# Prove .999 repeating = 1

Mark44
Mentor
I've had HW threads I've posted runaway like this with lots of experts arguing the "finer points" of the proof/problem and I did not find it very helpful as the student.
We've had lots of threads on exactly the same subject: whether 0.999... = 1. The most common misconceptions in threads of this nature are a flawed understanding of what it means mathematically for a sequence of numbers to converge, as well as a misunderstanding about how a limit is defined mathematically.

jack action
Gold Member
I must admit that all the "proofs" I've seen until now tend to say ##0.999... \triangleq 1## therefore ##0.999... = 1##.

But I finally found an explanation that satisfies me:
https://en.wikipedia.org/wiki/0.999...#Infinitesimals said:
Some proofs that 0.999... = 1 rely on the Archimedean property of the real numbers: that there are no nonzero infinitesimals. Specifically, the difference 1 − 0.999... must be smaller than any positive rational number, so it must be an infinitesimal; but since the reals do not contain nonzero infinitesimals, the difference is therefore zero, and therefore the two values are the same.
I didn't know about the infinitesimals concept (I guess this is one for the Today I learned thread), but explained this way, it makes sense when there is a distinction between real numbers and infinitesimals, with zero as the only common number in the two sets.

I think this is what the equation in post #33 by @Mark44 states, I'm just not fluent enough in that type of notation to understand it.

Mark44
Mentor
From the wiki article:
Specifically, the difference 1 − 0.999... must be smaller than any positive rational number
I asked a question similar to this a couple times: if 0.999... is not equal to 1, then what do you get from the difference 1 - 0.999... ?
I think this is what the equation in post #33 by @Mark44 states, I'm just not fluent enough in that type of notation to understand it.
I didn't say anything about infinitesimals. What I said in that post was that the difference between 1 and ##\sum_{i = 1}^N \frac 9 {10^i}## can be made as small as anyone demands, by choosing an appropriate value for N.
If for a given N the difference isn't as small as you would like, I'll choose a larger value of N.
If for that N the difference isn't as small as you would like, I'll choose an even larger value of N.
If for that N the difference isn't as small as you would like, I'll choose an even larger value of N.
If for that N the difference isn't as small as you would like, I'll choose an even larger value of N.
and so on until you tire of the game.

I asked a question similar to this a couple times: if 0.999... is not equal to 1, then what do you get from the difference 1 - 0.999... ?
##1-0.\dot9=0.\dot0 1##

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weirdoguy
jack action
Gold Member
I didn't say anything about infinitesimals. What I said in that post was that the difference between 1 and ##\sum_{i = 1}^N \frac 9 {10^i}## can be made as small as anyone demands, by choosing an appropriate value for N.
If for a given N the difference isn't as small as you would like, I'll choose a larger value of N.
If for that N the difference isn't as small as you would like, I'll choose an even larger value of N.
If for that N the difference isn't as small as you would like, I'll choose an even larger value of N.
If for that N the difference isn't as small as you would like, I'll choose an even larger value of N.
and so on until you tire of the game.
The problem with that is that there is no end of the game: there is always a difference. A difference necessarily means not equal to zero. This is why you need the concept of infinitesimals, i.e there exist a number that is indivisible. That may put an end to the game.

No matter what, you will have to define something as being true. My previous source has a citation that I find interesting:
https://en.wikipedia.org/wiki/0.999...#In_alternative_number_systems said:
Although the real numbers form an extremely useful number system, the decision to interpret the notation "0.999..." as naming a real number is ultimately a convention, and Timothy Gowers argues in Mathematics: A Very Short Introduction that the resulting identity 0.999... = 1 is a convention as well:
However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic.
I asked a question similar to this a couple times: if 0.999... is not equal to 1, then what do you get from the difference 1 - 0.999... ?
Let's begin with numbers that are finite. You get:
$$1 - 0.(9)_n = \frac{1}{10^n}$$
Where ##n## is a positive natural number. Even for ##n=0##, it works: ##1 - 0 = 1##. No matter what is the value of ##n##, there is always a difference that is non-zero.

