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Complex numbers are used widely in quantum mechanics and the waveform, is there a physical interpretation of what this means about the structure of the universe? Why is it not used in macro physics?

Do physicists really think that calling imaginary numbers a 90 degree rotation is a good enough answer? It seems to be used in many areas to mean similar things.

Is there an explanation to do with dimensions as I have tried in this conversation a better way to understand them?

OzOz
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    Possible duplicate of What is the need of complex functions in wave analysis? – Brick Nov 05 '19 at 15:20
  • Related: https://physics.stackexchange.com/q/8062/2451 , https://physics.stackexchange.com/q/11396/2451 , https://physics.stackexchange.com/q/32422/2451 and links therein. – Qmechanic Nov 05 '19 at 20:12
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    I think you should reformulate your question as : "what is the physical interpretation of probability amplitudes?". And the honest answer to that would be : "we don't know." – user140255 Nov 05 '19 at 22:32
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    They are used for "macro-physics" (depending on what mean by this term) such as to work with AC voltage in electrical systems. – Steeven Nov 06 '19 at 12:39
  • This is probably better asked on maths. Physicists are notorius for having a "shut up and calculate!" point of view... – Stian Nov 06 '19 at 13:57
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    @Stian Yttervik- I think the reason for that is that a LOT of people (not the OP on this question) both online and in real life approach physics with some really weird unreal stuff: "what if particles are held together by flows of wavefunction energy" was one I was once asked. The best way of dealing with such people is not to tell them what they are saying doesn't make sense (they will argue back). But to say "Sure, cool theory. Can you predict or calculate something for me with it?". This immediately highlights to them that their "theory" doesn't qualify as science as it is "not even wrong". – Dast Nov 06 '19 at 16:00
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    @Dast Yeah, I agree, and some of the things you learn in physics are impossible to learn untill you are familiar with the equations and the only way to get there is to... shut up and calculate. It is self reinforcing. Certainly, if you are able to both have that familiarity with the equations but simultaneously have a mental model of the universe that allows for you to create simplifications and explanation models, you end up being quite revered – Stian Nov 06 '19 at 19:15
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    I am definitely not a maths or physics wiz in any way but it seems to me that the question is backwards. The universe IS. We have created numbers and math based on those numbers. And in our system we had to add 'imaginary numbers' to make our equations better match the universe we're trying to understand. Had we used some other method of quantifying our physical universe they might not be imaginary numbers at all - they could be very important ones that are incredibly useful in describing the physical phenomenon that we observe, measure, and model. – CramerTV Nov 07 '19 at 00:11
  • The more this conversation has gone on, the more poor that the description of complex numbers being 90 degrees is. While not being wrong, it fails to describe that complex numbers are reducing a many dimension systems to only two dimensions, with the imaginary being how the other dimensions effect the real part? Anyone else think this is more accurate? Or, is it totally wrong? – OzOz Nov 07 '19 at 10:42
