Implications of Imaginary Numbers?

[Sobeita]

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.

I found one page that tried to show how imaginary numbers could be used for calculating fast mental rotations, but it turned out they had just memorized points on a circle, just as you could with normal trigonometry.

I have heard they are used in fluid dynamics, in electricity, in astrophysics, and any number of other settings, but I just haven't found an example. If someone can show me one, I can finally prove to the "math skeptics" that they're wrong! Cheers.

 

[Tac-Tics]

Complex numbers do not yield anything new you can't already do with geometry. Their use is an implementation detail. Choosing between complex numbers and 2-vectors is like choosing between flat-head and Phillips-head screws. They both get the job done, and neither are "more correct" than the other. It just really matters what kind of screw driver you have on hand.

So you could say, one of the major reasons we use complex numbers is everyone is used to them by now. They are well institutionalized in physics and engineering literature. They are easy enough to manipulate on paper. There is no standard notation for multiplying, exponentiating, taking the conjugate of vectors. There is no reason you could define such operations, but if you have to communicate your ideas to other people, it makes a lot of sense to use what people are already familiar with.

As far as physical quantities go, there's nothing mysterious about the word "imaginary". If I have "a quantity with a real part and an imaginary part", the "imaginary" part isn't somehow un-real. It's not some ethereal, intangible, unmeasurable enigma the universe has. It's just a frickin' number. The deal is that these complex quantities are just ORDERED PAIRS of regular, boring old, real-valued quantities.

Take electric impedance for instance. It's a complex number. The real part is the resistance. The imaginary part is the reactance. You mush them together into this single complex number, and the formulas turn out nicely. And it's not surprising either, because you're dealing with sinosoid voltages (circles) and two quantities (two-dimensions) and derivatives (exponentials) and when you mix those three things together, you will inevitably end up with something that works exactly like complex numbers.

 

[matt_crouch]

imaginary numbers just allow us to solve equations that have root of a minus number basically. we don't know the number so we use this symbol "i" which allows us to get a real number.
=]

 

[Gerenuk]

Stricly speaking you could do most calculations with real numbers only, but then you'd have more equations which are more difficult to deal with. So people like using complex numbers as a package. Complex numbers are indeed are very important tool in electromagnetism, quantum physics, wave physics, fluid flow and many more.

I'll give some thought and I hope someone else will write a longer explanation.

Complex numbers can represent rotations in 2D. Some people use them for geometry.

Also they can simplify the maths whenever periodic wave phenomena are considered. That's why they are used for electromagnetic waves for example. Instead of writing A\cos(\omega t+\phi) it's much more mathematically convenient to use A\exp(i(\omega t+\phi)) and assume that you want to use the real part only. Then advances in phase be be written with a multiplication \exp(i\phi)\cdot\exp(i\theta)=\exp(i(\phi+\theta)) whereas no such simple thing is possible for the real representation \cos(\phi)\quad\to?\quad\cos(\phi+\theta).

Also complex numbers are a fundamental part of quantum mechanics as the Schroedinger equation is written (in short notation)
\nabla^2 \psi+V\psi=-i\psi_t[/itex]<br /> where i is the imaginary unit and \psi the complex wavefunction.<br /> <br /> In fluid dynamics and electrostatics complex numbers can be used through &quot;conformal mapping&quot; to solve 2D problems. With conformal mapping problems of complicating shape (for example electric field in a polygonial region) can be transformed to equivalent problems in a much simpler region (a circle for example) and so solving one problem gives the solution to another.<br /> <br /> In electric circuits complex numbers are used to deal with passive elements like capacitors or coils. The impedance (something equivalent to resistance) of a resistor is R, of a capacitor Z=1/i\omega C and of a coil L=i\omega L. This way you can use Ohm&#039;s law and calculate the effect of an applied sinusoidal voltage.<br /> <br /> For oscillating systems (anything that vibrates and is driven by a periodic force) the solutions involve square roots. Under some circumstances the value under the square root becomes negative and then you need complex numbers to make sense of you solution.<br /> <br /> I hope someone can give good references for all the topics... <img src="https://cdn.jsdelivr.net/joypixels/assets/8.0/png/unicode/64/1f642.png" class="smilie smilie--emoji" loading="lazy" width="64" height="64" alt=":smile:" title="Smile :smile:" data-smilie="1"data-shortname=":smile:" /><br /> <br /> The &quot;maths skeptics&quot; of yours can&#039;t be very clever. It&#039;s a big mistake to say something is useless just because you haven&#039;t looked at the applications.

