$\mathbb{C}=\mathbb{R}[x]\mathrm{/}(x^2+1)$ as the Euclidean plane and as a normed real algebra: what the identification preserves

The complex field used here is the quotient $\mathbb{C}=\mathbb{R}[x]\mathrm{/}(x^2+1)$ of The complex numbers as $\mathbb{R}[x]\mathrm{/}(x^2+1)$, with the real embedding and imaginary unit $i$, with $i$ the class of $x$. The bijection $\mathrm{\Phi }(a+bi)=(a,b)$ of $\mathbb{C}$ is the real coordinate plane, with coordinate arithmetic carries addition and real scalar multiplication to the coordinatewise operations on ${\mathbb{R}}^2$, while complex multiplication becomes

$(a,b)(u,v)=(au-bv,av+bu)\mathrm{.}$

Thus $\mathbb{C}$ is the Euclidean plane as a real vector space, together with the additional bilinear operation of complex multiplication. A general real-linear map of the plane need not respect that operation and therefore need not be complex-linear.

The definitions in Real and imaginary parts, complex conjugation, and modulus make conjugation the reflection $(a,b)\mapsto (a,-b)$ and give $\mathrm{\mid }a+bi\mathrm{\mid }=\sqrt{a^2+b^2}$. In particular

$\mathrm{\mid }z-w\mathrm{\mid }=\parallel \mathrm{\Phi }(z)-\mathrm{\Phi }(w){\parallel }_2,$

so the metric, convergence, Cauchy, and continuity notions of The Euclidean metric, convergence, Cauchy sequences, and continuity on the complex plane are exactly their Euclidean-plane counterparts. The metric topology is consequently the usual topology on ${\mathbb{R}}^2$, in accordance with The product, Euclidean-metric and norm topologies on ${\mathbb{R}}^n$ agree, and for $n=1$ they agree with the real-line topology. Openness, connectedness, and real total differentiability will always be read through this identification.

The identification supplies no compatible field order. Indeed $i^2=-1$, while in any ordered field a square is nonnegative and $1$ is positive. This obstruction concerns the multiplication, not the Euclidean geometry: the plane still has its inner product and orientation, but neither orders $\mathbb{C}$ as a field.