Complex Numbers as Vectors: An Exploration of Their Properties and Applications

[7777777]

I am confused if complex numbers really are vectors. They seem to behave as vectors
in addition, but not in multiplication. So why are the complex numbers defined to be vectors
although they don't follow the same principles always.

Another confusing thing for me is the "complex vector", is this entity some kind of combination
of a normal vector and a complex number? For example if z is a complex number and r
is an ordinary vector, what is then z*r? Is it a complex vector?

 

[Pythagorean]

I believe complex numbers are not the same as vectors, but you can sometimes treat vector quantities with complex notation.

Interestingly, you can see i as a transform on a quantity that puts it in the orthogonal plane (a 90 degree rotation). Thus i^2 is a 180 degree roration (-1). And in his way, you can see an eigenbasis (x-hat, y-hat) as (x-hat, i*x-hat).

 

[jedishrfu]

Historically complex numbers were developed to address quadratic polynomial solutions that were not defined under real numbers.

Later, Hamilton extended complex numbers to quaternions using the notation a+bi+cj+dk and tried to reformulate physics using quaternions.

However, some physicists felt quaternions were too complicated for what was needed at the time and so extracted out and simplified quaternions into vectors keeping the i,j,k notation interpreting it as unit vectors in the x,y,z directions.

http://en.wikipedia.org/wiki/A_History_of_Vector_Analysis

More recently, people are revisiting quaternions because of their elegant rotational properties.

http://en.wikipedia.org/wiki/Quaternions

So the similarity of vectors and complex numbers is not simply a coincidence.

There are also complex vector spaces showing that complex numbers can be embedded inside a vector as the components of the vector.

 

[7777777]

I did some research myself and found that a complex vector is a vector with complex coefficients,
for example:
ae1+ be2+ ce3 where a, b and c are complex (I have used "e1", "e2", and "e3" as basis vectors rather than "i", "j", and "k" so as not to confuse the basis vector "i" with the imaginary unit, "i").

I think that this means that a complex number should be understood to be a number (as its name implies) because it can be a coefficient for a vector. It is hard for me to think that a vector could be a coefficient of another vector.

 

[jedishrfu]

I think that this means that a complex number should be understood to be a number (as its name implies) because it can be a coefficient for a vector. It is hard for me to think that a vector could be a coefficient of another vector.

In a practical sense, that is true. However, often mathematicians will also view complex numbers as an entity over an R x R space with special operations... in the interests of group theory and abstract algebra.

 

[Number Nine]

I did some research myself and found that a complex vector is a vector with complex coefficients,
for example:
ae1+ be2+ ce3 where a, b and c are complex (I have used "e1", "e2", and "e3" as basis vectors rather than "i", "j", and "k" so as not to confuse the basis vector "i" with the imaginary unit, "i").

I think that this means that a complex number should be understood to be a number (as its name implies) because it can be a coefficient for a vector. It is hard for me to think that a vector could be a coefficient of another vector.

Do you know the formal definition of a vector space? The complex numbers form a field. They can also be treated as a vector space over themselves (since they are an abelian group). Just as \mathbb{R} can be treated as a one dimensional vector space over itself.

 

[7777777]

Do you know the formal definition of a vector space? The complex numbers form a field. They can also be treated as a vector space over themselves (since they are an abelian group). Just as \mathbb{R} can be treated as a one dimensional vector space over itself.

I don't usually care about formal definitions if they don't match common sense. I can try to
consider them if they make sense.
Wikipedia says that "When the scalar field F is the real numbers R, the vector space is called a real vector space. When the scalar field is the complex numbers \mathbb{C}, it is called a complex vector space".

It seems to me that this definition is contradictory, it assumes that complex numbers are
scalars, but you can define them to be vectors, just like the real numbers which are also
scalars but one can call them vectors by defining them as vectors (although they are not vectors but scalars). You can define a cat to be a dog, but it does not make a cat into a dog.

The problem is, take any complex number z=b+c*i, the real part is b, just a real number, and not a vector. But does this real number b become a vector just by defining
\mathbb{C} to be a complex vector space?

The point is again the same as my original question, can you define a complex number to
be a vector? It seems to me that you can define whatever you want, but there should
be reasons for it, and in this case there does not seem to be enough reasons to call or define
a complex number as a vector.

 

[Mark44]

I don't usually care about formal definitions if they don't match common sense.

What are the "common sense" notions about vector spaces?

I can try to
consider them if they make sense.
Wikipedia says that "When the scalar field F is the real numbers R, the vector space is called a real vector space. When the scalar field is the complex numbers \mathbb{C}, it is called a complex vector space".

