Measuring Complex Numbers: Is it Possible?

[rbj]

You know, I really don't appreciate the "You're wrong, I'm right." style of debate.

Hurkyl, I'm not trying to get in a sort of demeaning "your argument is stupid" debate and, to be clear, i respect you and the other physicists here greatly. don't get me (or my attitude) wrong about this. but i mean what i said: except for concepts that i have never learned or previously heard of (and i took quite a bit of applied math inc. Real Analysis, Complex Analysis, Functional Analysis, etc. as an undergrad and graduate student of EE, 30 years ago) such as "hyperreal numbers", every other part of your argument is, essentially, sophistry. utterly unpersuasive. i'll touch on some (somewhat re-ordered):

Why do you think physical quantities even can be expressed as real numbers?

uh, because they have been.

we count out "9192631770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom" and call that period of time a "second". during that same period of time, we measure the distance, in terms of the spacing between two little scratch marks on a special piece of platinum/iridium that we had been calling a "meter", that light travels in vacuo and that distance is 299792458 of those meters. (later we redefine what we mean by "meter" to guarantee that same number.) we measure electrical current by counting electrons passing some surface boundary (not really the right way, but they may redefine the kilogram so that this could be the right way). we measure electrical voltage by measuring in some manner how much mean energy these electrons have at different places. we measure how much energy by how much they move something (perhaps themselves) against a force. we measure force by ...

these are the physical quantities that we measure, not complex impedance. we construct conceptually a "measure" of complex impedance of something by making real measurements of real quantites. but there is no (instantaneous or mean) voltage or energy or number of electrons that was measured to be complex. all of these measurements were real, and, because of finite precision, rational.

Here is a picture of my ruler. It measures in centimeters.

View attachment 8707

It sure looks like I'm getting an irrational number when I use it to measure a length.

this doesn't say anything.

Furthermore, why do you think physical quantity can only be expressed as a real number? The imaginary axis of the complexes is exactly as good as the real axis for, for example, measuring lengths. Here's a picture of another of my rulers: it measures in imcentimeters. (The conversion between centimeters and imcentimeters is that 1 cm = i imcm)

View attachment 8708

no, you don't have such a ruler, you have a ruler of real length with markings on it that you apparently want to be interpreted as imaginary numbers. but there are no imaginary lengths. if you use that ruler to measure real lengths (since it is a real ruler) you will get real results, despite the markings.

One important thing to note: for example, if there "exist", two incommensurate lengths (their ratio is not rational), then it is mathematically impossible to represent length as natural number multiples of some fundamental length.

well, sure, i guess. you can't represent it exactly. but neither can you measure (or perceive) it exactly. that's why our measurement (or perception, which is just measurement with the devices inherent in our senses) of physical quantity are rational numbers times some reference quantity. but even though i can't measure the diagonal of a perfect square (of unit length on a side) to be exactly the \sqrt{2}, i can accurately theorize that such would be the length and confirm with an actual ("real") measurement that results in a rational approximation to the "theoretical value" to as close as the technology allows.

And you've already stated that measuring devices can measure complex quantities:

i didn't say that. i said that fundamentally measuring devices measure real quantities.

The instrument measured gain.

naw, it measures input and output voltages and computes gain.

Gain is a physical quantity.

not in the sense we are debating. "Gain", as a magnitude, is a ratio of physical quantities. the complex gain that we signal processing engineers talk about is an abstraction that incorporates the real magnitude gain along with phase shift for a restricted class of input and output signals we call sinusoids. this complex gain, as a number, has no meaning for more general signals. for an LTI (linear and time-invariant) system a complex gain that is a function of frequency can be used to relate the general input to output.

Gain is complex. Ergo, the instrument measured a complex quantity.

(What is "gain" in this context? phase difference? Phase difference certainly isn't a real quantity)

time difference is. that's all phase difference is, normalized to the period of the sinusoidal function, another real quantity. so how is phase difference not real?

(back to the recent post...)

1. Real numbers "real"?
Just to be sure we're on the same page, recall that my point here is that our scientific theories do not support the notion that real numbers are "real", nor do they support the notion that complex numbers are not "real".

i think i agree with you that this is your point and, in that sense, we're on the same page. but i disagree with that point.

1.1. Reduction
One of the main themes in your argument is the notion of reduction. You look at a concept like impedance, which our physical theories describe with complex numbers, and choose to reduce it into different concepts which can be described with real numbers. You look at a concept like momentum, which our physical theories describe with vectors, and choose to reduce it to a triple of real numbers. You even look at the complex numbers themselves, and choose to reduce them into pairs of real numbers.

But none of these support your point. These explain how a person who rejects the complex numbers and vectors could still manage to do physics despite that handicap. They do not explain why the real numbers should be considered "real", and why complex numbers and vectors should not be considered "real".

physical law ultimately comes from empirical observation of "real" events, of reality. we see certain events happen in such a repeatable and systematic manner that we infer a law that, hopefully in a simpler language, describes these events consistently to within the degree of accuracy that we observe or measure. we rely on that law until we find exceptions to what the law says, and then look for a better law (and hope that correspondance principle is satisfied). the real numbers are considered "real" because those are the numbers we fundamentally use to measure and describe, quantitatively, the physical phenonmena we are observing and codifying into laws. now, for simplicity and elegance, the law may preferably be stated in such a way that vectors and/or complex numbers represent quantites that the law refers to, but when that law is used to make a prediction that we hope to verify in experiment, a reduction to real quantities is made (and is part of the law, such as expectations or probability densities in the case of Schrodinger, or real changes in magnitude and phases for a sinusoidal electrical system) and measurements of the corresponding quantities in the real physical world are made and compared. perhaps those real measurements or real quantities are combined in some manner to create complex or vector quantities for use in our conceptualization (BTW, i didn't say vectors with real components aren't real, and i don't think you can successful use that as evidence of an inconsistancy - vectors are different than complex numbers) .