 

[Sobeita]

Thank you, Matt. I was asking about physical representations, though. When would you find the root of a "minus number" in real life?

Ah, Tic-Tacs, that makes more sense. I didn't realize they were fully interchangeable like that. Thank you!

Edit: @Gerenuk: Wow, that was a very detailed and helpful post! Thank you very much. As far as the math skeptics, I know where they're coming from - how can you respect a number which is labeled imaginary by the people who study them? That's why some people choose to call them "complex" (and I imagine this is probably the correct term by now.)

 

[Gerenuk]

Complex numbers do not yield anything new you can't already do with geometry. Their use is an implementation detail. Choosing between complex numbers and 2-vectors is like choosing between flat-head and Phillips-head screws.

You must be a mathematician

I'd like to see you calculate waves and quantum mechanics with the split notation.

 

[matt_crouch]

lots of physical things have a root of a minus number like the gerenuk said waves and quantum mechanics are a good place to find them =]
x

 

[Tac-Tics]

You must be a mathematician

The situation is worth than that. I'm a programmer.

 

[DaveC426913]

...some people choose to call them "complex" (and I imagine this is probably the correct term by now.)

Nitpick: 'Imaginary numbers' and 'complex numbers' are not synonymous.

A complex number is composed of a real number and an imaginary number: (a + bi).

 

[Sobeita]

Yes, I know. It's getting late here. :P I've dealt with them before. I did some interesting work a while ago with complex numbers creating fractals. ( http://soulfox.com/images/images.php )

If the math skeptics ever laughed, it was when they heard about Friendly Numbers... ( http://xkcd.com/410/ )

 

[mikelepore]

I enjoyed this fact about sinusoidal steady state circuits. Multiplying by i represents a 90 degree phase shift. Therefore, multiplying by i-squared represents a 180 degree phase shift. True, shifting a sinusoid by 180 degrees is the same thing as flipping it upside down, or multiplying by -1.

But imaginary numbers existing in the real world? I'd say no. They are abstractions that we use because doing so causes us to arrive at the right answers to problems. No further justification is required.

 

[Pengwuino]

Of course, imaginary numbers are useful mathematical tools, but you never run into something physical that is imaginary. You can build your theory using complex numbers, but at the end of the day, when you look at what is physical, you don't have imaginary parts. I've never seen an imaginary electric field that is.

 

[Gerenuk]

Of course, imaginary numbers are useful mathematical tools, but you never run into something physical that is imaginary. You can build your theory using complex numbers, but at the end of the day, when you look at what is physical, you don't have imaginary parts. I've never seen an imaginary electric field that is.

First of all quantum mechanics needs complex numbers.

Of course again one can split them into real and imaginary part, but this time the equations become more complicating, so it's reasonable to assume that complex numbers are essential. Just because they fit perfectly the requirements.

It's not straightforward to say what's "real" and what's just a mathematical construct. Once you have studied enough theories you come to the conclusion that in the end all of physics is only a mathematical construct and the only thing that matters is the simplicity of the equation. And often complex numbers are easier.

Only if someone claims that physics is only what can be measured with a ruler, then he might think in real numbers only. But he is slower at solving equations. I'd say both pictures are not "real", so let's at least take the more comfortable one.

 

[Hurkyl]

But imaginary numbers existing in the real world? I'd say no. They are abstractions that we use because doing so causes us to arrive at the right answers to problems. No further justification is required.

And, for the record, the real numbers, rational numbers, and integers all enjoy the same state of affairs.

 

[mikelepore]

I would say that the real numbers exist in the physical sense. If I earn $7 per hour, and my dinner costs $21, then I have to work for 3 hours to pay for my dinner. That seems like a material reality to me. I don't know of any ordinary experience where we would find imaginary numbers.

 

[Gerenuk]

I would say that the real numbers exist in the physical sense. If I earn $7 per hour, and my dinner costs $21, then I have to work for 3 hours to pay for my dinner. That seems like a material reality to me. I don't know of any ordinary experience where we would find imaginary numbers.

