If $0.999\cdots = 1$ Then Does $\frac{1}{10^\infty} = 0$?

Question

Recently I stumbled across a, to me, rather strange idea. I was messing around with the proof of $0.999... = 1$, when I figured that what $0.999...$ means is that those are all nines. That way I came upon a weird idea. Say $a = 0.999...$, then $a = 1 - x$. Yet, how do we define what $x$ is? It should say $x = 0$, but my theory did not. If $0.999...$ is just an endless sequence of nines, then why can't we say $x$ just is an endless sequence of zeroes, ending with a 1, like $\frac{1}{10^\infty}$? If we take the equation $$n = 0.9$$ for example, then, what would $1 - n$ be? Yes indeed, $0.1$. Following that theory, can't we say that $$0.999... = 1 - \frac{1}{10^\infty}$$Now, if $0.999... = 1$ this would be impossible. Go figure. $$1 = 1 - \frac{1}{10^\infty}$$ then $$\frac{1}{10^\infty} = 0$$ but that is impossible, because we cannot say $$10^\infty * 0 = 1$$When I came to this point, I really got stuck, because, in my head everything I did was right, however, it is impossible. Can somebody please explain to me what mistakes I may have made, and enlighten me about what else I did wrong?

Thanks in advance. Sjoerd Dorrestijn.

EDT: I prefer $\frac{1}{10^\infty}$ to use as an indication of $0.000...01$, even though $10^\infty = \infty$ in some way, I just seem to find this more clear.

EDT2: Just to be clear, I read a proof that $0.999...9 = 1$ because it'd be $1 - 0.000...0 = 1 - 0 = 1$. What I tried to prove here is that it is not equal to $1$, because otherwise maths would collapse. My question was whether I am right or wrong. Since the original statement uses infinity (an infinite amount of nines) I think it is a must to use infinity as well. So, the question is if either the original statement is false, or if I made a mistake somewhere.

Answer 1

The (proven true) statement that $0.999\ldots=1$ means that $\lim_{n\to\infty}0.\underbrace{999\ldots9}_n=1$. That in turn is equivalent to $\lim_{n\to\infty}\left(1-0.\underbrace{999\ldots9}_n\right)=0$. Now note that $1-0.\underbrace{999\ldots9}_n=\frac{1}{10^n}$, so we have $\lim_{n\to\infty}\frac{1}{10^n}=0$.

However, although the above reasoning works, it is easier and more straightforward to conclude that $\lim_{n\to\infty}\frac{1}{10^n}=0$ directly from the definition of the limit, without ever considering $0.999\ldots$.

Finally, it is advisable to avoid the notation $\frac{1}{10^\infty}$ altogether, as it is not defined given the usual definition of its constituent symbols.

Answer 2

The problem you have with $10^{\infty}$ arises if we treat somewhat imprecise notation as precise mathematical statements. What people (mathematicians) usually mean when they say that $0.99\dots=1$ is that the sequence $0.9 , 0.99 , 0.999 , \dots$ approaches $1$, i.e. the limit $\lim_{n\to\infty}{\sum_{i=0}^{n}0.9\cdot 10^{-i}}$ is equal to $1$. Turning this equation around we get $\lim_{n\to\infty}(1-\frac{1}{10^n})=1$, or in other words that $\lim_{n\to\infty}{\frac{1}{10^n}}=0$. The notation $10^\infty$ supresses this limiting process and suggests that one can use this in the same way as usual real numbers. However limits only behave nicely together with multiplication and addition if they exist. The sequence $10^2, 10^3, 10^4,\dots$ does not have a limit (in the real numbers), so computing with $10^\infty$ leads to mathematical nonsense. A more imprecise way of seeing this would be to say that $10^\infty=\infty$ and $\infty\cdot 0 $ is undefined. This notation usually leads to wrong calculations and/or confusion as you have noticed yourself and should generally be avoided.

Answer 3

can't we say that $0.9999...={1\over 10^\infty}$

Not if we don't define our terms. Obviously $10^\infty$ is a term that needs defining, but let me go one step back - how do we define $0.999...$? Or rather, how do we define decimal notation in general?

The answer is via limits. Think of an expression like "$0.999...$" as a name for a number - it's not the number itself, it's how we write it down, and there may be other names for the same number. Decimal notation describes a number as a limit of simpler numbers: e.g. when we say "$\pi=3.14159...$" what we mean is that $\pi$ is the limit of the sequence $$3, 3.1, 3.14, 3.141, ...$$

Now, precisely defining what a "limit" is takes serious work, but let me sidestep that for the moment. What you've observed is

For each $n$, $1-0.99...99$ ($n$ many $9$s) equals $1\over 10^n$.

From this, we can argue$^*$ that the limits are the same: $$\lim_{n\rightarrow\infty} (1-0.99...99)=\lim_{n\rightarrow\infty}{1\over 10^n}.$$ Let's look at the left hand side for a moment; we can argue$^*$ that $\lim (A-B)=\lim A-\lim B$, and $\lim_{n\rightarrow\infty}1=1$ clearly$^*$ and $\lim_{n\rightarrow\infty}=0.9999...$ since that's what "$0.9999...$" means. Finally, it's not hard to show$^*$ that $\lim_{n\rightarrow\infty}{1\over 10^n}=0$. So this all translates to:

$$1-0.99999...=0.$$ This handles most of your question; the rest boils down to essentially: why can't we say $10^\infty\cdot {1\over 10^\infty}=1$? Well, you're basically asking why we can't divide by zero. Even though zero is a limit of things we can divide by (we can divide by $1$, we can divide by $1\over 10$, we can divide by $1\over 100$, ...), that doesn't mean that it itself is something we can divide by. That is, sometimes we can't swap arithmetic operations and limits: it can happen that $\lim {A\over B}$ is well-defined but $\lim A\over \lim B$ is not, and this is just something we have to be careful of - whenever you do something with limits, you have to prove that you actually can.

Finally, let me point out that notation like "${1\over 10^\infty}$" is highly discouraged, precisely because it suggests that we can manipulate $\infty$ just like a real number. And this is wildly false, and the source of many confusions.

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$^*$We can argue these things, or prove these things, or these things become clear, once we've worked with limits a bit. But I want to give the big picture rather than obscure the details.

Answer 4

What is $\infty$ really? It is not a number, you can't put it in equation and expect to get something, but what you can do is to take a number, let's say $n$ and make it grow larger and larger till it approach $\infty$(we say $\lim\limits_{n\to\infty}$ in this case). Now what happened when you take $h=\frac1n$? $h$ get smaller and smaller till it approuch $0$(we say $\lim\limits_{h\to0}$ in this case), so:

When you say $\lim\limits_{n\to\infty}\frac1n=0$ what you really have $\lim\limits_{n\to\infty}\frac1n=\lim\limits_{h\to 0}h$ and multiplying by $n$ will give you an indeterminate form.