True or False: $i^2 = -1$, $\mathbb{C} = \mathbb{R}^2$

Question

The complex numbers are typically defined as the set of all ordered pairs $(x,y),$ where $x,y \in \mathbb{R},$ along with the usual operations of addition and multiplication (which I won't write out here.)

The standard argument then goes that, by defining $i = (0,1),$ the rule of multiplication on the complex numbers gives $i^2 =(0,1)(0,1) = (-1,0);$ therefore $i^2 = -1.$ But this seems to me to just be $\it wrong.$ Although there clearly exists an isomorphism that maps $i^2$ to $-1,$ it is in fact the case that $i^2$ is an ordered pair; not a real number.

Am I missing something here? Are mathematicians just being lazy/careless when saying that $i^2 = -1?$ This is my first question.

My second question has to do with the definition of $\mathbb C:$

I have always made a distinction between the $\it set$ $\mathbb R$ and the $\it field \ \mathbb R.$ I make a similar distinction with $\mathbb R^2.$ With this in mind, I am slightly confused about $\mathbb C.$ Is $\mathbb C$ only a field, and not a set? Does such a thing as "the set of all complex numbers" even exist? For the underlying set in the field $\mathbb C$ is clearly $\mathbb R^2,$ when $\mathbb C$ is defined as above.

Thank you for your responses.

$\bf Edit:$ Please read my question. Nobody who responded seems to be paying attention to anything I wrote beyond the title of my post.

Answer 1

The idea that $\mathbb{C} = \mathbb{R}^2$ is false. In $\mathbb{R}^2$ there is no meaning to $(x_1,x_2)\cdot(y_1,y_2)$. It is true that $\mathbb{C}$ satisfies all of the axioms for a field (associative, commutative, etc.) so it is a field.

I feel that this excerpt from Conway's "Functions of One Complex Variable" answers your other question better than I ever could.

We will write $a$ for the complex number $(a,0)$. This mapping $a \to (a,0)$ defines a field isomorphism of $\mathbb{R}$ into $\mathbb{C}$ so we may consider $\mathbb{R}$ a subset of $\mathbb{C}$ (Page 1).

Answer 2

There is no such thing as the set of complex (or real) numbers. One can say that there exists, unique up to isomorphism, field of complex (or real numbers). These fields have many models. Besides the one you know, one can model complex numbers for instance by certain real 2-by-2 matrices. I can write more details if this is unclear.

Edit: Incidentally, there is no such thing as a complex number $i$ until you fix a model of complex numbers. What is well defined, however, is the set of roots of the polynomial $z^2+1$.

Personally, my favorite definition of C is as the algebraic closure of R. Once you have the fundamental theorem of algebra, you can then identify C with the set of pairs of real numbers (as a model).