You can even find ##\frac{1}{3}## with something similar:
$$3 \times 0.(3)_n = 0.(9)_n = 1 - \frac{1}{10^n} = 1 \times \left(1 - \frac{1}{10^n}\right)$$
$$\frac{1}{3} = \frac{0.(3)_n}{1 - \frac{1}{10^n}}$$
The difference between ##\frac{1}{3}## and ##0.(3)_n## can be found this way:
$$\frac{1}{3} - 0.(3)_n = \frac{0.(3)_n}{1 - \frac{1}{10^n}} - 0.(3)_n = \frac{1}{3 \times 10^n}$$
And since for each value of ##n## there is a value ##n+1##, it is possible that the difference get always smaller.

Now let's go into infinitesimals. If there is always a difference with finite number, then it seems logical that there is also one with infinite number. But it has to be smaller. Let's say ##\epsilon## is indivisible, thus the closest number to zero. One could define the difference as:
$$1 - 0. \bar 9 \triangleq \epsilon$$
If we use that definition to find ##\frac{1}{3}## as we did before, we get:
$$\frac{1}{3} = \frac{0.\bar 3}{1 - \epsilon}$$
Similarly, the difference between ##\frac{1}{3}## and ##0.\bar 3## should be found this way:
$$\frac{1}{3} - 0.\bar 3 = \frac{0.\bar 3}{1 - \epsilon} - 0.\bar 3 = \frac{\epsilon}{3}$$
Now we have a problem, since ##\epsilon## is indivisible by definition.

But if one defines the difference as:
$$1 - 0. \bar 9 \triangleq 0$$
Then repeating everything above yields:
$$\frac{1}{3} = 0.\bar 3$$
And:
$$\frac{1}{3} - 0.\bar 3 = 0$$
That is coherent and it is elegant. So by convention (as stated in my previous quote), accepting that ##1 - 0. \bar 9 \triangleq 0## is OK.

But what if ones defines the difference as:
$$1 - 0. \bar 9 \triangleq \lim_{n \rightarrow \infty} n\epsilon$$
I don't know if it causes other problems, but I know it solves my ##\frac{1}{3}## problem:
$$\frac{1}{3} = \frac{0.\bar 3}{1 - \lim_{n \rightarrow \infty} n\epsilon}$$
and:
$$\frac{1}{3} - 0.\bar 3 = \frac{0.\bar 3}{1 - \lim_{n \rightarrow \infty} n\epsilon} - 0.\bar 3 = \frac{\lim_{n \rightarrow \infty} n\epsilon}{3}$$
Because for a very large value of ##n##, there is always a ##\frac{n}{3}## that exists, then it is coherent.

But without looking further, that is already - as stated in my previous quote - a strange new object.

So, it makes sense to define ##1 - 0. \bar 9 \triangleq 0##, but I think the ##\triangleq## is important here, it cannot be proven, it is a definition. So the question «What else could it be equals to?» may very well have an answer if we insist that there must be a difference. But - when compared to not having a difference - would it add something without complicating everything? Probably not.

weirdoguy and hmmm27
jbriggs444
Homework Helper
2019 Award
##1-0.\dot9=0.\dot0 1##
But that fix is worse than the problem it is intended to solve.

Edit: here we have @Mark44 anticipating you:
Please do not tell me that the value is 0.00000...1, where there are an infinite number of 0's ahead of the 1.
The 9's on the left hand side are presumably indexed by the natural numbers in their standard order. The digits on the right hand side are indexed by some other set [or at least a different order type]. The numbers are incommensurate.

The problem with trying to make an arithmetic complete by adding new numbers presented with new and baroque syntaxes is that you then have to think about how you do addition, subtraction, multiplication and division with the new numbers. Instead of solving problems, one is creating new problems. It is not as easy a game as one might think.

It is also problematic when one throws around notation without defining it first. What exactly is the order type for the digit positions in ##0.\dot0 1## ?