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".

 

[A.T.]

I would say that the real numbers exist in the physical sense. If I earn $7 per hour, and my dinner costs $21, then I have to work for 3 hours to pay for my dinner. That seems like a material reality to me.

The dollar notes are material reality. But not the numbers we made up to quantify their value.

I don't know of any ordinary experience where we would find imaginary numbers.

"Ordinary experience" is highly subjective, and doesn't qualify as criteria. We make up new types of numbers to have solutions for all operations we can do on numbers we have allready.

First we invented natural numbers to count objects. Then we invented fractions to have solutions for all divisions. Then negative numbers to have solutions for all subtractions. Then irrational numbers to have solutions for some roots etc. And finally imaginary numbers to have solutions for all roots.

I think we're done for now, unless we invent some operations with no solutions within the current numbers.

 

[Pengwuino]

First of all quantum mechanics needs complex numbers.
.
.
.
Only if someone claims that physics is only what can be measured with a ruler, then he might think in real numbers only. But he is slower at solving equations. I'd say both pictures are not "real", so let's at least take the more comfortable one.

Yes but my point stands, at the end of the day, all you see in the physical world is real numbers. Even in QM, you need real observable quantities.

 

[Gerenuk]

Yes but my point stands, at the end of the day, all you see in the physical world is real numbers. Even in QM, you need real observable quantities.

I can sort of follow what you mean. I just think the "real world" can include all concepts, not just something that is a touchable object. I cannot think of a good example right now. Maybe interest rates in a bank? In any case it's just a matter of taste. I believe you too would use complex numbers to solve equations and that's all that matters. It not so important if one calls something "real" or not

 

[GoldPheonix]

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.

I found one page that tried to show how imaginary numbers could be used for calculating fast mental rotations, but it turned out they had just memorized points on a circle, just as you could with normal trigonometry.

I have heard they are used in fluid dynamics, in electricity, in astrophysics, and any number of other settings, but I just haven't found an example. If someone can show me one, I can finally prove to the "math skeptics" that they're wrong! Cheers.

This is going to be a much less satisfying, though much more correct, answer than I think you were hoping for.

Numbers --imaginary, real, rational, integer quaternion, complex, split-complex, p-adic, et cetera-- do not exist in the physical universe.There's no number 2 floating out and about in the universe, waiting to get discovered, anymore than there is a square root of minus one waiting to be discovered. Mathematics isn't something that you discover (at least, not in the physics sense), mathematics is simply the procedure labeling an abstract object (real numbers, functions, etc) and defining its logical/mathematical properties (commutativity, associativity, identity, idempotency, etc).

So when a physicist (improperly) says, "imaginary numbers appear in the real world", what he or she means is that "Imaginary numbers can be used to model the real world." As it has been pointed out, you could arbitrarily select a different (but mathematically equivalent, or isomorphic) numerical system/structure to do the job.
The best example of using complex (and thus imaginary), by far, however, is in quantum mechanics. Schrodinger's Equation, the quantum mechanical definition of momentum and energy, the definition of probability currents, and most wave functions involve complex numbers.

http://en.wikipedia.org/wiki/Schrodinger's_equation#The_Schr.C3.B6dinger_equation
http://en.wikipedia.org/wiki/Momentum_operator
http://en.wikipedia.org/wiki/Hamiltonian_operator (Hamiltonian operator is a version of the energy operator)Another good example comes from special relativity. While it's not often discussed, the geometry which describes special relativity is that of Minkowski space. And points in Minkowski space are actually themselves objects which are known as split-quaternions. The split-quaternions have a different imaginary unit (in fact, they have three distinct imaginary units) than the square root of minus one, but they are still imaginary numbers.

http://en.wikipedia.org/wiki/Minkowski_spacetime#Structure

Yes but my point stands, at the end of the day, all you see in the physical world is real numbers. Even in QM, you need real observable quantities.

This is sort of true, but from a mathematical standpoint, all you're doing is thinking of an object which is isomorphic to the set of complex (or split-complex, quaternion, split-quaternion, or what have you) numbers.

