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I am trying to understand the difference between a sizeless point and an infinitely short line segment. When I arrive to the notion coming from different perspectives I find in the mathematical community, I arrive to conflicting conclusions, meaning that either the mathematical community is providing conflicting information (not very likely) or that I don't understand the information provided (very likely).

If I think of a sizeless point, there are no preferential directions in it because it is sizeless in all directions. So when I try to think of a line tangent to it, I get an infinite number of them because any orientation seems acceptable. In other words, while it makes sense to talk about the line tangent to a curve at a point, I don't think it makes sense to talk about the line tangent to an isolated sizeless point.

However, if I think of an infinitely short line segment, I think of one in which both ends are separated by an infinitely short but greater than zero distance, and in that case I don't have any trouble visualising the line tangent to it because I already have a tiny line with one specific direction. I can extend infinitely both ends of the segment, keeping the same direction the line already has, and I've got myself a line tangent to the first one at any point within its length.

What this suggests to me is that sizeless points are not the same notions as infinitely short line segments. 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.

But then I got to the topic about 0.999... being equal to 1, which suggests just the opposite. In order to try to understand the claim, I decided to subtract one from itself and subtract 0.999... from one, hoping to arrive to the same result. Now, subtracting an ellipsis from a number seems difficult, so I started by performing simpler subtractions, to see if I could learn anything from them.

1.0 - 0.9 = 0.1

That was quite easy. Let's now add a decimal 9:

1.00 - 0.99 = 0.01

That was almost as easy, and a pattern seems to be emerging. Let's try with another one:

1.000 - 0.999 = 0.001

See the pattern?

1.0000 - 0.9999 = 0.0001

I always get a number that starts with '0.' and ends with '1', with a variable number of zeros in between, as many as the number of decimal 9s being subtracted, minus one. With that in mind, and thinking of the ellipsis as adding decimal 9s forever, the number I would expect to get if I performed the subtraction would look something like this:

1.000... - 0.999... = 0.000...1

So if I never stop adding decimal 9s to the number being subtracted, I never get to place that decimal 1 at the end, because the ellipsis means that I never get to the end. So in that sense, I might understand how 0.999... = 1.

However, using the same logic:

1.000... - 1.000... = 0.000...0

Note how there is no decimal 1 after the ellipsis in the result. Even though both numbers might be considered equal because there cannot be anything after an ellipsis representing an infinite number of decimal digits, the thing is both numbers cannot be expressed in exactly the same way. It seems to me that 0.000...1 describes the length of an infinitely short line segment while 0.000...0 describes the length of a sizeless point. And indeed, if I consider the values in the subtraction as lengths along an axis, then 1 - x, as x approaches 1, yields an infinitely short line segment, not a sizeless point.

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?

Thanks!

EDIT:

I would like to conclude by "summarising" in my own non-mathematical terms what I think I may have learned from reading the answers to my question. Thanks to everyone who has participated!

Regarding infinitely short line segments and sizeless points, it seems that they are indeed different notions; one appears to reflect an entity with the same dimension as the interval it makes up (1) while the other reflects an entity with a lower dimension (0). In more geometrical terms (which I find easier to visualise) I interpret that as meaning that an infinitely short line segment represents a distance along one axis, while a sizeless point represents no distance at all.

Regarding infinitesimals, it turns out most of them are not real, that is, most of them are not part of the set of real numbers; they are numbers whose absolute value is smaller than any positive real number. But there is an integer -a number without any fractional part-, therefore also a real number, whose absolute value is smaller than any positive real number and that is zero, of course; zero does not have a fractional part and it is smaller than any positive real number. Hence, zero is also an infinitesimal. But not necessarily exactly like other infinitesimals, because it seems you cannot add zero to itself any number of times and arrive to anything other than zero, while you can add other infinitesimals to themselves and arrive to real values.

