickbait since it was written in '93!
  - topaz0 5y ago
    Fair enough!
wrycoder 5y ago
You can make some progress by thinking of the imaginary numbers as being a two dimensional vector space over the reals, equipped with a very peculiar multiplication operator that is a combination of the usual stretch with a rotation. Addition is the usual addition of components, but the multiplication (using polar coordinates) looks like C = ABexp(T1+T2), where the T are the phases of A and B. To multiply two positive reals, since their phases are zero, it’s only a stretch: AB To multiply two negative reals, since their phases are pi, it’s ABexp(pi+pi). But exp(2pi) is unity, so we have AB again, a positive number. To find the square root of a positive number D, find a positive real (zero phase) that, when multiplied by itself, gives D. AA, for example. So, real number arithmetic is a subset of complex number arithmetic, with zero phase. And therefore zero rotation. Now, find the square root of a negative number, say -4. To do this, we have to step off the real axis for the first time: out into the Argand plane! -4 is represented as 4exp(pi) OK? So we are looking for a number out here on the Argand plane that when multiplied by itself following our rule gives 4exp(pi). And that number is 2exp(pi/2), which is two units out on the “imaginary” axis of our Argand plane. 2exp(pi/2)2exp(pi/2) equals 4exp(pi/2+pi/2) equals 4exp(pi) QED

viktorcode 5y ago

I highly recommend video by Sabine Hossenfelder [1] on this topic. She's explaining the topic in layman terms, starting from what does it mean to exist (or be real) in physical sense. Complex numbers, as useful as they are, just an abstraction, a tool which can be replaced with different forms of calculations. That makes the question of "reality" rather complex (pardon the pun). [1] https://www.youtube.com/watch?v=ALc8CBYOfkw https://www.youtube.com/watch?v=ALc8CBYOfkw

Psyladine 5y ago
For those interested I also highly recommend Veritasium's video[1] as an excellent explainer as well, especially regarding the roots of imaginary/complex numbers in geometry. [1] https://www.youtube.com/watch?v=cUzklzVXJwo https://www.youtube.com/watch?v=cUzklzVXJwo

1970-01-01 5y ago

I've been calling them invisible numbers most of my life. It makes much more sense.

adityapatadia 5y ago

This is one video which shows history of imaginary numbers and their usefulness in current times. It's from Veritasium: https://www.youtube.com/watch?v=cUzklzVXJwo https://www.youtube.com/watch?v=cUzklzVXJwo

dboreham 5y ago

This is only mentioned in passing in the article, but the penny dropped for me when I read somewhere else that you should first think about negative numbers. They seem pretty real, but they're not. There's no negative quantity in nature. You can't have -5 rocks. Negative numbers are just the solution to equations of the form x + n = 0. Now after you let that sink in, imaginary numbers don't seem so weird after all. Just another kind of number defined as the solution to some equation.

mettamage 5y ago
I disagree. > You can't have -5 rocks You can have 5 rocks destroyed in the future. That to me, is what -5 physically means: a guarantee that the object associated with the number will be eliminated out of existence in the future and decrement the negative number by 1. Some people call it debt, but to me, emotionally that word feels too financial. So I prefer “a guarantee to be eliminated out of existence in the future “, or something like that. Conversely, 5 cows means: 5 cows currently in existence. 5 cows - 5 cows means: I see 5 cows and now they don’t exist any more and there is nothing. I wish my math skills were better, I am optimistic that I’d find a similar thing for imaginary numbers and maybe even complex numbers. With that said, I do get where you’re coming from and I find it a compelling perspective as well. It’s simply that I feel the perspective I described as well.
- easywood 5y ago
  >> You can have 5 rocks destroyed in the future You have 5 rocks now, and zero rocks in the future. At no point are there negative rock-shaped holes in the fabric of space. The parent's comment still stands, he is talking about reality, not our mathematical interpretation of it.
- marcus_cemes 5y ago
  > You can have 5 rocks destroyed in the future. Exactly, you use it to store some information that has no real quantity but may be converted to a real quantity in the future through some other process. I'm a polytechnic university student, we use imaginary numbers extensively in all sorts of places, especially whenever there is any oscillatory behaviour, such as an electrical signal or a light wave. A complex number is just a two-dimensional vector with real/imaginary components, whcih provides an amplitude and a phase (angle). An oscillating sinusoidal signal/wave may appear to be zero and completely static if you freeze time at the right moment, but as time progresses, it will continue oscillating, like a swing in a park. In a way, the magnitude represents the built up "momentum" of the system, whilst the real quantity is the immediate physical value at any given point in time (given by the phase). The amplitude is always the same at any given moment, even when the swing is vertical, it has momentum which will help it reach its maximum height. Personally, I still think they are just "invented", but I think the vast majority of engineers much prefer them to the alternative, manipulating trigonometric functions (every engineer's nightmare). They're a neat way to represent the exchange of potential and mechanical/electrical energy with a single value and some simplified mathematics (this is an engineer's, not a mathematician's, point of view). Like negative numbers, we could have chosen to have two positive quantities, balance and debt, instead we find use in merging these definitions, whether negative values make sense or not. We have become used to to negative numbers representing the "inverse" action, which makes sense when representing a phyiscal quantity such as velocity.