  • Re, "Do physicists really think that calling imaginary numbers a 90 degree rotation is a good enough answer?" Here's an experiment you can try at home: Plot a random assortment of numbers in the complex plane. Then, make another plot in which you've multiplied all of the original numbers by $i$. Compare the two plots. (Hint: if you physically rotate the first plot counter-clockwise by 90 degrees, it may help you to see where this is going.) – Solomon Slow Nov 07 '19 at 17:04
  • @Solomon: Of course we have all studied complex numbers and agree with this... But it's not a very useful or full explanation. Vectors also do exactly the same thing. Are complex numbers and vectors identical? We have had a wide variety of answers here. It's not wrong, it's just not the full answer. – OzOz Nov 07 '19 at 18:23
  • @Solomon: Let me ask this: if it is just a 2d rotation, then does that mean that whenever it is used to calculate something that it can only be used on a 2d system? So, lots of quantum mechanics is about rotation in a 2d system? What happens when we use complex numbers in a larger dimension space? Just saying it's a rotation through 90 really leaves many people scratching their heads as to why is is used so much when it all it is is a simple vector rotation? Why even use it if it's just a vector rotation? – OzOz Nov 07 '19 at 18:28
  • @OzOz, I suggest that you read the first several chapters of this book: https://www.amazon.com/Road-Reality-Complete-Guide-Universe/dp/0679776311 This one might also be helpful: https://www.amazon.com/Linear-Algebra-Right-Undergraduate-Mathematics/dp/0387982582 – Solomon Slow Nov 07 '19 at 19:59
  • @Solomon: Thank you very much for your answer! I have if course read many books on the topic. You couldn't even try to explain? – OzOz Nov 07 '19 at 22:49
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    The "two dimensions" in the complex number plane of a probability amplitude do not correspond to two spatial dimensions in our universe. A wavefunction $\psi$ of a particle in 3D space is a mapping $\mathbb{R}^3 \to \mathbb{C}$. – Scott Centoni Nov 07 '19 at 23:57
  • @Scott: This is one of the big questions I have. When we make this mapping, what happens to the third dimension? I assume we have lost information? Then how can we still fully describe the system? The more this conversation has gone on the more I think that complex numbers are a way of studying one dimension of a multi dimensional system (the real component) by compressing the information from the other dimensions into the imaginary by describing how the other dimensions effect the main real dim only, their own internal effects removed. But, people don't seem to like this! – OzOz Nov 08 '19 at 02:05
  • @OzOz, I too have read many books, but I never could appreciate complex numbers until I read "The Road to Reality." You'll have to work your way through several chapters to see it. I can't fit that into this comment thread, nor do I have the time. I'll give you this hint though: Every practical calculation can be approximated to "close enough" using nothing but ratios of integers. The reason complex numbers are so compelling is not that they are any more powerful than rationals. It's the algebra. The written formulas are cleaner, simpler, when the symbols stand for complex values. – Solomon Slow Nov 08 '19 at 15:35
  • A temperature field is a mapping $\mathbb{R}^3 \to \mathbb{R}$. Do you still see this as a problem? Are you still worried about the second and third dimension? – Scott Centoni Nov 26 '19 at 05:35