In fact, there is a simple reductio ad absurdum argument to demonstrate that reduction cannot be used to support your assertions. Suppose that the form of argument

Anything we can do with X, we can do with Y. Therefore, Y is more "real" than X.​

is a valid one. Then, we would be able to argue

Anything we can do with complex numbers, we can do with real numbers. Therefore, the reals are more "real" than the complexes.​

But, we would also be able to argue

Anything we can do with real numbers, we can do with complex numbers. Therefore, the complexes are more "real" than the reals.​

Voilà: a contradiction. There are, in fact, many different structures to which one can reduce the reals.

utterly unpersuasive. this doesn't say anything are physical quantity.

1.2. Abstraction
Everything notion in a physical theory is an abstract. This follows directly from the very definition of abstract. Impedance, momentum, length, time, mass; these are all abstract notions, constructed by humans.

impedance, perhaps. but not length or time or mass or charge. those are real things. they're real stuff. you can go to Stuff-Mart and buy them. i do not concede that these things are mere constucts of humans (unless we're in The Truman Show). stars are spaced apart by real distances (that change in time) and have real mass. this stuff is real and the kind of numbers we use to measure it and describe it are real numbers.

Complex number, real number, natural number: also abstracts.

the numbers and strict meaning to us for what they represent are abtracts, but the physical quantities that we measure are real things and the kind of numbers that are used to measure these real things are real numbers (actually a subset).

You can see I've attached another picture of a ruler.

You tried to argue that measuring in rational multiples of pi is not measuring an irrational number. (Which is silly, because a rational multiple of pi is an irrational number)

but the mutliple of \pi is not. the number multiplying it is rational (by axiom).

Well, I've attached another example of a ruler that measures in the irrationals to block that particular argument.

this doesn't convince at all. it says nothing.

Also, when you talk about finite precision implying we measure in rational numbers -- that's not because you think significant figures are meaningful, is it?

there are degrees of meaningfulness. but this is off-topic.

Hurkyl, you still haven't gone beyond sophistry here. Real numbers are what we use to measure and describe fundamental physical quantity in reality. Imaginary and Complex numbers are not used for such. Real numbers are qualitatively different. the unit in the real number line (i.e. the number "1") is the multiplicative identity and is qualitatively different than its negative "-1". but the unit in the imaginary line, the imaginary unit that we conceive and label i is qualitatively no different than its negative -i. both have equal claim to square to be -1. if every textbook and math/science/technical paper in the world was rewritten and -i was substituted in for i (which would also have the effect of replacing every occurace of -i with +i), all facts and theorems would remain equally true. the same cannot be said for +1 and -1. they are not interchangable.

 

[Crosson]

It is not that difficult to see why people question the need for complex numbers in physics. The SI base units are sufficient for expressing all known measurable physical quantities, and these units always (Kilograms, meters, coulombs, etc) refer to non-imaginary quantities.

My response is that imaginary numbers are not ever needed to discuss the results of measurements, but in some cases they are needed to discuss the dynamics that underlie the observations.

 

[rbj]

It is not that difficult to see why people question the need for complex numbers in physics. The SI base units are sufficient for expressing all known measurable physical quantities, and these units always (Kilograms, meters, coulombs, etc) refer to non-imaginary quantities.

My response is that imaginary numbers are not ever needed to discuss the results of measurements, but in some cases they are needed to discuss the dynamics that underlie the observations.

i think that complex numbers and variables and techniques (integration on paths, residues, etc.) are very, very, very useful (to within \epsilon of "needed") for description of the dynamics, but since every complex number can be split into its representation of two real numbers and every complex equation can be split into two equations equating the real and imaginary parts, complex numbers aren't absolutely needed, but expressing such equations in the complex form is far more compact and elucidating and easier to solve than than the counterpart pairs of real equations.

 

[Swapnil]

since every complex number can be split into its representation of two real numbers and every complex equation can be split into two equations equating the real and imaginary parts, complex numbers aren't absolutely needed, but expressing such equations in the complex form is far more compact and elucidating and easier to solve than than the counterpart pairs of real equations.

But can't you give the same argument for real numbers. If you forget about irratinal numbers, then real numbers can be written as a pair of natural numbers to give you rational numbers (for example, (a,b) = a/b ).
(Don't know what we'll do about irrational numbers or negative numbers though...)

naw, it measures input and output voltages and computes gain.

I agree. But HOW does it measure input and output voltages? Perhaps there are another set of quantities that are really "measured" and used to compute input and output voltages...

 

[rbj]

But can't you give the same argument for real numbers. If you forget about irratinal numbers, then real numbers can be written as a pair of natural numbers to give you rational numbers (for example, (a,b) = a/b ).

i s'pose but i don't see what you mean by making the "same argument". might be OT.

I agree. But HOW does it measure input and output voltages? Perhaps there are another set of quantities that are really "measured" and used to compute input and output voltages...

how gets to be more and more technical and messy, but ultimately it gets down to counting charge accumulating somewhere and knowing (quantiatively) the nature of materials that result in a property we call "resistivity" or "conductivity" as well as some geometric properties of parts (we call "resistors").

 

[Hurkyl]

BTW, i didn't say vectors with real components aren't real, and i don't think you can successful use that as evidence of an inconsistancy - vectors are different than complex numbers

Okay, then can explain to me why your argument that the complexes are not "real" fails to apply just as well to vectors?

(This question, incidentally, is the one I'm most interested in hearing answered)

so how is phase difference not real?

Phase is circular. The reals are linear. Circles are not lines. 0 and 360 are different real numbers, but 0° and 360° are identical phase differences.

Real numbers are qualitatively different. the unit in the real number line (i.e. the number "1") is the multiplicative identity and is qualitatively different than its negative "-1". but the unit in the imaginary line, the imaginary unit that we conceive and label i is qualitatively no different than its negative -i. both have equal claim to square to be -1. if every textbook and math/science/technical paper in the world was rewritten and -i was substituted in for i (which would also have the effect of replacing every occurace of -i with +i), all facts and theorems would remain equally true. the same cannot be said for +1 and -1. they are not

What "real" meaning does the unit have? I can't think of a "physical quantity" where +1 has an absolute meaning: the + is simply denoting an orientation with respect to a chosen convention, and the 1 is simply denoting a scale with respect to a chosen unit.

