Implications of Imaginary Numbers?

[Bob S]

Our understanding of causal relations in complex systems is based in part on the Kramers Kronig (dispersion) relations, that are based on analytic functions (real and imaginary functions). One surprising result (to me at least) is an application in electrical circuits called real part sufficiency theorem, that allows prediction of the imaginary component impedance of a circuit in the frequency domain, if only the real part impedance of the circuit is known.

Bob S

 

[Tao-Fu]

Correction -- descriptions of real world phenomena we opt to describe with real numbers use real numbers.

Okay. Granted.

My point being, any measurement is simply counting, which does not involve any thing more exotic than a real number line (or even natural numbers in many cases). I can certainly describe my observations non-quantitatively and this obviously would not involve real numbers (or units for that matter).

 

[Hurkyl]

The great power of mathematics is that it allows you to bring many different perspectives to bear on a problem. One unfortunate side-effect is that it enables people to wedge everything into their favorite box, and never want to look outside of it.

I've seen people who want to insist that everything "physical" is a real number -- they insist that complex numbers are just a mathematical bookkeeping trick -- so they mentally reject any description of reality in terms of complex numbers, instead dissolving it into real components (maybe real and imaginary part, maybe magnitude and phase, whatever).

I've seen people do similar things, rejecting the negative reals in favor of magnitude and sign. Rejecting the reals because all measurements are just rational numbers. Rejecting rationals because we should be thinking in terms of ratios of integers.

(Oddly, I don't think I've ever seen someone reject integers in favor of the act of counting)

It looked like you were starting to push one of these "thou shouldst thinketh inside this box" perspectives on physics, which is why I felt the need to reply.


Incidentally, I find it unfortunate this biased thought has crept into quantum mechanical foundations -- i.e. only operators whose imaginary part is zero are allowed to be called "observables".

 

[xlines]

I've seen people who want to insist that everything "physical" is a real number -- they insist that complex numbers are just a mathematical bookkeeping trick -- so they mentally reject any description of reality in terms of complex numbers, instead dissolving it into real components (maybe real and imaginary part, maybe magnitude and phase, whatever).

I've seen people do similar things, rejecting the negative reals in favor of magnitude and sign. Rejecting the reals because all measurements are just rational numbers. Rejecting rationals because we should be thinking in terms of ratios of integers.

And we should reject integers in favor to Peano successor function S(n) and number 0. :-)

I find it disturbing when someone insists that complex numbers are somehow more "unphysical" unlike "real" numbers. For one thing, if someone wants to measure part of an apple, all that is needed is small subset of rational numbers, since all rationals not belonging to that subset require you to break nucleons to quarks or worse :) It seems to me that this kind of numeral solipsism should not be allowed to limit our insight of the reality.

Economy and elegance of one's calculation is a good reason to use complex numbers since set C is algebraically closed set and there exist mighty theorems on analytic function which often provide shortcut to results that would otherwise require much more work.

I'm not sure about this one,but it may be possible to derive Schrodinger equation as a pair of coupled equation, one for real module and one for real phase which are equivalent to starting equation. If so, that's a high price for ideology, IMHO.

(Oddly, I don't think I've ever seen someone reject integers in favor of the act of counting)

LOL! :)

It looked like you were starting to push one of these "thou shouldst thinketh inside this box" perspectives on physics, which is why I felt the need to reply.

Incidentally, I find it unfortunate this biased thought has crept into quantum mechanical foundations -- i.e. only operators whose imaginary part is zero are allowed to be called "observables".

I agree, as long as people are aware of limits of their description, I find such limitations void. Now, I'm off to do some rotations with quaternions. Don't try to stop me! ;-)

 

[bjacoby]

Hello,
I have a quick question that I imagine anyone who has studied physics or math at a university can answer rather easily. If not, I apologize in advance for the effort!

What is the physical significance of imaginary numbers? I have heard repeatedly that imaginary numbers are relevant, that they do appear in the real world, and so on, but I haven't found a good example. I'm convinced that they're telling the truth, but I don't have anything to back that up.

If by "physical significance" you mean where do we find imaginary numbers in nature, the answer is nowhere. We find mathematics nowhere in nature. Mathematics are abstract creations of mind. They are self-consistent systems of thought. They have no basis in physical reality. When it comes to mathematical systems you can have valid systems where one is based on one idea and another is based on the diametric opposite of that idea (example parallel lines meet at infinty or they don't meet ever). You can have systems based on nonsense such as the square root of -1 (imaginary numbers...or more correctly complex numbers) reality does not matter. With mathematics only self-consistency of a given system matters.

