Implications of Imaginary Numbers?

1. Nov 5, 2009

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.

2. Nov 5, 2009

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.

3. Nov 5, 2009

matt_crouch

imaginary numbers just allow us to solve equations that have root of a minus number basically. we dont know the number so we use this symbol "i" which allows us to get a real number.
=]

4. Nov 5, 2009

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 wanna 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] where i is the imaginary unit and $\psi$ the complex wavefunction. In fluid dynamics and electrostatics complex numbers can be used through "conformal mapping" 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. 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's law and calculate the effect of an applied sinusoidal voltage. 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. I hope someone can give good references for all the topics... The "maths skeptics" of yours can't be very clever. It's a big mistake to say something is useless just because you haven't looked at the applications. 5. Nov 5, 2009 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.) Last edited: Nov 5, 2009 6. Nov 5, 2009 Gerenuk You must be a mathematician I'd like to see you calculate waves and quantum mechanics with the split notation. 7. Nov 5, 2009 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 8. Nov 5, 2009 Tac-Tics The situation is worth than that. I'm a programmer. 9. Nov 5, 2009 DaveC426913 Nitpick: 'Imaginary numbers' and 'complex numbers' are not synonymous. A complex number is composed of a real number and an imaginary number: (a + bi). 10. Nov 5, 2009 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/ ) Last edited: Nov 5, 2009 11. Nov 6, 2009 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. 12. Nov 6, 2009 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. 13. Nov 6, 2009 Gerenuk 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. 14. Nov 6, 2009 Hurkyl Staff Emeritus And, for the record, the real numbers, rational numbers, and integers all enjoy the same state of affairs. 15. Nov 6, 2009 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. 16. Nov 6, 2009 Gerenuk 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". 17. Nov 6, 2009 A.T. The dollar notes are material reality. But not the numbers we made up to quantify their value. "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. 18. Nov 6, 2009 Pengwuino 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. 19. Nov 6, 2009 Gerenuk 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 20. Nov 6, 2009 GoldPheonix 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 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. Last edited: Nov 6, 2009 21. Nov 6, 2009 Pengwuino Well, I just hope to god no one builds my car's air bag to deploy with a pressure of 300 + 500i PSI. 22. Nov 7, 2009 Redbelly98 Staff Emeritus 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. Last edited: Nov 7, 2009 23. Feb 14, 2010 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. 24. Feb 14, 2010 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. [tex] 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.

25. Feb 14, 2010

Hurkyl

Staff Emeritus
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.