Multiplication does not have an absolute physical meaning: every physical law involving a multiplication also includes a physical constant that modifies the equation so that the scale and orientation turn out correct.

If we change the units on certain measurements, the physical constants adjust to compensate.

If we change our sign convention, the physical constants adjust to compensate.

We can even change quantities from purely real to purely imaginary, and the physical constants adjust to compensate.

(Algebraically, all three of these amount to exactly the same operation)

e.g. I could select three oriented lengths, one along the up-down axis, one along the East-West axis, and one along the North-South axis. I could then declare that the first one is +1, the second is -3, and the third is 3+2i, and all of physics still works, all physical laws retain their form: the physical constants simply take on values with respect to these new units.


(In summary... of course you can swap -1 and +1! I can also replace +1 with +2 and -1 with -2. I can even replace +1 with +i and -1 with -i, if I choose. It's just that these symmetries are a little bit more involved than that of swapping +i and -i at every single occurence)

 

[Swapnil]

i s'pose but i don't see what you mean by making the "same argument". might be OT.

since every real number can be split into its representation of two natural numbers* and every real equation can be split into two equations equating the denominator and numerator, real numbers aren't absolutely needed, but expressing such equations in the real form is far more compact and elucidating and easier to solve than than the counterpart pairs of natural equations.

*I know I am ignoring two important subsets of real numbers -- the irrationals and negative numbers. So my argument doesn't do much unless there is a way to represent irrational numbers and negative numbers as a pair of natural numbers.

 

[rbj]

since every real number can be split into its representation of two natural numbers*

no, only rationals. not every real.

and every real equation can be split into two equations equating the denominator and numerator,

that's not true either. the fact is that

z_1 = z_2

where these are possibly complex, this is the same as

\mathrm{Re}(z_1) = \mathrm{Re}(z_2)
and
\mathrm{Im}(z_1) = \mathrm{Im}(z_2)

if the top equation is true, the bottom pair must be. and vice versa.

but it's not true that:

\frac{y_1}{x_1} = \frac{y_2}{x_2}

means that:

y_1 = y_2
and
x_1 = x_2

it could be true, but it wouldn't have to be. but the converse is true.

so i don't see the parallel here. this is about different stuff.

real numbers aren't absolutely needed, but expressing such equations in the real form is far more compact and elucidating and easier to solve than than the counterpart pairs of natural equations.

*I know I am ignoring two important subsets of real numbers -- the irrationals and negative numbers. So my argument doesn't do much unless there is a way to represent irrational numbers and negative numbers as a pair of natural numbers.

it's not the same.

 

[Manchot]

There are many instruments that display complex quantities. Network analyzers, for one.

 

[rbj]

There are many instruments that display complex quantities. Network analyzers, for one.

surely. still these complex quantities are caculated from a series (at least a pair) of real measurements. there is no real place were you clip on a pair leads onto an unknown voltage and read out, from a single instantaneous measurement, a voltage that is complex.

but with concepts like the Hilbert Transform, we can certainly construct, in the mind of a computer using the mathematical rules of complex numbers (which are just the reasonable analytic extensions of the same rules for reals), create a "complex" quantity and treat it as such.

 

[LeonhardEuler]

Here is an example that I think might help. It's been said here that only natural numbers can be used to count things, but this is not true. If you want a number system to be used only for the purpose of counting things, the integer multiples of imaginary numbers will do. This is because counting requires only a notion of addition and not of multiplication and the natural multiples of imaginaries will behave in every way exactly like the naturals themselves for the purposes of addition. . So while it may seem natural in our culture to say that you have 7 apples and absurd to say that you have 7i apples, it is only a matter of convention. If you have 7i apples and take 2i away, you have 5i apples. The only reason you wouldn't normally say this is that we have grown up so accustomed to representing quantities of things in natural numbers that the fact that this itself is merely an abstraction is lost on us. We think there really are 7 apples, when in fact the quantity of apples is just represented by the number 7 in our convention.

 

[Hurkyl]

Here is an example that I think might help.
...So while it may seem natural in our culture to say that you have 7 apples and absurd to say that you have 7i apples, it is only a matter of convention. If you have 7i apples and take 2i away, you have 5i apples.
...

Actually, it is perfectly accurate to say that what you describe actually is a system of natural numbers: if we define the successor of x as x+i, then it satisfies Peano's axioms.

(And happily, you show that the addition of purely imaginary numbers agrees with the usual addition operation defined on a system of natural numbers)

 

[LeonhardEuler]

Actually, it is perfectly accurate to say that what you describe actually is a system of natural numbers: if we define the successor of x as x+i, then it satisfies Peano's axioms.

(And happily, you show that the addition of purely imaginary numbers agrees with the usual addition operation defined on a system of natural numbers)

Are you sure? To be clear, even though multiplication is not used in the application I'm talking about, I'm still assuming these are ordinary imaginary numbers with multiplication defined in the usual way so that i*i=-1, which means that this system is not closed under multiplication. I'm not sure if the set I described actually is the natural numbers because mathworld defines them as the set of positive integers, clearly ruling out the possibility of this set being the natural numbers, but wikipedia (which first of all includes 0 as a natural number) gives a construction in which only the properties of addition are assumed, and then multiplication is defined in a way that gives the normal properties of the natural numbers, from which it is possible to derive statements like i*i=i in this case.

Regardless, though, this does show an example of counting things with non-real numbers.

 

[vanesch]

surely. still these complex quantities are caculated from a series (at least a pair) of real measurements.

