How to correctly write 1.0000[infinite zeros]01

Question

NOTE: I am not a "Math Guy", so my question may not use correct terminology.

I was helping my daughter with her math homework and I realized I did not recall the correct way to write something.

If I want to write a number that is infinitly close to 1, approaching from "less than" (0.99999999999999999999 on and on forever) I write: $0.\bar9$

What if I was coming from "greater than"? How would I write that? Something like $1.\bar01$? (I don't think that is right.)

Answer 1

It's subtle. Unless you are doing non-standard analysis, where "infinitesimals" are given real meanings, the notion of a number that is "greater than $1$ but 'infinitely close' to $1$" is not really immediately available.

The idea of $0.\overline{9}$ is meaningful because it represents $$ \sum_{n=1}^\infty \frac9{10^n} $$ which you can explain to your daughter as $$ \frac9{10}+\frac9{100}+\frac9{1000} + \cdots $$ But there is no corresponding concept for "coming from greater than" (other than $1.\overline{0}$ which is exactly $1$ with no fanfare).

I guess you could get at what you want by describing it as $$ 2.0 - 0.\overline{9} = 2 - \frac9{10}-\frac9{100}-\frac9{1000} - \cdots $$

So unless you are helping your daughter with an advanced upper class college math course, that is about the best you are going to do.

Answer 2

Two things: $0.\bar9$ is actually $1$. If you approach infinitely close, you will get one.

Second, there really isn't a way to represent something if you are coming from "greater than". Take $1.\bar01$ for example. Write out the first few digits: $1.000000000000000000000000000...$ The bar means that there are infinite zeros in the decimal representations. This implies that you will never get to the final "$1$" you put at the end.

Answer 3

I'm afraid the online preview stops before the quote I want. Milo keeps asking about the biggest number there is. On page 191, we get

"Just follow that line forever," said the Mathemagician, "and when you reach the end, turn left. There you'll find the land of Infinity, where the tallest, the shortest, the smallest, and the most and least of everything are kept."

Alright, they do show the end of that line:

Answer 4

The usual notations are $1^-$ and $1^+$.

Generally you will see it used in limits expressions like $\lim\limits_{x\to1^+}$ it means $x\to 1$ and $x\gt 1$.

In the same way $1^-$ means $x\to 1$ and $x<1$.

So this is equivalent to say we are going to $1$ from the left side or from the right side, and actually we speak about a left side limit and a right side limit.

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Since I'm downvoted, here is some defense :

First, thanks for explaining your reason for downvoting, many don't.

Yet I think the OP question was about how to express something infinitesimally close to $1$ but less than $1$, and the only valid answers in my opinion are $1^-$ and $1-\epsilon\ $ with $\epsilon$ being the one of non-standard analysis (or any extension of the reals like hyperreals, surreals, and co.)

All other answers that consider $0.99999...$ and $1.0000...1$ as stated by Jeff are achieved limits and in that sense their value is simply $1$, not infinitesimals in any way.

So yes $1^-$ and $1^+$ are not real numbers, and it is the way it has to be, it is a notation for something that carries a meaning greater than a number.

In the same way that we do some calculation with some $o(x)$ for instance, while $o(\cdot)$ not being a function, this is a notation of a concept, and consequently I believe I'm answering the request of Vaccano.