# Some remarks on complex numbers

by ClamShell
Tags: complex, complex numbers, imaginary numbers, numbers, remarks, u-bit, ubit
P: 190
 Quote by SteveL27 The number i is a gadget that represents a counterclockwise quarter turn of the plane.
Why don't we have gadgets to represent turns of the plane in different dimensions? If i moves it counterclockwise as the viewer sees it, where's the 'imaginary' number to tilt it anterior?
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P: 6,387
 Quote by oneamp Why don't we have gadgets to represent turns of the plane in different dimensions?
'And Hamilton said, "Let there be quaternions". And Hamilton saw the quaternions, and behold, they were very good...'

Sorry, I got carried away by homeomorphic's "not-god-given" interpretations.
 P: 221 Notations seem to possess qualities of "protocol". Real numbers seem to possess some "material", "down-home" quality.
P: 1,049
 Why don't we have gadgets to represent turns of the plane in different dimensions? If i moves it counterclockwise as the viewer sees it, where's the 'imaginary' number to tilt it anterior?
We do. They are called matrices (specifically, orthogonal matrices, those that preserve distances, if you just want rotations with no distortion). As we have mentioned, you can do everything i does by using a matrix.

If you wanted something a little bit more like complex numbers, that gets a little trickier. In general, what you would get in higher dimensions are called Clifford algebras, which also include complex numbers and quaternions as special cases. These are combinations of different square roots of -1 (in the most basic version--there are more general formulations). However, Clifford algebras have a lot of quirks that make them not quite like complex numbers. For one thing, they are not division algebras, so things don't always have inverses anymore. So you can't divide by guys in the Clifford algebra, like you can with complex numbers.

Also, the geometric interpretation of how they operate isn't so straight forward in general. You can still use them to describe rotations in higher dimensions, though. The square roots of -1 in the Clifford algebra act by reflection across a coordinate hyper-plane--a funny "spinorial" reflection, in which you have to reflect 4 times to get back to where you started because although the square is -1, which acts on the space by doing nothing, the Clifford algebra secretly knows that something is different until you square it once more and get back to 1. It's a bit of a long story.

Another thing about Clifford algebras is that, while you could think of them as a higher dimensional space because they are just vector spaces with some kind of extra multiplication, it's not as natural to think of them that way as it is with complex numbers and the 2-dimensional plane. You have to throw in all the products of the square roots of -1 which boost the dimension up way higher than the space they act on (2^n for n-dimensional space), and algebraically, it's a lot more messy. Very different from the complex plane which acts on itself.
P: 221
 Quote by ClamShell Notations seem to possess qualities of "protocol". Real numbers seem to possess some "material", "down-home" quality.
Yikes, this is probably close to what Pythagoras thought about rationals.
Par day m'wah.
 P: 221 OK, let me collect my thoughts again..."Where did I leave them, in the refrigerator again?" This relates back to my first post, "sqrt(-1) enters the picture when we attempt to factor a sum." It would seem to me that trying to get rid of "i", after it appears, such as by "substituting" a two by two matrix for "i", is absolutely TOO trivial a thing to do to purge "i" from the analysis. The two by two matrix is just a "wolf in sheep's clothing"; an isomorphism as revealed above. We've got to stop its introduction BEFORE it appears, so lets start thinking of removing the factoring of sums(polynomials) from the analysis. I know...then how will we find out the zero crossings and those precious eigenvalues. I'm thinkin' that Dr. Wootters isn't merely performing an isomorphism on QM, but it is more like a reformulation avoiding "i" by not ever seeing it in the first place. This could be done by avoiding factoring polynomials, but rightfully so...I could not really understand Dr. Wooters' lecture. So how about avoiding the factoring of polynomials as a way to avoid "i"? Maybe subtractions factoring into a conjugate real pair would be OK, but real conjugate pairs might need to be avoided too, for consistency. Could analysis even be performed without this factoring? Could reality not even know how to factor, and a model that does factor, is expecting too much from Mother Nature or some other deity? Is the real meaning of "i", simply that "No FISHING IS ALLOWED"... I mean "no factoring is allowed"; dag-nab keyboard.
 P: 221 And might not the removal of factoring from the picture be similar to the "Pythagorean Dream" of: "NO IRRATIONAL NUMBERS" After all, cannot every number with a finite number of digits be represented as a ratio of integers? I'm thinking that the removal of irrationals via the removal of factoring might yield a peculiar integer mechanics of its own; just sayin'.
 Mentor P: 7,292 This is getting a bit tiresome. Have you learned anything in any of the posts of this thread? Are you even interested in learning? These forums are for learning, if you are not here to learn then this thread is pointless.
 P: 1,049 If you don't allow factoring of some polynomials, you can't let i in because i is going to factor everything. But you have to start somewhere. If you allow factoring of a certain polynomial, you get what's called its splitting field, which is everything you need to factor that polynomial, but no more. You have to start with something, though, so it's not just the splitting field, it's a splitting field over some base field like the real numbers. The splitting field of x^2 + 1 over the real numbers is the complex numbers. Splitting fields are nothing special, though. You can always get them by throwing in enough stuff, rather than requiring a polynomial to factor. So, you can start with the rational numbers and throw in square roots of 2 (and all resulting multiples, etc.) or you can require that x^2-2 factors. Either way, you get the same result. So, no, there's nothing particularly special about allowing or not allowing things to factor. It's the same as throwing stuff in or kicking it out.
P: 221
 Quote by Integral This is getting a bit tiresome. Have you learned anything in any of the posts of this thread? Are you even interested in learning? These forums are for learning, if you are not here to learn then this thread is pointless.
Yes, what I've learned so far is that substituting a two-two matrix for "I"
is a bit meaningless, so another direction is called for.