So why do so may physicists worship mathematics as if it were somehow a super-reality? It is because over the years humans have discovered that mathematics is often useful for more or less predicting things in advance. A very useful tool to have. And mathematics is therefore used as a MODEL of reality. The assumption (not always true by the way) is that whatever is going on in our model will also be found in reality. Models are adjusted over and over to insure that this is a true as possible given current understanding.

So the correct way to phrase your question would be to ask "what is the utility of imaginary numbers in physical science?" And the true answer goes something like this. There are many phenomena that are modeled by sinusoidal functions. Sin, Cos Tan and the like. Therefore those functions and the associated mathematics are very useful predicting many phenomena. Now mathematically speaking, there is a connection between imaginary or complex numbers and the trigonometric functions. Even though imaginary numbers aren't real and are based on total imagination, it turns out that the self-consistent relationship connecting sinusoidal and complex numbers is very useful for modeling. And the reason is that complex variables represent a more compact representation of the given calculation.

So if one is calculating say a Fourier transform using normal trignometric methods one finds TWO terms one which is a sin and one which is a cos. But doing the same thing with complex variables yields only ONE answer which of course is split into real and imaginary parts and indeed can also be transformed into the former sin and cos.

The advantage is that humans are not computers. We can handle ideas, but not lots of detail. In tensor analysis for example, if we can take large matrices of data and huge sets of equations and represent the SETS by single variables, it is a great thinking tool. Since the relationship of our simplified variables helps give us the idea of relationships involved and yet by simple rules we (or a computer) can work backward from the IDEA of the thing back to the ACTUAL NUMBERS involved. This is exactly what models using complex numbers are all about.

OK?

 

[LouieHussey]

Bjacoby, you write: “You can have systems based on nonsense such as the square root of -1 (imaginary numbers...or more correctly complex numbers) reality does not matter.”

Just out of curiosity, why do you describe imaginary numbers as “nonsense”? Your statement seems to imply that it is because they are not based in “reality” (meaning physical reality). Is that correct?

And what about negative numbers, do you also consider them to be “nonsense”?

Are there any numbers that you consider to be “not nonsense”?

Are the positive whole numbers legitimate in your mind?

And are positive whole numbers expressed in one particular base more legitimate than those expressed in other bases?

I need to point out, in passing, that the “nonsensical” imaginary numbers you refer to were “invented” in order that the natural number system would be closed under the inverse operation of raising to powers. Otherwise there would be no solution to certain equations, and that usefulness of mathematics that you refer to would be considerably diminished from what it is today. Each of the expansions of the natural number system has resulted in increased usefulness of the number system.

I read that resistance against the negative numbers was still alive and well when the strange new imaginary numbers were introduced and began to be used. Your comments inform me that resistance (and even outright resentment) to the introduction of the new numbers seems never to have died out in some quarters, in these quarters there is only a begrudging tolerance of these numbers because of their usefulness.

 

[HarryWertM]

Much thanks to LouieHussey for outstanding post introducing me to quaternions. A quarrel with Hurkyl: You described Louie's number systems extensions as "linear extensions"? I would say they are more accurately "regressive" or "recursive" definitions.
-harry wertmuller

 

[kcdodd]

There is also a geometrical interpretation to complex numbers and quaternions that I find enlightening, which is based on the algebra of multi-vectors. It is also related to the matrix representation of those quantities. If you define a vector basis, then you can define multi-vectors as the product of those basis creating directed area, volume, etc, in addition to directed length. The property of the vector multiplication is inherently anti-commutative, and is understood geometrically as changing the direction of the area, or volume.

For example, in 2-D, you may have the basis:

$$e_1, e_2$$

Any 2-D vector can be represented by a linear combination:

$$X = a e_1 + b e_2$$

The product of the basis gives the highest order multi-vector of the space, a bivector, which is anti-commutative.

$$B = e_1e_2 = -e_2e_1$$

This has the property that the bi-vector times itself is -1, since a (unit) basis times itself is one.

$$B^2 = e_1e_2e_1e_2 = -e_2e_1e_1e_2 = -1$$

Hmm, very similar to the imaginary unit. If you notice, you can take a normal 2D vector and multiply it by one of the basis, and create scalar and a bi-vector, which has the same algebra as complex numbers.

$$e_1X = a e_1e_1 + b e_1e_2 = a + bB$$

All we have done is "prefer", or project out, the e_1 part. Equate e_1 to the "real" line and e_2 to the "imaginary" line.

The complex conjugate is just multiplying the vector from the other side:

$$Xe_1 = a e_1e_1 + b e_2e_1 = a - bB$$

The quaternion is similar, but with three bi-vectors:

$$W = a + b e_2e_3 + c e_1e_3 + d e_1e_2$$

where

$$i = e_2e_3, j = e_1e_3, k = e_1e_2$$

EG

$$ij = e_2e_3e_1e_3 = - e_2e_3e_3e_1 = -e_2e_1 = e_1e_2 = k$$

You can work out the rest to show it's equivalent.

