# Boxing Pythagoras

## Yet another failed attempt at showing 0.999…≠1

I’ve discussed before how mathematics can sometimes lead to very counterintuitive results. One of the most common, and famous, of these counterintuitive properties of math is that the number 0.999… (that is, zero point nine, nine, nine, repeating) is equal to 1. This one is so well known that it is fairly often taught even to Elementary and High School students. If you are unfamiliar with this discussion, I highly recommend that you watch this video from Vi Hart, in which she discusses 10 different reasons to accept this concept. Additionally, you may have fun watching this video, in which she lampoons the common objections to the concept.

Despite the fact that it is fairly simple to prove that 0.999…=1, the concept is so counterintuitive that I find people try to struggle against it– even when they know and accept the reasoning behind the equality. One such attempt comes from Presh Talwalkar. In the following video, Mr. Talwalkar attempts to demonstrate that on the Surreal number system, 0.999…≠1.

Unfortunately for Mr. Talwalkar, he is wrong. Even on the Surreals, it is still true that 0.999…=1.

In the video, Mr. Talwalkar acknowledges that it is absolutely true that 0.999…=1 on the Real numbers. However, he then asserts that it is not true that 0.999…=1 on the Surreal numbers. Right away, this should look fairly suspect to anyone familiar with the Surreals. The reason for that is that the Surreals are a superset which contains the Real numbers. Anything which is true for a number which exists within the Real numbers will similarly be true for that number in the Surreals. It is therefore entirely incoherent to claim that 0.999… is a different number in the Surreals than it is in the Reals.

Most of Mr. Talwalkar’s video is fairly accurate, though I would prefer a more rigorous treatment of its subject matter. The point where it goes wrong, however, comes when he attempts to discuss some “weird numbers” which can be constructed on the surreals. He begins by discussing $\{0|1,\frac{1}{2},\frac{1}{4},\frac{1}{8},...\}=\epsilon$, which he erroneously claims to be “1 divided by infinity” and “point zero repeating, with a one at the end.” Neither of these descriptions is even coherent. Infinity is not a number. You cannot divide 1 by infinity any more than you can divide 1 by Blue, or by Sweet, or by Alexander Hamilton. The latter description “point zero repeating, with a one at the end” is quite obviously self-contradictory. If we have “point zero repeating,” then there is no “end” at which to write a “one.” That said, the Surreal number which Mr. Talwalkar defines here, $\epsilon$, is an actual number and can be utilized in mathematics. As defined, $\epsilon$ represents a number which is greater than zero, but smaller than all of the Real numbers. That is to say, for any Real number $r$, it is true that $0 <\epsilon .

He continues by defining another "weird number," $\{ 0,\frac{1}{2},\frac{3}{4},\frac{7}{8},...|1\}=1-\epsilon$. Again, this number is correctly defined, and it is an actual number. It is therefore possible to deduce certain properties which are possessed by $1-\epsilon$. For example, since we know that $\epsilon> 0$, we can know with certainty that $1-\epsilon <1$. However, Mr. Talwalkar oversteps the bounds of logic when he baldly asserts that "you can think about it as point nine repeating." He gives absolutely no justification for asserting that $1-\epsilon=0.999...$, and it is actually quite simple to prove that this assertion is, in fact, entirely untrue. I shall do so, now, by Reductio ad Absurdum.

Let’s start by assuming Mr. Talwalkar’s assertion to be true. Then, by exploring the properties of the numbers, I’ll show that this assertion leads to a logical contradiction, and that it therefore cannot be true.

1. $1-\epsilon=0.999...$
2. $\epsilon>0$
3. Given (1) and (2), $1-\epsilon\neq 1$ and therefore $0.999...\neq 1$
4. $\frac{1}{3}=0.333...$
5. $\frac{1}{3}\times 3=1$
6. $0.333... \times 3=0.999...$
7. Given (4) and (6), $\frac {1}{3}\times 3=0.999...$
8. Given (5) and (7), $0.999...=1$
9. Given (3) and (8), $0.999... \neq 0.999...$

This, of course, is nonsensical. A number must always equal itself. Therefore, our premise (1) cannot be true.

