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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.)

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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.

If desired you can think about things in terms of isomorphism classes of set-theoretic objects that have the desired properties. This can be a bit hard to make rigorous, though, since these classes need not necessarily be sets.

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  • $\begingroup$ 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 $\mathbb C$)? $\endgroup$ – Aaron Hall Jan 13 '16 at 1:03
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    $\begingroup$ @AaronHall If all you ever use is addition, subtraction, multiplication, division, and all the notions that come out of the metric, there is no reason to consider them different things at all. (Note that the order is not on this list!) $\endgroup$ – Ian Jan 13 '16 at 1:05

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