TITLE: What allows us to use imaginary numbers?
QUESTION [34 upvotes]: What axiom or definition says that mathematical operations like +, -, /, and * operate on imaginary numbers?
In the beginning, when there were just reals, these operations were defined for them. Then, i was created, literally a number whose value is undefined, like e.g. one divided by zero is undefined.
Does anyone know about how mathematical operations' ranges and domains were expanded to include imaginaries?
EDIT: An interesting comment notes a first use of complex numbers where,

those values would cancel in the end.

But can I refute that with, "from an inconsistency, anything is provable"?

A corollary question: Could I define a new number z which is 1/0 and simply begin using it? Seems ludicrous.

REPLY [2 votes]: Several answers have already explained about the definitions involved in setting up the complex number system and its arithmetical operations. I'd like to add a little about the word "use" in the title of your question, because using numbers can involve lots of different things. For example, when dealing with real numbers, we can invoke various algebraic laws (associative, commutative, distributive, etc.) to prove things like $a^2-b^2=(a+b)(a-b)$. And we can invoke laws about the ordering relation and multiplication to prove things like $a^2\geq0$. Some of these things continue to work with complex numbers (for example $a^2-b^2=(a+b)(a-b)$) but others don't ($i$ is a counterexample to $a^2\geq0$). 
So the title of your question should really be "which of the familiar laws for the real number system continue to work for complex numbers, and why?" The answer to "which ... continue to work" is essentially that laws that are exclusively about addition, subtraction, multiplication, and division continue to work whereas laws about the ordering relation ($<$ or $\leq$) do not. (One doesn't normally even try to define, say, whether $i<1$ or $i\geq1$.)  
And why do the laws about $+,-,\times,/$ continue to work? That needs to be proved --- and it has been proved (a few centuries ago). 
If you think about all the formulas about $+,-,\times,/$ that you had to learn in high-school algebra, proving them all again for complex numbers might look like an awful lot of work. But fortunately, they can all be deduced from a rather small subcollection of laws (as I mentioned above for $a^2-b^2=(a+b)(a-b)$). So only that subcollection needs to be checked specifically for the case of complex numbers; the rest of the high-school laws then follow for complex numbers just as they did for real numbers with no additional work. The relevant subcollection, from which the other laws about $+,-,\times,/$ follow, is called the field axioms, and that's why several of the previous answers talk about the complex numbers being a field, i.e., satisfying the field axioms.