It doesn’t matter whether we are talking about the Rational numbers or the Real numbers or the Hyperreal numbers or the Surreal numbers. The simple fact of the matter is that 0.999… is equal to 1. For some reason, many people find this very difficult to accept, but it is absolutely true. Presh Talwalkar’s attempt to show otherwise fails just as surely as all those which came before it. To quote Vi Hart on the issue:

If you’re having Math problems, I feel bad for you, son.
I got 98.999… problems, but 0.999… equals 1.

## 29 thoughts on “Yet another failed attempt at showing 0.999…≠1”

1. It is perhaps unfortunate that infinite decimal expansions of 1/3, 1/7, etcetera are the first exposure to kids of “the infinite”, and we hide the handwaving while happily subtracting 0.9999…. from 9.99999… to get 9, without actually performing the calculation.
Strictly speaking an infinite decimal is an infinite series, and has to be seen as a limit of partial sums of the terms of the corresponding infinite sequence 0.9, 0.09, 0.009,…..
It doesn’t take too long to establish the limit as 1

What is the value of pi – e ?????

2. Ignostic Dave on said:

It seems to me that he could have skipped the business with surreal numbers and said, “Look at all those nines, it’s clearly not equal to one.”

• The funny thing is that it is extremely easy to find a way to legitimately make 0.999… not equal to 1. Just change the base of your number system to anything greater than 10.

For example, 0.999… in Hexadecimal is exactly 9/F (that is, 9/15 in Decimal). This is CLEARLY not equal to 1.

3. In 2015 it was proved that 0.999… does NOT equal 1. I will explain why here, but you may prefer to watch the two videos by ‘Karma Peny’ on YouTube.

(1) If we can prove 0.333… = 1/3 then it follows that 0.999… = 1, so all we need to do is prove 0.333… = 1/3.

(2) We start with 1.000… divided by 3.000…. (we start by assuming all our ‘real’ numbers have ‘infinitely many’ digits to the right of the decimal point).

(3) Next is the tricky bit. We need to explain how the division process can complete, because if it does not complete then we have not got a fixed value. This argument assumes numbers have fixed values and are not constantly changing their value.
If we stop the division process after n decimal places have been processed, then to the right of the decimal point we will have n iterations of the of the digit ‘3’ plus an expression for the remainder part, which is 1/(3 times (10 to the power n)). The n digits plus the remainder expression equals 1/3 exactly.

(4) Now, however large we make n, there will still be this remainder expression which has a positive non-zero value.

(5) We could assert that after ‘infinitely many’ iterations, the remainder part is somehow no longer relevant and we will have a fixed value that equates to 1/3. But equally we could assert that after ‘infinitely many’ iterations there are still ‘infinitely many’ more non-zero terms that need to be processed by the division process. Here the remainder expression still represents a non-zero value (even though we appear to have lost the ability to talk about it accurately because we have replaced ‘n’ by ‘infinitely many’).

(6) Now we need a clear understanding of what ‘infinitely many’ means in order to resolve this issue. We need to show how ‘infinitely many’ can be achieved because the two assertions made above directly contradict one another.

(7) We find we cannot show how the division process can end, and so we cannot resolve the problem of how the reminder expression can be ignored. In short, ‘infinitely many’ cannot overcome the contradiction it creates.

(8) Traditional proofs like the 10x -x proof use ‘bad mathematics’ because they informally take terms ‘from infinity’ in order to produce the result they want to produce. If you do this proof with rigour (as shown in the ‘Karma Peny’ video), then the flaw in the so-called proof is obvious.

(9) The ‘epsilon-delta’ argument says that 0.999… must equal 1 because there is no number between the two. But this argument starts with the assumption that 0.999… can exist in ‘entirety’ and does have a static value. That is to say, its starting assumption is that ‘infinitely many’ is a valid concept. But we have already seen above that this concept results in contradiction.

In short, if we model ‘endlessness’ instead of trying to realise ‘infinity’ then mathematics will work without paradoxes and without contradictions.

• As I mentioned in my reply on the BP Facebook page, this argument does not seem so much to be that 0.333… does not equal 1/3 (and, therefore, 0.999… does not equal 1) so much as it is the claim that 0.333… and 0.999… are not actually numbers. You seem to justify this position by rejecting the concept of a completed infinite set.

As I see no reason to accept that there cannot be a completed infinite set, and since such a position would require the rejection of ALL Real numbers, your argument does not seem very convincing.

