Principal difference between complex numbers and 2D vectors revisited

[SVN]

I know this topic was raised many times at numerous forums and I read some of these discussions. However, I did not manage to find an answer for the following principal question.

I gather one deals with the same set in both cases equipped it with two different structures (it is obvious if one regards the product operation; it has rather dramatic consequences, for example, a complex number is in some sense 1D object and 2D vectors are, well, 2D). Complex numbers can be regarded as specific case of vectors, but it is not quite helpful, for this specific case is so special that it generates a separate mathematical discipline.

I presume one might say complex numbers and vectors are two different representations for some abstract object.

And that leads straight to my question. What are the criteria to choose one representation over the other? In other words what is the condition that would allow one to say «No, it is incorrect (or pointless, for it won't help to find the solution of this specific problem) to use complex numbers in this case, it is an absolute must to work with vectors here».

P. S. I am not sure this question is a correct one. Quite possible it is not. But even if it is so, it still may give some idea of what I can't understand.

 

[FactChecker]

Complex numbers have a form of multiplication defined that tie directly to the addition of their angles. That gives us division (rotate the other way), square roots (cut the angle in half), etc. The consequences are profound.

If you are considering an application where that definition of multiplication makes sense, take advantage of it.

 

[member 587159]

$\mathbb{R}^2$ is the same as $\mathbb{C}$ when you consider these as $\mathbb{R}$-vector spaces. You get $\mathbb{C}$ if you put a multiplication on $\mathbb{R}^2$ that makes $\mathbb{R}^2$ into a field. A tuple $(a,b)$ is then later written as $a+bi$ because it is less tedious to work with. So, I would say $\mathbb{C}$ is just the field you get by putting the multiplication on $\mathbb{R}^2$ you are used to.

 

[SVN]

Complex numbers have a form of multiplication defined that tie directly to the addition of their angles. That gives us division (rotate the other way), square roots (cut the angle in half), etc. The consequences are profound.

If you are considering an application where that definition of multiplication makes sense, take advantage of it.

Well, let's consider a specific problem (the way I see it is still my original question, just reformulated for specific situation).

There is a 2D regular surface in Euclidean 3D space. If one talks about tangent space in any of its point, he necessarily use concept of 2D linear space. Vectors, not complex numbers (at least, I have never heard of using complex numbers in this case)! Why? What makes it impossible (or, again, pointless) to use the latter concept in this specific case? In theory, one can switch from one formalism to another. But in many cases it just does not make sense because it leads nowhere.

Some rule like «If you talk about anything that can be either positive or negative, use real numbers and be happy. However, if what you are trying to describe can rotate, you should give chance to complex numbers».

 

[fresh_42]

What are the criteria to choose one representation over the other?

This is already fundamentally wrong. They are not two representations of the same thing. They are two different things obeying different rules. It is more than just representation.

In other words what is the condition that would allow one to say «No, it is incorrect (or pointless, for it won't help to find the solution of this specific problem) to use complex numbers in this case, it is an absolute must to work with vectors here».

There is an inclusion: complex numbers build a two dimensional real vector space, but not the other way around. Hence everything which needs the vector space structure can as well be written with complex numbers; so there cannot be such a case you're looking for. The other way around does not hold: you cannot divide in a vector space.

 

[FactChecker]

If you are talking about an application that is embedded in higher dimensions, then you should be careful. The advantage of complex numbers does not make much sense in higher dimensions. The subject of geometric algebra is an attempt to retain those advantages in higher dimensions.

 

[SVN]

$\mathbb{R}^2$ is the same as $\mathbb{C}$ when you consider these as $\mathbb{R}$-vector spaces. You get $\mathbb{C}$ if you put a multiplication on $\mathbb{R}^2$ that makes $\mathbb{R}^2$ into a field. A tuple $(a,b)$ is then later written as $a+bi$ because it is less tedious to work with. So, I would say $\mathbb{C}$ is just the field you get by putting the multiplication on $\mathbb{R}^2$ you are used to.

I can't argue with that. But I fail seeing how it answers my question, sorry. I realize the difference results of this extra algebraic structure. My question is how to understand whether this extra is helpful for solving a task (thus choosing between the two formalisms to work with the problem at hands, for example, vector analysis and complex analysis).

 

[SVN]

If you are talking about an application that is embedded in higher dimensions, then you should be careful. The advantage of complex numbers does not make much sense in higher dimensions. The subject of geometric algebra is an attempt to retain those advantages in higher dimensions.

It was not about higher dimensions for me, actually. It is just an example. If one talks about 2D tangent space, he uses vectors, not complex numbers. It is probably possible to consider tangent plane as Argand plane, but no one is doing that (AFAIK). There must be a reason for that.