And, as I argued earlier, these "real" measurements are in fact calculated from naturals (like ADC outputs and the like) - and even not naturals, but a finite subset of it. So, all we're doing when we are measuring, is *counting*. All the rest is abstraction and modeling, whether or not we are aware of it. Real numbers are abstractions, vectors are abstractions, complex numbers are abstractions, manifolds are abstractions. The only thing we do is counting, in that we compare integer multiples of reference concepts with to-be-measured concepts. In many cases, we tend to believe that we are *really* measuring some genuine quantity, but that is because we take our abstract models "for real" - which can be a good thing, but then we should take also our other abstractions (such as complex numbers, vectors,...) for real.
There is no manner in which the abstraction of the real numbers is in any way "more real" than other abstractions. Now, it is true that other abstractions often have the structure of a differentiable manifold, and hence are describable by (sets of) real numbers (which is then called a coordinate system), but there's 1) no reason why this description has any more reality to it than the structure itself (rather the opposite: usually we give more "geometrical existence" to the manifold structure than to its coordinates, which have an arbitrariness to it) 2) no reason why all mathematical structures we need or will need in physics will be of the differentiable manifold kind.

There's another way to show why "real numbers" are "less real" than abstract manifold structures: consider change of units!
The real number "1" has a special meaning in the structure of real numbers: it is the unit element of multiplication. There's no other real number satisfying this property. Given that "1" is special, this would mean that, if there were genuine physical reality to real numbers, there would be a special physical meaning to a physical quantity which corresponds to the real number "1". But if we measure something to be 1 meter long, then this simply means that, compared to our measuring reference (meter), we find that the thing we measure seems to be "equal" concerning the observation of length. If we change our reference to "feet", we get another number (of the order of 3 something). It is not "1" anymore. So there's nothing special to "1" in our measurements, while the REAL NUMBER 1 is special and unique.
So the actual mathematical object we are using to model our measurements with, is not "the real numbers", but a 1-dimensional linear differentiable manifold, over which we can define different coordinate systems (corresponding to different units), consisting of mappings from the real numbers into our "abstract but real" manifold. As such, we see that, if we want to assign some "reality" to some mathematical structure, it is not the set of real numbers itself, but rather the differentiable manifold itself.
But if we go through this abstraction, then there's no reason to prefer one-dimensional manifolds over multidimensional manifolds. Some of those can have extra structure, such as distance, or curvature, or complex or symplectic structure or whatever. There's then no reason to give more reality to a single real coordinate over that structure, than to the structure itself.

 

[rbj]

And, as I argued earlier, these "real" measurements are in fact calculated from naturals (like ADC outputs and the like) - and even not naturals, but a finite subset of it. So, all we're doing when we are measuring, is *counting*. All the rest is abstraction and modeling, whether or not we are aware of it. Real numbers are abstractions, vectors are abstractions, complex numbers are abstractions, manifolds are abstractions. The only thing we do is counting, in that we compare integer multiples of reference concepts with to-be-measured concepts. In many cases, we tend to believe that we are *really* measuring some genuine quantity, but that is because we take our abstract models "for real" - which can be a good thing, but then we should take also our other abstractions (such as complex numbers, vectors,...) for real.

i think i agree with that, but...

There is no manner in which the abstraction of the real numbers is in any way "more real" than other abstractions.

i don't agree with this. a counter-example would have to do with a real and idealized geometry in 2-space. a perfect square of unit length on the side. even though i measure the length of the diagonal to be rational (my finite precision measuring instrument only has integer values of the tiny ticks at the limit of its precision), i know that, in the ideal, these rational measurents of the diagonal will only approach this theoretical irrational value as my precision gets better and better. they will not be approaching a rational value in the idealized limit.

but this is not the case for complex or imaginary numbers. there ain't no square out there with the length of diagonal of 5 i using any units of reality.

i'll try to crack the remaining point you made.

 

[vanesch]

Proof by color

i don't agree with this. a counter-example would have to do with a real and idealized geometry in 2-space. a perfect square of unit length on the side. even though i measure the length of the diagonal to be rational (my finite precision measuring instrument only has integer values of the tiny ticks at the limit of its precision), i know that, in the ideal, these rational measurents of the diagonal will only approach this theoretical irrational value as my precision gets better and better.

Clearly you are setting up a theoretical idealized mathematical entity which you give the status of "real", but you have absolutely no guarantee that your idealized concept of "perfect square" is any more real than the other hypotheses you made. There is absolutely no requirement for perfect squares to have any fundamental reality, any more than any other theoretical and formal concept (such as Newton's equation of motion or the like).
You have already "taken for granted" that Euclidean space is somehow "reality" before you can even consider the "reality" of a perfect square.
And then we're back to square ( ) one: IF you assign (by hypothesis) some reality to the Euclidean space, then you should do so too to any other theoretical/formal construction of the physical theory at hand ; if that theory uses vectors, or manifolds, or complex numbers, then that's also just as much part of it, and hence of its (hypothetical) "reality".

I know that it is difficult to let go, but "Euclidean space" is just as well a hypothetical and formal construction as any other part of a physical theory, and it is only within such a formal structure that a concept like "perfect square" makes sense. As such, the irrationality of the diagonal is only a product of our formal structure, which is the Euclidean space, and there is no way to verify this empirically. To illustrate this, consider the possibility that what we call space, is actually a discrete structure of some kind (a finite set of things we could call "points"), but which is so fine-grained, that for all practical purposes, every Euclidean manipulation is correct up to, say, 50 decimals. We can make very good approximations to a square in such a structure, but the concept of "perfect square" as in Euclidean space, is meaningless: it would be just a "set of 4 elements of the finite set of points".
In such a view, there would be no problem of "the irrationality of the diagonal". It is only in the formal model of Euclidean geometry that such a thing arises. It's a good formal model, which makes good predictions for measurements of what we call "distance", with say, 30 decimals. But it is only that: a formal model.

but this is not the case for complex or imaginary numbers. there ain't no square out there with the length of diagonal of 5 i using any units of reality.