10 Answers10

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Complex numbers are used in all of mathematics, and therefore by extension they are used in other fields that require math; not just physics, but also engineering and other fields. Trying to assign a "physical interpretation" to a complex number would be like assigning a physical interpretation to a real number, such as the number 5.

A complex number is just an extension of a real number. Many of us were taught about the "number line" in elementary school, which is just a line that (to quote Wikipedia) serves as an abstraction for real numbers. Being a line, it is 1-dimensional. Complex numbers are the same, except they are 2-dimensional: instead of being described by a 1-dimensional real number line, they are described by a 2-dimensional "complex number plane". Using $i$ for the imaginary axis (where $i^2 = -1$) is a mathematical convenience that makes the 2-dimensional complex numbers extraordinarily useful.

BioPhysicist
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Richter65
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Complex numbers are used in "macro" physics. They are used in analysis of electrical circuits (especially when AC is involved) and in fluid dynamics. Solution of differential equations is simplified if complex numbers are used, as is Fourier analysis. Any scenario that involves periodic or cyclic functions can be modeled using complex numbers.

gandalf61
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    they are fundamental to “micro” physics as well since one cannot do quantum mechanics without complex numbers. – ZeroTheHero Nov 05 '19 at 13:40
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    @Zero Depends on the meaning of "one cannot do quantum mechanics without complex numbers". Relevant https://physics.stackexchange.com/questions/32422/qm-without-complex-numbers – jinawee Nov 05 '19 at 13:44
  • Thank you, but still unsure, -x has a physical interpretation of going backwards, what is the physical interpretation of ix? – OzOz Nov 05 '19 at 13:46
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    @OzOz $x+iy$ is a handy way of combining two real state parameters into a single quantity. For a pendulum these parameters could be position and momentum. For an electrical circuit they could be current and voltage. – gandalf61 Nov 05 '19 at 13:58
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    $ix$ has no easy physical interpretation, however $i\frac{d}{dx}$ has (negative momentum). That shows that the complex plane gives us an easy way to merge position and momentum probability distributions into a single function (the wavefunction). – Martin Peschel Nov 05 '19 at 14:00
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    You definitely could do QM without complex numbers, @ZeroTheHero. Whether you'd want to do it that way is a different question. Schrodinger equation has a real formulation - one complex equation goes to two real equations. You could do the same for any other field theory, but it probably be a slog. – Brick Nov 05 '19 at 15:23
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    $ix$ has a perfectly good physical interpretation, when if $x$ represents a vector in a complex plane: $ix$ represents the vector you get by rotating $x$ clockwise by $90^\circ$. – Lee Mosher Nov 06 '19 at 01:33
  • "especially when AC is involved" -- True. @OzOz Compare this question. – Kamil Maciorowski Nov 06 '19 at 06:56
  • Interesting answers. I remember some number theory that said that, in order to describe the real line properly, you actually had to use complex numbers. The reason being something like because the real line has uncountable infinities, in order to describe the shape of objects that are countably infinitely small on the real line, you had to have a new variable to describe this shape. These infinitely small objects did not exist, but they could expand or move at speed and have an effect real objects. The physical interpretation of momentum only thus makes sense? – OzOz Nov 06 '19 at 10:09
  • Sorry, that answer was confused! I guess I mean: complex numbers are definitely something to do with forming a second dimension? What sort of dimension and why? I think it has something to do with keeping shape in another space that has no shape in the previous dimension, but I know how wooly that is, can anyone explain this better? – OzOz Nov 06 '19 at 10:11
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    When complex numbers are used in electrodynamics there is just one perpendicular thing, the magnetic field... In quantum mechanics i has many perpendicular dimensions. – OzOz Nov 06 '19 at 14:09
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    The complex numbers are actually talking about a system where dimensions are nested within each other, one normal size with shape, but also a smaller dimension, which also has shape, but only to itself, not to an observer in the normal sized dimension. Only neighbouring dimensions can interact, and they do, one to the other and back again, in wavelike manner. The imaginary dimension cannot be seen but over time it's change can make it increase in size into the normal sized dimension, this is why so many say i d/dx is real. It is also a rotation from the normal dimension into a new dimension. – OzOz Nov 06 '19 at 14:14
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    @OzOz Perspective from a non-physicist mathematician: when you have vectors, you can model them using linear algebra or complex numbers. Complex differential functions have a lot of magic properties, which makes them more convenient when you need to do a lot of calculus. – BlueRaja - Danny Pflughoeft Nov 06 '19 at 21:28
  • Can you add in something that addresses the main question of having a physical interpretation of imaginary numbers? – CramerTV Nov 07 '19 at 00:01
  • @LeeMosher ITYM multiplying by i rotates the vector anticlockwise. – Jens Nov 07 '19 at 20:28
  • Good point. Was thinking "positive direction" and wrote clockwise. – Lee Mosher Nov 07 '19 at 21:37
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The fundamental object in quantum mechanics is the amplitude, which encodes information about how a system transitions from one state to another state. For example, if you are doing a double slit experiment you might care about how an electron transitions from the incoming pre-slit state to a state where it hits a certain location $x$ on the detector. For each different outcome state there would be a different amplitude $\mathcal{M}_x$.

We care about amplitudes because they can tell us about probabilities. According to the Born rule the probability that the electron ends up at location $x$ is given by absolute value of the square of the amplitude, $P(x) = |\mathcal{M}_x|^2$.

The probability is a non-negative real number, but what kind of object should represent the amplitude? A positive real number? Any real number? A pair of real numbers? A complex number? Some even more abstract mathematical object?

This paper addresses the question by noticing that since amplitudes correspond to different experiments, and experiments can be chained together in various ways, we have to be able to combine two amplitudes to get a third amplitude, and we have to be able to combine them in at least two distinct ways. The paper then proves that, if you choose to represent amplitudes as pairs of real numbers, the operations that correspond to combining experiments end up acting exactly like complex addition and complex multiplication.

