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I've often heard that $dfrac{1}{3} = 0.overline{3} $, and $dfrac{1}{3} cdot 3 = 1$, yet $0.overline{3}cdot 3 = 0.overline{9} neq 1$. There is an $epsilon$ (infinitesimal) thrown in there as well. How do you understand these extremely small values and what do I need to do to account for them when calculating very precise values with them? I know that they are too small to make a difference when dealing with smaller numbers, but when does it start to impact your results?

Actually, $$0.overline{9} = 0.overline{3}cdot3 = frac{3}{3} = 1$$

For a more rigorous way, note that $0.overline{9} = frac{9}{10} + frac{9}{100} + ldots$. By the formula for the sum of an infinite series, $0.overline{9} = 1$.

Essentially, we need to prove that $lim_{ntoinfty} left(dfrac{1}{10^n}right) = 0$. By the squeeze theorem, this is clearly $0$.

For another way, let $S = 0.overline{9}$. Then $10S = 9.overline{9}$. This means that $9S = 9 implies S = 1$

Therefore, we have established (in three different ways) that $0.overline9=1$.

To convince you that there is no infinitesimal, try to construct a number between $0.overline{9}$ and $1$.

What do you mean by $0.overline 9$? Most mathematicians would agree that you mean this limit:
$$
lim_{Ntoinfty}sum_{k=1}^Nfrac{9}{10^k}.
$$
It has been shown many times on StackExchange that this limit is equal to $1$. The above limit is all we mean by that symbol $0.overline 9$ in mathematics: nothing more, nothing less. There is no "infinitesimal" between $0.overline 9$ and $1$; they are identical.

If $x<y$ then $x<frac{x+y}{2}<y$. So two distinct real numbers always have a third real number strictly between them.

So, what is the real number strictly between $0.overline{9}$ and $1$?

While it is possible to create number systems with infinitesimals, that does not validate the intuition that you've stated, that somehow $0.overline{9}$ and $1$ differ by an infinitesimal. You lose a lot of nice properties of the real numbers when you add infinitesimals.

For example, either infinitesimals are not invertible, or you give up the Archimedean property of the real numbers.

The real key is to know that there are no infinitesimals in the real number line. Thus, you can safely state that they are unimportant if you are limiting yourself to analysis of real numbers.

What I find the key to an intuitive understanding of $0.overline{9} = 1$ is to recognize that $0.overline{9}$ and $1$ are not real numbers per. se. They are symbols which represent real numbers. In particular, $0.overline{9}$ is a symbol which represents a real number that can be found via an infinite construction process. In that way of thinking, instead of just saying $0.overline{9} = 1$, one might say "$0.overline{9}$ and $1$ are the same number, written different ways." Think of it as similar to saying $frac{1}{2}$ and $frac{2}{4}$ are the same number, even though they are written differently. I find the transition from talking about equality to talking about sameness (or identity) helps my intuition. There are many domains where this concept of identity is a deeper concept than equality (computer programming being one domain), so thinking in terms of "sameness" helps break my mind free from the assumptions I make about how the number line "must" work.

So as for "When do they start to impact your results," the answer is an obvious "never," because they are simply the same number. There is no delta to cause impact with, not even an infintessimal one.

There are number systems that do truly have infinitesimals. However, not all of the assumptions you can make about real numbers apply in these number systems.

Also note that there is a concept of "limits," which are used in a few of these answers. They are an approach which involves using what are sometimes informally called "infinitesimals," when they are in fact not. Limits are carefully constructed in a way which permits a lot of infintessimal-like behaviors but confined such that they can operate over real numbers. In my opinion, their real beauty is taming infintessimals enough to solve problems such a Zeno's Paradox without requiring true infintessimals at all.

When people say $0.bar{9} neq 1$, it is some combination of the following two categories:

• They are simply wrong


• They are not talking about what you think they're talking about

The fact of the matter is that that $0.bar{9}$ is the same number as $1$; there is no difference between the two numbers.

While the usual meaning of $0.bar{9}$ is the real number whose decimal expansion has a $9$ in every place to the right of the decimal place, two examples of different things people often have in mind can be formalized as:

• They do not think of a single number, but instead of the sequence of numbers $0.9$, $0.99$, $0.999$, $ldots$


• They are thinking about the hyperreal numbers, and they are thinking of a terminating decimal expansion $0.underbrace{99ldots99}_{H text{ nines}} $ where $H$ is some infinite hyperinteger.

As other responders have noted, it depends on what you mean when you speak of a number with an infinite tail of $3$'s after the decimal point. One possible interpretation is a tail with a specific infinite number, say $H$, of $3$'s. Such an infinite terminating tail $0.333ldots3$ cannot be designed in the real number system $mathbb R$ since $mathbb R$ contains no infinitesimals. On the other hand, the hyperreal number system ${}^{ast}mathbb R$ contains both infinite numbers and infinitesimals, and there you can rigorously speak about a number with an infinite tail of $3$'s falling infinitesimally short of $frac13$, or of a number with an infinite tail of $9$'s falling infinitesimally short of $1$, precisely as you have heard. Such student intuitions are not erroneous but rather unrealizable in the real number system. Robert Ely in his 2010 publication in a leading education journal describes such intuitions as nonstandard.