 

[Pengwuino]

I can sort of follow what you mean. I just think the "real world" can include all concepts, not just something that is a touchable object. I cannot think of a good example right now. Maybe interest rates in a bank? In any case it's just a matter of taste. I believe you too would use complex numbers to solve equations and that's all that matters. It not so important if one calls something "real" or not

Well, I just hope to god no one builds my car's air bag to deploy with a pressure of 300 + 500i PSI.

 

[Redbelly98]

In skimming through this thread, I didn't find any explanation as to why it is useful to use complex numbers. So I'll provide one.

I think at the root of it, we often encounter differential equations that describe oscillatory phenomena. Familiar examples of this are the Schrodinger Equation, and the voltage & current in capacitors and inductors.

What makes complex numbers convenient is the fact that the derivative of exp(iωt) is proportional to exp(iωt), which greatly simplifies the solving of linear differential equations. Since sin(iωt) and cos(iωt) do not have this property, it is advantageous to use exp(iωt) instead.

In the end, we obtain real-valued answers by either taking the real part of the answer (in the case of voltages and currents) or multiplying by a complex conjugate (in the case of quantum mechanics). The use of complex numbers is just an intermediate mathematical tool towards finding real-valued answers.

 

[LouieHussey]

This is a reply to Sobeita’s question as to whether imaginary numbers have physical significance, and whether there are physical examples of imaginary numbers in nature.

Sobeita, I preface my remarks by saying that the labels “real” and “imaginary” in regards to numbers can, and often do, cause confusion among the uninitiated, because they are not descriptive of what they connote. Because of the confusion that it sometimes causes, the terminology is, in my opinion, unfortunate. As already pointed out in these replies, “imaginary” numbers are just as real as “real” numbers are, but they are a different kind of number.

Now for your question. The answer is yes, imaginary numbers have physical significance in physics and in the geometry of space-time, examples of which I will give in a moment. They have no physical significance in regards to money, however, because all money is “real”, mathematically speaking.

Before I get to the examples, let me give some background that is relevant to the more general question of the relation of mathematics to physical reality. There is a intimate relationship between the Natural Number System (a description of which I will give in a moment), space-time, and physics. Einstein believed that physics could ultimately be reduced to space-time geometry (and there is reason to believe that this is the case, although it has only partially been achieved to date--it is still a work in progress). And the Natural Number System, when applied to space-time with analytical geometry, describes space-time (they are isomorphic to each other, if you will). So it is not surprising to find that the numbers of the Natural Number System have physical significance in the geometry of space-time and in physics.

Mathematicians have extended the mathematical operations of the Natural Number System over the years. First there was addition, then multiplication, then raising to powers. They have also extended the numbers of the Natural Number System over the years (the number types). First there were counting numbers. Then with the discovery of negative numbers, counting numbers (positive whole numbers) were extended into integers; then with the discovery of fractional numbers, integers were extended into real numbers, then with the discovery of imaginary numbers, real numbers were extended into complex numbers; then with the discovery of hyper-imaginary numbers, complex numbers were extended into quaternions. And this successive extension of the Natural Number System continues to this day. It is a work in progress.

Now a curious thing has been noted in the literature: that the new numbers arising out of any extension of the natural number system can always be defined as ordered pairs of the preexisting numbers. For example, complex numbers, which are an extension of the real number system, can be defined as an ordered pair of real numbers. And the extension can be brought about with the following general procedure: beginning with a natural number system that is closed under its existing mathematical operations, its highest order mathematical binary operation is first extended (always defining the new operation as a repetition of the old; if the highest order mathematical operation is addition, for example, then its extension, which is multiplication, is defined as repeated addition), and then defining a new class of numbers as an ordered pair of the preexisting numbers, where mathematical operations on those numbers are defined in such a way that it will make the system closed under the inverse operations of the new extended operation while at the same time preserving the mathematical laws of the former operations for the former numbers. So the natural number system and its mathematical operations are extended together, for the extension of the mathematical operations requires the extension of the number system itself in order to preserve closure.