The paper doesn't answer the question of why amplitudes should be pairs of real numbers instead of single real numbers, or triples or something more complex, but it's a good starting point for seeing how complex arithmetic falls out of the logic of quantum experiments.

P.S. Using single real numbers for amplitudes cannot explain the single slit / double slit experiment, where adding a second slit results in zeros in the probability distribution that weren't present in the single slit probability distribution. Using a pair of real numbers (or one complex number) is the next simplest system that can explain this behavior.

Luke Pritchett
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  • That is very interesting, will take some time to understand. – OzOz Nov 05 '19 at 23:12
  • "Using single real numbers for amplitudes cannot explain the single slit / double slit experiment" I don't think so. Please see my answer at https://physics.stackexchange.com/questions/32422/qm-without-complex-numbers – akhmeteli Nov 06 '19 at 03:07
  • @akhmeteli My argument is pretty simple. The probabiltiy distribution with one slit is $\psi_1(x)^2$, and has no zeros. The probability distribution with two slits is $(\psi_1(x)+\psi_1(x+a))^2$ where $a$ is the slit separation. The two slit distribution does have zeros. There is no continuous real function that behaves this way when added to a shifted version of itself. Ultimately it is because to pass from positive reals to negative reals you must pass through zero (which isn't true for complex numbers).

    You can define QM with real wavefunctions, but it can't do the above.

     – Luke Pritchett Nov 06 '19 at 13:57
  • So, to reword this answer, the way to understanding the "physical interpretation of complex numbers" (as asked by OP) is by understanding the physical interpretation of the operations on complex numbers as amplitudes. And then one can realize that the features of the amplitudes give a definition of complex numbers. Sounds neat! – JiK Nov 06 '19 at 14:38
  • @LukePritchett : And why do I need to do the above? – akhmeteli Nov 07 '19 at 02:26
  • @akhmeteli Because it is an experimental fact that the probability distribution for the single slit experiment has no zeros and the probability distribution for the double slit experiment has zeros? If you believe that probability distributions should come from squares of continuous wavefunctions then you must use complex wavefunctions, otherwise you won't be able to predict the zeros correctly. – Luke Pritchett Nov 07 '19 at 15:39
  • @LukePritchett : Your argument does not seem to hold water: Schrödinger explained in his 1952 article that any solution with a (scalar) complex wave function is physically equivalent to a solution with a real wave function, which can be obtained from the initial solution by a gauge transformation. Schrödinger wrote: “That the wave function of [the Klein-Gordon equation] can be made real by a change of gauge is but a truism, though it contradicts the widespread belief about `charged’ fields requiring complex representation.” – akhmeteli Nov 08 '19 at 03:31
  • @akhmeteli I'm not talking about the Klein-Gordon equation though. Solutions to the Klein-Gordon equation famously don't work as wavefunctions, right? But in the end, show me a real wavefunction that gives the correct single- and double-slit distributions and I'll believe you. – Luke Pritchett Nov 08 '19 at 12:43
  • @LukePritchett : Schrödinger's approach works for the original Schrödinger equation as well. I don't need you to believe me or Schrödinger, but if you have a complex solution of a diffraction problem $\rho\exp(i\varphi)$ in 4-potential $A_\mu$, then you have a real solution $\rho$ of a problem in 4-potential $A_\mu+\partial_\mu \varphi$, which produces the same electromagnetic field (I neglect signs and constant factors in the formulas). – akhmeteli Nov 08 '19 at 15:27
  • @akhmeteli But I'm not talking about a vector field. I'm talking about the wavefunction of a non-relativistic electron. A four-potential has nothing to do with the experiment I'm talking about. Here: $e^{ikr}/r$ is a wavefunction for an electron through a single slit. Added to itself shifted the two-slit wavefunction is $e^{ikr}/r + e^{ik|\vec{r}-a\hat{x}|}/|\vec{r}-a\hat{x}|$. These two give the correct probability distributions on a detector far from the slits. What real wavefunction should I use instead of $e^{ikr}/r$ to get the same distributions? – Luke Pritchett Nov 08 '19 at 15:39
  • @akhmeteli It is possible to write both of my example wavefunctions as real functions times a phase function. But my question is how do I start with the real version of the first wavefunction and get the second wavefunction? The phase difference between the two summands is crucial to the interference pattern, but the right phase difference is not obvious at all when using only real wavefunctions. Complex addition has to happen somewhere, even if it is hidden by taking away overall phases at the end. – Luke Pritchett Nov 08 '19 at 15:59
  • @LukePritchett : "But I'm not talking about a vector field. I'm talking about the wavefunction of a non-relativistic electron. A four-potential has nothing to do with the experiment I'm talking about." I don't understand that. I am talking about a non-relativistic electron described by a scalar wave function $\varphi$, however, the electron moves in external electromagnetic field $A^\mu$ (if you don't have external field, you don't have the double slit experiment). You can choose a gauge transformation involving $\varphi$ and $A^\mu$ in such a way that $\varphi$ becomes real. – akhmeteli Nov 09 '19 at 04:06
  • @LukePritchett : Just a correction to my previous comment: I should have denoted the scalar wave function by letter $\psi$ instead of $\varphi$, to avoid confusion with the phase of the wave function. – akhmeteli Nov 09 '19 at 16:23
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Complex number as any number alone does not say anything about physics at all. It has to be bound to some measurement unit(s) or have a well-defined definition in physics.

