# The relationship between the real and complex numbers from a set theoretic perspective

From a set theoretic perspective, how is the set of real numbers $\mathbb R$ and the set of complex numbers $\mathbb C$ related? If we look at complex numbers as numbers of the form $z = a + bi$ with $a,b \in \mathbb R$ and $i^2 = -1$, one might be inclined to conclude that $\mathbb R \subset \mathbb C$ because every real number $n$ is a valid complex number of the form $n + 0i$. On the other hand, we can also look at complex numbers as vectors or 2-tuples of the form $z = (a, b)$, with $a = Re(z)$ and $b = Im(z)$. In this case, a real number $n$ is not an element of $\mathbb C$ because $n \neq (n, 0)$, and we would conclude that $\mathbb R \not\subset \mathbb C$ (in fact, $\mathbb R$ and $\mathbb C$ would be disjoint sets). So which is it? Do we consider one notation/perspective to be the "correct" one and treat the other as a somewhat inaccurate alternative, or does the perspective (and thus whether or not $\mathbb R \subset \mathbb C$) vary depending on the context? (I'm not ruling out the possibility that I misunderstand one or both of the perspectives.)

• Jan 13, 2016 at 1:43
• @HenningMakholm Huh, interesting. I liked the first answer's reminder that "informal math" can often be quite different than formal math. Jan 13, 2016 at 1:47

We typically work agnostic of the exact set-theoretic structure. For instance, we have containment of $\mathbb{R}$ in $\mathbb{C}$ in the sense that there is an embedding, by which I mean an injective, bicontinuous field homomorphism. Whether we have containment in the sense of set theory doesn't really matter. Note that this agnosticism is not specific to the containment of $\mathbb{R}$ in $\mathbb{C}$. For instance, equivalence classes of Cauchy sequences of rationals and Dedekind cuts provide two different set-theoretic constructions of $\mathbb{R}$, even if your set-theoretic construction of $\mathbb{Q}$ is already fixed.
• Makes sense. Is there ever a time where a real number $n$ and its equivalent complex number $n + 0i$ are viewed as different objects related only by isomorphism, or are they always considered the same thing (even if we don't consider $\mathbb R$ to be a subset of $\mathbb C$)? Jan 13, 2016 at 1:03