The extension of the natural number system under this general procedure yields a self consistent mathematical system that is unique. So you are not arbitrarily “inventing” new numbers, but rather “discovering” new numbers along with the properties that they must have in order to meet the conditions of closure laid out above, just as the laws of physics are not arbitrarily invented, but are rather discovered. It should also be noted that the natural number system extensions made to date were not necessarily derived in this manner historically, but they can be. Now can the natural number system be extended indefinitely in this manner, or will it eventually close--that is, will the extension of the mathematical operations of the system eventually result in an operation whose inverses are already closed under the preexisting numbers of the system, hence requiring no additional extension of the number system? I, myself, do not know. I am aware of nothing that has been discovered to date that would settle the question (but it may be possible to use a technique similar to finite induction to look ahead in these extensions to see whether there is any possibility of ultimate closure). Eventual closure of the procedure would imply that there are a limited number of physical dimensions and space-times in the universe.

Note: extending the natural number system by hand with this procedure tends to become more and more cumbersome the further the number system extensions go (because there are more and more pairs to manipulate, and more permutations to consider), but it should be practical to program modern day computers to quickly and automatically accomplish a series of extensions, which would enable us to make a quantum leap in the advancement of the natural number system and in the advancement of physics too, since new natural number system mathematics points to new physics. Perhaps some researcher who is experienced in computer programming will see this note and decide to take up this project.

The mathematical system derivable by this procedure is sometimes referred to in the literature as “The Natural Number System”, the Number System of Nature, for the numbers of this system (negative numbers, real numbers, complex numbers, quaternions, hyper-quaternions, and so on) have a physical meaning in physical space-time, and when the natural number system is applied to analytical geometry for the study of space-time, there is something like a one-to-one correspondence, if you will, between the properties of the Natural Number System and those of space-time. There is no mystery in this, by the way, it is not numerology or anything like that, it is because the fundamental laws of nature are determinate, and therefore require a determinate mathematical system for their complete description, determinate in the sense that its equations, written in terms of its mathematical operations, have solutions. [Note: some physicists today would argue that nature is indeterminate, but some very notable physicists down through the years, including Newton, Einstein, and de Broglie, believed that nature is determinate, and that the fundamental laws of physics are not statistical in nature, but are determinate]. The implication of the correspondence between the natural number system and physical space-time is that expansion of the number system automatically points to new physics, and visa versa.

The following sequential extensions have been made to the natural number system to date. The closure of the natural number system under the inverse of addition (subtraction) necessitated the “invention” of the negative numbers, giving rise to the integers, otherwise there would be no solution within the number system to the problem of subtracting a larger number from a smaller number. The closure of the natural number system under the inverse of multiplication (division) necessitated the “invention” of the fractional numbers, giving rise to the real numbers, otherwise there would be no solution within the number system to the problem of dividing a smaller number by a larger number. And the closure of the number system under the inverses of raising to powers (under the taking of roots for example) necessitated the “invention” of the imaginary numbers, giving rise to the complex numbers, otherwise there would be no solution within the number system to the problem of taking even roots of negative numbers. Lastly, the complex numbers were extended by Hamilton in 1843 to the quaternions (which are hyper-complex numbers that can be mathematically defined as an ordered pair of complex numbers). Although Hamilton did not do it this way, it can be shown that a new mathematical operation m * n can be defined as m appearing as a “factor” n times in a series of factors in which m is raised to the mth power a successive number of times, and that the quaternions make the mathematical system closed under the inverses of this new operation. Attempts have more recently been made to extend the quaternions to the hyper-quaternions, but those efforts have largely been speculative in nature and not guided by the procedure outlined here. [There are various clues that can guide one to correctly extending the natural number system without first extending the mathematical operations of the system, but when one does that, there are interesting new mathematical relations that are missed, very beautiful and useful relations, relations that have an analogue in the properties of space-time.].

The introduction of each of these strange, new numbers ignited a lengthy period of resistance and controversy, during which philosophical questions concerning their mathematical legitimacy, meaning, and physical significance were debated, and during which their use was often avoided if there were more conventional methods of arriving at the same conclusion, until the power and utility of these new numbers in mathematical analysis and in the physical world finally compelled mathematicians and physicists to accept them as legitimate numbers and legitimate extensions of the number system--forcing their general acceptance. We are presently still in the period of resistance and controversy against the last extension of the number system to be made, that of the quaternions. (Note: three dimensional vectors were invented by physicists towards the beginning of the resistance as an alternative to using quaternions--today, these three-vectors have been expanded to four dimensions, and are used by many physicists for the study of space-time and relativity physics; these four-vectors, which are better suited to the study of relativity physics than three-vectors, are more nearly like quaternions, but are still not identical to quaternions).