 

[PeroK]

It was not about higher dimensions for me, actually. It is just an example. If one talks about 2D tangent space, he uses vectors, not complex numbers. It is probably possible to consider tangent plane as Argand plane, but no one is doing that (AFAIK). There must be a reason for that.

The complex numbers form a very specific structure: an algebraically closed field. If you are dealing with 2D vectors, you could use complex numbers instead and neglect the algebraic properties. You could study 2D motion in a horizontal plane with one axis as the real axis and the one axis as the imaginary axis. And, up to a point it might make some kind of sense.

But, 2D vecors are generally just a subset of 3D space. It's not clear how you would generalise your complex numbers to have a third, vertical axis?

When you come to define angular momentum of an object in your 2D space, that is a cross product and is represented by a vector in the z-direction. Again, you can't so this with complex numbers. There is no concept of a cross product.

Moreover, for example, you have the equation:

$\frac{dE}{dt} = \vec F . \vec v = F_1v_1 + F_2 v_2$

There is no analogue of the dot product in complex numbers.

The point to note is that the extra algebraic properties of complex numbers come at a price. There isn't an embedding of the complex numbers in a 3D space. The algebraic properties add nothing to the concept of a vector and just get in the way.

 

[FactChecker]

It is probably possible to consider tangent plane as Argand plane, but no one is doing that (AFAIK). There must be a reason for that.

Yes. If the subject matter has a concept of multiplication and division, then the complex plane and number system may offer significant advantages. Otherwise, it is wrong to imply that there is some meaning to the multiplication and division as defined by the complex number system.

 

[Dr.D]

If describing real things, I think it is much better to keep it real.

 

[FactChecker]

If describing real things, I think it is much better to keep it real.

IMHO, that makes it sound like cyclic behavior, oscillations, power series expansions, etc. should be done without complex numbers. Nothing could be farther from the truth.

 

[Dr.D]

IMHO, that makes it sound like cyclic behavior, oscillations, power series expansions, etc. should be done without complex numbers. Nothing could be farther from the truth.

Strange as it may sound, I get by just fine in the real world.

 

[FactChecker]

Strange as it may sound, I get by just fine in the real world.

Of course, there are a lot of subjects that should not use complex numbers. But I consider the problems of stability and control to be "real world" and that is just one of the many real-world subjects that I would hate to work on without complex numbers and analysis.

 

[Dr.D]

that is just one of the many real-world subjects that I would hate to work on without complex numbers and analysis.

Isn't it nice that you have that option!

 

[lavinia]

I know this topic was raised many times at numerous forums and I read some of these discussions. However, I did not manage to find an answer for the following principal question.

I gather one deals with the same set in both cases equipped it with two different structures (it is obvious if one regards the product operation; it has rather dramatic consequences, for example, a complex number is in some sense 1D object and 2D vectors are, well, 2D). Complex numbers can be regarded as specific case of vectors, but it is not quite helpful, for this specific case is so special that it generates a separate mathematical discipline.

I presume one might say complex numbers and vectors are two different representations for some abstract object.

And that leads straight to my question. What are the criteria to choose one representation over the other? In other words what is the condition that would allow one to say «No, it is incorrect (or pointless, for it won't help to find the solution of this specific problem) to use complex numbers in this case, it is an absolute must to work with vectors here».

P. S. I am not sure this question is a correct one. Quite possible it is not. But even if it is so, it still may give some idea of what I can't understand.

- You could ask the same question about the real numbers. They may be thought of as a 1 dimensional vector space or as an algebraic field.

- Complex numbers are a 2 dimensional vector space under addition. That is: addition of complex numbers is the same as vector addition. A common way to think of the difference is that the complex numbers are a two dimensional vector space (with real numbers as the scalars) that has an additional structure which is a multiplication that turns it into an algebraic field.

- One situation that distinguishes complex numbers is in calculus. A function is differentiable in $R^2$ if its derivative exists in any direction. In forming Newton quotients one uses only vector increments and their lengths. The function is defined on the two dimensional vector space not on the complex numbers.

A function is Complex differentiable if increments are understood as complex numbers. In a Newton quotient dividing by the increment means dividing by a complex number. A complex differentiable function is defined on the complex numbers.

 

[fresh_42]

You could ask the same question about the real numbers.

Or go even a step beyond: the real numbers are an infinite dimensional rational vector space. But nobody would consider to replace the reals by those vectors.

 

[pbuk]

Complex numbers have a form of multiplication defined that tie directly to the addition of their angles.