For example complex refractive index is defined in physics as :

$$ {\displaystyle {\underline {n}}=n+i\kappa .} $$

Here imaginary part $\kappa$ is defined as attenuation coefficient - matterial resistivity to penetration of light waves

EDIT

Complex numbers are used intensively in describing any kind of waves, because you can put wave amplitude and wave phase into a single complex-valued wave amplitude:

$$ Z = Ae^{i\phi} $$

So most things which is related to waves can be, at least theoretically, expressed in complex numbers.
For example,- complex refractive index can be traced back to other wave properties in such way : $$ \underline{k} = 2\pi \underline{n}/λ_0 $$ where $\underline{k}$ is complex wavenumber

BONUS

Another reason why complex plane is attractive - you can do more math if you are not bound to real numbers. For example, you can even take a natural logarithm of negative real number : $$ \ln(-x) = \ln(x) + \pi \space \textrm{i} $$

which results in complex number ! So, never trust your pocket calculator

Agnius Vasiliauskas
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  • I feel this is getting closer to an actual answer, but it is too specific, is there any way this could be generalized? – OzOz Nov 06 '19 at 07:04
  • see edit, I tried to do some sort of generalizations – Agnius Vasiliauskas Nov 06 '19 at 08:32
  • Interesting, so a complex number can be used to express any system in which there are two independent properties that can be measured? Which could be a second dimension or a different physical property? – OzOz Nov 06 '19 at 09:30
  • Is this any different to introducing an entirely new variable? I would assume that the i^2=-1 identity would not apply if they were fully independent dimensions or fully independent physical properties? What does this connection mean physically? It is similar to adding a new dimension, but there is a difference, a connection between the imaginary and real dimension? I feel it might be similar to the difference between the radial dimension (goes to infinity, does not repeat) and a radial dimension (limited distance, repeats, length is set by the radial dimension, it is based on the radial)? – OzOz Nov 06 '19 at 09:39
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    @OzOz If we take positive number and multiply it by i we end up with an imaginary one which is like a phase shift of 90 degrees and is completely orthogonal. So there yes, it appears to be just like an independent variable, but then, if we multiply by i again, we now have a phase shift of 180 degrees, which is negation along the original real axis. – Rodney Nov 06 '19 at 14:21
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    @OzOz, Re, "[can a] complex number...be used to express any system in which there are two independent properties...?" a two dimensional vector can do that, and the complex plane is a two-dimensional vector space, but some problems (e.g., mathematical descriptions of periodic functions and wave motion) are an especially good fit for the peculiar algebraic properties of complex numbers. – Solomon Slow Nov 07 '19 at 17:27
  • @Solomon Why is it a good fit for periodic functions? Besides the fact that periodic functions move between two things, which could be described using vectors? If it is just that it is easier then how and why? But, other people say it's because of it being easier with calculus, still more say it's because of the complex analysis Cauchy proofs work, still more say they are exactly the same as vectors. It really feels that this is poorly understood? Could you explain further and clear it up? Can anybody? – OzOz Nov 07 '19 at 19:36
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Complex numbers are just a convenient way of representing a 2-dimensional vector. They are used in all sorts of everyday situations where you have an X and a Y component, or a magnitude and a phase.

