Great. Now you’re going to have them asking, “So why i for current?” and so forth.
Let’s just say, even though there’s no evidence that EM fields cause brain dysfunction, we’ve seen some pretty strange behavior from electrical engineers and researchers over the year. Tesla was supposedly a pretty normal guy before he started running 70,000v charges through his forebrain. 
Stranger
The best explanation I could find was this.
I’m finishing up my EE degree and just went through an entire “complex analysis” course, which was two quarters of nothing but relearning all of algebra and calculus extended to complex space.
It was fun 
We had this spooky ex-MIT genius who was on another plane in his lectures, but the class was invaluable in helping straighten out this “what is i?” thing.
Mathematically, i, e, euler’s equation, and their ilk are all a natural extension when imaginary numbers are assumed as a result of having an additional vector space on our typically math system.
So basically, your one-dimensional real number line has an additional orthoganal vector space, the R axis and C axis that people have talked about. But it is important to treat it not as an added math crutch that gets slapped on, but an inherent part of the number space.
Now, if you describe your space in terms of 2x2 vectors, with all real numbers being a multiple of the identity matrix, and all complex numbers being a multiple of of the imaginary unit (0 -1, 1 0) then, with a spooky kind of magic, all of the math (roots, phase angles, euler, all kinds of transforms) just elegantly falls into place and makes sense. I had many a class where I would just come home and stare into a lava lamp for hours…“dude…it all makes sense…”
Oddly, Wikipedia gives only a passing mention of the concept (see the matrix representation portion), leading with the phrase “While usually not useful…”, I found it to be the only way that I have been able to finally see the “Man Behind the Curtain” when it comes to complex numbers.
Yes, sorry, I’m mixing metaphors. Actually, by complex number, I do mean those numbers with a real part and an imaginary part, so I think we’re all on the same page there. Where I think I’ve confused some people is with the term “familiar numbers,” by which I mean the reals.
I learn more and more of the terminology by reading others’ posts, actually. In other words, to say that something “has a solution in C but not in R”…not having the most solid math background, I have to work to keep up. 
Right, which is why I asked: when one considers the system before we had the complex numbers, would it be true to say that we were simply unaware of the second, imaginary dimension? In other words, when you talk about powers and roots, R is incomplete, or not closed. So part of my question was, why, exactly, is it incomplete? Was the system closed until the inclusion of negative numbers, and when we invented the negatives, we suddenly had an unclosed system that forced us to invent the “imaginary” component? When you say it’s important to consider it an inherent part of the number space, that, to me, suggests that imaginary numbers were there all along–it just took us time to recognize them.
See…that’s where I’m having problem: I don’t see why the natural numbers wouldn’t be natural concepts. I can see how i is an intellectual construct, but not the number 2. I think I understand your point about humans’ need to classify objects, but I don’t understand why that means the counting system wasn’t there in the first place, before we had numbers and symbols. Take the classification component out of it for a moment: let’s just say we’re talking about apples. Nine apples is three times as many apples as three apples, not because we need it to be, but because it is. It was always that way, before we had the labels “three” or “nine.”
Maybe this is a poor way to make this point, but consider Sagan’s Contact: the aliens choose the sequence of primes to identify their signal as intelligent. They didn’t do that on the off chance that we invented an identical system–they did it because it’s there, in nature. It’s absolute. It’s an inescapable truth, no matter what the labels or rules you use are.
Obviously, that’s fiction…but I think the point is valid enough: there is a number system in nature that’s observable, that’s inherent. We formalized it, but we didn’t invent it. At some point, we started inventing numbers and systems and we’ve never stopped, because we’re way past counting sheep, and we need those systems. Obviously, we’re not just pulling them out of our asses–we can demonstrate how they’re real and useful and perhaps necessary for our various purposes. We can also demonstrate how they can be used to describe nature. But nature didn’t invent them–we did.
The whole idea I’m interested in here is the conceptual leap one must take to work with complex numbers. It’s analogous to the leap one takes to work with fractions or negatives or even zero. As a student, I was encouraged just to make the leap without thinking about it too much, and I’ve been able to do that, but that doesn’t mean I’m not interested in going back and looking at it.
As mentioned, it takes a certain leap to visualize one-half of a sheep, but you can do it. You can imagine, in your head, what half a sheep looks like, as unpleasant as it might be. It takes a greater leap to imagine a negative quantity in your head, but you can do it: I could show you a group of two sheep and another group of five and ask you to count how many less sheep are in the first group.
