I was somewhat deliberate in not actually saying what a "vector" is, since (as I mentioned) vectors are actually a fairly abstract concept that involve pulling in several other abstract concepts to define.

The big conceptual leap to make is that mathematicians tend to avoid defining things by what they are; but instead define things by what properties they poses. So, an ordered pair such as (c, d) is not a vector except in a context where you also defined operations on it that satisfy the properties that we want vectors to satisfy. I'll put a more complete description on what those properties are later in this post.

> We write a complex number as c+di why not simply write it as (c, d)?

As I mentioned in my previous post, it is also possible to define complex numbers not as an ordered sequence of 2 real numbers, but instead as a 2x2 matrix of the form:

a b
-b a

By making complex numbers a matrix, we can re-use many facts we already know about matrices when studying complex numbers.

It is also possible to define complex numbers as a type of modular arithmetic of polynomials, denoted by R[x] / <x^2 - 2>. The R[x] here says to start with all polynomials with real coefficients, and the /<x^2 + 1> says to consider them mod x^2 + 1 [0]. For instance, we have:

x^3 = x(x^2 + 1) - x mod x^2 +1
x^3 = x(0) - x mod x^2 +1
x^3 = -x mod x^2 +1

Similarly, you have:

x^2 = (x^2 + 1) - 1 mod x^2 + 1
x^2 = 0 - 1 mod x^2 + 1
x^2 = -1 mod x^2 + 1

So "x" is (almost by definition) a square root of -1. This structure also gives you the complex numbers.

For the most part, we don't actually care what construction of the complex numbers we are using. As long as we know that all the possible constructions are equivalent, we can switch between them freely based on what is most convenient.

When you write (c,d) you are committing to thinking of complex numbers as an ordered pair of 2 real numbers. This makes it harder to both think about complex numbers as their own abstract object, or as matrices, or as polynomials. However, in some circumstances it is useful to think of complex numbers in those other ways.

In contrast, when you write complex numbers as c+di, you are making no statement about what complex numbers actual are. As long as someone's idea of complex numbers includes:

1) some square root of -1 that is written by "i"
2) An obvious way of turning a real number into a complex one ( e.g, x -> (x,0) ).
3) Addition/subtraction/multiplication/division that work as expected

They can use the above notation.

Additionally, the c+di notation allows you to use complex numbers without decomposing them. For instance, you can say: x=c+di, then carry around x as a single term.

You could, in theory, also say x=(c,d) and carry x around, but then you are just a few short steps away from someone saying:

i = (0,1)
1 = (1,0)
2 = (2,0)
3.7 = (3.7, 0)
r = (r, 0) for all real numbers r.

At which point you are right back to the c+di notation.

> Then a complex number's two parts could both be complex numbers too right?

Kind of. In all of the definitions of complex numbers I gave, it was assumed that each individual component is a real number. Letting components themselves be complex means that you are defining complex numbers in terms of complex numbers. Circular/recursive definitions like this are not impossible, but are more difficult to reason about (and I'm not sure how well it would work in the case of complex numbers).

In terms of the c+di notation, really c,d,i are all just complex numbers, and c+di is what you get when you multiply d with i and add c. We tend to write them with c and d being real numbers because that is the "simplest" form. For instance, if we had c=(x+yi) and d=a+bi, we would have:

c+di = (x+yi) + (a+bi) = (x+a) + (y+b)i

At which point we might as well compute m=x+a and n=y+b and just write:

m + ni

> Multiplying such numbers could mean simply multiplying their element complex numbers.

This doesn't work. Or, at least, does not give you the complex numbers. If the complex numbers are C, the structure you just described is called C x C, or the cartesian product of C by itself. To see why this is not equivelent to the complex numbers consider the following mulitplication in C x C:

(0+0i, 0+1i) * (0+1i, 0+0i) = (0,i) * (i,0)

Multiplying these elements indiviually gives us:

(0,i) * (i,0) = (0,0)

However, in the complex numbers, the only way to get to 0 by multiplication is if one of the factors is also 0.

> Just to be sure you would say that a vector is an ordered sequence of numbers or vectors?

Not quite. At this point, I will just link to the definition of vector [1]. Essentially, it is just a list of properties that a structure must have. Any structure with those properties is a vector.

Any ordered sequence of "numbers" (assuming a suitable definition of "number") becomes a vector when you attach to it:

A) the obvious definition of addition: (a,b,...,k) + (m,n,...z) = (a+m, b+n, k+z)

B) and the obvious definition of multiplying a sequence of numbers by a single number: a(m,n,...,z) = (am,an, ..., az)

A sequence of 2 real numbers is therefore a vector over the real numbers when you attach the above notions of addition and multiplication.

Notice that the above says nothing about multiplying 2 vectors together. It does not always make sense to multiply 2 vectors together; and when it does, there are often different notions of multiplication you can use. The defining feature of complex numbers is arguably how they behave under multiplication with other complex numbers, which has nothing to do with them being vectors.

Even though every ordered sequence of "numbers" is a vector, not every vector is a ordered sequence of "numbers"

For example, the set of all real functions, { f : R -> R } for a vector-space relative to the real numbers, with addition and multiplication defined as follows:

(f + g)(x) = f(x) + g(x)
(a * f)(x) = a * f(x)

However, it is impossible to express all such functions as sequence of numbers.

> Just to be sure you would say that a vector is an ordered sequence of ... vectors?

Technically, any ordered sequence of vectors is also an ordered sequence of numbers. E.g ((a,b), (x,y)) can be thought of as just (a,b,x,y). So an ordered sequence of vectors is itself a vector.

However, it seems like you are thinking of something more specific. Specifically, Normally, when we think of a vector as an ordered sequence, each component of that ordered sequence is the same type of object as a scalar is. This only works for an ordered sequence of vectors if the vectors themselves can also be scalars. As it turns out, the complex numbers (with their particular notion of multiplication) happen to fit the requirements to be a scalar as well, but most vectors do not.

> Why do we need BOTH vectors AND complex numbers? Or would it make it easier to say that complex numbers are a subset of vectors?

I think I answered this, but to make it explicit. The complex numbers are an example of vectors, but not all vectors are complex numbers.
The complex numbers have interesting properties that are unrelated to them being vectors.
Vectors have proven properties that can be applied to any example of vectors. This allows us to apply everything we know about vectors to the complex numbers; even things we learned about vectors while studying very different types of vectors.

[0] Technically, the notation in R[x] / <x^2 - 2> is a bit more general then I am describing here, but not by much.

[1] https://en.wikipedia.org/wiki/Vector_space#Definition