Are infinitesimals equal to zero?

Question

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 1

"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 $1$.

Answer 2

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.

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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.

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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**.

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* 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.

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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.

Answer 3

"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.

Answer 4

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 $0$.

Answer 5

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.

Answer 6

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.

Answer 7

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.

2. …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.

Answer 8

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 $1$

Answer 9

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.

Answer 10

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 $0.\overline{3}$.