The main problem with your position comes in (3) through (5). You are here admitting that:


..which is an infinite sum. You then rightly say that any nth partial sum of this series would differ from 1/3 by exactly:


Interestingly, this seems to be a tacit admission that 0.333…=1/3, since we could replace the 1/3 in this expression with 0.333… and it would be exactly as true.

However, you assert that “after ‘infinitely many’ iterations there are still ‘infinitely many’ non-zero terms that need to be processed,” which does not seem to be true. In fact, it seems fairly trivial to see that the cardinality of the terms in our infinite series is Aleph Null, which is the smallest possible cardinality an infinite series can have. Therefore, if it is true that we have a completed, infinite set of all of the terms in the sum, then it cannot be true that we have infinitely many more terms which are not yet accounted for.

• By the way, you are correct to say that I am claiming that 0.333… and 0.999… are not actually numbers, if a number is considered to have a fixed/static value.

Also,
1/1 + 1/2 + 1/3 + 1/4 + … is not a number

1+1+1+1+… is not a number
1+2+3+4+…. is not a number
1-1+1-1+1-1+… is not a number

But if you accept ‘infinitely many’ combined with the idea of ‘convergence at infinity’ then you will think 0.333… is a number,

and you will think
1/2 + 1/4 + 1/8 + 1/16 + … is a number

All these objects are essentially of the same fundamental type; they are endless series were we have an expression to define the nth term.

You would not think that parallel lines meet at ‘infinity’ because we can prove with simple logic that the two paths will not meet at any distance.

We can also prove using simple logic that 0.999… can not equate to 1 however far we go. But the wordplay trick of ‘convergence’ is to forget simple logic and take it that the paths somehow magically meet at a mystical place called infinity.

• Also, my position does not reject ALL real numbers, just those that are claimed to contain ‘infinitely many’ non-zero terms.

Note that Pi, the square root of 2, and one third can all be represented in entirety on a computer (or in written form) in finite terms.

This finite representation includes a finite numeric part (e.g. diameter = 1) and the definition of an algorithm (e.g. algorithm to calculate the circumference, or the division algorithm).

But when we apply the algorithm, either it needs to stop on its own, producing a finite numeric result, or after n terms we include an expression for the remainder part. This way we maintain equality, we have no ’rounding errors’, and most importantly we maintain mathematical rigour.

What we should not do is invent a mystical place called infinity and pretend everything will magically end and not end at the same time at this strange place.

• You would not think that parallel lines meet at ‘infinity’ because we can prove with simple logic that the two paths will not meet at any distance.

Interesting that you would choose this example, as the truth of the matter is actually precisely the opposite of what you claim. This problem is known as the Parallel Postulate, and it stumped the world’s greatest mathematicians for 2500 years. The simple fact of the matter is that it is not possible to prove with “simple logic” that the two paths will not meet at any distance. We have to take this position axiomatically, in Euclidean geometry.

It was the realization of this exact point which led to the exploration of non-Euclidean geometries in the 19th Century.

We can also prove using simple logic that 0.999… can not equate to 1 however far we go.

The “no matter how far we go” is the problem with this. This presumes a finite exploration of the number. I will absolutely agree that:


…for any finite Natural number i. This does not imply that the same thing holds for, say, an infinite Hyperreal number i.

Yes, if you hold to an a priori rejection of the concept of the infinite, then there is no such thing as an infinite decimal expansion– in which case, the whole question is moot. If 0.999… is not a number, then it makes no more sense to proclaim, “0.999… does not equal 1” than to proclaim, “Blue does not equal 1.” Personally, I see no good reason to reject the concept of the infinite; and the successes of infinite and infinitesimal mathematics more than justify its adoption, in my eyes.

Also, my position does not reject ALL real numbers, just those that are claimed to contain ‘infinitely many’ non-zero terms.

So, you do not reject all Real numbers, just all Real numbers which are not Rational numbers.

Note that Pi, the square root of 2, and one third can all be represented in entirety on a computer (or in written form) in finite terms.

As a computer scientist and mathematician, I’d be incredibly interested in how you propose this is possible– presuming you mean that decimal expansions of these numbers can be represented in their entirety in finite terms.

What we should not do is invent a mystical place called infinity and pretend everything will magically end and not end at the same time at this strange place.

Infinity is not a place, mystical or otherwise. It is a property of numbers.

