What's the difference between complex numbers and vectors?

[djosey]

Well it's all in the title: i don't understand why we define vectors and complex numbers differently, with different properties (eg vector dot product and cross product and complex multiplication). After all, all a complex number is is a 2-uplet of real numbers, but that's exactly the same as a 2 dimensional vector... Is a complex number a 2-dimensional vectors, and vectors only the generalisation of complex numbers to n dimensions? Or is there some fundamental difference between those two concepts that i don't get?

It's just that at the moment, I'm studying those two objects in my lectures totally independently from each other, while they seem to be more than linked.

edit: not sure that's the right forum for this post, put it here because both complex numbers and vectors can be thought of as a matrix

 

[Fredrik]

You should think of them as two different mathematical structures with the same underlying set. See post #11 in this thread.

 

[lavinia]

Well it's all in the title: i don't understand why we define vectors and complex numbers differently.

every field is a vector space over itself and over every one of its subfields.

the complex numbers are a one dimensional vector space over themselves and are a 2 dimensional vector space over the subfield of real numbers.

over the subfield of rational numbers they are an infinite dimensional vector space.

the real numbers are a 1 dimensional vector space over themselves but are infinite dimensional over the rational numbers.

 

[AlephZero]

You more or less answered your own question when you said "we define them differently with different properties". In other words, they are two completely different things.

I would guess from the question these courses may be your first introduction to a more "abstract" approach to math, rather than thinking of numbers as "things in the real world" which have "obvious" properties, When you start learning math (or rather arithmetic), the "rules" are kind of "obvious", because of facts like "taking 2 apples and then 3 more apples" is the same as "taking 3 apples and then 2 more apples". It takes a while to get used to the idea that "numbers" in math don't really have anything at all to do with counting apples.

Of course you can represent the real and imaginary parts of a complex number as a point on a plane (the Argand diagram) and you can do the same for the components of a 2-dimensional vector. Therefore complex numbers and 2-dimensional vectors will have some "geometrical" properties that are similar. But as you go further into using complex numbers in calculus (for example "analytic functions"), and study things like infinite-dimensional vector spaces where the elements of the vectors are not even numbers at all, you will find there are many more differences than similarities.

 

[HallsofIvy]

Do you understand that R₃, the set of ordered triples of real numbers with "component wise" additon and scalar multiplication is a vector space? What does that have to do with complex numbers? The set of all polynomials is a vector space. What does that have to do with complex numbers?

You seem to be fixed on the idea that we represent complex numbers with two numbers and we represent vectors in R² by two numbers but:
1) the complex numbers and vectors in R² have very different operations defined and
2) there exist vector spaces of different dimensions than 2.

 

[pbandjay]

The gist of it is that the complex numbers form a field (just like the reals), so are equipped with the same properties as the reals (associative, commutative, closure under an addition and a multiplication, existence of identities w.r.t. the operations, inverses, distributive).

A vector space is a set with an underlying field that has many "analogous" properties as well, but the properties are built around the interaction of the vector space and its underlying field.

Most notably, the vector space R^2 doesn't have a function that takes in two 2-tuples and gives another 2-tuple (the vector space R^2 isn't closed under any type of "vector" multiplication, only scalar multiplication). The complex numbers do.

It's a pretty neat thing once you start to really understand fields, vector spaces, algebras, etc...

 

[Deveno]

hmm. as a vector space, over the real field, there's not much difference between R^2 and C. the complex number a + bi can be seen as a linear combination of the the basis {1,i} just as a point in R^2 can be seen as a linear combination of the basis {(1,0), (0,1)}.

but vector spaces, in general, do not have a multiplication: VxV--->V that obeys the distributive law. some do, these are interesting because of that.

some examples of vector spaces WITH such a multiplication:

R[x], the vector space of all polynomials in x (if you consider only polynomials of degree 1 or less (linear polynomials and constant polynomials), this also "looks like R^2 and C", you can assign 1 = (1,0) and x = (0,1), but then the (usual) multiplication of polynomials is not closed since x^2 is not in the space).

