Why can’t infinitesimal quantities be included in the set of real numbers? I’ve recently read in my calculus text book that infinitesimal quantities cannot be defined in the set of real numbers unless extended in some way. However the author of this text book was pretty vague about this and didn’t elaborate on this at all which I thought was pretty unfortunate. 
So concretely, my question is: why can’t infinitesimal quantities be defined in ℝ and how would one go about expanding ℝ to allow for infinitesimals?
 A: I suspect that this is a point on which the text's language is somewhat unclear. So first let me clearly state the negative situation:

There is - up to isomorphism - a unique complete ordered field.


*

*Of course this takes proof! Uniqueness is pretty easy, but existence is actually hard if we don't want to just outright assume it as an axiom (which many texts do). But I'm not going to treat that here.

(See here for a definition of ordered field; "complete" means that every bounded nonempty subset of the field has a least upper bound. Roughly speaking, a complete ordered field is a thing satisfying the basic rules of addition/subtraction/multiplication/division and ordering and which also doesn't have any "holes.")
Ignoring isomorphism issues, this is the thing we call $\mathbb{R}$ by definition. We can then define the natural numbers (roughly, $\mathbb{N}$ is the smallest subset of $\mathbb{R}$ containing $1$ and satisfying $x\in\mathbb{N}\implies x+1\in\mathbb{N}$). And with $\mathbb{N}$ in hand, we can finally define what it means to be infinitesimal:

$x$ is infinitesimal iff $x>0$ but for all nonzero $n\in\mathbb{N}$ we have $x<{1\over n}$.

Style note: there are actually a few different definitions of infinitesimals. One difference you'll see is whether $0$ is or is not considered infinitesimal; I've chosen to not consider it infinitesimal, but some texts - although in my experience a minority - do allow it. Another difference you'll see is the positivity requirement being dropped; that's actually fairly common, but I'm sticking with positive infinitesimals since I think they make for a clearer picture at first.

We can now prove the claim in question:
Theorem: There are no infinitesimal elements of $\mathbb{R}$.
Proof: Suppose otherwise. Let $I$ be the set of all infinitesimal elements of $\mathbb{R}$. By assumption $I$ is nonempty, and $I$ is clearly bounded, so $I$ has a least upper bound $\eta$ (since $\mathbb{R}$ is complete), and moreover we have $\eta>0$. Now we ask: is $\eta$ infinitesimal?

*

*If $\eta$ is infinitesimal, we have a problem: $2\eta$ must also be infinitesimal! (If $2\eta>{1\over k}$ then $\eta>{1\over 2k}$.) But $2\eta>\eta$, contradicting the assumption that $\eta$ is an upper bound of $I$.


*But if $\eta$ is not infinitesimal, we also have a problem! Since $\eta$ is not infinitesimal we must have ${1\over n}\le\eta$ for somenonzero $n\in\mathbb{N}$. But then ${1\over 2n}<\eta$, and every infinitesimal is $<{1\over 2n}$, so $\eta$ is not the least upper bound of $I$.
So we have a contradiction; this tells us that there are no infinitesimals in $\mathbb{R}$.

This isn't the end of the story, though. The above stemmed from the particular way we defined $\mathbb{R}$. But there are other ordered fields out there, and in particular there are ordered fields larger than $\mathbb{R}$ which necessarily contain infinitesimals (they fail to be complete, however). We can reasonably ask, "Why do we work with $\mathbb{R}$ instead of one of these other fields?" And even if we prefer $\mathbb{R}$, we could still reasonably ask, "Might we learn something interesting by looking at these other fields?"
The former question is of course subjective, but the latter has a definitive answer: even if we are primarily interested in $\mathbb{R}$, studying larger ordered fields can be quite useful. Even in the context of "classical" questions about real numbers this is true, a key term here being nonstandard analysis (and "hyperreals"), which shows that in a precise sense theorems about $\mathbb{R}$ can be proved by (carefully) using infinitesimals in (certain) larger fields!
This is what is being referred to by "expanding $\mathbb{R}$" - we're not literally changing $\mathbb{R}$, but rather studying a new, larger structure (or class of structures) either in place of or in addition to $\mathbb{R}$.