And regarding 0.999... being exactly 1, I think I now understand what is going on. First, I apologise for my use of a rather unconventional notation, so unconventional that even I didn't know exactly what it meant. The expressions '0.999...', '1.000...' and '0.000...' do not represent numerical values but procedures that can be followed in order to construct a numerical value. For example, in a different context, '0.9...' might be read as:

1) Start with '0.'
2) Add decimal '9'
3) Goto #2.

And the key thing is the endless loop.

The problem lied in the geometrical interpretation in my mind of the series of subtractions I presented; I started with a one-unit-long segment with a notch at 90% the distance between both ends, representing a left sub-segment of 0.9 units of length and a right sub-segment of 0.1 units. I then moved the notch 90% closer to the right end, making the left sub-segment 0.99 units long and the right 0.01. I then zoomed my mind into the right sub-segment and again moved the notch to cover 90% of the remaining distance, getting 0.999 units of length on one side and 0.001 on the other. A few more iterations led me to erroneously conclude that the remaining distance is always greater than zero, regardless of the number of times I zoomed in and moved the notch.

What I had not realised is that every time I stopped to examine in my mind the remaining distance on the right of the notch, I was examining the effects of a finite number of iterations. First it was one, then it was two, then three and so on, but in none of those occasions I had performed an infinite number of iterations prior to examining the result. Every time I stopped to think, I was breaking instruction #3. So what I got was not a geometrical interpretation of '0.999...' but a geometrical interpretation of '0.' followed by an undetermined but finite number of decimal 9s. Not the same thing.

I now see how it does not matter what you write after the ellipsis, because you never get there. It is just like writing a fourth and subsequent instructions in the little program I am comparing the notation to:

1) Start with '0.'
2) Add decimal '9'
3) Goto #2.
4) Do something super awesome
5) Do something amazing

It doesn't matter what those amazing and super awesome things may be, because they will never be performed; they are information with no relevance whatsoever. Thus,

0.000...1 = 0.000...0 = 0.000...helloiambob

And therefore,

(1.000... - 0.999...) = 0.000...1 = 0.000...0 = (1.000... - 1.000...)

Probably not very conventional, but at least I understand it. Thanks again for all your help!

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    $\begingroup$ I suggest you look for a real definition (= rigourous) of what's an infinitesimal. But an infinitesimal is not a real number (= not an element of $\Bbb R$) $\endgroup$ – Tryss Aug 17 '16 at 11:18
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    $\begingroup$ You need to clarify your understanding to concept of the limit. To note, your reasoning of infinitesimals forms the definition of the limit in the non-standard analysis. $\endgroup$ – Hanul Jeon Aug 17 '16 at 11:19
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    $\begingroup$ By far the most direct way to talk about "infinitely short line segments" is to use nonstandard analysis. In standard mathematics, there are various ways to make sense of 'infinitesimal' geometry, which is basically what calculus is secretly doing, and what differential geometry does more explicitly. You add an extent to points not by doing anything to the points themselves, but by remembering more information about everything else. e.g. if you have a function $f$, then at a "sizeless" point you only remember the value $f(P)$. But to give it some extent you remember both $f(P)$ and $f'(P)$. $\endgroup$ – Hurkyl Aug 17 '16 at 11:54
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    $\begingroup$ "When I arrive to the notion coming from different perspectives I find in the mathematical community, I arrive to conflicting conclusions, meaning that either the mathematical community is providing conflicting information (not very likely) or that I don't understand the information provided (very likely)." ← there is a third option, which is that "some of the perspectives I find that claim (explicitly or implicitly) to be from the mathematical community are not, and are actually from confused/wrong people instead" $\endgroup$ – Ben Millwood Aug 17 '16 at 15:22
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    $\begingroup$ The limit does not always have same properties as objects converging to it. Reciprocals of natural numbers are positive, but their limit is not: $\lim_{n\to\infty}(1/n) = 0$. Numbers $(1+1/n)$ are rational for natural $n$'s, and so are their natural powers $(1+1/n)^n$, however their limit $\lim_{n\to\infty} (1+1/n)^n$ is irrational (and even transcendental). All the real functions $x\mapsto \sin(ax)/a$ for $a\ne 0$ are non-zero almost everywhere with derivative $\cos(ax)$, bouncing between $-1$ and $1$ – however their limit for $a\to\infty$ is a constant function, zero everywhere. $\endgroup$ – CiaPan Aug 18 '16 at 5:13

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"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 $1$.