In the same way as there are no distances equal to -5.6 meters. And if you are going to tell me that we CAN have that in a coordinate system, then I can say that we CAN have "distances" of "3.0 + 5.2 i" meters in the Euclidean plane, according to a very similar construction.

"distances" are not "more or less real" than other theoretical constructions in physical theories. Turns out that Euclidean geometry is very closely related, as a formal structure, to R^3. You can call Euclidean geometry "the theory of measurements of distances and angles".

But other physical theories use other formal structures, and "the theory of electrical networks" for instance, uses complex numbers in a natural way.

 

[rbj]

vanesch (and Hurkl), i won't be able to keep up because some of what you guys say needs more digestion before response which i may end up not having time for. but, i'll try to respond to the "easy" stuff.

In the same way as there are no distances equal to -5.6 meters.

sure, distances or positions are relative. and a distance metric in a metric space is always real and non-negative. and we always measure it to be real, rational, and non-negative. but negative quantities do exist in nature. electric charge is one. i don't care which you choose as "positive" charge (proton vs. electron), but whichever you choose as positive, the other must be negative. negative quantities of some "stuff" are real things in nature. we can construct other quantities, like rate of change, which will necessarily have negative values. it's perfectly appropriate to include the negative numbers with "real" numbers.

And if you are going to tell me that we CAN have that in a coordinate system, then I can say that we CAN have "distances" of "3.0 + 5.2 i" meters in the Euclidean plane, according to a very similar construction.

you can say it, but it's not true. not in reality. there is no qualitative difference in the dimension that you're defining as "imaginary" and the dimension with real coordinates. but there is a qualitative difference between imaginary quantities and real quantities. square a real quantity and you have similar real animal. square an imaginary quantity, and the result is no longer imaginary.

"distances" are not "more or less real" than other theoretical constructions in physical theories.

but they are or may be in reality. a voltage of 5 volts is a real physical thing (that can be measured to some precision), but a "phasor" (the EE kind, not the Star Trek weapon) of 4 + 3i is not a real physical thing, but an abstraction that makes our life easier.

Turns out that Euclidean geometry is very closely related, as a formal structure, to R^3. You can call Euclidean geometry "the theory of measurements of distances and angles".

that's pretty damn closely related to the 3-dimensional space we, and the milky way galaxy, live in.

But other physical theories use other formal structures, and "the theory of electrical networks" for instance, uses complex numbers in a natural way.

i'm quite familiar with the theory of electrical networks. i do DSP for a living and deal with complex quantities or variables all the time. they're abstrations. they have some correspondance to "real" quantities (quantities we perceive or measure in reality), but they are not the real quantities.

 

[Crosson]

I am not sure there is anyone in this thread is an opponent of complex numbers, but I noticed some arguments in favor that are choir-preaching-to-choir type i.e. "if we demand closure under this algebraic/analytical/geometrical operation, why not this one?"

Most opponents to complex numbers are coming from an anthropocentric perspective, they are only motivated to learn things that immediately relate to humans. The only way to show them that mathematics provides another type of motivation is to put their perspective in context: talk to students about why they "believe" in various number systems (whole, integer, rational, real) so that they see that the reasons for believing in complex numbers are equally compelling, from a particular point of view.

 

[rbj]

just to be clear: i am no opponent to complex numbers (to the concept of their existence, or to their use either in theory or practice). i am a big proponent of complex numbers.

i am only an opponent to the concepts that quantities of raw physical stuff is measured as complex (we construct complex parameters from multiple, at least 2, "real" measurements) or that what we call "imaginary numbers" are every bit as "real" (meaning having to do with actual quantities of stuff in physical reality) as what we call "real numbers".

but, when doing math, i treat complex/imaginary numbers with every amount of care that i treat real numbers. it's just that i think the terms "real numbers" and "imaginary numbers" are appropriate terms because fundamental physical quantities in reality are real numbers. so what better term can you think of for those other kinds of numbers of values that no basic physical quantities would take on, but we can imagine that when squared, result in a negative (real) number?

 

[Hurkyl]

there is no qualitative difference in the dimension that you're defining as "imaginary" and the dimension with real coordinates.

Doesn't "realness" count as a quality?

a voltage of 5 volts is a real physical thing

This surprises me even more than you thinking that 3-vectors are "real". I honestly cannot see any consistent pattern to what you believe is "real" and what is not "real".

Actually, that's not entirely true -- there is the obvious pattern that you label something "real" if and only if it doesn't invoke complex numbers. But I'm assuming that is merely a correlation and not causation.

I'm still looking forward to an explanation of why 3-vectors are "real", and now I'm even more looking forward to an explanation of why electric potential is "real".

But now I'm curious as to if there are any quantities measured by real numbers that you would consider not "real".

 

[vanesch]

sure, distances or positions are relative. and a distance metric in a metric space is always real and non-negative. and we always measure it to be real, rational, and non-negative.

Right, but I thought you took "measuring distances" as the litmus test for "reality".

but negative quantities do exist in nature. electric charge is one. i don't care which you choose as "positive" charge (proton vs. electron), but whichever you choose as positive, the other must be negative. negative quantities of some "stuff" are real things in nature.

Ok, but if "electric charge" is now included into the arena of "real" things, then why not phasors for instance ? Charge is the ratio of electrical attraction wrt a reference charge and it turns out that we only need a real number for that (even just an element of Z if we do it right !). But that is not clear a priori! After all, charge is characterised by a force, which is normally a vector quantity. It is only because for electric Coulomb interaction, this force always points along the axis between the test charge and the charge under test, that we can limit this vector quantity to a single real number. If charge were such that the direction of the force changed, we would not be able to express the comparison between the test charge and the charge under test as a single real number (but as the ratio between two vectors)!

The ratio of two sinusoidal oscillations cannot be expressed simply by a real number: it is evidently represented by a positive real number and an angle, which is naturally a complex number.

we can construct other quantities, like rate of change, which will necessarily have negative values. it's perfectly appropriate to include the negative numbers with "real" numbers.