M.W.
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Complex numbers do two obvious things. If you think of them as 2D vectors on a plane, starting at your arbitrary point (0,0), then adding complex numbers is vector addition.

And if you think of them as angles off an arbitrary polar-coordinate angle (0,1), then when you multiply two of them you get the sum of the angles (and the product of the magnitudes).

That can be useful whenever you have something that works like a 2D plane, where you want to do vector addition or addition of angles.

So for example, a pendulum can have kinetic energy and potential energy, and mostly the sum of them is constant. They are two different things so you can represent them on a 2D plane, as a circle whose radius is the total energy. When you convert from one to the other it moves around the circle. You can represent its motion with complex numbers.

You can do that with anything that converts back and forth between two forms, but sometimes it will involve easier complex number math than other times.

Sometimes things fit rotations in 4 dimensions, and then you can use quaternions like you'd use complex numbers for 2 dimensions. You can easily represent elliptical orbits with quaternions -- even easier than you can use them for 3D rotations. For any angle along the orbit, you can get the 3D position and also the time -- how far it's ahead or behind the time it would reach that angle in a circular orbit.

Use the math wherever it fits.

J Thomas
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  • Agreed, use the maths where it fits, for generations, then one day you have enough places where it fits and then someone comes along and understands it and puts it all into a concise theory like Newton did for calculus. Maybe it but it really feels like the explanations I get to complex numbers are lacking somewhere. – OzOz Nov 06 '19 at 17:22
  • I explained the concise theory. You get to do vector addition and rotations, between two things that are independent and so can be considered different dimensions. That's it. It's useful anywhere it makes sense to do either or both of those. – J Thomas Nov 06 '19 at 19:41
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If positive real numbers are forwards numbers and negative real numbers are backwards numbers then imaginary numbers are sideways numbers.

In terms of angles, positive real numbers could be thought of as having an angle of 0°, negative numbers have an angle of 180° and sideways or imaginary numbers are at ±90°. This is useful in electrical engineering when quoting impedances. An impedance is the AC version of resistance in a DC circuit. It has a restive component which does not change the phase angle between current and voltage and a reactance which changes the angle between them by ±90°. (The sign depends on whether the reactance is capacitance or inductance.)

If you want to combine the two into one “number” you can use complex numbers where the real part is the resistance and the imaginary part becomes the reactance. Formulas then continue to work just like the simple Ohm’s Law ones using resistance but with complex numbers instead. Both resistance and reactance are taken into account at the same time.

Basically, anywhere you have things that are 90° apart in some way then imaginary numbers could be useful. That could be x and y coordinates or where both sine and cosine waves occur.

So, if you need two dimensional numbers they could be the way to go. For three or more dimensional numbers you would probably move on to tensors.

Tony Dick
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  • +1. You could extend this example with ideas like phase of a varying current shown in a complex number. Adding two complex currents is meaningful (e.g. if they are parallel). Multiplying a complex current by a complex impedance is also meaningful to give a complex voltage, which may be out of phase with the the current. Sometimes you may want to consider your complex numbers in polar form, which will emphases the amplitude and phase – Henry Nov 07 '19 at 14:22
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It's a good idea to think of the imaginary numbers as numbers perpendicular to the reals. Multiplying a real by -1 "rotates" it over 180° on the line of real numbers. Multiplying a real by i rotates it over 90° so it lands on the imaginary line. Multiplying again by i rotates it 90 degrees further so it lands on the real axis again. Hence i*i=-1. This is all incredibly basic but it's the way I like to approach complex numbers in more complicated scenarios involving complex exponentials and differential equations etc.

