.999 = 1?

Factual Questions
1931

Schooner26 This is for you.
I am sure that one of the issues here is one of notation - that is, when you write 0.999… you mean something different by the symbols than what I mean. I think that is what is going on. I would love you to clarify a couple of things for me.
Let me make it easier by listing them.

Item 1 - Use of the word “Limit”
You have said a couple of times that the limit of 0.999… is not the same as 0.999…
Can you please explain what you mean by this. I am confused.
As far as I know, one does not obtain the limit of a number. Nor does one obtain the limit of a representation of a number.

I can find the limit of a function.
I can find the limit of a sequence.
I can find the limit of a series.
(I can even find the limit of my patience.)

But I never find the limit of a number - unless (trivially) it is that number. That is, limit of 7 is 7. Following this through, that would suggest that limit of 0.999… is equal to 0.999… You seem to object to this line of reasoning.
What am I missing here?
Item 2 - Use of ellipsis
When I write 0.33333… I mean the digital representation of a number (base 10 is assumed) where all of the digits following the decimal point are "3"s. Furthermore, the number of "3"s is equal to the cardinality of the natural numbers. (Which is well defined by Cantor.) Let’s avoid the word, “infinity” here. Although it is true that mathematicians refer to the term “countably infinite” to refer to the cardinality of natural numbers.
Do you agree with this definition?

(I am somewhat confused because you keep talking about some remainder. I never get up to a remainder because there are a countably infinite 3’s to write first.)
Item 3 - recurring decimals and their relation to infinite series
This is something of a combination of items 1 and 2, but indulge me here.

Consider the number 0.7777777… (see what I did there!)
I interpret this as the representation of a real number - a number that can also be represented by a series as follows:
0.7777777… = 7/10 + 7/100 + 7/1000 + 7/10000 + 7/100000 + 7/1000000 + 7/10000000 + …
The ellipsis on the right hand side refers to the concept that the number of terms is identical to the cardinality of the natural numbers.

This corresponds exactly to the standard definition of a geometric series:
S = a + ar + ar[sup]2[/sup] + ar[sup]3[/sup] + ar[sup]4[/sup] + ar[sup]5[/sup] + …
For this particular case, a=0.7 and r=0.1

For the number at the focus of this thread, a=0.9 and r=0.1. And again the number of terms is the same as the cardinality of the natural numbers.
Do you agree with this interpretation of the notation? Or do you infer something different?
(I ask because you have been invited several times to discuss the matter in terms of an infinite geometric series but you haven’t really responded to this invitation.)
Item 4 - Summation of an infinite geometric series
I note that you have used the summation formula a couple of times. I am just checking here that you mean the same thing by it as what I do and that you agree with me on the validity of its derivation.

I would love your thoughts on this – whether or not you agree with me up to this point. I rather suspect that the point where you disagree with me is upstairs from here. I think we could discuss Item 4 without mentioning 0.999… at all.
Step 1, by definition
S = a + ar + ar[sup]2[/sup] + ar[sup]3[/sup] + ar[sup]4[/sup] + ar[sup]5[/sup] + …

Step 2, multiply both sides by r
Sr = ar + ar[sup]2[/sup] + ar[sup]3[/sup] + ar[sup]4[/sup] + ar[sup]5[/sup] +ar[sup]6[/sup] + …

Step 3, subtract line 2 from line 1


S–Sr = a + ar + ar[sup]2[/sup] +  ar[sup]3[/sup] +  ar[sup]4[/sup] +  ar[sup]5[/sup] + ...
         – ar – ar[sup]2[/sup] –  ar[sup]3[/sup] –  ar[sup]4[/sup] –  ar[sup]5[/sup] – ...

Step 4, simplifying line 3
S–Sr =a

Note on step 4
If there is any point where you disagree with me in this derivation then I suspect it will be this one here. I maintain that this simplification is valid.
It is not because of the limit (represented by the ellipsis)
It is not because of the infinity of the terms – it is not even necessary to consider all infinity terms.
The simplification is valid because there is a one to one correspondence between the terms in the two sequences which can be readily seen when terms with the same power are lined up one above the other.

(Note 2 on step 4
I suppose greater mathematical rigour would be required in the case of a divergent series, that is when |r|≥1. I believe that comes under the heading of Borel Summation. But I would be close to the limit of my mathematical understanding. (See what I did there!) (Twice in one post, no less.))

