I recall (vaguely) in college taking a class called Complex Variables, which involved basically calculus using complex numbers (with both a real and imaginary part). One of the problems we solved had to do with the transfer of heat. We were able to describe how heat travelled through a substance over time when one side was adjacent to a heat source.
Fractals, many of them are plotting with the imaginary part being used for the vertical axis.
Fractals have all sorts of real life applications, the least (greatest?) of which is that they look awfully cool. 
I use imaginary numbers to balance my checkbook. My accountant has recommended dropping that practice…
Of course. But that’s true of all numbers, including your friendly neighborhood positive integers. Anything that you need imaginary numbers for, can be expressed instead in terms of functions of real numbers; anything you need real numbers for can be expressed instead in terms of (possibly infinite) sequences of (functions of) rational numbers; anything you need rational numbers for can be expressed instead as ratios of integers; anything you need negative integers for can be expressed instead in terms of positive integers and addition/subtraction; and anything you need positive integers for can be expressed in terms of generalized set theory and logic. So really, there aren’t any such things as numbers-they’re all abstractions of one sort or another. Naturally-who’s ever seen the number one?
We tend to think of the positive integers as more ‘real’ than imaginary numbers because they’re more directly useful and familiar. But the fact that you can’t count something with a number doesn’t make it useless. Heck, forget the square root of -1; just look at -1. My six-year-old son could tell you that there’s no such number. It’s seven minus six, right? But you can’t take away a bigger number from a smaller number! (Then he’d say ‘Duh!’ I’m trying to fix that little habit of his …) Maybe so-but try doing accounting without it. Pain in the neck, if you ask me.
I suppose it’s possible to do electrical engineering without using i. But I strongly doubt that the equations used to describe AC circuits could have been developed without the use of that handy little number. And that’s as ‘real’ as I need my numbers to be, thank you!
(If you can find it, check out the late Isaac Asimov’s anecdote on this subject in On Numbers. As is typical for him, he delivers a rhetorically entertaining but logically empty ‘zinger’. But entertainment is always worthwhile, I think …)
OK-everyone knows I meant to type ‘six minus seven’. Right?
:smack: Preview-preview-preview …
Ok, SCSimmons but consider a spatial analogy. One could plot the position of a car along a road as a function of its past location (zero, say) and how far it has gone forward (+) or in reverse (-).
Which is to say that although I cannot imagine a negative amount of apples, I can certainly imagine an object moving along a demarcated line. Or grid, for that matter.
OTOH, as I am not a physicist or electrical engineer, the usefulness of imaginary numbers still escapes me. Square root of negative one? Huh?
Alright, here’s a spatial analogy. A generalized complex number can identify a point in a plane-something which requires two real numbers to do. (The x axis is the real numbers, the y axis the imaginaries.) This is the ultimate reason behind the use of complex numbers in engineering and most of physics, actually. Any simple harmonic motion can be expressed as the real portion of rotational motion in the complex plane. The equation of a circular path in rectangular coordinates in this complex plane (r cos theta + i * r sin theta) become both easier to express and easier to deal with in calculus when you convert it to polar coordinates (r * e ^ [i * theta]). It’s almost impossible to explain how much easier life becomes with this model … Integrating combinations of sines and cosines is, frankly, a pain in the neck-not to mention how hairy-looking the equations become. Integrating exponentials is trivial-the integral of e^x is e^x plus a constant. And multiplying pairs of binomials with sin and cos components makes for very complicated equations; multiplying exponentials involves adding the exponents. Representing simple harmonic motion (or an AC current, for another example) as a rotation in a complex plane makes the math easier to do, and the situation easier to visualize.
Well, real life applications, yes… but they’re much more of mathematical abstractions than anything truly practical. Don’t get me wrong, I love fractals, I just don’t necessarily see them as being particularly useful in the real-world in the same sense that the complex numbers themselves are useful. There are plenty of real-world situations where complex numbers are useful: in doing conformal mappings, calculating residues (for duing some otherwise nasty integrals), or proofs of basic theorems about harmonic, elliptical, and other fascinating functions. Certain chaos theories and attempts at probing randomness have what I would term oblique references to fractals, but as far as I know they really are just succint geometrical ways of illustrating one’s point rather than particularly useful mappings. Of course, teaching a computer to do a fractal is a useful exercize in programming, but I don’t know that it has much meaning beyond that.
OTOH, if you really want to see where some complex sequence converges (like Newton’s method on the complex plane), fractals are pretty useful. I suppose in some numerical regimes the forms may become important. However, their actual shapes are pretty much impractical in that they “wow” us and don’t exactly teach us anything practical about what they’re picturing. This is just my subjective-self talking though, so take it with a grain of salt.
Well, real life applications, yes… but they’re much more of mathematical abstractions than anything truly practical. Don’t get me wrong, I love fractals, I just don’t necessarily see them as being particularly useful in the real-world in the same sense that the complex numbers themselves are useful. There are plenty of real-world situations where complex numbers are useful: in doing conformal mappings, calculating residues (for duing some otherwise nasty integrals), or proofs of basic theorems about harmonic, elliptical, and other fascinating functions. Certain chaos theories and attempts at probing randomness have what I would term oblique references to fractals, but as far as I know they really are just succint geometrical ways of illustrating one’s point rather than particularly useful mappings. Of course, teaching a computer to do a fractal is a useful exercize in programming, but I don’t know that it has much meaning beyond that.
OTOH, if you really want to see where some complex sequence converges (like Newton’s method on the complex plane), fractals are pretty useful. I suppose in some numerical regimes the forms may become important. However, their actual shapes are pretty much impractical in that they “wow” us and don’t exactly teach us anything practical about what they’re picturing. This is just my subjective-self talking though, so take it with a grain of salt.