TITLE: Are the reals genuinely a subset of the complex numbers?
QUESTION [21 upvotes]: In Michael Spivak's Calculus, he defines a complex number as an ordered pair of real numbers: $z=(a,b)$ with $a,b \in \mathbb{R}$. The imaginary unit $i$ is then just a shorthand for the ordered pair $(0,1)$. Spivak goes on to say

When complex numbers were first introduced, it was understood that real numbers were, in particular, complex numbers; if our definition is to be taken seriously then this is not true—a real number is not a pair of real numbers, after all.

Although the complex number $(a,0)$ behaves in pretty much the same way as the real number $a$, they are still not identical. There seem to be some non-trivial differences, as well: we can't write $(5,0)>(3,0)$ in the way we can write $5>3$. With this in mind, I ask the following questions:

Can the real numbers be said to be a subset of the complex numbers if complex numbers are defined as ordered pairs?
If the complex numbers are constructed in some other way, then is it meaningful to write $\mathbb R \subset \mathbb C$?

REPLY [9 votes]: There are a lot of "inclusions" of a similar character, for example $\mathbb N \subset \mathbb Z \subset \mathbb Q \subset \mathbb R$. It depends on your interpretation of what these sets are whether you regard $\subset$ as a genuine inclusion or if you regard $\subset$ as a notation indicating that there is a canonical embedding of the object on  left hand side as a subobject of the object on the right hand side.
Frequently one starts on the level of axiomatic set theory and then successively constructs $\mathbb  N$, $\mathbb Z$, $\mathbb Q $, $\mathbb R$ etc. The usual constructions produce an "ascending  sequence" of algebraic objects which are not related by genuine inclusions, but the respective construction yields a new object plus an embedding of the original object into the new one. Recall the construction of $\mathbb Q$ as a set of equivalence classes of pairs in $\mathbb Z \times (\mathbb Z \setminus \{0\})$.
However, usually there are various alternative constructions supplying specific models of the sets $\mathbb N, \mathbb Z, \mathbb Q, \mathbb R$ etc. But I think  most people have a more abstract (or, if you want,  more intuitive) understanding of these objects. We know what their purpose is, which operations can be performed and which rules are satisfied. The specific construction is then fairly irrelevant. I would call that an axiomatic point of view. For example, the reals can be constructed based on $\mathbb Q$ via Dedekind cuts,  via nested intervals or via Cauchy sequences in  $\mathbb Q$, and certainly there are even more approaches. But I doubt the anybody imagines a real number as a Dedekind cut or as an equivalence class of Cauchy sequences.
Therefore the concrete model of an object as $\mathbb R$ is fairly uninteresting, it can be replaced without problems by any other model having the same properties.
In your case we start with some model of $\mathbb R$ and construct $\mathbb C$ as the set of real pairs. Then $\mathbb R' = \mathbb R \times \{0\}$ is a subfield of $\mathbb C$ which is canonically isomorphic to the original $\mathbb R$ and we may now work with the new model $\mathbb R'$ which is a genuine subset of $\mathbb C$.
If you do not like this replacement, be aware that the standard construction of $\mathbb C$ is just one possible approach (but a very transparent one). It can be easily modified as follows. The simple idea is to add new (non-real) numbers to $\mathbb R$ instead of embedding $\mathbb R$ into $\mathbb R^2$ (which produces the isomorphic subfield $\mathbb R' = \mathbb R \times \{0\} \subset \mathbb C$ and causes the "philosophical discussion" whether, or in what sense, $\mathbb R$ is a genuine subset of $\mathbb C$).
Given $\mathbb R$, the sets $\mathbb R$ and  $\mathbb R \times \mathbb R^*$ (where $\mathbb R^* =  \mathbb R \setminus \{0\}$) are disjoint. Define
$$\mathbb C  = \mathbb R \cup (\mathbb R \times \mathbb R^*)$$
and
$$\phi : \mathbb C \to \mathbb R \times \mathbb R, \phi(\zeta) = \begin{cases} (x,0) & \zeta = x \in \mathbb R \\ \zeta & \zeta \in \mathbb R \times \mathbb R^*\end{cases} $$
This is a bijection. The usual complex field structure on $\mathbb R \times \mathbb R$ is then transferred via $\phi$ to our modified $\mathbb C$. Doing so, we get a genuine inclusion $\mathbb R \subset \mathbb C$. But, to be honest, I can't see that this has an added value. Anyway, our model of $\mathbb C$ is a real vector space with basis $\mathcal B  = \{1 \in \mathbb R, i = (0,1) \in \mathbb R \times \mathbb R^* \}$. Noting that $x\cdot 1 = x$ for $x \in \mathbb R$ and writing $y \cdot i = i y$ for $y \in \mathbb R$ we get
$$\mathbb C = \{x\cdot 1 + y\cdot i = x + iy  \mid x,y \in \mathbb R\} .$$
Here the complex number $x + i0$ is literally the same as the real number $x$.
Using similar methods we can of course construct a chain of genuine set inclusions
$$\mathbb N \subset \mathbb Z \subset \mathbb Q \subset \mathbb R \subset \mathbb C .$$