In the end imaginary numbers are no more "unphysical" than negative numbers. Negative numbers extend the line of positive reals by adding some numbers to the left and imaginary numbers extend the reals by adding some numbers perpendicularly. The use of both negative and imaginary numbers could be eliminated from equations but it would make them a lot less convenient.

Milan
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In dealing with sinusoidal functions which are out of phase as in AC circuits or waves, its usually possible to put the equations into a form which resembles the addition of the x components of two or more vectors to get the x component of the resulting vector. The vectors can be thought of as rotating in a 2D plane. Its often more convenient to work with the vectors than with the components. If the vectors are visualized in an xy plane, only the x components are significant. If they are visualized in the complex number plane, they are handily represented by complex functions, but again in most cases, only the real components of the vectors have physical significance.

R.W. Bird
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Sorry for a long story, which only address the title of your question (and not the inner questions).

I remember the first time I was introduced to complex numbers at school. The teacher (of mathematics, not physics) was explaining us how to solve quadratic equations (a.x^2+b.x+c=0). After giving us the method, he ended up with the well known solution for the roots: $$x=\frac{-b\pm\sqrt{b^2-4ac} }{2a}$$

Of course it didn't take long for a bright student to tell the teacher: "Hey, but then what happen if the expression in the square root is negative?" For example solve x^2+1=0, your roots will be: $$x=\frac{\pm\sqrt{-4} }{2}$$

All (or most of) the class understood the conundrum and started to scratch their head as they knew for sure that no number could be squared and retain a negative sign ...

The teacher continued, completely undisturbed, "It's not a problem, we can make tools for that. Let's just use a quantity i defined such as i^2=-1". And he went on to introduce the complex numbers and the rules in the complex plane.

Again it wasn't long before a voice from the baffled audience shouted "So this is a actually a convoluted way to bypass rules you taught us previsouly (like a squared number will always be positive). What use is that? why go to such complexity ? (no pun intended, although I now wonder how the complex numbers got their name from initially).

So the teacher put it this way:

There are many physics equations which follows a quadratic law, or even more complicated laws where the solutions involved square roots of potentially negative numbers, and (before the Complex numbers) the physicians couldn't solve their system fully so they asked mathematicians to define a new domain (larger than the Real domain) where these systems would be solvable. The Complex numbers are the tool mathematicians came up with.

By now my understanding of the complex numbers is a bit deeper, but this simple description still holds true. The Complex numbers are just a mathematical tool. A complex number do not have other physical equivalent than the one you give to them.

Same can be said about the Real numbers. I work with a multi sensor tool which measure 10 different parameters in parallel. The output for anyone is just a list of numbers, only myself knows that:

  • the first number represents a Weight, in [N],
  • the second is a Moment, in [N.m]
  • the third is an acceleration, in [G]
  • and so on ...

All different physical dimensions, yet on my screen they're all just numbers, only in my head do I know this one represent this, this one represent that...

For complex numbers, you have 2 components. Each may represent a different physical dimension (electric field and magnetic field for EM). The i part is only the mathematical tool allowing you to handle these numbers in a more graceful form (because you could also describe each components separately with real numbers only, but the equations become real ugly). The i in itself means nothing physically.

Hoki
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  • I agree, complex numbers are a mathematical tool. But I don't like the description of a rotation through 90 degrees... I think they are a mathematical tool where you take a main dimension (which you call the real numbers, usually something you can measure), then you put all the other dimensions in the imaginary part. The imaginary part is how all the other dimensions effect the part you have chosen to be real. Earlier in physics there was just one dimension in the imaginary side, now with quantum there are many dims and the maths is complicated because there are multiple dimensions I could be. – OzOz Nov 06 '19 at 13:30