How do we Distinguish the Complex Numbers from Euclidean $\mathbb R^2 $?

Question

We know the two are of equal Cardinality $\mathbb c \times \mathbb c $, diffeomorphic as Real 2-manifolds, though I am not sure that Euclidean 2-Space is a Complex 1-manifold. The two are homeomorphic per the map $(x,y) \rightarrow x+iy $ I guess the Complexes are an algebra ( over the Reals) and the Euclidean Reals are not? I am also confused as to why/how z=x+iy is a single (Complex) variable: what "binds" x,y (both Real numbers) into the (single) variable $x+iy$? Thanks.

Answer 1

You got it. $\mathbb C$ is an $\mathbb R$-algebra, while $\mathbb R^2$ is not (naturally speaking; c.f. Noah Schweber's comment). In fact you can define $\mathbb C$ as $\mathbb R^2$ with a fancy operation added onto it, namely $$(a,b)\star(c,d)=(ac-bd,ad+bc)$$

Answer 2

As you say, they are indeed very similar objects! We're always thinking of ordered pairs of real numbers, $(x,y)$; the only difference is how we think about their structure. If you like, each of the definitions goes beyond the set of pairs, there is some baggage associated with it. When we speak of the complexes, the definition includes the field structure (the operations $+$, $-$, $\times$, $\div$ except by $0$). When we think of $\mathbb R^2$ we do it as a vector space, so there is a linear structure on it. The addition and subtraction operators are also present in the complexes, but now you have a scalar (dot) product, and often we also the matrix algebra ($2\times 2$ matrices) associated with linear maps.

Can you always turn complexes into $\mathbb R^2$ and vice versa? Of course! We just choose whichever setting is more convenient or natural. If you are thinking of Euclidean geometry or linear algebra, you might use vectors and matrices. If you are thinking of polynomial algebra, you might prefer to use the complexes. But there is nothing stopping you from expressing the complex-valued solutions to a quadratic equation as vectors in $\mathbb R^2$. It's just that many of us prefer the complex notation because that's how we were taught.

Hope this helps!