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    $\begingroup$ Thank you Eff for a helpful answer. So if I have understood you correctly (please be patient with my choice of words because I am neither a mathematician nor a native English speaker), 0.999... would only be smaller than 1 if the ellipsis were taken to mean any finite number of decimal places, which is not the case. In other words, in my subtraction, as soon as I decide to write a trailing '1' in the result, I am settling on a finite number of decimal places and thus the number being subtracted stops behaving like 1. Would that make sense? It would to me. Thanks again! $\endgroup$ – Carvo Loco Aug 17 '16 at 12:27
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    $\begingroup$ @MikhailKatz The question was, in my opinion, clearly about understanding limits. Tangent lines, $0.999... = 1$, et cetera. It seemed quite obvious that the OP was not actually asking about "infinitesimals" from nonstandard analysis, but it was simply misunderstanding of a normal limit where something tends to zero, i.e. from real analysis. $\endgroup$ – Eff Aug 17 '16 at 14:34
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    $\begingroup$ @MikhailKatz I don't think Eff's answer shows a dismissive attitude at all, and the OP seems fine with limiting the context to real numbers given that they've accepted this answer. Just because someone doesn't find nonstandard analysis to be as mathematically fundamental as real analysis, doesn't mean they're dismissive of it. (I don't, and I still find it very mathematically interesting.) $\endgroup$ – Noah Schweber Aug 17 '16 at 14:47
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    $\begingroup$ @MikhailKatz I guess I still don't see how their comment is dismissive. They are dismissing the possibility that the OP is asking about nonstandard analysis, but that seems grounded in fact (given that the OP accepted this answer, leaving aside interpretation of the question - on which point I agree with Eff by the way). And one has to do one's best to interpret any not-perfectly-precise question, and favoring real analysis over nonstandard analysis seems a pretty mild interpretive bias to have. But this sort of thing is subjective, so I'll bow out here. $\endgroup$ – Noah Schweber Aug 17 '16 at 14:53
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    $\begingroup$ @CarvoLoco Don't worry about non-standard analysis for now. Everything you've asked about can be understood in the framework of standard analysis. From high school to all the way through university, you can usually get by without ever knowing about non-standard analysis. $\endgroup$ – Eff Aug 17 '16 at 17:31
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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.
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  • $\begingroup$ Don't forget the surreals they have the advantage that they are fairly easy to explain, and even more general. $\endgroup$ – Oscar Smith Aug 17 '16 at 19:32
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    $\begingroup$ As an aside, every hyperreal does have a decimal representation where the places are indexed by hyperintegers rather than integers. $\endgroup$ – Hurkyl Aug 18 '16 at 0:51
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    $\begingroup$ Greg, you should probably clarify what "matches up with the standard definition" means. $\endgroup$ – Mikhail Katz Aug 18 '16 at 7:02
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    $\begingroup$ @GregRos Good point! I suppose I should have spoken about the absolute value, shouldn't I? What about 0≤|a|<|r|? Would that hold true? I think I can see how 0 would be smaller than any positive real number, but I am unsure of whether that implies it behaves just like any other infinitesimal, or whether it is different from other infinitesimals in any way. For example, and please forgive me if this doesn't make any mathematical sense, if I could add up the same a>0 infinitesimal over and over again, would I ever get to a real number? $\endgroup$ – Carvo Loco Aug 18 '16 at 10:58
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    $\begingroup$ @CarvoLoco I recommend you read a textbook on the subject. A good one is Calculus: An infinitesimal Approach. It answers all of those questions and shows how infinitesimals can be used in practice. It doesn't go into the complex details of how to construct them and how to prove they are well-behaved, which can be quite complicated. $\endgroup$ – GregRos Aug 18 '16 at 11:32
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"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.

