I’m surprised that after 60 posts, no one has mentioned that time is multiplied by i in spacetime, i.e. Minkowski space.
(I do know how to use Wikipedia, but hope some Doper has a pithy intuitive explanation of this.)
So I gave enough pointer to git 'er done, and not enough to spoil the fun? I’m getting better at this. 
I can’t believe I forget to mention Cauchy Integration:
Using contour integration and the Cauchy integration formula, you can evaluate a lot of integrals over the infinite domain that don’t yield to more traditional methods of integration, and for which the final formulae might not involve complex numbers at all. Getting a simple, closed-form solution to a knotty integral can be very practical. Furthermore, this is a case of the use of imaginaries that doesn’t involve either rotation or the square roots of negative numbers at all*.
*I realize that the contours taken around the poles in the Argand plane are traditionally taken to be circular, and the limit is taken as the radius goes to zero. But they don’t have to be – the result will be the same if I built little squares or other polygons around the poles and then shrank them. It’s just a lot easier to use circles.
I’m a big fan of Laplace transforms (like several other people on this thread, I was an electrical engineering major).
I think the most widely-used method for representing rotations and scalings in 3D space is transformation matrices. Which can also be used for 2D; it just so happens that the set of 2D transformation matrices with the appropriate properties corresponds exactly to the set of complex numbers.
It depends, quaternions are used extensively in computer rendering, because rotating a vector with a quaternion is fewer operations than doing it with a transformation matrix. Though perhaps tellingly, the rotations themselves are almost invariably described with transformation matrices or euler angles and then converted to quaternions before use.
Engineer-guy here. I prefer ‘i’ when described as a coordinate vector, but that is just a repackaging of the x,y plane concept.
Basically, ‘i’ contains the extra dimensional information that is otherwise not being expressed when you are describing a quantity in a dimension that is missing that extra dimension.
Less-babbly …the x number line seems to contain all of the attributes needed to manipulate the quantities of anything. 1 dog + 1 dog = 2 dogs. No problem. But there are phenomena in the world that don’t behave this way, we discover something extra has been hidden from view. As mentioned, in anything with an oscillating time-phase component, it becomes quite common.
So, using dogs, suppose you returned to your arithmetic lab room (hah!) and found that 1 dog + 1 dog = 3 dogs. What happened? Did mathematics fail? Are the very fundamentals of arithmetic unsound?
Well, it turns out you were actually counting your dogs over time, and dogs can reproduce. So if you introduce a time-phased vector to represent pregnancy state of the dog, let’s call it “i”, then you get an “imaginary” number that manifests itself on the arithmetic number line in certain cases.
It isn’t that your number system isn’t correct, it’s just that you’ve been taught a simplified version that has left out “i” because you normally don’t need to worry about that (and other) attribute that the real world doesn’t make obvious.
My experience has been the reverse. The model’s skeleton was specified in quaternion form at the application level and was transformed into transformation matrices when sending geometry into the rendering pipe because that’s what the rendering engine accepted.
Can you graph the many values?
Quaternions are more common in CPU-side computations. Also keep in mind that a Quaternion can be easily implemented on top of a 4D vector, so as long as your shader language has a vec4 it’s trivial to implement the 3-4 functions you really need to work with Quaternions (hell, most of the time all you need is qTransform which is qvqInverse. You also may need interpolation, but that’s usually CPU-side and works exactly the same regardless of whether it’s a quat or vector).
Someone upthread linked to quaternions.
It uses complex space and imaginary numbers to elegantly solve rotation vectors not just in engineering and navigation, but even in computer science and software engineering. This sounds esoteric and only of use to a small fraction of people, but as one who entered the graphics, animation and gaming industry, imaginary space comes into play everyday, so much so that I take it for granted, and it’s become almost second nature.
In CGI and 3D modeling, it comes in very handy if you ever need to define a vector in space around a local or world origin.
To the young or cynical: Despite a 1:1 application between the real numbers and however many fish you have, we build, create, invent, imagine, and live with an abstract mind. Not just mathematics itself, but numbers are an abstract notion; so much more can be done when you begin to think beyond just whole integers or how many fish you have. How about how many fish you don’t have, and how can you find them? That requires some… imagination…
But still has nothing to do with “imaginary” numbers.
Not literally, sure, but like zero, there was a time it wasn’t considered a number, or was seen as a useless abstraction.
“Pshhh…” They’d laugh, “when will I ever have to count how many fish I don’t have?”
Negative, zero and imaginary numbers are having the last laugh.
If they had mouths.
And could laugh.
Sort of hard to have a use of imaginaries that doesn’t involve square roots of negative numbers at all…
[I mean, if it really didn’t make any use of imaginaries qua square roots of negative numbers, then it could just as well be understood as about something other than imaginaries, specifically. And thus probably not provide much ammo for the OP.]
The final result of many Cauchy integrations are formulas that only involve real quantities, with real results. They’re derived using an equation that does itself use the imaginary number I, but it disappears from the final result. So you have something that, in its final form, doesn’t seem to use the square roots of negative numbers, but was achieved using them. Supports the OP perfectly.
Gotcha. I was just misunderstanding what you meant by “doesn’t involve either rotation or the square roots of negative numbers at all”. I see now you were speaking only in terms of the ends and not the means.
This is similar to what I used when I taught. I basically said Complex numbers are 2-dimensional and sometimes that extra dimension is useful.
Whoa … we’re talking about a 14-year-old … i times a tensor isn’t what they need to hear.
Time to introduce the lad or lassy to the concept of “Faith” … don’t worry what it’s for, just know it shows up occasionally and this is how you deal with it. I can’t imagine them seeing it again until long past the delta/epsilon proofs.
“faith” is the worst way to teach mathematics, and there’s absolutely no reason to delay understanding of complex numbers until after delta/epsilon proof, for a child who quite probably finds the former more interesting and readily graspable than the latter.
Whoa … the question is what complex numbers are good for … understanding them is easy, it’s taught to 14-year-olds. The sciences where it’s used is generally taught after Calculus. At least I don’t remember complex numbers turning up in High School physics, chemistry or biology.