"Just because you don't know the answer, you don't have to get mad", said the
lion to the elephant. Please don't throw me into the Mediterranean, like Hippasus.
I'm not a magazine salesman, nor do I have some personal theory. I'm just trying
to figure out ways to avoid "I", like the New Scientist article wants too. I think
not factoring it out in the first place is a fertile not futile endeavor.
P: 221
 Quote by homeomorphic If you don't allow factoring of some polynomials, you can't let i in because i is going to factor everything. But you have to start somewhere. If you allow factoring of a certain polynomial, you get what's called its splitting field, which is everything you need to factor that polynomial, but no more. You have to start with something, though, so it's not just the splitting field, it's a splitting field over some base field like the real numbers. The splitting field of x^2 + 1 over the real numbers is the complex numbers. Splitting fields are nothing special, though. You can always get them by throwing in enough stuff, rather than requiring a polynomial to factor. So, you can start with the rational numbers and throw in square roots of 2 (and all resulting multiples, etc.) or you can require that x^2-2 factors. Either way, you get the same result. So, no, there's nothing particularly special about allowing or not allowing things to factor. It's the same as throwing stuff in or kicking it out.
Thank you Homeomorphic, answers are always good, even the ones with bad news.
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P: 6,387
 Quote by ClamShell I'm just trying to figure out ways to avoid "I".
Whatever floats your boat, but that seems a rather pointless endeavour. If you accept that the integers obey Peano's axioms (or start from ZFC if you prefer!), the concept of "i" exists as a consequence of that assumption (and so does the concept of an irrational number), even if you personally refuse to give it a name and/or talk about it.
Mentor
P: 7,292
 Quote by ClamShell Yes, what I've learned so far is that substituting a two-two matrix for "I" is a bit meaningless, so another direction is called for. "Just because you don't know the answer, you don't have to get mad", said the lion to the elephant. Please don't throw me into the Mediterranean, like Hippasus. I'm not a magazine salesman, nor do I have some personal theory. I'm just trying to figure out ways to avoid "I", like the New Scientist article wants too. I think not factoring it out in the first place is a fertile not futile endeavor.
You will be way better off to learn to use and appreciate complex numbers.