rthomas6 5y ago

What finally made imaginary numbers intuitively make sense to me is realizing that the number i just represents a 90 degree rotation on the complex plane. That's why i*i = -1 (it's rotating 180 degrees), why imaginary numbers are orthogonal to real numbers, why e^ix = cos(x) + i*sin(x) makes sense, everything. When you are dealing with 2 dimensions, complex numbers are a kind of hack for representing both dimensions without any kind of vectors or pairs or anything, just numbers.

starwind 5y ago
I majored in math and took a course on waves through the physics department. Everyone struggled with why imaginary numbers would show up and I was just like "we took the square root of this negative number, and the reals aren't closed when we take square roots, therefore imaginary." I missed out on a lot of physical intuition and had basically no concept of what the imaginary numbers actually meant
ThrustVectoring 5y ago
More relevant and less hacky, IMO, is that the real numbers aren't algebraically closed - there are polynomials with real coefficients that you cannot factor into getting real zeros. This is a big problem when you have something like a spring-mass-damper system. The springiness, damping, and mass of the system generate coefficients of a degree-two polynomial, and the zeros of that polynomial correspond to how the system moves over time.
Epa095 5y ago
With this intuition, why is it i*i=-1 and not i+i (90+90=180 after all)?
- rthomas6 5y ago
  Because with complex numbers you are implicitly defining a point on the complex plane. So how on the Cartesian plane you can have 2x + 3y, and that defines the point (2,3), you can have 2 + 3i on the complex plane defining the same point. If you want to rotate the Cartesian coordinate -90 degrees, x' = x*cos(π) - y*sin(π) = -y y' = y*cos(π) + x*sin(π) = x 2x + 3y => -3x + 2y So check it out: (2 + 3i)*i = 2i - 3 It's the same as rotation!

deleted 5y ago

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nuncanada 5y ago

Mathematicians have been constantly been giving bad names do their concepts... Imaginary Numbers is one of the atrocious ones, Imaginary Numbers are part of the fabric of our reality... Meanwhile Real Number were axiomatized in a way that creates completely unrealistic numbers, "ghost-like" numbers, numbers that can never be given a name... The first course in Analysis is ridden with examples of Continuous Functions that are not continuous in any sense humans would consider ever it...

openfuture 5y ago
Completely and utterly disagree, it may have been the case in the 1800s or whatever but the naming sense of modern mathematics is pretty on point and actually incredibly insightful. Consider the definition of a "connected" topological space and tell me it doesn't tell you something about society.

Agamus 5y ago

I'm not an expert here, but truly interested to hear responses to this question. To say that 1+1=2 is "true", does that not require a corollary in "reality" to something fundamental that can be called a "one" object? I believe this is called mathematical constructivism. Imagine, hypothetically, that we cannot identify something that is physically fundamental and individual. My question is whether any mathematics in that scenario could be considered "true" without such constructivism, in other words, without a physical correspondence to an unquestionably, physically fundamental "one" object.

phendrenad2 5y ago

Imaginary numbers are just a shorthand way of representing things in nature that relate to one another via the sine function, for example, charged particles in a magnetic field. You can bust out calculus to describe the motion, or you can use a convenient set of rules that represents the partially-solved equation.

Khaine 5y ago

Veritasium has a good video on how complex numbers were invented https://www.youtube.com/watch?v=cUzklzVXJwo https://www.youtube.com/watch?v=cUzklzVXJwo

abecedarius 5y ago

Details on Heron's missed sqrt(-63): https://nrich.maths.org/13402 https://nrich.maths.org/13402 Apparently there's not much to say about how that mistake came up, though.

gweinberg 5y ago

Quaternions still suck though. Octonions even more so.

analog31 5y ago

Here's what I wonder: If somebody is hung up on complex numbers, will getting them over that hump get them any closer to understanding how or why complex numbers are used in physics or engineering?

dataflow 5y ago
I think they'll understand it if you get them to solve a few Fourier series/transforms with sinusoids, and then with complex exponentials, and observe the difference in difficulty.
ganzuul 5y ago
The behavior of complex numbers under certain operations is seemingly the same mechanism as is seen in physical systems. The style of notation used in engineering usually denotes a periodic motion or a compound of many waves. Of interest for physics is that without the imaginary part, real numbers alone are a leaky abstraction. This structure extends to quaterions and octonions.
- analog31 5y ago
  Indeed, I'm a physicist with a background in electronics, so I'm fluent in that stuff. The question is whether explaining complex numbers to a non physicist is really going to bring them any closer to physics.
  - ganzuul 5y ago
    Considering complex numbers lead to group theory and group theory may model atomic structure, this metaphysical correlation is at least worthy of note. It is almost as if we are living in a distortion of truth.