Yes, and in the same way, complex numbers, no ?

you can say it, but it's not true. not in reality. there is no qualitative difference in the dimension that you're defining as "imaginary" and the dimension with real coordinates. but there is a qualitative difference between imaginary quantities and real quantities.

Why couldn't I have a preferred direction (say, the north) and express, with my complex numbers, the distance and orientation wrt this preferred direction ?

square a real quantity and you have similar real animal. square an imaginary quantity, and the result is no longer imaginary.

Well, the squaring the complex number comes down to doubling the angle with the preferred direction.

but they are or may be in reality. a voltage of 5 volts is a real physical thing (that can be measured to some precision), but a "phasor" (the EE kind, not the Star Trek weapon) of 4 + 3i is not a real physical thing, but an abstraction that makes our life easier.

This is what I don't understand. What is your procedure to declare a thing "a real physical thing" ?
Both the voltage and the phasor are things that can be measured with devices, both are in fact derived from "natural numbers". Both have a formal existence within some theoretical framework which is assumed to be adequate for the problem at hand (otherwise the measurement apparatus would not make sense).

 

[rbj]

Doesn't "realness" count as a quality?

sure. don't talk to me about that. vanesch said that then I can say that we CAN have "distances" of "3.0 + 5.2 i" meters in the Euclidean plane, according to a very similar construction. he/she is attributing a qualitative difference (real vs. imaginary) between 2 dimensions of Euclidian space where there is no such qualitative difference. both dimensions of Euclidian space have the quality "real". distances in either dimension, when squared, result in non-negative numbers.

This surprises me even more than you thinking that 3-vectors are "real". I honestly cannot see any consistent pattern to what you believe is "real" and what is not "real".

Actually, that's not entirely true -- there is the obvious pattern that you label something "real" if and only if it doesn't invoke complex numbers.

that is a misrepresentation/misinterpretation of what i said. i said that fundamental physical quantities that we experience and measure in reality are real numbers.

But I'm assuming that is merely a correlation and not causation.

I'm still looking forward to an explanation of why 3-vectors are "real", and now I'm even more looking forward to an explanation of why electric potential is "real".

But now I'm curious as to if there are any quantities measured by real numbers that you would consider not "real".

what are you talking about?

listen, perhaps, if the string theorists are correct, there are several more dimensions of space than 3. perhaps, if Einstien is correct, this 3 space we experience in reality is only an approximation (being that no nasty black holes are in our neighborhood) to a warped or curved non-Euclidian. it's not the point. when we measure or perceive quantity of stuff in reality, we measure such as real (and rational) numbers. but just because we cannot construct, out of materials, a perfectly square object (of unit side length with a diagonal that is exactly \sqrt{2}) we can expect that *if* such geometry existed, the rational measurements we made to the diagonal would approach, as the precision of measurement got better and better, without limit, that it would approach this irrational \sqrt{2} and no rational value. "realness" of a physical quantity is not about the precision of measurement. for all we know, fundamental physical quantities of irrational value exist, but our measurements of them, with finite precision, will always be rational.

certainly we can use complex numbers to abstract and describe real processes, at least in EE, when we do that we split a real quantity into the sum of two complex conjugate quantities and we deal with one knowing that superposition applies and the conjugate counterpart gets dealt with similarly. but, to be strict, you would have to pass both complex signals through the system (one at a time), getting two complex outputs that should also be conjugates, and add (superimpose) those outputs. if the result isn't purely real, you know you did something wrong.

 

[Hurkyl]

what are you talking about?

Short answer: if you explain why you think vectors are "real", then maybe in the process you will clarify the key difference that allows vectors to be "real" but the complexes not.

Similarly, explaining why you think electric potential to be real might clarify things.

Also, coming up with a real-valued physical quantity you consider not to be "real" would clarify things in a different way.

he/she is attributing a qualitative difference (real vs. imaginary)
...
distances in either dimension, when squared, result in non-negative numbers

There is no qualitative difference between a real-valued distance and an imaginary-valued distance: multiplication plays no part in the concept of distance.

 

[rbj]

Short answer: if you explain why you think vectors are "real", then maybe in the process you will clarify the key difference that allows vectors to be "real" but the complexes not.

vectors are a construct. direction is real.

Similarly, explaining why you think electric potential to be real might clarify things.

is energy real? is electric charge real?

Also, coming up with a real-valued physical quantity you consider not to be "real" would clarify things in a different way.

why would i do that? that's not my burden.

There is no qualitative difference between a real-valued distance and an imaginary-valued distance: multiplication plays no part in the concept of distance.

i guess that's an opinion of yours and you certainly have a right to it. doesn't make any sense or have any support in fact from what i can observe in reality.

 

[rbj]

i think i might run out of time to continue to argue about this from a theoretical/philosophical POV, but i would like to say why, in a practical and pedagogical sense, this differentiation between "real quantity" and "imaginary quantity" is useful, at least for an engineer working on real signals. this is a specific example but i am trying to elude to how this might happen in the more general case.

when i think of a real signal that we view with a real oscilloscope and ostensibly has no (visible) imaginary part, i (and Fourier and many others) think of it as a sum of sinusoids of which one sinusoidal component might be:

x(t) = A \cos(\omega t + \phi)

i think of

x(t) = \left( \frac{A}{2} e^{i \phi} \right) e^{i \omega t} + \left( \frac{A}{2} e^{-i \phi} \right) e^{-i \omega t}

the first term ostensibly has a "positive frequency" of \omega and the second term has a "negative frequency" of -\omega.

because all of the derivatives of an exponential are also exponentials with the same "\alpha", then the exponential function can easily be understood to be an eigenfunction for a linear, time-invariant system (one where the convolution operator defines the input-output relationship). if i were to split each real sinusoid into the two exponential functions that necessarily have imaginary exponents that are negatives of each other, then using superposition, i can analyze this system with one of those complex "sinusoids" quite simply because it is a common factor to all additive terms and, being an exponential and never zero, we can divide it out and that's when we have the "frequency-domain" counterpart to the original time-domain equations which are simpler.

but, to do this correctly, in an analytical sense, we have to repeat this same this with the other term with the "negative frequency", get the output due to that input, add it to the output due to the "positive frequency" component to get our actual output.

now here's the rub: if that final output we get from adding the components, both the positive and negative frequencies, if that final output result on paper is not a purely real number (that is, the imaginary part must be zero for all t), then we know we did something wrong, there is a goof somewhere. driving a real system, built with real components, with a real input signal, better the hell result in a real output signal, even if we ventured into "imaginary" or "complex" domain to get there. if any of the total net output is imaginary, we cannot expect to hook up a scope and see such imaginary voltages. they ain't there.

