These explanations about rotations are great, but really, you could do all of that with vectors and special operators and you wouldn’t ever need i.
The only place (so I’ve been told) where i is uniquely needed to describe physical situations and you couldn’t get around it any other way because it’s the only math we have that works, is quantum mechanics. The quantum mechanical states/operators depend on i being its own unique thing and not just a clever way of rotating things. Or so I was told by a number of physics professors.
This is a good point. When you start to think of mathematics as building blocks to build conceptual models of the real world, you see that you don’t need every block for every model even though most of the blocks end up getting used in some model or another. In some models, it makes sense to have half of some amount or a negative amount of it. In others, it doesn’t make sense.
They have as much meaning as complex numbers, which is the point. I was responding to a post claiming that complex numbers “have no real world meaning”. That’s clearly not true, or if it is true, just as true for irrationals.
I challenge you to find anything in the real world with a ratio of precisely pi and NOT simply a rational number whose value is close to what we call “pi” up to the limits of our measuring equipment.
In other words, we can’t measure truly irrational numbers. No fundamental constants can be physically measured to infinite precision.
We only get to infinite precision if we jump into theory. And if we’re jumping into theory anyway, complex numbers are in the same boat.
You can indeed handle rotations with other math, but in many situations the answer you need falls out naturally and obviously with complex notation, especially when there’s phase shifting or a decay envelope (imaginary terms in your oscillation frequency translate into decay terms). Fourier analysis can be done with separate sine and cosine terms, but it’s much easier and concise to do it with complex expressions. You can describe polarized light using 4 X 4 real-valued Mueller Matrices 9which are non-obvious and a pain to derive), or you can do it with intuitively obvious 2 X 2 Jones matrices that I can write down from memory.
It’s notb that you need complex numbers to do this. It’s that using them is more simple and direct. the n roots of x[sup]n*/sup] = B are obviously equally-distributed points around a circle of radius B[sup]1/2[/sup]plotted in the Argand plane, but only a collection of weird complex terms if you write them out
Likewise, we can get around the use of complex numbers in electrical impedance, too, but the result is unintuitive math and a lot of unnecessary complexity.
And for quantum mechanics as well. As Indistinguishable says, really all that complex numbers are is just two-dimensional rotations. Anything that uses complex numbers, you can replace them with any other means of describing two-dimensional rotations. The reason we use complex numbers specifically, in electrical impedance and quantum mechanics and Fourier analysis and all the other places we use them, is that they’re a nice simple way to do it, that follows in a natural way from other things we do on a regular basis.
You can actually do quantum mechanics using only real numbers, but you have to impose a special rule (a so-called ‘superselection rule’) according to which all operators must commute with a special operator which in a simple system (a qubit) takes the form of the matrix
( 0 1 0 0)
J = (-1 0 0 0)
( 0 0 0 1)
( 0 0-1 0)
(The state space of a qubit is two-complex dimensional, that is, two complex numbers are required to describe its wave function; since a complex number can be capture using two real numbers, its state space is four dimensional in real number quantum mechanics.) This requirement was first formulated by Stückelberg in the '60s, I think. More recently, William Wootters has worked on a theory in which Stückelberg’s rule would emerge dynamically (by adding a ‘universal re(al)-bit’ that essentially accounts for the global phase degree of freedom, for the experts), and which thus is also a theory equivalent to QM done with real numbers.
One might think, however, that this is a kind of cheat, because in a way, using the operator J reintroduces complex numbers ‘through the backdoor’ in a sense, since the square of J is equal to
(-1 0 0 0)
J²= ( 0-1 0 0)
( 0 0-1 0)
( 0 0 0-1)
Which is basically -1 for 4x4 matrices. But still, one can make a case that this theory can teach us something about the world: first of all, time reversal in quantum mechanics is related to complex conjugation (changing i to -i everywhere, basically). In the real theory, this is related to the superselection rule: any such rule effectively ‘decomposes’ your theory into certain fragments, which for instance can’t enter into superposition, etc. So the real theory can incorporate time reversal symmetry in a more natural way than complex QM. Secondly, the operator J is related to interference, which is in many ways the central phenomenon of QM.
Nevertheless, it’s something that’s not widely used (or known), and perhaps more of a peculiarity than anything else.
Mentioned in post #46, where I point out that it’s not strictly necessary, but is an awful lot easier if you simply have complex exponentials rather than separate sine and cosine functions.
I know I’m being massively oversimplistic here, but if your using objects with the algebraic properties of the complex/imaginary numbers it’s difficult to say that you have avoided using them (especially when you are selecting said objects for their algebraic properties).
There’s no difference between doing it “with vectors and special operators” and doing it "with complex numbers. Whether you call something “the 90 degree turn operator” or “i” or “j” or what have you, it has exactly the same properties: applying it twice to a value is the same as negating that value. Call it a complex number, or a matrix, or a linear operator, or whatever you like… the label makes no difference, except pyschologically.
In the same way, one might say “Why do we need fractions? What can you do with fractions that you can’t do with just a pair of whole numbers, and suitable rules for manipulating these as ratios?”… Well, yes. That’s what fractions are. They’re a pair of whole numbers representing a ratio.
For that matter, we don’t “need” any numbers… Instead of saying “I have 3 apples”, you could always just say “I have an apple, and another apple, and yet another apple”. But there isn’t really a difference between these! The former is just another way of saying the latter (one which in its terseness is convenient if you’ll be saying that sort of thing a lot).
Quaternions don’t directly describe scaling-and-rotation in 3d the way complex numbers directly describe scaling-and-rotation in 2d; rather, quaternions directly describe particular kinds of scaling-and-rotation in 4d (specifically, for rotations on 4d space which move every vector by the same angle [aka, “isoclinic rotations”], there is a natural notion of “orientation”, of which there are two possibilities; the quaternions, then, will describe the combinations of scaling and rotation of a particular orientation). It just so happens that, if you have a distinguished way to decompose a 4d space into perpendicular 3d and 1d subspaces, then you can represent every rotation of the 3d subspace as a composition of two “mirror-image” isoclinic rotations of the 4d space, and thus represent 3d rotations by quaternions as well. But you’ll find that the additive properties of quaternions don’t match those of rotations on 3d space thought of as linear operators, because quaternions aren’t fundamentally for 3d space, but for 4d space.
Another way to understand what’s going on here is that, in some sense, the reason the complex numbers are so nice is because, for any two directions in a plane, there’s a unique rotation of the plane which takes the one to the other, so that we can think of complex numbers as “ratios” of 2d vectors. The same is no longer true for two directions in 3d space (for example, there are infinitely many ways to rotate the Earth so that the North pole is sent to the North pole). However, in 4d space, we can make it true again IF we add in the demands that A) the rotation must move all directions by the same angle [as automatically happens in the 2d case], and B) the rotation must have a particular “orientation”. And thus, we get the quaternions as ratios of vectors in 4d space the same way the complex numbers are ratios of vectors in 2d space and the real numbers are ratios of vectors in 1d space, but there is no analogous 3d structure.
