Are infinitesimals equal to zero?

"I think of one in which both ends are separated by an infinitely short but greater than zero distance"

That does not exist within the real numbers. So what you think of "infinitely short line segment" does not exist within the context of real numbers.

"And if I can remember correctly from my school years, when I was taking limits I could assume that, as a variable approached some value, it never quite got to the value so I could simplify expressions that would otherwise result in 0/0 indeterminations by assuming that they represented tiny non-zero values divided by themselves and thus yielding 1. That, again, suggests to me that infinitesimals are not equal to zero."

When taking a limit $\lim_{x\to 0} f(x)$, $x$ is not some "infinitesimal." You're most likely stuck because you only have an intuitive notion of the limit, I suggest you look up a rigorous definition of limit. Furthermore, regarding the simplifications of indeterminate expressions, here are some questions that might help you: here and here.

Regarding $0.9$, $0.99$, etc.

It is true that for any finite number of nines, you end with a one at the end, i.e. $$1 - 0.\underbrace{99...99}_{n} = 0.\underbrace{00...00}_{n-1}1.$$

However, we are not talking about some finite number of nines, we're talking about the limit which can be rigorously proven to be $1$, i.e.

$$\lim\limits_{n\to\infty} 0.\underbrace{99...99}_n = 1.$$

"So what is it? Is the distance between the points $(0.999..., 0)$ and $(1.000..., 0)$ equal to zero, or is it only slightly greater than zero?"

It is (exactly!) zero because we define $0.999...$ as the limit of the sequence $(0.9, 0.99, 0.999,...)$, which happens to be $1$.


Let's put the question like this.

Is there a number $a$ that holds $0 < a < r$ for every positive real ("regular") number $r$?

But part of the question is missing: where are we looking for this number? If we're looking for it in the set of counting numbers, $ℕ$, then the answer is quite obviously, no. If we're looking for it in the set of real numbers, the answer is still no.

So where else can we look for it? Or possibly, how can we define such a number in a way that makes sense?

Mathematics is all about coming up with very careful definitions for things. There are all kinds of mathematical entities that might be called infinitely large (here is a rather dizzying introduction, though I kind of disagree with some of the characterizations), and sometimes these entities even have the same notation. Some examples of symbols for infinite entities are $ℵ_0$, $ε_0$, and $∞$ (which in particular can represent several different mathematical entities). These entities aren't necessarily "larger" or "smaller" to one another, though can be very different. None of them are real numbers, although they have some number-like traits.

The notation $0.9999....$ is generally taken to be equivalent to the following formula:

$$\sum_{n=1}^{∞}9⋅{1\over 10^n} = 0.9 + 0.09 + 0.009 + \cdots$$

Which, given the right definitions for infinite sums, exactly equals $1$. If you don't like the definition for the symbol $0.9999...$ (or the definition for an infinite sum), then it might mean something else to you. But then you'd be speaking a different language than the rest of us.

The notation $0.0000...1$ does not really have a well-defined meaning, and it's hard to give it one that makes sense. (How many $0$s are there before the $1$? Infinitely many? What does that even mean?). In a certain light, you can view it as the limit of the sequence:

$$0.1, 0.01, 0.001, ...$$

Then it equals $0$ (given the right definitions for limit and sequence and so on). But I don't think that notation should be used because it's very confusing and unnecessary. And if you don't define that notation, it doesn't mean anything.


Now, it is possible to define a bunch of entities that are infinitely small but are different from $0$. It's not very easy to define (people only worked out how to do it properly in the 20th century), but the result is very intuitive and behaves very well.

They are called the Hyperreal numbers. This set also includes infinitely large numbers, and somewhat answers the question of what happens when you multiply them with each other.

In the system of the hyperreals, there exist infinitesimals (often denoted $\epsilon$) which hold $0 < \epsilon < r$ for every positive real ("regular") number $r$. So it's smaller than any member of the sequence:

$$0.1, 0.01, 0.001, ...$$

But is still larger than $0$, which is the limit of that sequence. But in any case, the number $\epsilon$ and its fellows aren't really related to real numbers directly. They're like another special kind of number that we snuck in between them. They don't exactly have a decimal representation*, and indeed, we can't say much about them other than they can exist, and if you pick one it behaves in a certain intuitive manner.

If hyperreal numbers are okay, then the answer is yes. There are a few other special varieties of number that can be considered too. Basically, an intuition for "infinitely small quantities" can be made to make sense.


You're definitely right that there can be an infinite number of lines "tangent" to a point. But usually we talk about tangent to a function or curve at a point, which is visually kind of intuitive, even though trying to phrase it in technical terms can be a bit tricky.

In can make sense to denote length using hyperreal infinitesimals, and you can have a line segment of infinitesimal length. In fact, non-standard analysis, the principle application of hyperreal numbers, defines things like derivatives and limits using infinitesimals in a way that is equivalent to the standard definition**.


* Actually hyper-reals do have a decimal representation, but it has all kinds of unintuitive qualities, and I feel that mentioning it would detract from the issue at hand.

** i.e. the limit $\lim_{x → k} f(x)$ is the same whether you use one method or the other, and it exists using one definition if and only if it exists using the other.


Edits:
  • Mentioned @Hurkyl's point about hyper-reals having a decimal expansion, but it's somewhat complicated and I don't want to get into it here.
  • Cleared up @MikhailKatz's issue with the phrase matches up with and changed it to something clearer.

"the points (0.999..., 0) and (1.000..., 0)" are one and the same point in $\mathbb{R}^2$. Just like $(3^2, (5-1)/2)$ and $(9,2)$ are the same point.

To reiterate $0.999\dots$ is nothing but another way to represent the number $1$.

Yet $0.000...1$ just has no commonly agreed upon meaning. To me the notation is undefined. You can assign a meaning to it if you want. Then, this notation might be intuitive and useful or not. But before we can discuss this you must give it some meaning.

You just cannot start from a string of symbols and try to derive what it might mean. You need to assign a meaning to the string or use the meaning others assigned to it.