M2[R], the vector space of all 2x2 real matrices, with matrix multipliciation. this multiplication is not commutative, AB is usually not the same as BA. it is interesting that one can actually view the complex numbers as a SUBSET of this vector space,by a+bi --->

[ a b]
[-b a]

C, the complex numbers itself.

vector spaces that have this "additional structure" are given a special name, "(associative) algebra".

of the three algebras listed above, only one, the complex numbers, possesses multiplicative inverses for every non-zero element (R[x] has inverses for constant polynomials only, and M2[R] has only has inverses for matrices with non-zero determinant). this again is somewhat special such algebras are called "division algebras". division algebras are relatively scarce in the world of mathematics, if one desires that a division algebra also be a vector space over the field of real numbers, one finds that the choice of dimensions is severely restricted (for example, there is no 3-dimensional division algebra over the reals, a fact that has vexed physicists for quite some time).

complex numbers represent a fork in the road in terms of how to view higher dimensional structures. to the left, one tries to keep the algebraic properties as much as one can, leading eventually to quaternions, octonions, and various clifford algebras. to the right, one tries to keep the higher dimensional structures the same as the reals, "only more copies", leading to hilbert spaces and differential manifolds (amongst other things).

in other words, the complex numbers have a rather unique collection of properties. they are a structure in which math is particularly nice to do. complex numbers can be viewed as numbers, as matrices, as vectors. they have nice algebraic properties, they are well-suited for solving equations in, and posses desireable spatial and geometric properties as well.

 

[djosey]

Thanks for all the answers!

This is quite fascinating, i had the feeling that the linear algebra i did was kinda opening the way for a much more generalized and abstract approach to numbers, that could shed a new light on many other topics, but i haven't gone to that point yet. I am also conforted in my intuition that complex numbers and vectors have at least some properties in common, and that they are different parts of a more inclusive set.

This:

M2[R], the vector space of all 2x2 real matrices, with matrix multipliciation. this multiplication is not commutative, AB is usually not the same as BA. it is interesting that one can actually view the complex numbers as a SUBSET of this vector space,by a+bi --->

[ a b]
[-b a]

C, the complex numbers itself.

is also very interesting, it's what i was trying to get at. I'll go delve deeper into all this.

 

[Fredrik]

is also very interesting, it's what i was trying to get at. I'll go delve deeper into all this.

Those things are pretty easy. I mean, it's hard to understand the definition of the field of real numbers, and the proof that a field with those properties exist in ZFC set theory, but it's easy to construct the complex numbers from the real numbers. ℝ is defined as an ordered field such that every set that's bounded from above has a least upper bound. (There are many such ordered fields, but they're all isomorphic, so it doesn't matter which one of them you call "the" field of real numbers. Just pick one of them and call it ℝ).

ℂ can be constructed as the field with underlying set ℝ², and addition and multiplication defined by

(a,b)+(c,d)=(a+c,b+d)
(a,b)(c,d)=(ac-bd,ad+bc)

It's easy to verify that this defines a field, and that the set of all members of the form (x,0) is a subfield. If we define a strict total order on that subfield by

(x,0)<(y,0) if and only if x<y

this subfield is isomorphic to ℝ. That means that its members satisfy all the requirements from the definition of the real numbers, so we might as well forget about the ℝ we started with, and use the symbol ℝ for this subfield of ℂ instead. Then we can call its members "real numbers".

Note that if we simplify the notation for (x,0) to just x, and define i=(0,1), we have (x,y)=x+iy.

 

[Studiot]

I don't know if you are coming at this from a physics/engineering point of view or a mathematical one.

One thing that is often done in physics/engineering is to chop the origin (or perhaps some other point) out of the complex plane. You can't do this with vectors.
This is done to avoid poles where a function, plotted on a complex plane goes to infinity.

Knowledge of poles is important because they are characteristic of the function and often lie on the boundary between different regions of behaviour in the real world system that the function represents.