 

[citizen_x]

Sorry for digging this up after a year, but there's one thing I'm curious about. What kind of vector product are we talking here, exactly? I've been playing a little with the math but I can't get one thing.

If it's cross product, then:

$$e_1e_1 = 0$$
And not 1. So:

$$e_1X = a e_1e_1 + b e_1e_2 = a\cdot 0 + b \cdot B = b \cdot B$$

which is not the same as

$$e_1X = a e_1e_1 + b e_1e_2 = a + bB$$

obviously. And even if one takes a different combination of the [itex] e_i [\itex] base vectors and a different [itex] B [\itex] vector, it still won't result in an entity consisting of a scalar + bivector. It'll be just rotating around in 3-space. To get scalar + bivector, one would have to have a kind of multiplication that pops me out of my vector 3-space.

If it's scalar product, then it should be commutative. And it's not.

It's 03:06 in my time zone, yup I'm a night reader, so I might be missing something.
Maybe a little more abstract kind of multiplication was assumed here?

Which brings me to a second question, how this would look written in matrices? Because I'm thinking it all looks much more clear that way.

The line of thinking I'm reffering to:

(...)
For example, in 2-D, you may have the basis:

$$e_1, e_2$$

Any 2-D vector can be represented by a linear combination:

$$X = a e_1 + b e_2$$

The product of the basis gives the highest order multi-vector of the space, a bivector, which is anti-commutative.

$$B = e_1e_2 = -e_2e_1$$

This has the property that the bi-vector times itself is -1, since a (unit) basis times itself is one.

$$B^2 = e_1e_2e_1e_2 = -e_2e_1e_1e_2 = -1$$

Hmm, very similar to the imaginary unit. If you notice, you can take a normal 2D vector and multiply it by one of the basis, and create scalar and a bi-vector, which has the same algebra as complex numbers.

$$e_1X = a e_1e_1 + b e_1e_2 = a + bB$$

All we have done is "prefer", or project out, the e_1 part. Equate e_1 to the "real" line and e_2 to the "imaginary" line.

The complex conjugate is just multiplying the vector from the other side:

$$Xe_1 = a e_1e_1 + b e_2e_1 = a - bB$$
(...)

By the way, pleasure to write here.

 

[kcdodd]

It's in effect both an inner product and a "cross-product" (actually the outer product). When you take the geometric product of two vectors, what you get is a combination of the two products. For vectors a and b:

$$ab = a\cdot b + a\wedge b$$

Now, this results in an object which is not just a scalar nor just a bi-vector, but both. Now, basis vectors are chosen to be orthonormal such that

$$e_i \cdot e_i = 1$$
$$e_i \cdot e_j = 0, i \neq j$$
$$e_i \wedge e_i = 0$$

So that when you take the geometrical product of the basis with itself, only the scalar part survives. And when you take the geometric product of two orthogonal bases only the bivector part survives.

$$e_i e_i = e_i \cdot e_i + e_i \wedge e_i = 1$$
$$e_i e_j = e_i \cdot e_j + e_i \wedge e_j = e_i \wedge e_j, i \neq j$$

Is that satisfactory?

 

[Rap]

That's only because you personally need applications with real numbers only. An engineer might appreciate complex numbers in many places and for him complex numbers would be just as natural and real as other numbers.

Hurkyl is quite right. All of the numbers are artificial. Especially irrational numbers like the diagonal of a square.

Really the only difference is that only real numbers are used to measure "lengths" and "amounts".

Actually, real numbers are never found in the "real" world. Every measurement you make is at most a rational number. There is not and never will be a computer in existence that performs concrete calculations on real numbers. The real number system is an extremely useful way to deal with a rational number world, just as complex numbers are an extremely useful way to deal with some aspects of a real number world. And regarding the idea that you cannot do quantum mechanics without using complex numbers, that is false. All you have to do is use pairs of real numbers and modify your operations accordingly.

 

[Studiot]

concrete calculations on real numbers.

Bogue equations anyone?

 

[citizen_x]

Is that satisfactory?

Yup, that's satisfactory. Thank you very much.

 

[Claude Bile]

I find complex numbers fascinating. I view (erroneously or non-erroneously; I make no guarantees) a complex number as holding "encoded" information. To access the information in the form of a real number, we must "decode" the complex number in an appropriate way. To get the magnitude we must use Pythagoras. To get the phase we must use trigonometry.

Also, I like that complex numbers can represent 2D rotations while quarternions can represent 3D rotations. It's quite a succinct way of describing what these number systems can do.

Can Octonions represent 4D rotations?

In any case, I think dismissing complex numbers merely as something we put up with because we find them useful ignores a deeper, more beautiful symmetry that exists in these number systems.

Claude.