Before anyone else calls me on it, I just noticed that my example wasn’t the best. The cube root of -1 has a perfectly obvious, real value.
But there are also two complex ones, so please pretend I was talking about that.
Being called “imaginary” was my NUMBER ONE problem learning imaginary numbers. If these are “imaginary”, why are we wasting time with them? What’s the point? (Of course, I could learn the rules in manipulating i^2=-1 like any other monkey, but I didn’t truly learn their usefulness until much later. Later in life, I thought it was cool that MS Excel had functions for magnitudes of complex numbers without having to use Pythagorean theorem. Well, if MS Excel has them, they must not be so “imaginary” after all.)
It would have been much easier if my math teacher on day 1 forcefully explained, “today we’re going to learn about imaginary numbers but let me emphasize that ‘imaginary’ is a terrible horrible stupid name for them.”
Oddly, I never had this speed bump with “irrational” numbers. I never wondered if 3.1459 had a psychotic neurosis disorder that the rational number like 3.5 (7/2) did not.
I guess the word “imaginary” triggers more confusion than the word “irrational.”
Perhaps because “imaginary” in the context of imaginary numbers was indeed coined to indicate imaginary in precisely the unfortunate sense people take it, while “irrational” in the context of irrational numbers was only ever meant to indicate “not given by a ratio” rather than “not conforming to reason”, the conflation between the two being an unintentional pun. You can easily see “Oh, yeah, irrational just means ‘not a ratio’”, but you can’t easily explain away the term “imaginary” as similarly devoid of derision.
I should have been more precise. Even a cubic with all real roots cannot be solved without the use of complex numbers. There is a theorem to that effect, proved using Galois theory.
The average person couldn’t do long division in those days. Come to think of it, I suspect that is still true. But the point is correct. Even sophisticated mathematicians of that era did not have the so-called Argand plane picture to work with.
A DOUBLE HELIX. Seriously, think about it as far as the number line definition and the abstract, metaphorical angle that the question is predicated on.
No, it wasn’t originally an unintentional pun. The Greek word a-logos, which was translated into Latin as “irrational”, did have negative connotations of senselessness and illogicality (“illogicality” is in fact a cognate of alogos), just as “imaginary” has negative connotations in English.
The fact that some quantities like the square root of 2 obviously could exist as magnitudes (for instance, sqrt-2 is the length of the diagonal of a square of side 1), but could not be articulated with a reasonable (rational) numerical expression designating an integer or a ratio of integers, definitely weirded out the early Greek number theorists. The name alogos applied to such numbers conveyed a certain philosophical disparagement, not just a neutral technical designation.
Huh? What you just said is harder to understand than complex numbers, not easier.
EDIT: That was in reply to Sped. Teacher.
Shucks, and I was hoping I’d said something really profound. 
A) I’m afraid I’m responsible for the quote you attributed to Ruminator
B) In its defense: While there were famously many among the Greeks who were greatly uncomfortable with irrational numbers, and while “ratio” as in fraction is indeed etymologically cognate to “reason” and does indeed derive from “logos” which is indeed cognate to “logical” and so on, I can’t find any evidence that the use of “logos” for fraction (as apparently coined by Euclid) was meant to explicitly carry with it the meaning of “reasonable”/“logical” as well; all I can find is that it indicated something like “the result of a reasoning (i.e., a reckoning, a calculation)”. But I have no expertise in the matter and I’d be interested to see whatever evidence there is that the pun (or even full-on identification of the two senses) was intentional in the coinage.
C) Regardless, to a modern speaker, it remains true that one readily distinguishes “ratio” from “reason” and thus can explain away “irrational” and “rational” in a way which one cannot so easily do with “imaginary” and “real”
D) Nothing to do with you, but to Sped. Teacher: Huh?
:DI think pete83 deserves a metaphor, so lets give a metaphor instead of endless description. Plots on a line, when flipped, etc., 3D, please! Visualize, as you would metaphors that explain aspects of shapes we dont fully understand. Think about planets and their similarities to atoms. Come on, I’m not making this stuff up. Does he not want a metaphor?