It seems to me that this definition is contradictory, it assumes that complex numbers are
scalars, but you can define them to be vectors, just like the real numbers which are also
scalars but one can call them vectors by defining them as vectors (although they are not vectors but scalars). You can define a cat to be a dog, but it does not make a cat into a dog.

Context is everything here. A scalar field can consist of real numbers or complex numbers or other numbers. For a given vector space, the coordinates of the vector come from the scalar field.

As you note, the real numbers can also be considered a vector space, as can the complex numbers.

The problem is, take any complex number z=b+c*i, the real part is b, just a real number, and not a vector. But does this real number b become a vector just by defining
\mathbb{C} to be a complex vector space?

A vector is anything that belongs to a vector space, a term that is defined in terms of two operations (vector addition and scalar multiplication) and a set of axioms that the operations must satisfy. If the focus of attention is complex numbers, then it is not useful to think of the components (the real part and the imaginary part) as vectors themselves.

The point is again the same as my original question, can you define a complex number to
be a vector?

You can't define something to be a vector, but if it satisfies the 10 or so axioms of a vector space, then it is a vector. Whether it's useful to think about that thing as a vector depends on what you're trying to do.

It seems to me that you can define whatever you want, but there should
be reasons for it, and in this case there does not seem to be enough reasons to call or define
a complex number as a vector.

 

[Number Nine]

I don't usually care about formal definitions if they don't match common sense. I can try to
consider them if they make sense.

I'm sorry, but this is mathematics; "vector space" has a precise definition. You either know the correct definition or you don't. Maybe you can favour us with a "common sense" definition of the stalk of a sheaf of rings on a topological space?

It seems to me that this definition is contradictory, it assumes that complex numbers are
scalars, but you can define them to be vectors, just like the real numbers which are also
scalars but one can call them vectors by defining them as vectors (although they are not vectors but scalars). You can define a cat to be a dog, but it does not make a cat into a dog.

This is precisely why you need to understand the proper definition. A vector space is constructed by considering a field S alongside an abelian group V, with an operation S \times V \rightarrow V such that

1a) s(v + w) = sv + sw
1b) (v + w)s = vs + ws
2b) (s + r)v = sv + rv
2b) v(s + r) = vs + vr
3) s(rv) = (sr)v
4) 1v = v

Where s,r \in S, v,w \in V and 1 is the multiplicative identity in S.

The elements of S are called scalars, the elements of V are called vectors, and we say that V is a vector space over S. Now, taking both S and V to be the complex numbers, we see that \mathbb{C} can be treated as a vector space over itself.

 

[1MileCrash]

The complex numbers certainly form a 2 dimensional vector space over the reals, you can view them that way if that's how you want to. It's how I view them.

What Mark44 and Number Nine are telling you is that a "vector" can be anything, as long as its containing set qualifies as a vector space. Saying that X is a "vector" doesn't tell me much about what kind of object X is at all, really. It could be a number, an ordered pair, a polynomial, a matrix, or maybe a horse.

 

[PSarkar]

I don't usually care about formal definitions if they don't match common sense. I can try to
consider them if they make sense.
Wikipedia says that "When the scalar field F is the real numbers R, the vector space is called a real vector space. When the scalar field is the complex numbers \mathbb{C}, it is called a complex vector space".

It seems to me that this definition is contradictory, it assumes that complex numbers are
scalars, but you can define them to be vectors, just like the real numbers which are also
scalars but one can call them vectors by defining them as vectors (although they are not vectors but scalars). You can define a cat to be a dog, but it does not make a cat into a dog.

Its true a cat is not a dog. But why? Because they have some definition and it would be a contradiction if a cat is a dog. Its not as if there is some "common sense" notion of cat and dog which we some how just knew when we came into this world. We had to learn what a cat and dog is.

Same thing for scalars and vectors. They have definitions. So what is a vector really? The answer depends on what level of math you are studying. In lower levels, it will be taught as something which has magnitude and directions etc. In higher levels, the actual definition (which is used by mathematicians) is much more general and precise. A vector is just an element of a vector space (over some field) and a scalar is an element of the corresponding field. Vector spaces and fields have precise definitions which other users have provided. Given these definitions, real numbers do indeed form a vector space over the field of real numbers. So in this case, both vectors and scalars are real numbers. Also, the actual mathematical objects that are vectors and scalars vary depending on the vector space and field you working with, and they can even be the same thing as we just saw. But there is no contradiction as you claimed. It probably seemed so because you did not know the actual definition or did not understand it. Anyway, I hope it's clear now.

 

[7777777]

I found out that when you multiply two complex numbers the result will be another
complex number.
Whereas, when you multiply two vectors, the result will not be another vector.

