Today was the first day of my summer school math class, and the teacher gave a bit of a description of imaginary numbers, such as the square root of -4. But he didn’t say what you would use imaginary numbers for. Well, he did, but all he said was that engineering students would use them.

And of course physics would be all but hopelessly intractable without the darned things. Basically, any time something rotates or oscillates, imaginary numbers provide a clean and compact representation.

Cellphone circuit design. Most audio & video equipment. Radio transmitters. If the electronics is not all digital, then it was probably made using imaginary number math. Remember Ohm’s law? When dealing with capacitors and coils, Ohm’s law expands into the imaginary number realm.

ultrafilter and g8rguy: they’re a neat trick for 2-D work, and that comes up often enough for them to be useful. But, as I’m sure you know, they aren’t extensible to higher dimensions. Maxwell’s system is, and it’s just an accident of history that we still use complex numbers for 2-D work.

You often encounter situations where an equation is non-analytic, i.e. non-differentiable in complex form, and so you’re forced to re-write as two real equations and proceed from there.

But for 4-D you can go to quaternions, with i[sup]2[/sup] = j[sup]2[/sup] = k[sup]2[/sup] = -1. And I think there is a 16-D version too. “Hyperquaternions” maybe?

(Nowadays quaternions are usually encountered as the Pauli spin matrices, and few people even use the term “quaternion.” The 16-D space is the space of the γ matrices of Dirac.)

Demostylus: quaternions can also be used to represent rotations of three-dimensional space, so quaternions have applications in computer graphics.

Also, complex numbers come up all the time in three-dimensional hyperbolic geometry. But sadly not everyone would consider that to be an “application” as such…

For 4-D you use what I called “Maxwell’s system”. It isn’t entirely due to Maxwell. Gauss, Green, Stokes, et al contributed. I guess it was Maxwell that finally said: “fuck it, that’s the way to go.”

The term “quaternion”, as I understand it, was originally coined for the 3-D analogue of a complex number, but no satisfactory artifact was ever found.

They are 16 matrices whose square is equal to the unit matrix. If you multiply some of them by i so that their there square is -1, you get a 16-d space that is analogous to the quaternions, with similar commutation rules. (Similarly, you have to multiply the Pauli spin matrices by i to get quaternions.)

Imaginary numbers is a misnomer…obviously, they have real applications. You might say it’s more like the rules of math we’ve invented to model the physical world around us has a slight flaw!

Jinx has the right of it. I would state it even more firmly.

Imaginary numbers are, themselves, not useful for anything.

In none of the applications mentioned so far is there any real significance to the fact that one of the components of these numbers happens to be the square root of -1. The “i” could be replaced with any kind of marker at all, and the marker doesn’t have to have an underlying meaning. It’s just a notational trick for denoting an axis in a two-dimensional system.

Consider a 2-dimensional cartesion coordinate system. We usually denote points (or vectors, which amount to the same thing) in that system as (x,y) tuples: e.g., (2.5, 7). But that notation does not lead to “natural” or “expressive” manipulation in many contexts. Another way to write the same thing is to introduce markers for the unit-length vectors along the x and y axes. These are usually indicated by writing ^ over the x and y, but I’ll just write them as x^ and y^ in this posting. So (2.5,7) can be written as 2.5x^ + 7y^. In that form, a lot of “ordinary” mathematical manipulation becomes natural. Addition of vectors, multiplication, etc. all work out the way you would expect if you simply treat x^ and y^ as arbitrary symbols that can’t be replaced or substituted for.

Now, take any mathematical formulation in terms of x^ and y^. Replace x^ by 1 and y^ by i. So (2.5,7) becomes 2.5x^ + 7y^ becomes 2.5+7i. Whatever mathematical steps you apply to the x^ y^ notation will work equally well with the complex number notation and the reverse is also true. The “i” is simply filling the role of an arbitrary symbol that can’t be replaced or substituted for. But sometimes the complex number approach leads to a more compact representation.

Nevertheless, there’s a subtle danger in using complex number notation in some circumstances. Too many people believe that the imaginary component either means something in the systems being described is imaginary/mysterious and maybe not really valid. Others may forget that the “i” direction often denotes something very “real”. For example, when complex numbers are used in graphics, thee “i” is just the “y” direction. When complex numbers are used to describe electromagnetic waves, the “real” part of the complex number is the electrical field and the imaginary part is the magnetic field.

Imaginary and complex numbers are a convenient notational device. But the way they get introduced in many math/science courses often seems to me to reflect a mistake in emphasis. A physical system does not become any more or less “real” simply because we choose to describe it using “imaginary” numbers. I suspect that many people would understand the underlying math of many systems better if it were presented in vector (x^, y^) form.

When I was introduced to imaginary numbers in high school they gave no explanation as to their use or purpose. It was just, “This is how to handle it and do it because we told you to.”

My mind utterly balked at this point. To me they were admitting they were making crap up to do things that were impossible. I never learned well by simple memorization…I needed to understand what was happening, the why of it as it were. Not that I was ever great at math and considered advanced mathematics important to what I thought I’d like to do when I grew-up but I can definitively say that imaginary numbers finished me on math for good.

I would have hoped by now schools would be better at teaching and explaining concepts but given the OP that sadly doesn’t seem to be the case.