Thanks everybody for your insight and patience; this is very helpful.
If the natural numbers exist–and by that, I mean anything that has all the properties that we would normally expect of what we call the natural numbers–then the complex numbers exist too.
This question is as old as Plato: Do the forms of things exist separate from their real-world examples, or are they simply a convenient mental picture? I’m glibly paraphrasing here, but one should note that if you come down on the side that the natural numbers are real entities independent of human thought, the same reasoning could be applied to complex numbers.
I suspect the dilemma you present lies not with the ‘naturality’ of, say, the number 2 itself, but with the ‘naturality’ of ideas associated with the use of these numbers, specifically countability and quantification. It is meaningless, for example, to say you have i sheep or a board that is i meters long, and therefore the number i does not exist in and of itself in the same way that the number 2 does.
But this shows a bias toward the concepts of countability and quantification as being more natural; that’s not bad, but this bias must be justified. I suspect the way many would justify this bias is by use of common real-world examples like the ones I used for i above. The examples used to justify this bias are often rather parochial–sheep, grain, the kinds of things that aren’t beyond your fingertips. If a more esoteric concept like “electrical impedance” is considered, then complex numbers naturally emerge to fill in for a notion of quantity. Furthermore, there are certainly items in the universe which do not yield to these ideas (i.e. they are not countable or quantifiable); hardly a criteria for raising them to a higher natural importance.
This discussion has turned somewhat philosophical, and for that I apologize.
Because c was already used for the speed of light!
This is the song that never ends / it just goes on and on my friends / some people started singing it not knowing what it was, and they’ll continue singing it because / this is the song that never ends …
Sequent writes:
> See…that’s where I’m having problem: I don’t see why the natural numbers
> wouldn’t be natural concepts. I can see how i is an intellectual construct, but
> not the number 2. I think I understand your point about humans’ need to
> classify objects, but I don’t understand why that means the counting system
> wasn’t there in the first place, before we had numbers and symbols. Take the
> classification component out of it for a moment: let’s just say we’re talking
> about apples. Nine apples is three times as many apples as three apples, not
> because we need it to be, but because it is. It was always that way, before we
> had the labels “three” or “nine.”
And by the same argument, i times i equals -1, not because we need it to be, but because it is. It always was, before we had the labels “i” or “-1”. See, it’s not just the number 2 that’s a mental construct, it’s the entire idea of numbers that’s a mental construct. Going back to my pile story, all the examples I gave are more “natural” sorts of mental constructs than numbers are, although arguably they aren’t completely natural either. The heights of piles are easier to notice than the number of objects in the piles, especially if it’s more than a few dozen. Similarly, the color or shape or hardness or total weight or size of the items in the piles are easier to notice than the number of objects.
Suppose you tell someone (who is also new to the world) that two piles, each with exactly 523 objects in them, are similar in some way. “What do you mean?” they say. “The objects in this pile are large, blue, squishy objects that are long and thin and the pile is tall and weighs a lot, while the objects in the other pile are small, green, hard objects that are sort of bumpy spheres and the pile is short and doesn’t weigh much. In what way are they the same?” So you show this person that the piles are similar in the only way you know of for someone who isn’t familar with the idea of number. You take one item out of each pile at a time and place those pairs of items, each with one item from one pile and one item from the other pile next to each other. When you’re done with this, you say, “So the two piles are similar in number.” And the other person says, “What is this arcane magical procedure you have of pulling items from two piles?” This is, after all, what number means, the ability to match up items from different piles, and it’s a lot less natural than some sorts of properties.
> Maybe this is a poor way to make this point, but consider Sagan’s Contact: the
> aliens choose the sequence of primes to identify their signal as intelligent. They
> didn’t do that on the off chance that we invented an identical system–they did
> it because it’s there, in nature. It’s absolute. It’s an inescapable truth, no
> matter what the labels or rules you use are.
Yes, it is a poor way to make your point, because, as you say later, that’s fiction. We have no idea if aliens even exist, and we sure don’t have any idea whether they use number. Even if it turns out that to go beyond some point in the intellectual and cultural evolution of a species it’s necessary to create the notion of number, that doesn’t show that the idea of number is “natural”. Perhaps it’s necessary to create the notion of written language to go beyond some point in intellectual and cultural evolution of a civilization, but that doesn’t make written language “natural,” it just makes it a necessary part of cultural evolution.