Edit: Backing up and taking a somewhat longer viewpoint...

We ask about the difference between 1.0 and 0.999... because we have an expectation that whenever we write down two numbers, the difference between the two will be another number. This is the principle of closure. We want our number system to be "closed" under the operation of subtraction.

We started with a numbering system where we are allowed to write down numbers with "..." notation at their ends. This numbering system seemed not to be closed under subtraction. So you've invented a notation where numbers can now end with "..." followed by a single digit.

What about the product of 1.000...1 and 1.000...1? Is it going to be 1.000...0002000...1? Where does the madness end?

It turns out that we already have an arithmetic that is closed (the real numbers) and a notation that works (decimal notation). The gotcha is that to make the notation match the arithmetic, we need to accept that both "0.999..." and "1" denote the same number. Now we have saved closure because 1 - 0.999 = 0.

Back when I was in school they used to teach the difference between numbers and numerals. Like everyone else, I did not see much point in the distinction. Why distinguish between a number and the name of a number? "Shut up and add already".

A number of people here now should be lamenting not paying attention to the distinction then.

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PeroK
Mark44
Mentor
The problem with that is that there is no end of the game: there is always a difference. A difference necessarily means not equal to zero. This is why you need the concept of infinitesimals
Sure, always a difference, for any finite string of decimal digits. But in the notation 0.999... there are an infinite number of 9 digits to the right of the decimal point.

The game that we both are referring to goes like this:
Loop:
You tell me a small number ##\epsilon > 0## such that ##1 - \sum_{k=1}^N \frac 9 {10^k} < \epsilon##.​
I choose a (large) integer N for which the inequality is true.​
If you are satisfied, goto Done​
Else goto Loop​
Done:
End​

As far as infinitesimals being needed, not really, which is why the concept of infinitesimals was superseded by and large by the concept of limits, and only revived by the relatively new studies of surreal numbers, hyperreal numbers, and superreal numbers.

Let's begin with numbers that are finite. You get:
##1 - 0.(9)_n = \frac{1}{10^n}##
No argument there.
But what if ones defines the difference as:
##1 - 0. \bar 9 \triangleq \lim_{n \rightarrow \infty} n\epsilon##
I don't know if it causes other problems,
##\frac{1}{3} = \frac{0.\bar 3}{1 - \lim_{n \rightarrow \infty} n\epsilon}##
The problem here is the limit: ##\lim_{n \to \infty} n \epsilon##.
Here we have n growing large without bound that is multiplying a number very close to zero. In ordinary mathematics of real numbers, this is what is called indeterminate.

WWGD
Gold Member
2019 Award
The problem with that is that there is no end of the game: there is always a difference. A difference necessarily means not equal to zero. This is why you need the concept of infinitesimals, i.e there exist a number that is indivisible. That may put an end to the game.

No matter what, you will have to define something as being true. My previous source has a citation that I find interesting:

Let's begin with numbers that are finite. You get:
$$1 - 0.(9)_n = \frac{1}{10^n}$$
Where ##n## is a positive natural number. Even for ##n=0##, it works: ##1 - 0 = 1##. No matter what is the value of ##n##, there is always a difference that is non-zero.

You can even find ##\frac{1}{3}## with something similar:
$$3 \times 0.(3)_n = 0.(9)_n = 1 - \frac{1}{10^n} = 1 \times \left(1 - \frac{1}{10^n}\right)$$
$$\frac{1}{3} = \frac{0.(3)_n}{1 - \frac{1}{10^n}}$$
The difference between ##\frac{1}{3}## and ##0.(3)_n## can be found this way:
$$\frac{1}{3} - 0.(3)_n = \frac{0.(3)_n}{1 - \frac{1}{10^n}} - 0.(3)_n = \frac{1}{3 \times 10^n}$$
And since for each value of ##n## there is a value ##n+1##, it is possible that the difference get always smaller.