gregfjohnson 5y ago

A very helpful intuition is to consider a notion of "Compass numbers"; one might say, "the nearest coffee shop is East 2 blocks and North 3 blocks". One might say the the real numbers are restricted to just two directions. Extend them to allow the "sign" of a number to be any direction. It is intuitive to imagine what addition of "Compass numbers" might look like, and what multiplication by a positive scalar real number might look like. The question is, what should multiplication of two compass numbers neither of which is real be? One of the earliest developments of complex numbers was done by a Danish cartographer, Caspar Wessel. He wrote a lovely paper that was published in an obscure forum in 1797. He is now credited as the first person to understand the correspondence of complex numbers and vectors on a plane. It seems quite natural that a cartographer would be interested in "numbers" that can point in any direction. If you posit the existence of a multiplicative identity and (arbitrarily) label it "1", and then take a compass number 90 degrees away from it and (arbitrarily) label it "i", (and furthermore assume field axioms), the formula "(1 + i) * (1 - i)" forces the conclusion that i * i is the compass number pointing in the opposite direction of the multiplicative identity.

gregfjohnson 5y ago
To elaborate slightly; (1 + i) * (1 - i) = 1 - i^2. The left side is somewhere on the complex plane on a circle of radius 2 centered at the origin. The right side is somewhere on a circle of radius 1 centered at 1. The circles meet at one and only one place, namely 2. So 2 = 1 - i^2, or i^2 = -1.

vishnugupta 5y ago

> So let’s picture multiplication by –1 as half a rotation, anticlockwise, around a circle (in our case, the circle passes through 1 and –1). It’s actually a rotation by 180 degrees. Sorry but this just didn't work for me. My mental model is to use +/- as directional indicators. For me multiplication by -1 is same as multiplication by +1 but in the opposite direction. I would rather think of "i" as rotation by 90°. In this realm +/- stand for anti/clockwise. So +/- continue to stand for direction and "i" tells me if I need to rotate or not. It neatly lines up with different operations. As an example (+i) × (-i) == -i² == 1. In terms of movements you are rotating 90° anti-clockwise followed by 90° clockwise bringing you back where you started. On the other hand, Nautilus's article is bit too much of mental gymnastics for me to follow their reasoning. > So let’s picture multiplication by –1 as half a rotation..It’s actually a rotation by 180 degrees..What happens if we only do half of this rotation? It’s halfway to multiplying by –1, which you can think of as the same as multiplying by √–1...

garmain 5y ago
I'm curious. I seem to be missing something, because I feel like what you're saying tracks very much with the "alternative interpretation". That is, I don't see the difference (in my mind's eye) at all. Is there maybe a different way to explain what you perceive to be critically different?
- vishnugupta 5y ago
  For me it is about expressing a concept as compactly as possible. In case of Nautilus they first ask us to imagine -1 as 180° and then proceed say half of it as 90° rotation and finally give it a name i. For some this may work fine as a way to understand. However it gets awkward to further explain. If 𝑖 stands for half of -1 (as they say) then does that mean 𝑖 == -(1/2)? If not then why not? and so on. Compare that with: 𝑖 is a unit of rotation which is 90°. That's it. Using this as a base it's easier to explain why 𝑖² = -1 (two units of anti-clockwise rotation) or why 𝑖³ = -i (three units of anti-clockwise rotation is same as one unit of clockwise rotation) and so on. More fundamentally, they ask us to imagine an operation on 1-d (-1) as happening through 2-d (180° rotation). It seems convoluted way to introduce 𝑖. I'd rather explain 𝑖 from the first principles and proceed to show its consequences such as 𝑖² = -1.

DarkCrusader2 5y ago

Can not recommend this series of videos from Welch Lab[1] enough on imaginary numbers. Deep dives into the history of how complex numbers were "invented" to enable mathematical representation which weren't possible by existing set of numbers. Also draws parallel to "invention" of negative numbers and zero to overcome the limitations of the then present number system (for eg. how to balance books without negative numbers). [1] https://youtube.com/playlist?list=PLiaHhY2iBX9g6KIvZ_703G3KJXapKkNaF https://youtube.com/playlist?list=PLiaHhY2iBX9g6KIvZ_703G3KJ...

j7ake 5y ago

I always imagine complex numbers by putting them on a Cartesian plane (2D linear space). Is there an extension of this for numbers that can be put on a 3D space?

lordgrenville 5y ago
That's my understanding of quaternions (but I might be way off).