> Obviously, that’s fiction…but I think the point is valid enough: there is a number
> system in nature that’s observable, that’s inherent. We formalized it, but we
> didn’t invent it. At some point, we started inventing numbers and systems and
> we’ve never stopped, because we’re way past counting sheep, and we need
> those systems. Obviously, we’re not just pulling them out of our asses–we can
> demonstrate how they’re real and useful and perhaps necessary for our various
> purposes. We can also demonstrate how they can be used to describe nature.
> But nature didn’t invent them–we did.
You’re arguing in circles here. The idea of numbers does not exist in nature. Things exist in nature which we can then apply the notion of numbers to, but then things exist in nature which we can then apply the notion of electrical current to, but that doesn’t mean that the number i exists in nature. Counting is a mental construct and it’s necessary to create it before you can speak of the numbers of items.
I think this really belongs in GD.
2-3. There are systems of numbers that cannot be identified with a subset of the complex numbers. I don’t really know anything about them, but you might want to look into hyppereals and p-adic numbers.
According to the usual set-theoretic notions, there are as many real numbers as complex numbers, but there are more reals than rationals. Specifically, there exist a bijection between complex numbers and real numbers, but while there is an injection from rationals to reals, there is no injection in the other direction.
Can’t you say the same thing about going from integers to rationals? What are rationals but pairs of integers that we use to express ratios. When you compute with rationals, you are really computing with integers, just as when you compute with complex numbers, you are really computing with pairs of reals. (Actually, no one computes with reals. If you think about it, floating-point numbers are not really reals. And humans don’t compute with reals either; we just have special symbols for the irrational numbers we use often, like pi, e, and sqrt(2).)
I should have read the rest of this thread first. I see that most of what I posted was already mentioned. I still think this belongs in GD, because it is more about the philosophy of mathematics than mathematics.
I think that natural numbers seem natural to us because of the world we live in. Imagine a world with very different physics than ours: a world without solids or discrete objects, only fluids that flow and merge. Beings in such a world might never invent natural numbers, although they might invent reals.
A nitpick, but for someone new to the world, if we were to attempt to demonstrate what counting is, I don’t think we’d start with two piles of different objects, do you? We’d use piles of the same objects, precisely because of this. But I get your point. Even with the same objects, who knows how someone new to the world, new to the very idea of an object would notice first?
I hadn’t thought about it that way, but it’s a good point. I suppose there are numerous mental constructs which have to be in place before the notion of counting or measuring would even make sense.
Yeah, I didn’t mean to turn this into a philosophical debate…(even though I think it’s a fascinating one)
I think that’s the point, like ultrafilter said, if one is for real, then so must be the other.
That’s a very good point, too. It turns the whole Contact analogy on its ear.
I suppose there is no real-world analog for i, and the sooner you make the conceptual leap the better, for the same reason you don’t encourage children to count using their fingers. It’s much easier for me to think of a complex number in terms of two-dimensional numbers vs. one-dimensional numbers. We measure, for lack of a better world, in two dimensions instead of one. And apples, sheep, and puppies are all one-dimensional objects in when we’re counting them. We only need one number to tell us how many apples there are.
Agreed that the floating-point representations computers use, or that we often write down on paper, should certainly not be confused with the set of real numbers. But I have to disagree that we never “really” compute with reals.
The square-root of 2 is well defined and ready for any computation you can think of, even though no positional notation system, such as standard decimal, can ever fully express it. For example, through a little on-paper computation I can tell you that sqrt(2) x sqrt(3) = sqrt(6), all without surrendering to any kind of approximation. I can also compute that cube-root(20) x cube-root(50) = cube-root(1000) = 10, which is a case where computation performed on irrational “unexpressible” numbers can yield a simple, easy-to-express integer. If this doesn’t count as computation, what does?
Perhaps you’d want to argue that sqrt(2), written like that, doesn’t quite give us a real number we can work with, because it leaves us with a “unfinished” computation. We have the 2, which is straightforward enough, and we have the square root function, but we don’t actually have in hand the value from applying that function on 2. Expanding the square-root of 2 in decimal of course only gives us an approximate value, to whatever precision we have enough patience to generate. But you could say the exact same thing about the number 1/7, whose most concise and most useful written form is, well, 1/7. We overlook that unfinished division operation, without qualms.
