Does anyone know a metaphor to use for imaginary numbers so I can understand them a little better?
Here’s the simplest, which other posters can expand further on:
First of all, we need an analogy for negative numbers. If you’re looking at a number line, then multiplying something by -1 is equivalent to flipping it around to the other side of the line. If you make two 180 degree turns, you’re back to where you started, which is why -1 squared equals 1.
But now we need something that, if you multiply it by itself, gives you -1. In our analogy, then, we want something that, if you do it twice, has the result of turning you around 180 degrees. Well, then, how about a 90 degree turn? If I turn 90 degrees left, and then turn 90 degrees left again, then I’ve turned around 180 degrees. So we have to turn our number line into a number plane, because now we’ve got two axes.
But wait, you might say, there are two different ways you can turn 90 degrees. You could turn left or right. But that’s OK, because i and -i both square to -1, so we can just say that turning in one direction is multiplying by i, and turning in the other direction is multiplying by -i.
This analogy is actually remarkably robust, and can be extended to explain just about everything about imaginary numbers and complex numbers (combinations of imaginary and real).
The above is pretty much exactly how electrical engineers use complex (real+imaginary) numbers.
j amounts to a 90 degree phase shift, and all the math works from there. Oh, yeah, EE’s use j =sqrt(-1) instead of i, because i is used for current.
Actually, the analogy I heard was the unit circle - imagine a circle of diameter 1 on the real-imaginary plane as described above. Turning 90deg is going imagineary, 180 deg is minus 1. Then -i is 270 degrees. Twice 20 is 180(+360)
To expand further on what Chronos said: Behind his analogy is the fact that we can (and mathematicians do) think of imaginary and complex numbers not only as operating on the plane, but as points in the plane itself. Traditionally, the real numbers are thought of as living on the horizontal axis and the imaginary numbers on the vertical axis, with i one unit up and -i one unit down from 0, which lives at the origin. The complex number a + bi is then the point with coordinates (a,b).
Multiplying a + bi times i gives -b + ai, and (-b,a) is the point you get by rotating (a,b) 90 degrees counterclockwise. If you multiply instead by -i, you get b - ai, and (b,-a) is the point you get by rotating 90 degrees clockwise.
Regarding the unit circle mentioned by md2000, those are the points in the plane at a distance of 1 from the origin, including 1, i, -1, and -i, but, of course, many others as well. If you have a complex number z on the unit circle, you can measure the angle from the positive real axis to z; multiplying by z rotates the whole plane by that angle.
If you multiply by a complex number not on the unit circle, you get not only a rotation of the plane but an expansion (if the number is outside the circle) or contraction (if the number is inside).
This is all more than an analogy. To many (most?) mathematicians, complex numbers are the points in a plane.
The only major point I’d emphasize on top of this is that every combination of scaling and rotation by some angle amounts to a complex number; in fact, complex numbers are simply the same thing as scaling and rotation. “get five times larger and rotate 37 degrees” is just as good a description of a complex number as “5 * cos(37 degrees) + 5 * sin(37 degrees) * i”.
(I’ve written a slightly more expository account of this on the boards before, which may be of some help in grasping the arithmetic details.)
Mathematicians bend the rules of math when it comes to taking the square root of a negative number. I guess they love numbers so much, they weren’t gonna let their own rules stop them from finding a solution. So, the solution to a negative square root yields an imaginary number. That’s the simple explanation. They go on to do a lot of weird things with imaginary numbers, so let’s leave it at that.
You need to understand that mathematics (at least as practiced now) is about playing around in a logical way. Invent some new object with well defined rules, then start proving theorems. If you come up with some easy to state, but hard to prove theorems, mathematicians will say, “Cool”. If you figure out that these theorems are useful in electrical engineering or physics, the mathematicians may not be very interested.
So with imaginary numbers, let’s invent an object, call it “i” which can be multiplied and added with other imaginary numbers or “real” numbers. For fun, let’s suppose that i*i = -1. Can we define the operations so there are no logical difficulties? Yes! Can we prove some cool theorems as defined above? Absolutely! Is it useful? Tremendously.
Since you asked for an analogy, let me give you one that is a bit complicated, but displays some of the spirit of this intellectual game called mathematics.
Imagine the Rubik’s Cube. Let’s invent the object L, which corresponds to twisting the left hand side 90 degrees clockwise. We’ll also invent objects R, T, B, F, and K, which correspond to twisting the right, top, bottom, front, or bacK by 90 degrees. Now let’s define multiplication of L by T to mean, first do T (twist the top), then do L (twist the left). We’ll write this as LT.
Now you can start playing and proving theorems, like L^4 = 1 (four twists brings you back to the start), or L^3*L = 1, which implies that L^3 is the multiplicative inverse of L, which we can write as L^3 = L^(-1). We can now start implementing complicated moves like LTL^(-1) and proving cool theorems. The mathematicians are now happy because they have invented some objects and proved some cool theorems. Rubik’s cube devotees might not be interested in the math, except when the math guys use those theorems to show them how to solve the Rubik’s cube.