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    $\begingroup$ No. A decimal expansion is indexed by natural numbers. The ellipsis usually means that the pattern continues from there on for all further natural numbers. There thus can come nothing after it if its a dec expansion. Of course, you can index digits by something other than the natural numbers and then the ellipsis notation would make sense, yet the string has no meaning yet, beyond being a string of digits. It's as if you would write a circle formed of seven digits and ask what it is. It does not mean anything beyond "a circle formed of seven digits" until you explain it to me. $\endgroup$ – quid Aug 17 '16 at 13:00
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    $\begingroup$ In the same way I do not know what $0.000\dots 1$ should mean as a mathematical object, a number. It has no meaning to me. I do not understand it. It is not part of the things defined in my mathematical universe. (And that's true for the overwhelming majority of mathematicians.) As I said, you can assign some meaning to it, but you have to do it. Until then, there is nothing to discuss about. @CarvoLoco $\endgroup$ – quid Aug 17 '16 at 17:13
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    $\begingroup$ @CarvoLoco my point was not to criticize you, but to explain by way of analogies that $0.000 \dots 1$ just does not have any particular meaning to me as a mathematical object (that is beyond an infinite string of 0 and then 1). Another analogy, if one writes some code one has to respect the syntax and restrictions of the language one uses. It is possible to use another language, or to extend the current one. But one has to do it. What is not possible is to use syntax similar to correct one, and to speculate what the code might do. If the syntax is undefined or incorrect, it just won't work. $\endgroup$ – quid Aug 17 '16 at 18:13
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    $\begingroup$ @CarvoLoco You wrote: "I have decided to assign the ellipsis in the expression $0.000\ldots1$ the meaning 'and so on without end, regardless of the decimal digits that may come after it.'" The problem is that this definition is contradictory: if the zeroes are "without end", then how can anything come after them? You would have to redefine the meaning of "end" for this to make sense. $\endgroup$ – Théophile Aug 17 '16 at 18:46
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    $\begingroup$ @quid And even if it were, I would still appreciate your criticism because you seem to be trying to help me understand, which is something that I can only thank you for. Your contribution has indeed helped me understand my inability to express myself in proper mathematical terms, but I already knew that. What your contribution has not helped me understand as much are the possible conceptual differences between an infinitely short line segment and a sizeless point. But you've made a good point about the importance of a precise description of terms, I do thank you for it. Have a good day! $\endgroup$ – Carvo Loco Aug 17 '16 at 23:10
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To answer your question "Are infinitesimals equal to zero?" one could mention the following. Leibniz used the equality symbol to denote the relation between two numbers that differ by an infinitely small number. In particular one could write that $\epsilon=0$ if $\epsilon$ is an infinitesimal. In this scheme of things an infinisimal indeed equals zero. Today we would use a different notation for such a relation. For example, we could write $a\approx b$ if $a-b$ is infinitesimal. Euler distinguished between two modes of comparison, which he called geometric and arithmetic. The one we denoted $\approx$ is what he would refer to as arithmetic. The geometric mode, denoted for convenience by $\;{}_{\ulcorner\!\urcorner}$, as in $a \;{}_{\ulcorner\!\urcorner}\; b$ corresponds to the ratio $\frac{a}{b}$ being infinitely close to $1$.

Similarly, one can have infinitely many $9$s in a decimal $0.999\ldots 9$ (with a final $9$ at an infinite rank) which is infinitely close to $1$ but still strictly smaller than $1$.