Having given this background, I finish by giving examples of the physical significance of imaginary numbers. Analytical geometry allows us to form a one-to-one correspondence, so to speak, between the natural number system and the geometry of space-time. A complex number, for example, can be used to represent position in two dimensional space-time: the real number part of the complex number represents temporal position (or the component of position in time), while the imaginary part represents spatial position (or the component of position in space). Quaternions, which are ordered pairs of complex numbers, and which contain one real term and three imaginary terms, can be used to represent position in four dimensional space-time. The imaginary basis of the quaternion (or that of the complex number in the case of two dimensional space-time) identify the physical dimensions of space-time, and positions of coordinate points along those dimensions--that is their physical significance when applied to the geometry of space-time. Multiplying a number by a real number produces a change in position or scale in space-time, while multiplying by a imaginary number produces a type of rotation in space-time, and that is the physical meaning of multiplying by a imaginary number.

Various other fundamental quantities in physics, aside from position in space-time, are also four dimensional and can be designated by quaternions (or by complex numbers in two dimensional space-time). Momentum, for example, when it is defined as the time derivative of the moment of mass, is actually four dimensional, having a time component that is represented by a real number, and three spatial components that are represented by three imaginary numbers (or if there is only one spatial component, by the imaginary term of a complex number). Here the real number has the physical meaning of energy (which is the time component of four-momentum), while the imaginary numbers have the physical meaning of momentum (which are the spatial components of four-momentum). If we are talking about physical fields, potential fields, for example, also have four components. The time component of the electromagnetic four-potential, for example, is the electric (or scalar) potential, and the spatial components of the electromagnetic four-potential are the three spatial components of the vector potential (or if there is only one spatial dimension, the real term of a complex number is the electric (or scalar) potential, and the imaginary term of the complex number is the vector potential). So in this case, the imaginary term in the complex number is physically the vector potential, and its real term is physically the scalar potential. To give one final example, force, when defined as the time derivative of the four-potential, is also four dimensional: its time component, which is the component of force in the time dimension, is a real number, and its three spatial components, which are the components of force in the three spatial dimensions, are imaginary numbers.

Each of the natural numbers generated by the natural number system extensions have a physical meaning in space-time (and therefore in physics). The sign of signed numbers, for example, identify direction along a line in space-time relative to the origin. The real numbers fill in the gaps between the integral positions along the line so that it forms a continuum, so their physical meaning pertains to the physical continuum of space-time. And the imaginary basis of complex numbers and quaternions (hyper-complex numbers) identify spatial dimensions of space-time, while the real basis identifies the time dimension.

 

[kcdodd]

My own interpretation of a complex number, at least when solving a differential equation, is keeping track of quantities which we don't care to measure. For instance, take a ball on a spring. It is described by a very simple differential equation.

m\ddot{x} = -k x

The solution to this equation something of the form:

x = Ae^{i\omega t} = A(cos(\omega t) + i sin(\omega t))

So, if the real part is the position, what is the imaginary part? Think about the physical system when x = 0 (ie when the real part is zero). There is something non-zero in the system to keep it oscillating. What? Kinetic energy, momentum, whatever. But we didn't care about what exactly it is when we formed our solution because we only care about x. The imaginary part is some quantity we really don't care about, but it is "real" in the physical system, to keep it going.

 

[Hurkyl]

Now a curious thing has been noted in the literature: that the new numbers arising out of any extension of the natural number system can always be defined as ordered pairs of the preexisting numbers.

That's just one scheme of making new systems. And your description has the unfortunate, incorrect implication that interesting number systems are linearly ordered -- e.g. the complexes contains the reals contains the rationals contains integers contains the natural numbers.

For example, one incredibly useful number system is the extended real numbers. This is the real numbers with two extra numbers: +infinity, and -infinity. This is, in some sense, the "right" number system for doing real calculus. However, +infinity and -infinity aren't complex numbers, and don't really have complex analogues.

Other examples are that it is often useful to treat polynomials as numbers, and the same with scalar fields.

Some interesting number systems are not formed by extending, but by reducing -- e.g. modular arithmetic.

 

[LouieHussey]

Hurkyl, there are indeed other schemes for inventing numbers, interesting and useful numbers, but these are outside the Natural Number System.