What’s interesting is that complex numbers actually have fewer logical difficulties than the real numbers that you’re familiar with. With just real numbers, there are a bunch of questions that you can’t give a meaningful answer to, like “What’s the square root of -1”, or “What number can I take the sine of and get 2”, or “What’s the logarithm of -1”. Complex numbers were invented to answer the first of those questions, but it turns out that once you have complex numbers, you can also answer the second and third questions without adding any more complexity to the system. You can even take square roots, logarithms, or trig functions of complex numbers, and you still just get complex numbers.
You do begin to add a little hassle, in that logarithms (even logarithms of positive real numbers) are no longer unique (indeed, except for 0 which has no logarithm, every complex number has infinitely many complex logarithms), and this non-uniqueness seeps all over the place, giving non-uniqueness of exponentiation (e.g., there are infinitely many complex-valued exponential functions with base 2), non-uniqueness of taking roots (e.g., there are three complex numbers which are cube roots of 2), and myriad other things.
But, of course, what counts as hassle and what doesn’t is in the eye of the beholder.
While this is all true, it’s not at all necessary to think of complex numbers as a purely formal abstraction. They did historically first get discovered this way, but history gets everything wrong.
As Chronos essentially pointed out, “complex number” is just another way of saying “combination of rotation and scaling” (just as “(possibly negative) real number” is just another way of saying “combination of scaling and 180 degree increment rotation”); this is a simple, fairly concrete geometric notion which is as much a part of everyday experience as any of the other basic mathematical concepts people consider mundane rather than exotic (e.g., natural numbers or negative numbers or non-whole ratios or vectors or what have you). Even small children have an intuitive understanding of complex numbers; they just don’t realize that that’s what their understanding of rotations and scalings is.
Complex numbers are no more exotic or removed from everyday experience than the integers. One doesn’t have to resort to electrical engineering to see them in the world around us; every time one takes two lefts to achieve a U-turn, they’re putting into action their knowledge of i^2 = -1.
Complex numbers are more than just playing games. You cannot solve cubic or fourth degree equations without them (and that was the first use of them). Ask any electrical engineer what he would do without them.
As for every number having infinitely many logarithms, that is not qualitatively different from the fact that every number has two square roots. Both 2 and -2 are square roots of 4 and both i and -i are square roots of -1.
While most mathematicians think of complex numbers as being the points of the plane, perhaps it is better, as an answer to the OP to think of them as rigid motions of the plane that preserve orientation. The latter means that clockwise remains clockwise.
You can’t solve many quadratic equations without them, e.g., x^2 + 1 = 0
What always seemed amazing about complex numbers, to me, is this. You can define this number “i” to be the square root of -1. Combinations of this and real numbers give you a two-dimensional number set, represented by the complex plane.
Now, what about if someone asks for the cube root of -1? My first, instinctive thought would be that this number would be some other abstraction, not able to be represented by a point on the plane defined by complex numbers with i. But, amazingly, the cube root of -1 can be formed by a combination of normal, real numbers, and that square root of -1 that we defined earlier.
Who’da thunk it?
Right, but the point is that in many cases solving cubic equations involves imaginary numbers, even if their solution is real. That’s why mathematicians first came up with them.
Or for a closer analogy, it’s even less qualitatively different than the fact that any given number has infinitely many arcsines.
Also potentially considered a new hassle, in comparison to the context of just the nonnegative real numbers (scaling with no rotation).
(Hell, in my experience TAing, I’d say this has to be one of the top five hassles for students (well, I suppose it’s not much of a hassle if you just constantly ignore it)…)
Indeed. As you say, it’s essentially just the same fact. But it does now (in the complex context) bleed through into a lot more situations than it did before, whereas previously, one could ignore this for logarithms, exponentiation, cube roots, etc.
Yes, this is nice. But let’s take the mystery out of it for anyone who’s unfamiliar by showing why it happens:
As noted before, complex numbers are just combinations of rotation and scaling, with multiplication being given by sequencing these; it’s easy to see that these already are closed under arbitrary nth roots (just divide the rotation angle by n, and take the nth root of the scaling factor).
So the interesting thing being pointed out is really that we can get arbitrary combinations of rotation and scaling for free, once we’ve introduced 90 degree rotation. That is, every combination of rotation and scaling is of the form (some scaling with no rotation) + (some scaling with 90 degree rotation) [more conventionally written a + b * i].
And why is this? Why, this is just the familiar fact that in two-dimensional space, given some reference vector, any other vector can be decomposed into the sum of components parallel and perpendicular to the reference vector (the parallel component being a scaling of the reference vector, and the perpendicular component being a scaling of the reference vector rotated 90 degrees). The basic and familiar geometric fact that lets us describe points in the plane with x and y coordinates. That’s all there is to it.
I think it’s easy to trivialize this connection now. However, it’s interesting to note that the rules for imaginary numbers were created 1572 but the idea for 2-dimensional complex plane to represent imaginary numbers geometrically wasn’t published until 1799. For more than 200 years the average person had to think about complex numbers without the use of geometric intuition.
Also, my high school math classes did not teach the complex plane even though we did endless drills with algebra equations with imaginary numbers.
Yeah. Like I said, history gets everything wrong and it takes forever to clean up the cruft and damage (e.g., names like “real” vs. “imaginary” (ironically, from the same Descartes who invented analytic geometry with the realization that pairs of coordinates describe points in the plane)).