Perhaps complex numbers should not be defined as vectors if they are not vectors.
Just like dogs are dogs, not cats, regardless where they live, even if they live
in a vector space.

 

[Pythagorean]

Whereas, when you multiply two vectors, the result will not be another vector.

That's not always true. "Multiply" takes on a new meaning in vectors and there is more than one way to multiply vectors, so people don't really use the term "multiply". They use terms like "inner product" and "outer product" to differentiate between the two main ways you multiply vectors.

the "cross product" of two vectors returns a vector.

 

[jedishrfu]

I found out that when you multiply two complex numbers the result will be another
complex number.
Whereas, when you multiply two vectors, the result will not be another vector.

Perhaps complex numbers should not be defined as vectors if they are not vectors.
Just like dogs are dogs, not cats, regardless where they live, even if they live
in a vector space.

You're trying to win an argument here when there is no disagreement. Mathematicians look at these from different angle to gain new insight. It makes sense from one perspective to treat complex numbers as numbers and it also makes sense to treat them as vectors. A mathematician limits the context to what needs to be studied.

Using your cats and dogs analogy, its easy to view them in this manner but you could also say they are 4-legged animals too and study them in that context.

With respect to multiplying complex numbers, yes you can either get another complex number or you can get a real number or an integer... but since integers are in the real number set and reals are in the complex number set
then you can say multiplying complex numbers produces complex numbers.

The same thing is true in with vectors and scalars, they are both can be considered tensors of rank 1 and 0 respectively. (see http://en.wikipedia.org/wiki/Tensor)

So try to see how both of these perspectives can be true and you will see deeper into the beauty of mathematics.

Historically vectors came from extending complex numbers into higher realms (quaternions and octonions). The difficulty of applying quaternions to physics led to the development of vectors.
Quaternions used the i,j,k notation that is also used in vector math with the i,j,k understood as sqrt(-1). There are a set rules for multiplying an i-term with a j-term...

 

[Pythagorean]

I think I could understand this better if someone helped me out with this:

Let's consider

(i+1)^2 = 1 + 2i + i^2 = 2i

the equivalent vector multiplication would be

(\hat{x} + \hat{y})^2 = (\hat{x} + \hat{y})?(\hat{x} + \hat{y}) = 2 \hat{y}

Does their exist a multiplying operation, ?, that returns that vector so that we can properly analogize vectors and complex numbers? The cross-product is ambiguous in 2D to me (if it even exists)...

 

[Jorriss]

Perhaps complex numbers should not be defined as vectors if they are not vectors.

What exactly is a vector to you?

 

[economicsnerd]

Here's an exercise:

Go to Wikipedia to see precise definitions, then verify the following:
i) F= \{0,1\} is a field, when addition is given by a+b = \begin{cases} 1 &amp;\text{ if } a\neq b \\<br /> 0 &amp;\text{ if } a=b\end{cases} and multiplication is given by ab = \begin{cases} 1 &amp;\text{ if } a=b=1 \\<br /> 0 &amp;\text{ otherwise.}\end{cases}
ii) V=\{\text{cat}, \text{dog}\} is vector space (over scalar field F) with addition given by v+w = \begin{cases} \text{cat} &amp;\text{ if } v\neq w \\<br /> \text{dog} &amp;\text{ if } v=w\end{cases} and scalar multiplication given by av = \begin{cases} v &amp; \text{ if } a=1 \\ <br /> \text{dog}&amp; \text{ if } a=0.\end{cases}

------------------

If you can't do this exercise, you don't know what a vector space is.

Given a vector space, we sometimes call its elements "vectors". So, in the above vector space (which you've by now proven to be a vector space), \text{cat} and \text{dog} are now examples of vectors.

What does it mean to multiply \text{cat} and \text{dog}? I couldn't tell you. In general, there's no multiplication built into the definition of a vector space.

 

[1MileCrash]

the "cross product" of two vectors returns a vector.

Except not really.

 

[Mark44]

I found out that when you multiply two complex numbers the result will be another
complex number.
Whereas, when you multiply two vectors, the result will not be another vector.

The only two operations that are required for things that belong to a vector space (i.e., vectors) are addition and scalar multiplication.

So when you add two complex numbers, the result is a complex number. When you multiply a complex number by a scalar, the result is a complex number. The fact that complex numbers can be multiplied while vectors in say, R³, can't, is just happenstance. It doesn't have anything to do with each of these kinds of objects being vectors.

Perhaps complex numbers should not be defined as vectors if they are not vectors.
Just like dogs are dogs, not cats, regardless where they live, even if they live
in a vector space.

Any object that belongs to a vector space (and therefore satisifies axioms of a vector space) is a vector, period.

 

[Mark44]

This thread has gone on long enough...