The difference between an infinitely short segments and sizeless points is that the former have the same dimension, namely $1$, as the interval (say, $[0,1]$) they make up, while the latter have smaller dimension, namely $0$.

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    $\begingroup$ I think it's important to clarify that the context(s) in which infinitesimals exist and are nonzero is not the usual real numbers. (Which is not to say they're uninteresting/unvaluable, just to forestall possible confusion on the OP's part.) $\endgroup$ – Noah Schweber Aug 17 '16 at 14:45
  • $\begingroup$ Schweber, Just because you are fixated on a certain type of numbers does not mean that he should be, sorry. $\endgroup$ – Mikhail Katz Aug 17 '16 at 14:51
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    $\begingroup$ I don't see how pointing out the unstated and subtle difference in context amounts to being fixated with one type of number over another. I never said that the reals are preferable in any way; I just think it does the OP a disservice to give an answer which is wildly out of line with what they will see in the vast majority of analysis texts, without even mentioning the discrepancy. $\endgroup$ – Noah Schweber Aug 17 '16 at 14:57
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    $\begingroup$ I agree with @Noah -- my biggest objection (only objection, really) to "hey look, you can talk about terminating decimals with infinitely many 9's" is when it crops up in the context of "I have misunderstandings about nonterminating decimals", and in my estimation, it furthers their misunderstanding. $\endgroup$ – Hurkyl Aug 17 '16 at 17:25
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    $\begingroup$ Even in nonstandard analysis, where I have seen people assign meaning to a notation such as $0.999\ldots9$ they consider it different from $0.999\ldots$; indeed they still say $0.999\ldots =1$. $\endgroup$ – David K Aug 17 '16 at 19:05
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What color is a widget? You can't answer because their is no definition of a widget. Is $0.\bar 9=1$ ? You can't answer unless you have a def'n of $0.\bar 9,$ and you can't define it as the least upper bound of $ \{0.9,0.99,0.999,...\} $ unless there IS such a least upper bound, and you can't prove that unless you define the algebraic structure $\mathbb R$ called the real number system, and define $0.\bar 9$ as the least upper bound, IN $\mathbb R,$ of $\{0.9,0.99,...\}.$

The system $\mathbb R$ can be expanded to bigger number systems. These other systems have numbers that are positive but smaller than any positive member of $\mathbb R.$ Their reciprocals are larger than any member of $\mathbb R.$ Their basic rules of arithmetic are the same rules as for $\mathbb Q$ and $\mathbb R.$ The main feature that $\mathbb R$ has that they don't is the existence of a least upper bound for every non-empty subset that has an upper bound. This is because $\mathbb R$ is defined as an ordered-arithmetic extension of $\mathbb Q$ with this property, and it is a theorem that only one such extension is possible.

In the extensions of $\mathbb R$, $0.\bar 9$ has no meaning and $\sup\{0.9,0.99,...\}$ does not exist.

What you need is to read about the axiomatic foundations of $\mathbb R.$ There is good treatment of this in many texts.... I recall a section in "Topology" by Choquet (which is not a book about topology but about analysis) and a section in the first chapter of "Fourier Series" by Carslaw. But there are many others, as this is necessary to know in order to make sense of calculus, for example.

