What exactly is a number? We've just been learning about complex numbers in class, and I don't really see why they're called numbers.
Originally, a number used to be a means of counting (natural numbers).
Then we extend these numbers to instances of owing other people money for instance (integers).
After that, we consider fractions when we need to split things like 2 pizzas between three people.
We then (skipping the algebraic numbers for the purposes of practicality), we use the real numbers to describe any length; e.g. the length of a diagonal of a unit square.
But this is when our original definition of a number fails to make sense- when we consider complex numbers, which brings me to my main question: what is a rigorous definition of 'number'?
Wikipedia claims that "A number is a mathematical object used to count, label, and measure", but this definition fails to make sense after we extend $\mathbb{R}$.

Does anyone know of any definition of number that can be generalised to complex numbers as well (and even higher order number systems like the quaternions)?


This doesn't exactly answer your question, but here are some characteristics all numbers I know of satisfy.(I know of counting numbers, integers, rational numbers, real numbers, complex numbers, quaternions and octonions).
Number 1: All the counting numbers seem to be equipped with two binary operations which receive the name of addition and multiplication.
Number 2: They are an extension of the counting numbers, in the sense that there is a function from the counting numbers to all the other number systems such that $f(a)+f(b)=f(a+b)$ and $f(a)\times f(b)=f(a\times b)$ for $a$ and $b$ counting numbers.
Unfortunately I'm still young inexperienced and those were the only ones I could think of.