 

[boit]

Is it too late to re-name imaginary numbers mirror number? I guess they are mirror image of real numbers. Maybe back of ones mind one should alway read mirror before imagery (imaginary). Funny the trully imaginary (existing in your imagination only in real life) ZERO earned a place in the real side of the divide.

 

[Hurkyl]

Gauss was responsible for the term "complex number". He was unsuccessful at trying to get people to use the term "lateral part" for "imaginary part".

AFAIK, strictly speaking, there is no such thing as an "imaginary number" -- the term only survives as a 'colloquialism'.

 

[bhauta]

I haven't gone through all the posts but it might not be much irrelevant if I digress to elementary mathematics:
If (-1)*(-1)=(-1) then there won't be any need for complex numbers. Then what to do with existing, useful 'complex' maths? Follow the link:
http://plus.maths.org/issue39/reviews/book1/index.html" which eventually takes you to Martinez's book 'negative math'.

 

[Hurkyl]

Well if you are interested in using something that isn't the integers, then you are unlikely to have use for the rationals, reals, or complexes.

Of course, this is just an academic exercise:

1. the ability to create a new number system doesn't change the fact the integers (and rationals, reals, and complexes) are appropriate for certain tasks
2. Until an application is found, it's just for fun

The review is perplexing -- it tries to claim this isn't an academic exercise on the grounds a completely unrelated algebraic system turns out to be useful?


P.S. there's nothing wrong with doing it for fun -- but it sounds like you haven't understood point #1 above

 

[Bob S]

For real world phisical significance of "imaginary" numbers, we should go back to at least Faraday's law:

∫E·dl = -(d/dt)∫B·n dA

because some way was needed to symbolize the 90 degree phase shift (in the frequency domain) between E and B, and in general the effect of the operator d/dt.

Bob S

 

[Matterwave]

A good practical way (I'm going to skip the philosophic or pure mathematical implications for now) of thinking about imaginary number is the analogue to negative numbers.

You don't ever have -5 of something. You can't have -5 oranges, or bananas; however, it's useful for other stuff.

So, say I throw a ball. I can designate my system to be if it's velocity is pointing up, I call that positive, and if it's velocity is pointing down, i call that negative. This would be easier to deal with than always having to say "up" and "down".

In the same way, imaginary numbers help us keep track of stuff.

 

[Tao-Fu]

lots of physical things have a root of a minus number like the gerenuk said waves and quantum mechanics are a good place to find them =]
x

Well, let's not mistake the cart for the horse. Real things don't have any numbers in them. Our descriptions of real things (or more accurately, of our observations of them) employ numbers.

Measurements of real world phenomena use real numbers. There's no physical meaning to a length of "i inches" for example. All physical measurements are simply counting of units, so they use ordinary, real numbers. Imaginary numbers allow us to invoke certain shortcuts in a higher dimensional space and then to cast the result back into a single dimension (for example, contour integration). They are a mechanical convenience that allow us to get results more easily. Of course, complex analysis is a fascinating and interesting branch of mathematics in its own right, it just doesn't directly correspond to real-world stuff. We have to convert a complex number into a real number before it can hope to have any descriptive relevance to actual observations we might make.

 

[Tao-Fu]

This is going to be a much less satisfying, though much more correct, answer than I think you were hoping for.
<snip>

Beautifully answered.

 

[Hurkyl]

Descriptions of real world phenomena use real numbers.

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

<br /> e_1, e_2<br />

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

<br /> X = a e_1 + b e_2<br />

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

<br /> B = e_1e_2 = -e_2e_1<br />

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

<br /> B^2 = e_1e_2e_1e_2 = -e_2e_1e_1e_2 = -1<br />

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.