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  • $\begingroup$ What do you mean??? Widgets are a vreeny shade of burble, everybody knows that! :-) Thank you, @user254665, I think I understand part of what you are saying, but unfortunately I am not a mathematician nor a native English speaker, so I have some difficulties following your reply. Could you re-state the idea "the main feature that RR has that they don't is the existence of a least upper bound for every non-empty subset that has an upper bound" using different words, please? Thanks for the reading suggestions! $\endgroup$ – Carvo Loco Aug 18 '16 at 14:55
  • $\begingroup$ One consequence is that no $x\in \mathbb R$ can be less than $1$ and be greater than every member of $S=\{0.9,0.99,0.999,...\}.$ Suppose otherwise : Then there would be a least such $x,$ which we can call $x_0.$ Note that $x_0$ is then the least upper bound of $S$. Let $y=2x_0-1 .$ Then for every $n\in \mathbb N$ we have $x_0>1-10^{-n-1}\implies y=2x_0-1>1-2\cdot 10^{-n-1}>1-10^{-n}.$ So $y$ is an upper bound for $S , $ and any upper bound for $S$ must be greater than or equal to $x_0.$ So $2x_0-1=y\geq x_0.$ But $ 2x_0-1\geq x_0\implies x_0\geq 1,$ contrary to $x_0<1.$ $\endgroup$ – DanielWainfleet Aug 18 '16 at 17:54
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There are no infinitesimals within the real numbers. Either you're talking about real numbers (and then you don't have infinitesimals), or you have infinitesimals (and then you're not talking about real numbers).

The real number denoted by 0.999... is indeed 1. This is easy to see - 0.999... is greater than $1-10^{-n}$ for any $n$, and there are no real numbers smaller than 1 with this property (if there was, the difference from 1 would be infinitesimal, and there are no infinitesimals in the reals), so it is at least 1; and of course it is at most 1, so it is equal to 1.

But if you're using a number system that has infinitesimals, you can have a number denoted by 0.999... which has a value lower than 1. But of course, that will not be a real number.

And if you build a geometry using such a system, then you can have infinitesimal line segments with a definite direction.

Since there are no real infinitesimals, you need to be careful when trying to employ them in the context of limits of real functions. You can define such limits by going through a system which does have infinitesimals, but often they are mentioned without a formal construction, as mere intuitive figures of speech or abuses of notation (which is not necessarily a bad thing; you just have to recognize them for what they are). The common definition of limit talks about $\epsilon$ and $\delta$, which are both real, positive, finite numbers.

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  • $\begingroup$ Meni, your comments on infinitesimals are well taken, but your comments about limits reveal a common misconception. Limits can be defined either via epsilon-delta or via infinitesimals. In the latter case one exploits the standard part function, as explained in numerous posts under tags infinitesimals and nonstandard-analysis. $\endgroup$ – Mikhail Katz Aug 18 '16 at 16:29
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    $\begingroup$ @MikhailKatz: Sure, I'm aware of that, but then you're using objects that are not real numbers - I meant a setting where we stay within the realm of real numbers. But I guess this makes my argument kind of circular. I'll rephrase the main point of that paragraph. $\endgroup$ – Meni Rosenfeld Aug 18 '16 at 22:32
  • $\begingroup$ @MikhailKatz: Changes: 1. Quantified that infinitesimals could be abuses of notation. 2. Added that they could also be employed rigorously. 3. Referred to epsilon-delta as the common definition, rather than just the definition. $\endgroup$ – Meni Rosenfeld Aug 18 '16 at 22:37
  • $\begingroup$ Meni, I appreciate your recognizing that the argument based on the assumption that the real numbers are the default background of a question on infinitesimals is "kind of circular". $\endgroup$ – Mikhail Katz Aug 19 '16 at 8:32
  • $\begingroup$ @MikhailKatz: To clarify, my assumption was that the real numbers are the default background for talking about limits of real functions. My original answer excluded the possibility of using hyperreals as a tool in studying real functions. It did not exclude, and in fact explicitly included, the possibility of studying infinitesimals on their own right - and indeed, that should be expected when answering a question about infinitesimals. $\endgroup$ – Meni Rosenfeld Aug 19 '16 at 8:47
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TL;DR: Are infinitesimals equal to zero? Yes*. and No.

Yes*, in the domain of real numbers $\mathbb{R}$. (*Strictly speaking, infinitesimals do no exist within real numbers, so there's no question of equality. The closest value to an infinitesimal in $\mathbb R$ is zero.)

