Some remarks on complex numbers

[DrewD]

Pretty weak conjecture, if I do say myself.

Does no one have an objection to this?

Yes. From a mathematical standpoint, it is meaningless to reformulate QM using real matrices since there is an isomorphism between the two representations.

From a computational standpoint, I wouldn't be surprised if Matlab already does complex computations using matrices (I don't know this, but I know that a professor of mine wrote some code that always used the matrix representation because it is easier to deal with on a computer)

Physically, things will be the same. Maybe some problem will be solved by accident while trying to reformulate QM.


Also, 17 is a grouping of pixels that we use to communicate the idea of a particular quantity. If I rewrite 17 as !&, math will not change. Similarly, if I rewrite $i\equiv\left[\begin{array}{cc} 0&1\\-1&0\end{array}\right]$, math will not change. The numbers we write exist only for equations (and theorems etc.). The algebraic relationship between elements of the set $\mathbb{C}$ exist with or without the notation, but $a+bi$ is only one of infinitely many ways to write those relationships, so why should it be especially meaningful? I side with lurflurf. At least it is a legitimate philosophical position even if you don't personally agree with it.

 

[ClamShell]

@DrewD. 17 balls in a hat; 10 white and 7 black. Looks like
probability of an observation has something to do with
"significance" too. We will never pick a sqrt(-1) ball from the
hat, even if someone says some of the balls are positive
sqrt(-1) balls. That's all I've got, maybe math was invented
and not discovered. I have often considered that math is entirely
notational. If you've got 17 pigs and I have 17 sheep, then maybe
both of us would be better off with 17 pigs and sheep each.

 

[homeomorphic]

@Homeomorph, now you need to explain the significance of 42.

Well, you can extrapolate to any number by counting. We have a system for generating numbers. If I say, give me 43 millions apples, we know what that means. You just count it out until you get to 43 million. That's all that is really needed. We have a way of referring to something physical that is out there. The only condition is that you have to be able to interpret what I am saying as more than just a string of letters or sounds.

The numbers we write exist only for equations (and theorems etc.)

It gets a little better if you add theorems, etc, but obviously there are real world uses for numbers, like money, counting things, etc. Not restricted to math-land.

If I rewrite 17 as !&, math will not change.

I don't think you want to just talk about the symbols, though. There are concepts behind the symbols. The notation isn't important, but it isn't a bunch of meaningless symbols. Meaningless symbols are for computers, not human beings. Human beings think in pictures or concepts, and we can manipulate symbols as well, but that's not the whole picture. For me, i is simply the point (0,1) on the complex plane and it acts on the plane by rotating it 90 degrees. This is an equivalent way to think about it to the algebra and is not in terms of equations. You can use this in any way you like. You can use it to think about actual planes and rotations, you can use it to think about wave functions, or you can use it as a visual representation of things that you could calculate with the arithmetic of complex numbers, using the formal rules. These points of view are all different sides of the same thing.

It doesn't matter if we use i or if we use the matrix, but that has nothing to do with whether they have physical significance. They both have the same physical significance. So, yes, it is superfluous to chose one or the other, but that is because they share the same physical significance, not because they have no physical significance. Complex numbers correspond to points on a plane. So do matrices for rotation and dilation. When you interpret them this way, that gives them physical significance. They can be used to symbolize something out there. If I give you a point on a plane, if we set up the right coordinate system, it could be used to symbolize an actual point out there. Like a point on the chalkboard, just as natural numbers can refer to a physical collection of apples. We could change the symbols around, but that misses the point.

Denying any physical or geometric significance would just be tying your hands behind your back, psychologically speaking. Even great mathematicians like Euler made retarded mistakes with complex numbers because they didn't have a geometric representation to help them to spot the errors. Once people started thinking about them as points on a plane, magically everything started making sense, and a lot of nonsense was finally put to rest. Let's not go back to the 1700s level of understanding of complex numbers. Once the complex plane was revealed, it was really sort of an embarrassment to the previous couple hundreds years of mathematicians how simple it actually was. I mean, what a cheap, stupid, dead obvious little trick. Points on a plane. But it made all the difference and suddenly there was huge progress in understanding complex numbers.

 

[homeomorphic]

@DrewD. 17 balls in a hat; 10 white and 7 black. Looks like
probability of an observation has something to do with
"significance" too. We will never pick a sqrt(-1) ball from the
hat, even if someone says some of the balls are positive
sqrt(-1) balls. That's all I've got, maybe math was invented
and not discovered. I have often considered that math is entirely
notational. If you've got 17 pigs and I have 17 sheep, then maybe
both of us would be better off with 17 pigs and sheep each.

Well, the problem here is not that i doesn't have a physical interpretation. It's just that it doesn't have one that extends the representation given to natural numbers by counting, unless perhaps i means 1 unit up, 2i means 2 units up, etc. So, it's sort of a 2-dimensional quantity, whereas the usual counting we 2 is only applicable to 1-dimensional quantities.

Maybe you can think of it this way. You can GIVE it a physical interpretation. It's not a God-given one. You can just use it to refer to stuff. It's conventional now to identify complex numbers with points on a plane. So, that's a standard interpretation that any mathematician will know.

 

[oneamp]

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?

 

[AlephZero]

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.

 

[ClamShell]

Notations seem to possesses qualities of "protocol".
Real numbers seem to possesses some "material", "down-home" quality.

 

[homeomorphic]

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.

 

[ClamShell]

Notations seem to possesses qualities of "protocol".
Real numbers seem to possesses some "material", "down-home" quality.

Yikes, this is probably close to what Pythagoras thought about rationals.
Par day m'wah.

 

[ClamShell]

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 let's 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.

 

[ClamShell]

No "fishing" allowed

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'.

 

[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.

 

[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.

 

[ClamShell]

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.

 

[ClamShell]

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.

 

[AlephZero]

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.

 

[Integral]

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.

Thread closed.