Step 5 rearranging step 4
S = a/(1–r)
Item 5 - Use of the formula above to determine the value of the number represented by 0.9999…
I note that you have actually posted this so I think we are in agreement here.

As previously discussed, 0.999… represents a geometric series where a=0.9 and r=0.1
Substituting,
S=0.9/(1–0.1)
S=0.9/0.9
S=1
Conclusion, if 0.999… means anything in the real number system then it means the sum to infinity of a geometric series where a=0.9 and r=1. Therefore 0.999… = 1.
Now, I know that you arrive at a different conclusion from me. I suspect that the difference arises in item 1, 2 or 3 where you treat ellipsis, limit or infinity in a way that I do not follow – You have not defined your meaning of these to my satisfaction and it seems that there is some difference between what you mean and what I mean by these concepts.

It has been stated that there are mathematical systems that do treat 0.999… as being unequal to 1. However, it must be noted that

  1. These systems are not the real number system that we are used to
  2. The standard interpretation of the notation of the OP implies the real number system
  3. These systems differ from the reals at an axiomatic level - the resultant numbers behave quite differently from the reals.
  4. There is more than one such system. If you are proposing to use one of these then you need to be rigorous and define your terms precisely so that you avoid contradiction in what you are saying.
  5. the reals are everywhere continuous. In this system there are no infinitesimals greater than zero (although infinitesimals are a useful tool applied to the reals.) These two concepts taken together are all that is required to conclude that in the Real Numbers, 0.9999…=1
1932

Phew. Longest post I have ever made here. Still, I have been something of a spectator and it’s about time I pulled my weight in this thread.
Schooner, I would love some feedback. If you would be gracious enough to respond to each of items 1, 2, 3, 4 and 5 – well, that would be grand. Like I said, I am sure the disagreement is one of interpretation of some notation and that because of your interpretation, you gravitate towards an understanding of numbers that is not the real number system. I can’t for the life of me work out what that difference is or what the properties are of the system that you arrive at.

1933

Item 4 violates schooner26’s law of series addition

Addition is not associative in schooner26 arithmetic.

1934

To be fair when summing infinite series addition isn’t generally commutable (which is the problem schooner sees (and not associativity)).

1935

Which is why it’s better to avoid that line of argument and deal directly with the infinite geometric series as the limit of the finite geometric series:

When we say

S = a + ar + ar[sup]2[/sup] + ar[sup]3[/sup] + …

we mean

S = lim[sub]n->∞[/sub] (a + ar + … + ar[sup]n[/sup]) = lim[sub]n->∞[/sub] (a/(1-r) - r[sup]n+1[/sup]/(1-r))

When |r|<1, the term r[sup]n+1[/sup] goes to 0 and n gets arbitrarily large (not necessarily meant to be obvious, but requires another proof omitted here), hence

S = a/(1-r).

ETA: Of course, I’m using the phrase “goes to 0 as n gets arbitrarily large” as shorthand for the epsilontic proof (or your other favorite way of making the notion of limits precise).

1936

schooner’s post made it quite clear that he objects to the “selective” summing that leads to
lim[sub]n->∞[/sub] (a/(1-r) - r[sup]n+1[/sup]/(1-r))

You should probably re-read his post here (starting at “Here is a summary of Euler’s proof for infinite sums”) to best catch his drift.

1937

I’m reading his objection as being to taking the difference of the two infinite series and rearranging terms (which is legitimately questionable). This is what you avoid by taking the limit of the finite sums.

1938

Yes that is (at least one of) his objection(s), commuting terms through infinite sums.

TBH I don’t get what you are saying now, because the formula that you used above

implicitly supposes that we can do that we can rearrange the expression:
a + ar + ar² + … + arⁿ + … - (ar + ar² + … + arⁿ + … ) to our “convenience” (which is exactly where schooner disagrees (not I)).

1939

Sorry, I’ve just got your point: forget I posted.

1940

Yet you are still infinitely away from zero, therefore meaningless.

Ok. But this is the problem I have. Infinity is allowed, but infinitely small is not, yet it still exists.
As you said: 1 - infinitesimal = 0.(9)
So the idea existed, but then it is set to 0 after it was used.
If it was 0, then we’d have 1 - 0 = 1 and not 0.(9)

Key word is “smaller”. Small is not 0 though.