No, in the domain of hyperreal numbers $\mathbb{{}^*R}$, and surreal numbers $\mathbb S$.

Long Answer:

I know why you feel confused. Because when a mathematician talks about such stuff, they assume you know the context (which you don't, and i didn't too). Let me state the assumptions (and axioms), so you don't get confused.

  1. What this suggests to me is that sizeless points are not the same notions as infinitely short line segments.

[This comes under geometry. There are many types of geometries. The most "famous" one is Cartesian geometry with real coordinates (which you used to explain your example), so let's go with that.]

Yes, your notion is correct. A line segment is bound by two end points. Each point on the line has no size, but they are contained within the bounds of the two end points. An infinitesimally short line still has two end points. But to think that there are no more than two points on this short line is false. (See part 2 of this answer.) (…Unless your end points coincide to be the same point, in which case you don't call it a line segment any more.) Since points don't have size, you can fit any number of them between two points on a line.

  1. …when I was taking limits…

…to the topic about 0.999… being equal to 1…

To help you overcome this confusion, you need to understand that there are various sets of numbers.

The most intuitive one to grasp is the enumerable set of natural numbers, denoted by $\mathbb{N} = \{1, 2, 3, 4,\ldots\}$. Elements have an order (e.g. $1$ comes before $2$, $4$ comes after $3$, et cetera), and you can pick two elements in successive order such that there is no element between them.

However, the case is very different for real numbers $\mathbb{R}$. Elements here have an order, but there is no notion of successive elements. For any two elements that you pick, you can always find an element that comes in order between them (like the "infinitesimal line segment").

When you said:

$\ \ \ \ \ \ \ \ \ \ \ \ 1.0 - 0.9 = 0.1\\\ \ \ \ \ \ \ \ 1.00 - 0.99 = 0.01\\\ \ \ \ 1.000 - 0.999 = 0.001\\\ \!1.0000 - 0.9999 = 0.0001$

…that's correct. But it's incorrect to say that:

$1.000\ldots - 0.999\ldots = 0.000\ldots1$

The LHS (left hand side) of the equation has and non-terminating decimal representation. But how did you manage to terminate the decimal representation on the RHS (right hand side)?

The correct representation would be: $1.000\ldots - 0.999\ldots = 0.000\ldots$.

(Now you see where we're heading, right?) Infinity is not a real number. Thus, there's no reason to think that there exists a real number that is the reciprocal of infinity (an infinitesimal). However, mathematicians will confuse you here with the "infinitesimal line segment" by taking two arbitrarily close real numbers.

You might want to read/watch more about surreal numbers and hyperreal numbers, and why the infinitesimal is not equal to zero. There exist hyperreal/surreal numbers whose value can lie between two real numbers $a$ and $b$, where there doesn't exist a real number between $a$ and $b$. (Basically, a convoluted way of saying that those elements are not in the set of real numbers.) IMO, the one who invented calculus spotted the weakness in this definition of the set of real numbers.

