Given that complex numbers are of the form a + bi where a and b are real and i is the imaginary unit, is it correct to say that:
- All imaginary numbers are also complex numbers where a = 0, and
- All real numbers are also complex numbers where b = 0
?
Yes, that’s technically correct.
That’s the best kind of correct! 
Thanks.
Moreover, that the reals are contained in the complex numbers is pretty much the point. They’re an extension of the system of real numbers that can be used to solve equations (and many, many other things) that otherwise couldn’t be done. That is, the “incompleteness” of the real numbers motivates the constuction of the complex numbers. When we don’t need them, we don’t care about them, but when we do they’re there, and with more or less the same mathematical rules as the reals that came before them.
To get really technical (sorry), the real numbers are isomorphic to the subset of complex numbers with 0 imaginary part. The relationship is exactly the same between natural numbers and integers, integers and rational numbers, rational numbers and real numbers, and, for that matter, complex numbers and quaternions.
That said, these distinctions are almost always ignored. But technically, the answer is “no”. For details on these questions, see Edmund Landau’s Foundations of Analysis, a dry-as-bones construction of all these numbers.
As well as that between the natural numbers and the natural numbers (for any two different constructions of the natural numbers), the integers and the integers (for any two different constructions of the integers), etc.
Which is to say, every relationship between separate structures in math is some kind of homomorphism or isomorphism, if one is thinking abstractly enough (that is, enough not to care about the particular names used for elements of those structures). The natural numbers as an abstract algebraic structure whose elements are considered labelled with English names is in some sense different from the natural numbers as an abstract algebraic structure whose elements are considered labelled with French names, sure, but they are isomorphic.
It’s also worth noting that what counts as an isomorphism depends on what operations, relations, etc., one is interested in the structure of. For example, in terms of purely the structure of addition, the real numbers are not only isomorphic to the complex numbers a + 0 * i, but also isomorphic to the complex numbers 0 + a * i, and the complex numbers a + a * i, and the complex numbers 3 a - 5 a * i, and so on; in each case, there’s a natural way (indeed, even more than one way) to correspond the real numbers with such complex numbers in such a way as that addition of the real numbers corresponds to addition of the corresponding complex numbers. But if one is interested in matching up other pieces of structure besides (multiplication, for example), then one will want different correspondences. One could even consider correspondences which match up real number addition with complex number multiplication or such things if one wanted [and, indeed, this is often a very useful thing to do].
(But anyway, all this is rather besides the point for the OP’s simple question, for which the best simple answer, as given above, is indeed “Yes”)
Personally, I prefer to say that the complex numbers (or the real numbers, or the integers, or whatever) are the structure found in all of those constructions, not the constructions themselves. That is to say, the real numbers are defined as “a set with the following properties…”, and that the whole thing with, say, Dedekind cuts is real numbers since it’s a set with those properties, but anything else with those properties is also real numbers.
Yes, that is certainly the cleanest thing to do, which is precisely the same as never talking about structure equality and only talking about structure isomorphism, with all definitions of structures being given up to canonical isomorphism.
The isomorphism between those two algebras is so natural and compelling that it becomes difficult to see whether there really is any difference between the two sets; is it not reasonable to say that “zwei”, “two”, “dos”, “deux”, etc. are not different objects, but different labels for the same object? When one talks about identifying a factor group with the kernel of a homomorphism, it is clear that two separate things are being considered, and that the proof of the theorem telling you they are isomorphic gives you the correspondence. It seems to me that the isomorphism of the first example is in some sense different than the isomorphism in the second example.
Maybe there’s something about this idea in the link you gave later, but it’s way over my head.
A) Nitpicky point which I address first only so I can use it as an example in the next part: “identifying a factor group with the kernel of a homomorphism”? Do you mean “Identifying a homomorphism’s image with its the factor group given by quotienting its domain by its kernel” (just as any function between sets induces a correspondence between its image and the quotienting of its domain by its kernel pair)? If not, I’m not sure to which theorem you are referring (perhaps one specifically about product groups, and the fact that the factor group in which a binary product is quotiented by the kernel of one projection is isomorphic to the kernel of the other projection).
B) Well, the thing about English and French labelled natural numbers is that not only are those structures isomorphic, but, presumably, the definition you will hear in an English classroom of what the natural numbers are is often isomorphic to the definition you will hear in a French classroom of what the natural numbers are. It’s not just isomorphic structures, but isomorphic definitions (in some sense), which I think is the sort of phenomenon you are observing, whereas, given a group homomorphism, the definition of its image may be very different, as definitions go, from the definition of the quotient of its domain by its kernel, even though we know the resulting structures to be isomorphic.
Today’s moral, friedo: to get a short, simple answer to a mathematical question, be sure to ask exactly one mathematician.
Preferably zero.