 

[Hurkyl]

vectors are a construct. direction is real.

So do you consider vectors "real" or not? You suggested before that you consider a vector to be "real".

Direction, though, is not a real number. You have argued that the complexes are not "real", and they are a construct, because we can represent a complex number by two real numbers.

Well, we can represent a direction with two real numbers, can't we? Why don't you consider direction to be a construct? Why do you consider direction to be real?

Was I wrong in my interpretation? You, in fact, do not consider the fact a complex number can be represented as two real numbers as proof that complexes are not "real"?

is energy real? is electric charge real?

Are we talking "rbj-real" or "Hurkyl-real"? For this reply, I'll assume "Hurkyl-real".

This is a good question; I hadn't really thought about it before. Surprisingly (even to me), the answer is not an automatic yes. It depends on the context -- in particular, which physical theory we are currently considering. If we're considering classical mechanics, I would consider several forms of energy not to be real, because they are coordiante-dependant quantities.

If I'm thining classical mechanics or special relativity, then the electric charge of a point particle is certainly "real". I consider i * {the electric charge of a point particle} to be "real" too. I want to reserve judgement on other situations, because I haven't thought them through.

Of course, I consider the real numbers not to be "real". Special relativity, for example, talks about things like mass, proper velocity, rest energy, and 4-current, which would be "real", and things like kinetic energy, coordinate velocity, and time dilation, which would not be "real"... but it doesn't talk about real numbers. The real numbers are a tool used to study SR, not one of its objects of study. (Interestingly, just like Euclidean geometry, one can define Minkowski geometry without ever invoking the idea of a number!)

why would i do that? that's not my burden.

You are trying to make a point. I think that if you could demonstrate a real-valued quantity that is not "real", it would help you make your point. It was just a piece of advice. *shrug*

 

[rbj]

why would i do that? that's not my burden.

You are trying to make a point. I think that if you could demonstrate a real-valued quantity that is not "real", it would help you make your point. It was just a piece of advice. *shrug*

well, the point i am trying to make is not that there are real-valued quanties that are not "real". now, certainly there are numbers that many not be any natural measure of any physical quantity in the universe, but that's because the universe, although big, is not as big as the real number line. i suppose that the number of protons, electrons, and whatever other particles in the universe is finite. but my point is that any fundamental physical quantity in reality is a real number, not the opposite.

 

[Hurkyl]

this differentiation between "real quantity" and "imaginary quantity" is useful, at least for an engineer working on real signals.

Certainly -- if you are describing something with real values, then you had better end up with a real-valued descriptio

now here's the rub: if that final output we get from adding the components, both the positive and negative frequencies, if that final output result on paper is not a purely real number (that is, the imaginary part must be zero for all t), then we know we did something wrong, there is a goof somewhere. driving a real system, built with real components, with a real input signal, better the censored[/color] result in a real output signal, even if we ventured into "imaginary" or "complex" domain to get there. if any of the total net output is imaginary, we cannot expect to hook up a scope and see such imaginary voltages. they ain't there.

Fallacy: equivocation. You are using at least two different meanings of the "real" in this paragraph, but you seem to be treating them all as the same word.

 

[rbj]

Fallacy: equivocation. You are using at least two different meanings of the "real" in this paragraph, but you seem to be treating them all as the same word.

so what if you (think) you did the math completely correctly, started with a physical system in our observable "reality" made a computation of a physical quantity that you measure with real instruments, and came up with an answer of "3+4i volts", or "5+12i meters" or something like that? are you getting your real voltmeter to measure the 3 volts or your real tape-measure to measure the 5 meters, and then get your imaginary voltmeter to measure the 4i volts or you imaginary tape-measure to measure your 12i meters? or maybe your voltmeter will measure 5 volts? or your tape measure 13 meters? is that what you expect?

i expect, that when i interpret some complex result that i will see the consequences of that in real quantities that are expressed and measured as real numbers (these numbers might be many, as a real function of time, etc.)

but going over this, through this, and around this, you still haven't cited a single fundamental physical quantity that exists in imaginary quantities. and the arguments you have made still appear to be sophistry. i don't know who it was that thought of the terms "real" and "imaginary" for these two qualitatively different kinds of numbers that comprise complex numbers, but the terms are very apt, and have more credance with the momentum of history than do your objections to the terms. the terms really do make sense. when you square real numbers, really obtained from some physical quantity in reality, those numbers never square to be negative. only imaginary numbers that we imagine (but never measure) do that.

Hurk, i think I'm crapped out with this debate. occasionally it crops up on the USENET group comp.dsp where i hang out a lot and have seen it multiple times before. we also fight about the nature of the Dirac delta function. (is it really a function? do we dare let it exist outside an integral? what is the dimension of the dependent variable of the Dirac delta if the dimension of the independent variable is time? is the "Dirac comb" a function? do we dare express it as a Fourier series?) or whether or not the Discrete Fourier Transform inherently maps a periodic sequence from one domain to another periodic sequence of the reciprocal domain (or is the DFT any different, in essence, from the Discrete Fourier Series) .

it's just navel-gazing but we fight about it.

