Kdtyjb

We refer to complex numbers as numbers. However we refer to vectors as arrays of numbers. There doesn’t seem to be anything that makes one more numeric than the other. Is this just a quirk of history and naming or is there something more fundamental?

The two fundamental operations for numbers are "addition" and "multiplication" which obey very nice "laws" of arithmetic. Taking powers is also important. You can do all of those things with complex numbers. You can add two vectors but the "dot" product of two vectors is not a vector and the "cross" product of two vectors does not satisfy the "nice laws". Neither the dot product nor the cross product of vectors can be used to define powers.

They're called "numbers" for historical reasons, since the motivation in the development of the complex numbers was solving polynomial equations. They were viewed as natural extensions of the real numbers. It's somehow quite natural and satisfying to say "every polynomial equation can be solved by some (complex) number". Is it more natural to regard $i$ as being a number which, when squared, is equal to $-1$, or is it more natural to regard $i$ as being some non-number thingamajig which when squared is equal to $-1$? Clearly the former.

"Higher" number system, like quaternions, aren't really called numbers very often, for the simple fact that they are not as intimately connected with number theory and analysis in the same way that complex numbers are.

Beyond these social conventions, I can't see any other reason. The word "number" doesn't have a strict or absolute definition in pure mathematics. $mathbb{N}$, $mathbb{Z}$, $mathbb{Q}$, $mathbb{R}$ and $mathbb{C}$ are technically just sets with a certain algebraic structure.

I partially disagree with the other answers which claim that complex numbers are numbers simply by virtue of the fact that you can add and multiply them. Well, if that's the rationale, is every ring also a set of numbers?

Complex numbers do not have the same properties as vectors but they have similar properties (see dot product). We can represent complex numbers as vectors, but that is just a representation.

Complex numbers are also called imaginary numbers because the first appearance of them was quite confusing. They appeared in the solution of the cubic equation for the case of three distinct real roots. It took quite a time until people understood that they could calculate with complex numbers like with normal numbers but will some additional rules e.g. $i^2=-1$.

Complex numbers also do not have a comparison operator like $<$ or $>$. For example, assume
$i<0$ then $icdot i > 0 implies -1>0$ which is wrong. $i=0$ makes no sense. And $i>0$ then $icdot i >0 implies -1 >0$ which is wrong.

Hence, the name makes sense as these numbers have some imaginary or complex behavior which normal numbers do not show.

Numbers appeared first when we started to count things. $1$ tree, $2$ trees, $3$ trees, and so forth. That make up the set of non-zero natural numbers: $mathbb{N}$. Afterwards, people started to "count backwards" to get $mathbb{Z}$ (I'm just kidding, you can do some research on how negative numbers appeared historically). With the urge to divide things without remainders, the set of rational numbers $mathbb{Q}$ made its way to the world. A certain idea of geometric continuity gives us $mathbb{R}$. Finally we want all equations to have a root, that's how $mathbb{C}$ comes into play.

I guess what is considered "numbers" is rather a social question. The use of $mathbb{C}$
in physics and its $2$-D representation must have given a good intuition for a large set of people to accept it's intuitive enough to be considered "numbers".

I think it's not that natural to think of $mathbb{C}$ as a $mathbb{R}$-vector space of dimension $2$, not more natural than to think of $mathbb{R}$ as an infinite-dimensional $mathbb{Q}$-vector space, nor of $mathbb{Q}$ as a non-infinitely-generated $mathbb{Z}$-module.