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!
Answer
"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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