 

[vanesch]

so what if you (think) you did the math completely correctly, started with a physical system in our observable "reality" made a computation of a physical quantity that you measure with real instruments, and came up with an answer of "3+4i volts", or "5+12i meters" or something like that? are you getting your real voltmeter to measure the 3 volts or your real tape-measure to measure the 5 meters, and then get your imaginary voltmeter to measure the 4i volts or you imaginary tape-measure to measure your 12i meters? or maybe your voltmeter will measure 5 volts? or your tape measure 13 meters? is that what you expect?

I think you are really turning around the issue. You claim that all "fundamental quantities in nature" are real numbers, and then you go on by deciding that what qualifies as a "fundamental quantity in nature" is what is represented by a real number.

In order to illustrate your point, you always take a few examples of quantities which are indeed real numbers, and point out that for those quantities, it doesn't make sense to talk about them as complex numbers. You did this for distances, and now you do it for the scalar electromagnetic potential. Of course it doesn't make sense for these specific quantities to be represented by complex numbers, in the sense that the specific complex structure hasn't any special meaning. But it is not because you have found a few examples, that this proves in all generality that all (fundamental?) physical quantities must be necessarily real numbers and can never be complex numbers.

Hurkyl and I are trying to make you see two things.
One: what qualifies as "real" in a physical theory depends on its formalism, which is based upon mathematical constructions, such as real numbers, vectors, complex numbers, manifolds, ... , and there is not really a fundamental difference between the mathematical construction of the real numbers, and those of other constructions such as the complex numbers. It is the physical interpretation of the construction at hand which tells you whether it is a sensible hypothesis to assign some kind of ontological reality to the construction or not.
Two: in the case you want to limit yourself to fundamental quantities always being observable quantities, then the real number system doesn't qualify either, because every thinkable observation is necessarily an element of a discrete set of possibilities (in other words, can be represented by natural numbers).

You seem to hop between both views: you recognize that genuine observations will at most result in (finite lists of) natural numbers, but you do seem to think that these observations are approximations to a hypothetical quantity (which can hence only exist ontologically in a theoretical sense) that must be a "real number", and hence attribute "ontological reality" to that theoretical modellisation of the ideal real number behind the observation. But if we are allowed to assign ontological reality to theoretical constructions such as the real number, then there's no reason NOT to assign any reality to the other theoretical constructions of a theory too, as long as they have an obvious observer-independent notion to them.

When we bring about examples of such constructions, you object to them that in order to measure them, we can do so by "approximate real number" measurements. When we point out that such measurements don't measure real numbers, but at best rational numbers, you say that there is a theoretical notion behind them that corresponds to "the actual quantity" which ought to be a real number. So you change your definition of what is ontologically real during the argument.

Now, given that a scalar volt meter measures a quantity which is, indeed, represented by a real number, it is totally silly to try to assign a complex number to it. True. Same holds for distance.

But consider a quantity like "force". Force has "magnitude and direction" to it. It doesn't make sense to assign simply 3 real numbers to a force, because those 3 real numbers wouldn't be the same for different observers. However, an abstract element in a 3-dim vector space does qualify. Mind you that a 3-dim vector space is NOT R^3! An element of R^3 is a set of 3 real numbers. There is a fundamental distinction between, say (sqrt(2),0,0) and (1,1,0). The first set contains two times the neutral element for addition, while the second set contains two times the neutral element for multiplication.
Nevertheless, both of these 3-tuples are a valid representation for the same vector, in different coordinate systems. So there is something "intrinsic" to this vector representation, which a list of 3 numbers doesn't have, and that intrinsic property is exactly what we wanted to have for the physical notion of "force". As such, the abstract vector is a mathematical object which comes closer to the ontological reality of the concept of force, than a list of 3 real numbers. Moreover, the physical interpretation of force needs exactly the structure provided for by a vector space: vectorial addition. Without introducing an arbitrary coordinate system, given two forces, their vectorial sum has an intrinsic meaning. But whether or not their corresponding "list of 3 numbers" contains the neutral element for addition once or twice, doesn't have the slightest bit of intrinsic physical meaning. So there's more reality to the abstract vector than to its list of 3 coordinates.

In the same way, complex numbers are abstract constructions which qualify sometimes more than does "a set of 2 real numbers". As I pointed out, a typical use is as "the ratio of two sinusoidal signals of same frequency", also called a phasor. Typical application: a monochromatic EM field (light beam). Now, I'm not saying that the *complex signal* has any ontological reality to it. I'm just saying that the concept of "ratio of two sinusoidal signals with same frequency" is most naturally represented by a complex number:
if s1(t) = A1 sin(w.t + phi1) and s2(t) = A2 sin(w.t + phi2), then it is quite natural to represent s2/s1 by the complex number (A2/A1) exp(i(phi2-phi1)).
This has the advantage that if we have 3 signals, there is a transitivity:

if c12 = s2/s1 and if c23 = s3/s2 then c13 = s3/s1 = c12.c23 with the product given by the complex multiplication (which is the only extra structure C has over R^2). Moreover, this phasor representation also respects addition:
if s3/s1 = c13 and s2/s1 = c12, then (s2+s3)/s1 = c12 + c13.
So there is a complete interpretation to the complex number representation, and in the same way as we can talk about "distance" between two points in space, we can also talk about the "phasor relationship" between two points in the monochromatic EM field, which is a kind of "electromagnetic distance". With it, we can calculate interference patterns and everything.

The error you are committing in refuting any ontological reality to complex quantities is a straw man argument: you are attacking a point which nobody disputes. True, the complex signal A1 exp(w.t + phi) doesn't have a genuine physical interpretation. But that's not what a phasor is about. The phasor A.exp(i phi1) just represents the ratio between two real sine functions of time (yes, they are real, because they represent, say a scalar EM potential which is a real quantity, and not a complex one). But in the same way as "distance between two points in space" has physical reality, there's an "electromagnetic distance between two points in an optical setup" which can naturally be described by a phasor, which is independent of any "coordinate system choice".

Now, all this is a bit artificial of course. The genuine reality of the complex number system appears most clearly in quantum theory.