Now let's go into infinitesimals. If there is always a difference with finite number, then it seems logical that there is also one with infinite number. But it has to be smaller. Let's say ##\epsilon## is indivisible, thus the closest number to zero. One could define the difference as:
$$1 - 0. \bar 9 \triangleq \epsilon$$
If we use that definition to find ##\frac{1}{3}## as we did before, we get:
$$\frac{1}{3} = \frac{0.\bar 3}{1 - \epsilon}$$
Similarly, the difference between ##\frac{1}{3}## and ##0.\bar 3## should be found this way:
$$\frac{1}{3} - 0.\bar 3 = \frac{0.\bar 3}{1 - \epsilon} - 0.\bar 3 = \frac{\epsilon}{3}$$
Now we have a problem, since ##\epsilon## is indivisible by definition.

But if one defines the difference as:
$$1 - 0. \bar 9 \triangleq 0$$
Then repeating everything above yields:
$$\frac{1}{3} = 0.\bar 3$$
And:
$$\frac{1}{3} - 0.\bar 3 = 0$$
That is coherent and it is elegant. So by convention (as stated in my previous quote), accepting that ##1 - 0. \bar 9 \triangleq 0## is OK.

But what if ones defines the difference as:
$$1 - 0. \bar 9 \triangleq \lim_{n \rightarrow \infty} n\epsilon$$
I don't know if it causes other problems, but I know it solves my ##\frac{1}{3}## problem:
$$\frac{1}{3} = \frac{0.\bar 3}{1 - \lim_{n \rightarrow \infty} n\epsilon}$$
and:
$$\frac{1}{3} - 0.\bar 3 = \frac{0.\bar 3}{1 - \lim_{n \rightarrow \infty} n\epsilon} - 0.\bar 3 = \frac{\lim_{n \rightarrow \infty} n\epsilon}{3}$$
Because for a very large value of ##n##, there is always a ##\frac{n}{3}## that exists, then it is coherent.

But without looking further, that is already - as stated in my previous quote - a strange new object.

So, it makes sense to define ##1 - 0. \bar 9 \triangleq 0##, but I think the ##\triangleq## is important here, it cannot be proven, it is a definition. So the question «What else could it be equals to?» may very well have an answer if we insist that there must be a difference. But - when compared to not having a difference - would it add something without complicating everything? Probably not.
Two("Standard") Reals cant be indefitely close to each other without being equal.

Dopamine
jack action
Gold Member
The thing I want to emphasize is that a value at infinity doesn't exist. For example, the following graph represents the equation ##1/ x## enlarged at a region close to ##x = \infty## (where ##\infty^-## is the closest point to ##\infty##). The important thing to notice is that the function is not defined at ##x = \infty##. Intuitively, it always approaches zero. But it is not defined, so nobody can tell the value for sure. Nobody can prove it.

<Quasi profanity edited by mentor> I'm calling it: ##\frac{1}{\infty} \triangleq 0##.» There are clever ways with infinitesimals or limits to make it more bearable to define the indefinable, but in the end that's what it is: somebody, somehow, declared: ##\frac{1}{\infty} \triangleq 0##. Then it becomes true because this is how it is defined.

But if this is true, then you will end up with ##1- 0.\bar 9 = 0##. The bar over the nine represents a form of infinity. But you could have defined ##1- 0.\bar 9 \triangleq 0## and starting with this point, you would end up with ##\frac{1}{\infty} = 0##.

This is why when someone asks to be shown the proof that ##1 = 0. \bar 9##, it cannot really be done, because it is true because someone defined it as is. I've seen the ##\triangleq## used in a few definitions, and I think it is a very important distinction: ##1= 0. \bar 9## because ##1 \triangleq 0. \bar 9##.