My point is: our number notation systems all depend on some amount of computation, with some small set of primitive values and with operators to build new values from old ones. (We obviously can’t design a unique symbol for every possible number.) In the decimal system, only the single digits 0 through 9 have a primitive meaning. Every multi-digit integer represents an implicit chain of computation, involving exponentiation, multiplication, and addition. And a rational number, expressed as a ratio of two integers, has all that plus a final division. The notation sqrt(2) isn’t really any different from this, not in kind anyway.
And of course symbols like pi and e can be thought of as short-hand for particular infinite sequences of computation. (There are probably other ways to think about them too.) That could be troubling, but arithmetic involving pi or e is still real arithmetic. Sometimes that infinite-ness goes away anyway, as it does when you compute e[sup]i pi[/sup] - 1.
I suppose this is a matter of perspective. Seven is only prime because we’ve defined our number system in a certain way, and we’ve defined “prime” in a certain way. But I suppose you could say that that’s BS, that we defined our number system in a way that is inherrently correct. But it’s only inherrently correct for what we want to do with it. Defining the natural numbers and arithmetic operations in the way that we did is useful for doing things like figuring out whether you have more apples than your neighbor, or whether combining your apples with your neighbor’s will give you more than the guy down the street. That is to say, we can identify a correspondence between each set of objects in the real world and a natural number (which we call “the number of elements in that set”) in such a way that performing operations on the numbers gives us correct information about the sets.
However, there are other operations than apple counting for which a different definition of numbers might be useful. Forget imaginary and complex numbers, just think about pi. Pi is worthless for counting apples, because you can’t have pi apples. How could you? There’s only so precisely you can cut an apple, and to have 0.14159265358979… of an apple (the fractional part of pi), you’d be splitting individual atoms and whatnot. (In practice, it’d be darn near impossible to even get exactly half an apple, but I suppose it could be done in theory.) And yet pi certainly corresponds to something physical – the ratio of the circumference of a circle to it’s diameter. (OK, so you can say it’s never possible to have a truly perfect circle, but the point is pi is useful for real-world calculations.)
So pi clearly relates to real physical quantities. The question remains, is it a number? Not if you define numbers as “things useful for counting apples”. However, I doubt many mathematicians would define the word “number” that way. A number is an element of a set of objects for which certain operations exist and certain rules are satisfied. Those rules are such that natural numbers are numbers, and the real numbers are numbers, and the complex numbers are numbers. Likewise, each of those is useful for some physical calculations. But if you insist on treating numbers as things which can fill in the blank in the sentence “I have ___ apples”, then no, 2+i isn’t a number, nor is pi, nor is - 3.
The bottom line: We didn’t discover that the set of numbers appropriate for counting apples also had a property called an immaginary part. We invented a new kind of number that had an immaginary part, and which is appropriate for certain physical calculations (and thus in some way corresponds to the real world, albeit not in the same way as the natural numbers.) However, I’d say that’s no different than what we did with the naturals: invented something that is useful for certain physical calculations, and thus in some way corresponds to the real world (or at least to our observations of it.)
The way I like to think of it (loose analogy warning to preempt pedantry) is in terms of rotation.
Conceptually, multiplying a number by i is taking a 2-d vector and rotating ccw by 90 degrees on the Real/Imaginary plane. Multiplying again rotates another 90 degrees. So multiplying a real number by i^2 is the same as multiplying by -1. Therefore, i^2 equals -1. But until a multiple of i collapses to a real number, it isn’t apparent if you are looking purely in terms of the real number coordinate system (although it exists in the not-so-apparent complex coordinate system).
Think of it (getting to my analogy here) like looking at a louvred window. From most angles, the louvres block your view of the outside world, and based on that you might intuitively conclude that there was no outside world beyond the louvres if those were the only angles you had ever viewed from. However from certain angles, the louvres effectively dissapear and the outside world is shown to in fact exist.
Actually, when I said that “we don’t really compute with reals”, I meant specifically that we cannot assign a name to every real number. It is impossible since we can regard names as strings of characters from some alphabet, and the set of strings is countable while the reals are not. This implies that there are certain real numbers that can never arise in our computations, simply because we have no way to express them. We use special notations for reals we find useful, such as e, pi, and sqrt(2). However, we are still restricted to computing with a countable subset of the reals. The situation differs with rationals, because we have a way of naming every rational. Each rational can potentially be the result of a computation we perform.
Assuming, of course, the alphabet is finite/countable.