We don't define multiplication of complex numbers by the addition of angles, this is simply a result of what happens when we represent ℂ by the Argand plane.

If you are talking about an application that is embedded in higher dimensions, then you should be careful. The advantage of complex numbers does not make much sense in higher dimensions.

Quaternions, octonions?

So, I would say $\mathbb{C}$ is just the field you get by putting the multiplication on $\mathbb{R}^2$ you are used to.

Yes, absolutely - every other property of the complex numbers stems from this, including that $i^2 = -1$ (the deriviation of which is IMHO one of the most simply elegant things in mathematics).

 

[FactChecker]

We don't define multiplication of complex numbers by the addition of angles, this is simply a result of what happens when we represent ℂ by the Argand plane.

They go hand-in-hand. Which came first, the chicken or the egg? I don't think that the Argand plane would ever be heard of if it did not go along with the basic arithmetic properties of the complex numbers.

Quaternions, octonions?

Yes, and more generally, algebraic geometry. But all of those are major extensions of the idea. I am not considering them to be identical to the complex numbers in the complex plane.

 

[lavinia]

We don't define multiplication of complex numbers by the addition of angles, this is simply a result of what happens when we represent ℂ by the Argand plane.

The two definitions are equivalent. If what you are saying is that ,historically, complex numbers were defined in terms of $i$ perhaps to be able to solve for roots of polynomials with no real roots then the Argand plane interprets this as a two dimensional vector space with a multiplication. But one can just as easily start with angle addition and length multiplication. This way is geometrically intuitive which IMO is always a good thing.

Quaternions, octonions?

- Complex vector spaces and complex manifolds exist in all even dimensions.

- The theorem is that the complex numbers and the reals are the only Euclidean spaces that are fields. The quaternions and octonians are not fields.

Yes, absolutely - every other property of the complex numbers stems from this, including that $i^2 = -1$ (the deriviation of which is IMHO one of the most simply elegant things in mathematics).

What is the derivation that you are thinking of?

 

[fresh_42]

Faites vos jeux!$^*$
$$
\mathbb{C} \cong_\text{ring} \mathbb{R}[x]/\langle x^2+1 \rangle \cong_\text{field}\mathbb{R}[ i ] \cong_\text{vector space} \mathbb{R}^2 \cong_\text{group}\mathbb{R}^+\times U(1)\cong_\text{algebra}\mathcal{Cl}(\mathbb{R},x^2)
$$
J. Dieudonné writes in his book about the history of mathematics:

The term "number" initially included ... furthermore the imaginary numbers, the bold creation of Italian algebraists of the sixteenth century, whose essence remained mysterious until around 1800, but which were increasingly used in algebra and analysis from the seventeenth century onwards, because ...

I guess that's why I like the ring isomorphism most. It's where they came from, although Franciscus Vieta's real name was François Viète and he wasn't Italian, but Cardano and Ferrari have been.

$^*)$ The discussion here is like a roulette ball jumping from one of these boxes to another.

 

[pbuk]

If what you are saying is that ,historically, complex numbers were defined in terms of $i$ perhaps to be able to solve for roots of polynomials with no real roots

No, that is definitely not what I am saying: complex numbers are defined as a field over ℝ² with the multiplicative operation $(a, b) \times (c, d) = (ac-bd, ad+bc)$. The imaginary number $i$ comes later as a consequence (see below).

The quaternions and octonians are not fields.

No, but I was responding to the assertion that "the advantage of complex numbers does not make much sense in higher dimensions". Just because quaternion multiplication is not commutative does not mean that it does not make sense.

What is the derivation that you are thinking of?

Start by imagining that we can represent an ordered pair of real numbers $(a, b)$ as $a + i b$, then solve the equation $(a + ib)^2 = -1$.

 

[PeroK]

No, but I was responding to the assertion that "the advantage of complex numbers does not make much sense in higher dimensions". Just because quaternion multiplication is not commutative does not mean that it does not make sense.

The original question was "why not use complex numbers instead of vectors?"

Your answer seems to be as long as you miss out the odd dimensions you can. Inconveniently, however, 3D vector calculus is quite important.

 

[pbuk]

The original question was "why not use complex numbers instead of vectors?"

Your answer seems to be as long as you miss out the odd dimensions you can.

I think that's quite a leap from what I actually wrote, but just to be clear all I have been trying to say is that just as it is sometimes (but not always) possible, and useful, to use $\mathbb C$ to model $\mathbb R^2$, it is sometimes (but not always) possible and useful to use quaternions to model $\mathbb R^4$.

 

[SVN]

The way I see it the discussion could now be closed. At least I got my answer and it certainly makes sense to me. Thanks a lot to everyone involved.