This is true:
$$1 - 0.(9)_n = \frac{1}{10^n}$$
This is also true, but it only states that both limits are the same (of the undefined form ##\frac{0}{0} = 1##):
$$\lim_{n \rightarrow \infty} (1 - 0.(9)_n) = \lim_{n \rightarrow \infty}\frac{1}{10^n}$$
And the left side is basically a definition for a new notation:
$$0.\bar 9 \triangleq \lim_{n \rightarrow \infty} 0.(9)_n$$
So stating the following is still a comparison of two limits that ends up being an undefined form of ##\frac{0}{0}##:
$$1 - 0.\bar 9 = \lim_{n \rightarrow \infty}\frac{1}{10^n}$$
$$0 = 0$$

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weirdoguy
jbriggs444
Homework Helper
2019 Award
But it is not defined, so nobody can tell the value for sure. Nobody can prove it.
And nobody is trying to. You are arguing against a straw man of your own making.

weirdoguy and PeroK
jack action
Gold Member
And nobody is trying to. You are arguing against a straw man of your own making.
I think I'm arguing about circular logic.

weirdoguy
jbriggs444
Homework Helper
2019 Award
I think I'm arguing about circular logic.
Again, any circularity is in the straw man. You are responsible for any arguments it makes.

PeroK
Homework Helper
Gold Member
I think I'm arguing about circular logic.
Your uncertainty about how real analysis works is really just a "beginner's" mistake. Rather than confront that misunderstanding, you assume that the whole edifice of standard analysis is fundamentally flawed and that the 19th century mathematicians who developed it have duped all the mathematicians of the 20th and 21st centuries into a basic oversight. That's not a credible position.

jbriggs444, PeterDonis and weirdoguy
Mark44
Mentor
The thing I want to emphasize is that a value at infinity doesn't exist.
Agreed.
For example, the following graph represents the equation 1/x enlarged at a region close to ##x = \infty## (where ##\infty-## is the closest point to ##\infty##).
You have two errors in understanding here: "close to ##\infty##" and ##\infty-## being the "closest" point to ##\infty##. However "close to" ##\infty## you can name a number, I can come up with a number that is larger. And "closest to" ##\infty## makes no more sense than to say the closest number to 5, just to pick a number out of a hat.

jack action said:
I'm calling it: ##\frac{1}{\infty} \triangleq 0##.» There are clever ways with infinitesimals or limits to make it more bearable to define the indefinable, but in the end that's what it is: somebody, somehow, declared: ##\frac{1}{\infty} \triangleq 0##. Then it becomes true because this is how it is defined.
No. ##\frac 1 \infty## is not defined. Period.
Can you cite exactly who it was who claimed that ##\frac{1}{\infty} = 0##?

jack action said:
But you could have defined ##1- 0.\bar 9 \triangleq 0## and starting with this point, you would end up with ##\frac{1}{\infty} = 0##.
Again, ##\frac 1 \infty## is undefined, so it makes no sense to say that it is equal to any number, let alone 0.

jbriggs444, weirdoguy and PeroK
fresh_42
Mentor
The problem with that is that there is no end of the game: there is always a difference. A difference necessarily means not equal to zero. This is why you need the concept of infinitesimals, i.e there exist a number that is indivisible. That may put an end to the game.
This is wrong. There is no need for infinitesimals and any attempt to derive it is nonsense, since we are perfectly fine without. Your thinking is way too procedural. There is no process, hence no infinitesimal place left.

The quotation of Wikipedia only illustrates that ##0.\bar{9}## isn't properly defined if we do not assume a hidden limit, hence a convention. Namely ##0.\bar{9}=\lim_{n \to \infty}9\sum_{k=1}^n (\frac{1}{10})^k## which can easily be proven to be equal to ##1## without using any infinitesimals. The moment you drop limit and sum is the moment where convention comes into play and this discussion starts.

This thread will be closed. The entire debate depends on notational and therewith semantic arguments, plus we have had it already so many times.

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etotheipi, weirdoguy, jbriggs444 and 2 others
jedishrfu
Mentor
Since we have exhausted this thread topic to death and are in the infinitesimal stage of ending the discussion, I feel it's a good time to close it.

Thank you all for participating here.

Jedi

SammyS