# 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)
[tex]\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