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  • $\begingroup$ Thank you @garyF for your long and helpful answer. But I have a question. You said that "for any two elements [of RR] that you pick, you can always find an element that comes in order between them" and then, later, "There exist surreal numbers whose value can lie between two real numbers aa and bb, where there doesn't exist a real number between aa and bb." But I thought you said that for any two elements [of RR] that I pick, I can always find an element that comes in order between them. So if aa and bb are real, why wouldn't there exist a real number between them? $\endgroup$ – Carvo Loco Aug 18 '16 at 15:06
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    $\begingroup$ @CarvoLoco That's because those numbers $a$ and $b$ "coincide" in the real set (they have the same value). This is where the difference between "intuition" and "definition" come into play. Everday (school) experience deals with integers and real numbers. But mathematicians can define (read: "let there be (light)") there to be numbers that create chasms between real numbers that have the same value. e.g. in $\mathbb S$, there are surreal numbers in between $0.999\ldots$ and $1.000\ldots$, where there are no real numbers between them. $\endgroup$ – garyF Aug 18 '16 at 15:10
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    $\begingroup$ Oh, I think I see what you mean. I hadn't realised both numbers where equal when compared in real terms, but different by a surreal amount. Can I put it that way? Thanks for the clarification. $\endgroup$ – Carvo Loco Aug 18 '16 at 15:24
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    $\begingroup$ @CarvoLoco Yes. Another example would be to take an extension of the real numbers, called the complex numbers $\mathbb C$. If two complex numbers $a$ and $b$ have the same real part, they coincide on the real axis, but are not necessarily the same point in the complex plane. Hope that helps!~ (BTW, i edited the Yes part of my answer with an asterisk, after you asked your question. :P ) $\endgroup$ – garyF Aug 18 '16 at 15:31
  • $\begingroup$ Oh yes, of course! Two numbers can be equal in real terms but different by an imaginary amount. Thanks, it does help. $\endgroup$ – Carvo Loco Aug 18 '16 at 15:39
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Regarding 0.9, 0.99, etc.

Say we have some decimal number with a number (possibly 0) of non-repeating digits and a number of repeating digits. It looks like $A.BCDEF\overline{GHI}$

Let's subtract out the non-repeating portion, because that isn't very interesting.

$$ A.BCDEF\overline{GHI} - A.BCDEF = 0.00000\overline{GHI} $$

So what number is $0.00000\overline{GHI}$?

We can express decimal that has the overline notation with a fraction (a more familiar notation).

$$0.00000\overline{GHI} = \frac{GHI}{(10^3 -1)(10^5)}$$

The $3$ is the number of repeating digits. The $5$ is the number of non-repeating fractional digits.

So, the number $1.3245\overline{456}$... can be written as

$$ 1.3245 + \frac{456}{(10^3-1)(10^4)} = 1.3245 + \frac{456}{(999)(10000)} $$

What happens when the repeating portion is just the digit $9$? Let's see

$$ 0.000\overline{9}... = 0 + \frac{9}{(10^1-1)(10^3)} = \frac{9}{(9)(1000)} = \frac{1}{1000} $$ $$ 0.00\overline{9}... = 0 + \frac{9}{(10^1-1)(10^2)} = \frac{9}{(9)(100)} = \frac{1}{100} $$ $$ 0.0\overline{9}... = 0 + \frac{9}{(10^1-1)(10^1)} = \frac{9}{(9)(10)} = \frac{1}{10} $$ $$ 0.\overline{9}... = 0 + \frac{9}{(10^1-1)(10^0)} = \frac{9}{(9)(1)} = 1 $$

It's tempting to think of a repeating decimal as repeating forever, but that implies a process happening in time, which isn't what that notation represents. $0.\overline{9}$ is quite literally just another notation for $1$

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If you are happy to do a kind of mathematics very different from the usual, then you can have a theory in which infinitesimals that are not provably equal (or unequal) to zero possibly exist. See smooth infinitesimal analysis. It addresses the difference between a point and an infinitesimally short line – a line, no matter how small, has a gradient. Furthermore, curves are made of lines, rather than points.

We don't know of any non-zero infinitesimals, the point being that we can't ever do so. However, keeping the possibility open means that we have to reason parametrically about them, and it just happens to give us what Leibniz wanted in a fairly intuitive way.

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Just because nobody's seemed to give the simple solution, here's the easy proof that convinced me that $0.\overline{9}=1$ (or $0.\overline{0}1=0$.)

$$x = 0.999...$$ $$10x = 9.999...$$ $$10x - x = 9.999... - 0.999...$$ $$9x = 9$$

I hope everyone sees the small leap there: they both have the same expansion. After that it's rather trivial:

$$x = 1$$

Which also means that an infinitely small number equals zero.

This is also why $\frac 13$ exactly (and not approximately) equals $0.\overline{3}$.

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