<br /> e_1X = a e_1e_1 + b e_1e_2 = a + bB<br />

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:

<br /> Xe_1 = a e_1e_1 + b e_2e_1 = a - bB<br />

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

<br /> W = a + b e_2e_3 + c e_1e_3 + d e_1e_2<br />

where

<br /> i = e_2e_3, j = e_1e_3, k = e_1e_2<br />

EG

<br /> ij = e_2e_3e_1e_3 = - e_2e_3e_3e_1 = -e_2e_1 = e_1e_2 = k<br />

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:

<br /> e_1e_1 = 0<br />
And not 1. So:

<br /> e_1X = a e_1e_1 + b e_1e_2 = a\cdot 0 + b \cdot B = b \cdot B<br />

which is not the same as

<br /> e_1X = a e_1e_1 + b e_1e_2 = a + bB<br />

obviously. And even if one takes a different combination of the e_i [\itex] base vectors and a different B [\itex] vector, it still won&amp;#039;t result in an entity consisting of a scalar + bivector. It&amp;#039;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.&lt;br /&gt; &lt;br /&gt; If it&amp;#039;s scalar product, then it should be commutative. And it&amp;#039;s not.&lt;br /&gt; &lt;br /&gt; It&amp;#039;s 03:06 in my time zone, yup I&amp;#039;m a night reader, so I might be missing something. &lt;br /&gt; Maybe a little more abstract kind of multiplication was assumed here? &lt;br /&gt; &lt;br /&gt; Which brings me to a second question, how this would look written in matrices? Because I&amp;#039;m thinking it all looks much more clear that way.&lt;br /&gt; &lt;br /&gt; The line of thinking I&amp;#039;m reffering to:&lt;br /&gt; &lt;br /&gt; &lt;blockquote data-attributes=&quot;&quot; data-quote=&quot;kcdodd&quot; data-source=&quot;post: 2590982&quot; class=&quot;bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch&quot;&gt; &lt;div class=&quot;bbCodeBlock-title&quot;&gt; kcdodd said: &lt;/div&gt; &lt;div class=&quot;bbCodeBlock-content&quot;&gt; &lt;div class=&quot;bbCodeBlock-expandContent js-expandContent &quot;&gt; (...)&lt;br /&gt; For example, in 2-D, you may have the basis:&lt;br /&gt; &lt;br /&gt; &amp;lt;br /&amp;gt; e_1, e_2&amp;lt;br /&amp;gt;&lt;br /&gt; &lt;br /&gt; Any 2-D vector can be represented by a linear combination:&lt;br /&gt; &lt;br /&gt; &amp;lt;br /&amp;gt; X = a e_1 + b e_2&amp;lt;br /&amp;gt;&lt;br /&gt; &lt;br /&gt; The product of the basis gives the highest order multi-vector of the space, a bivector, which is anti-commutative.&lt;br /&gt; &lt;br /&gt; &amp;lt;br /&amp;gt; B = e_1e_2 = -e_2e_1&amp;lt;br /&amp;gt;&lt;br /&gt; &lt;br /&gt; This has the property that the bi-vector times itself is -1, since a (unit) basis times itself is one.&lt;br /&gt; &lt;br /&gt; &amp;lt;br /&amp;gt; B^2 = e_1e_2e_1e_2 = -e_2e_1e_1e_2 = -1&amp;lt;br /&amp;gt;&lt;br /&gt; &lt;br /&gt; 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.&lt;br /&gt; &lt;br /&gt; &amp;lt;br /&amp;gt; e_1X = a e_1e_1 + b e_1e_2 = a + bB&amp;lt;br /&amp;gt;&lt;br /&gt; &lt;br /&gt; All we have done is &amp;quot;prefer&amp;quot;, or project out, the e_1 part. Equate e_1 to the &amp;quot;real&amp;quot; line and e_2 to the &amp;quot;imaginary&amp;quot; line.&lt;br /&gt; &lt;br /&gt; The complex conjugate is just multiplying the vector from the other side:&lt;br /&gt; &lt;br /&gt; &amp;lt;br /&amp;gt; Xe_1 = a e_1e_1 + b e_2e_1 = a - bB&amp;lt;br /&amp;gt;&lt;br /&gt; (...) &lt;/div&gt; &lt;/div&gt; &lt;/blockquote&gt;&lt;br /&gt; 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:

<br /> ab = a\cdot b + a\wedge b<br />

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

<br /> e_i \cdot e_i = 1<br />
<br /> e_i \cdot e_j = 0, i \neq j<br />
<br /> e_i \wedge e_i = 0<br />

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.

<br /> e_i e_i = e_i \cdot e_i + e_i \wedge e_i = 1<br />
<br /> e_i e_j = e_i \cdot e_j + e_i \wedge e_j = e_i \wedge e_j, i \neq j<br />

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.