Are the reals genuinely a subset of the complex numbers?

Here is slightly abstract point of view: When we say $A\subseteq B$, we don't actually mean that $A$ is literally a subset of $B$. The symbol $\subseteq$ and its cousins $\subset,\supset,\subsetneq,\supsetneq,\dots$ are sensitive to context. If $A$ and $B$ are considered as sets, then the subset symbols mean what you were taught that they mean. But when $A$ and $B$ bear some additional structure, like being groups, vector spaces, metric spaces, fields, etc., then $A\subseteq B$ means:

"There is a natural, i.e., in some sensible way unique, injective homomorphism $i:A\longrightarrow B$, and from now on we mean $i(A)$ when we write $A$."

What a homomorphism is depends on the exact kind of structure we're looking at. If it's groups, we mean a group homomorphism. If it's vector spaces, we mean linear maps. If it's topological fields (fields equipped with a topology such that addition, multiplication and inversion are continuous), then we mean continuous field homomorphism.

Some authors try to get around this overloading of $\subseteq$ by specifying that $\subseteq$ literally means a subset and $\leq$ means what I wrote above. But if you ask me, in the big picture, it's not at all helpful to distinguish between the two cases. I can't think of a single instance where we actually care that, for instance, $\mathbb R$ is not literally a subset of $\mathbb C$ according to the standard construction of $\mathbb C$. We only care about how things behave, not how they look. And $\mathbb C$ contains a unique subset which, if equipped with the restricted field operations, behaves exactly like $\mathbb R$. And it's just cumbersome to always write "$\mathbb C$ contains a subfield isomorphic to $\mathbb R$" instead of just writing "$\mathbb C$ contains $\mathbb R$". So we do the latter.

Also, here's a better, less ambiguous way to talk about the subject: let $F$ be a field. A subfield of $F$ is a pair $(E,i)$, where $E$ is a field and $i:E\to F$ is an injective field homomorphism (field homomorphisms are automatically injective, but injectivity is needed for other structures). "Classical" subfields in the sense of subsets which are also fields are subfields in this sense if we take the natural inclusion mapping as $i$. And if $i$ is clear from context, we just say $E$ to be a subfield of $F$. Take this definition and then just say $\mathbb R$ is a subfield of $\mathbb C$, without using the subset symbol. It's what we care about, anyway: that one is a subfield of the other.


This is actually one of the problems with the idea of building maths purely on sets alone as foundations. If we take it strictly and formally, we get various statements which may or may not be true, subject to just how we have or have not constructed a particular object. For example, it gets even worse than what you are talking about: in a purely set-theoretic construction, natural numbers are sets, too, and thus we can ask whether, say,

$$1 \subseteq 3$$

and the answer to this is "it depends on your set-theoretic construction"!

For me, what I suggest is that this problem is very reminiscent of one often seen in computer programming: in computers, we have something similar going on in that everything we work with - pictures, sound, text, whatever - ultimately gets represented by the same "stuff": bits. And thus, if one does not have safeguards in place, one can try to interpret, say, the bits corresponding to text as a picture, or a picture as text, or conversely. Of course, what you get will be mostly scramble and nonsense, but you can do it, and the computer won't care.

So to deal with this, we need some way to encode that semantic information - that these two pieces of bits are semantically different - into the language in question.

And the way that is handled in computer programming is to use programming languages that require a data type, to discourage the programmer from arbitrarily mixing of different sets of bits that are meant to represent different things. Data typing attaches a semantic tag to each bit of data to say that it should represent a picture or text or a number, say, and then you cannot, in the same program, freely mix the two.

Likewise, this concept is not unheard of in maths - "type theory" explores a whole array of foundational systems and languages that use something very similar, and indeed both of these fields of application are closely related - but it's not the "standard consensus" foundation for maths.

But were we to use a typed foundation, I'd suggest the answer would best be thought of as a "no, but": the real numbers are not a subset of complex numbers, but we have the "type coercion" rule

$$x \mapsto (x, 0)$$

which allows us to "upgrade" a real number, should it be combined with a complex number in an expression, to a complex number. Such rules often feature in programming languages as I just mentioned, too. In general, they must be defined along with the types in question, but are typically based on whether or not "natural" correspondences of the kind you are perceiving here, exist.


1.- As already stated in your question, no: formally we can't say the reals are even a subset of the complex numbers. Yet there are many way in which we can embed $\;\Bbb R\;$ into $\;\Bbb C\;$ in such a way that the basic characteristics of these two fiels are mantained, and one of the most usual isomorphisms (=embeddings respecting the algebraic structure) is precisely $\;\phi:\Bbb R\to\Bbb C\;,\;\;\phi(r):=r+0\cdot i\;$ , or if you prefer the other very usual definition, $\;\phi(r):=(r,0)\;$. In this manner the real numbers become not only a subset but, as said, a subfield of the field $\;\Bbb C\;$ .

2.- Yes, we can define $\;\Bbb C\;$ as the algebraic closure of $\;\Bbb R\;$ , getting a fields extension of degree two, and in which in a rather canonical way, the basis field $\;\Bbb R\;$ is embedded in a the extension field $\;\Bbb C\;$ . In the sense of a theorem by Artin, this is the only possible algebraic extension of degree two of a real closed field.

Tags:

Complex Numbers

Abstract Algebra

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