1941

And this is why I hang out in threads such as these.
I thought that both associativity and commutativity were both preserved under the Reals.
I understand that there are issues with commutativity and divergent series. I was not aware that there are any problems with reordering a convergent series; which is clearly what we are discussing in this thread.

1942

There are no problems in rearranging the terms of an absolutely convergent series, i.e., one which remains convergent if all the terms are made positive. However, if Sn is convergent but /Sn/ is divergent, then you can have problems rearranging the terms of Sn.

ETA: Of course, infinite decimal fractions are absolutely convergent, because the terms are all positive to start with.

1943

0.999… = Σ (n=1 to ∞) 9/10ⁿ
limit 0.999… = limit Σ (n=1 to ∞) 9/10ⁿ

I agree that if you tell me you mean 1/3 when you write 0.333… then I would understand what you are telling me. However they are not exactly equal

You are forgetting that in order to get each 3, there is a remainder which allows it to be generated. Use long division on a piece of paper for 3 into 1. The numbers you write on top are the quotient and the subtraction that you do results in the remainder. The reminder is never 0 (the subtraction that is done)

ok

Yep

On page 37 I addressed the a / (1 - r) concept.

Here is your error. You illegally shifted the series to do the subtraction.
Infinite series subtractions are done for equal indices of the integers.
. You see, here is the correct way of subtracting infinite converging series:
Σ { a_i - b_i } = Σ a_i - Σ b_i
But you see in your proof, you subtract like this: a_i+1 - b_i:
S - rS = (a - 0) + (ar - ar) + (ar² - ar²) + …

However, when done correctly, you get consistent results:
S = Σ a_i
rS = Σ r x a_i
S - rS = Σ {a_i - ra_i}

S = (a - arⁿ) / (1 - r)
limit S = limit (n→∞) (a - arⁿ) / (1 - r) = a / (1 - r) = 1

a / (1 - r) is the result of a limit which is glossed over by many people.

Not without stating “limit”

1944

Neither infinity nor infinitely small are part of the Real number system.
Useful concepts, but not numbers and not part of the Reals. The language that Mathematics has adopted to work with these concepts is the language of limits. Admittedly it took more than a century after Newton and Leibniz for the subtleties of this language to be ironed out and all terms to be rigorously defined.

If you want to work within a system that allows infinitesimals as numbers then go for it. Just make sure that your system is fully defined and don’t pretend that it is the answer to the central question of this OP which was posited within the Real number system.

1945

Ok. Understood.

The ellipsis I understand to be a shorthand notation for limit. You treat it as a shorthand notation for some finite partial sum and require the addition of the word, “limit” to achieve the same result as I get.

I grant that “…” is a convenient shorthand and not something that is part of formal mathematical language. However, given that this is the point of difference between you and a large number of professional mathematicians that have contributed to this thread, can you point me to any other person of repute who interprets the ellipsis in the same manner that you do?

1946

My point is simply that this ability to reorder absolutely convergent series is not obvious and needs to be proved. It’s not necessary to understanding geometric series, so better to avoid it when discussing how one rigorously deals with infinite decimal representations.

1947

Thanks for clarifying that (and also Giles). I didn’t think that I was on the wrong track.
I was searching for an explanation that avoided the use of the limit concept since this was where the contention appeared to be. Besides, I have not had much experience in working with divergent series and although I recognise the need for care and rigour, I would likely make a number of significant errors if I was to present the necessary proof.

As it stands, I think it is a case of … for me not meaning the same as … for schooner I note that he has painted himself into a contradiction with this as follows:

In the first case, … is not a limit; resulting in 1/3 not equal to 0.333.
In the second case, … is a limit to infinity. He said, “ok”.

1948

“ok” means that I am following what you are saying. :slight_smile:

1949

Here’s what I think is going on (you can correct me if I’m wrong).

When mathematicians see “1/3” or “0.333…” they think of it as referring to a number: a specific element of the set of real numbers, a particular point on the real number line.

You (schooner26) think of one or both of these as referring to a process: the process of dividing 1 by 3 (via long division, which takes forever because you keep getting a remainder), or the process of adding up an infinite series.

1950

When I see 1/3, I see 1/3

The process is when *you *try to convert it to a decimal (in base 10).

Without assuming 0.333… = 1/3, try and put 0.333… on a number line by doing plotting each term of the series.