Not everything abstract can be restated as a 3-dimensional physical.
What is infinity? What is the 3-D representation of infinity? How would you draw such a thing on a piece of paper?
Maybe a 3D picture involving imaginary numbers would be quaternion. However, that makes things harder, not easier.
Yes, I agree that Euclid’s sense of “logos” to mean “fraction” is a strictly technical term divorced from psychic connotations of reasonableness or logicality. But the use of “alogos” to characterize irrational numbers predates Euclid. As far as we can tell, it seems to have been introduced by the Pythagoreans, for whom “alogos” definitely did have a somewhat pejorative sense. If you want to see some better-informed discussion of the issue than I can provide, try here.
And apologies to you and Ruminator for my misattribution—oops!
D) Yeah, me too.
Oh, “alogos” predates “logos”? I had no idea. That’s really quite interesting. (And it makes me now suspect that perhaps Euclid was aware of the pun he was creating(?). I mean, if “alogos” was already around and had been used as a description of precisely the opposite of the sort of thing he was about to make “logos” mean, how could he not be? And yet, I see no remarks that suggest he was. I… need to read more.)
Well, remember that we don’t know much about Euclid, and we know even less about most of the pre-Euclidean Hellenistic mathematicians whose work was swept into historical oblivion by the fantastic popularity of Euclid’s work. IMO it’s likely that the word “logos” for fraction was kicking around for a while before Euclid himself—long enough to gain familiarity as a narrow technical term with no philosophical baggage about rationality—but Euclid’s work happens to be the first surviving example of its use.
The north pole of a Riemann Sphere?
Sure. And also, once you’ve defined i such that i[sup]2[/sup] = -1, what about some number that, when squared, yields i? Intuitively we would think “Oh no, now we need some even more imaginary numbers, and so on ad infinitum”, and it ain’t so. It turns out that the square root of an imaginary number is itself partly imaginary and partly real, and so we only needed to add one level of weirdness, not an infinitely recursive stack of it.
I like when the metaphors are more complicated than the referent. Here’s my answer: imaginary numbers are like your crazy uncle…there’s internal consistency to the things he’s mumbling, but it may not make much sense to outsiders.
Since nobody has mentioned the term yet, I’ll jump in: To say that adding i gives solutions to all these other equations is to say that the field of complex numbers is algebraically closed. Precisely, every polynomial with complex coefficients has a complex root. This is not true of the real numbers — the polynomial x[sup]2[/sup] + 1 has no real root. The amazing thing, as several people have pointed out, is that creating a root for this one polynomial creates roots for all other polynomials. This result is often called the Fundamental Theorem of Algebra. One interesting thing, tangential to the main discussion here, is that it’s very difficult to prove this using only algebra. My favorite proof, obviously, involves some topology. Another uses complex analysis. (That’s a technical term, not a descriptor. Here’s another somewhat unfortunate term: “Complex” here means something built out of simpler parts, not “complicated.”)
Eh, it’s impressive enough that introducing complex numbers completely closes the system with respect to polynomials, but I think it’s even more impressive that it also closes it with respect to things like inverse trig functions and logarithms. I imagine that this stems from the fact that trig functions and the like can be expressed in terms of infinite sums, but that still adds another layer of nontriviality.
More specifically, “complex” in a mathematical context refers to numbers with both a real and imaginary part.
Just for fun, by the way: Several others have mentioned that complex numbers suffice to find the square root of i, but I think it’s instructive to do the demonstration. Let x = 1/sqrt(2) + 1/sqrt(2)i . Now let’s take xx. We can do this the same way we’d multiply anything of the form (a+b)*(c+d):
xx = (1/sqrt(2) + 1/sqrt(2)i) * (1/sqrt(2) + 1/sqrt(2)i) = 1/2 + 1/2i + i1/2 + 1/2ii
Replacing the ii in the last term by -1, and combining the two middle terms, we have
xx = 1/2 + i - 1/2 = i.
So xx = i, which means that x is the square root of i.