• I did not impose a priori rejection of the concept of ‘infinitely many’ in my argument. I found that if I assume this concept is true, then it leads to a contradiction.

Your counter argument now seems to be that my reasoning holds for objects that consist of a finite amount of terms, but that my reasoning does not hold for objects with ‘infinitely many’ terms.

There are two problems with this counter argument.
1. I was allowing the possibility of ‘infinitely many’ when I highlighted the contradiction.
2. Here you are imposing your priori acceptance of the concept of ‘infinitely many’. You are effectively saying that because you already believe ‘infinitely many’ is valid, then any counter argument must therefore be invalid.

Regarding decimal expansions, we can express one third as 0.333… to the n-th 3, plus 1/(3(10 to the power n)). Thus we can express the decimal digits to any required level of accuracy for our user (an engineer, say), and completely without loss of accuracy. There are no rounding errors here. As you are a computer scientist, you are likely to be familiar with the algorithms behind vector graphics. The principal is very similar.

Is your objection to my found contradiction that there are no more terms after ‘infinitely many’ and thus the series is not endless after all?

Can you explain why there cannot be ‘infinitely many’ after ‘infinitely many’? This is a valid scenario according to the example of Hilbert’s Hotel (where the hotel has ‘infinitely many’ guests and can still accommodate ‘infinitely many’ more).

• I did not impose a priori rejection of the concept of ‘infinitely many’ in my argument. I found that if I assume this concept is true, then it leads to a contradiction.

When I pointed out the reasons why there is no contradiction, you replied that if you presume the contradiction to be applicable, then the properties of infinite sets cannot be utilized to show that the contradiction is inapplicable. I took this to be an a priori dismissal of actual infinites.

If, however, you want to assume that the concept of “infinitely many” is valid, then it would seem you should also assume that the properties of infinite sets– including cardinality– are valid. That is, of course, unless you can show some good reason for rejecting those properties.

Here you are imposing your priori acceptance of the concept of ‘infinitely many’. You are effectively saying that because you already believe ‘infinitely many’ is valid, then any counter argument must therefore be invalid.

That’s not what I’m saying. I’m saying that if we assume the validity of the concept of “infinitely many,” which (as you’ve said) is the scenario which you were adopting for this argument, then the alleged contradiction which you presented is not actually present. There may be other valid counter-arguments. I’m not aware of any, but they may exist. I’m only speaking to the specific argument which you laid out.

Regarding decimal expansions, we can express one third as 0.333… to the n-th 3, plus 1/(3(10 to the power n)). Thus we can express the decimal digits to any required level of accuracy for our user (an engineer, say), and completely without loss of accuracy.

It is not “completely without loss of accuracy,” in the least. There is a completely obvious loss of accuracy. You and I have already agreed that no finite decimal expansion of 0.333… will ever equal 1/3. Therefore, any finite decimal expansion of 0.333… will bear a loss of accuracy as compared to 1/3.

Can you explain why there cannot be ‘infinitely many’ after ‘infinitely many’? This is a valid scenario according to the example of Hilbert’s Hotel (where the hotel has ‘infinitely many’ guests and can still accommodate ‘infinitely many’ more).

Hilbert’s Hotel requires manipulating the arrangement of guests in rooms. If you manipulate the arrangement of decimal digits in the number 0.333… in an analogous manner, then you will no longer have the number 0.333…

Nor is the concept of “after” even coherent when discussing the digits of 0.333… The word “after,” in this context, means “subsequent to the final term of the series.” As there is no final term of the series, there cannot be an “after.”

• I am not admitting that 0.3 recurring equals ‘infinitely many’ terms, which is what you are saying with that statements that includes “from n= 1 to infinity”.

What I did was I assumed ‘infinitely many’ was possible, and then I showed that this leads to a contradiction. Therefore statements such as ‘for n=1 to infinity’ and ‘infinitely many’ are invalid.

If my logic is correct and there is a contradiction, then Stevin’s real numbers and Cantor’s infinite sets are invalid and cannot be used as arguments against the original contradiction (i.e. ‘infinite sets’ and Aleph Null are inadmissible).

This is a fundamental problem and it needs to be addressed using fundamental core logic. .

But I think the meaning of your reply can still be gleamed. Your counter argument appears to be that after ‘infinitely many’ terms there are no more terms. This is the same as the first assertion in point (5) of my original argument.

Therefore my original argument still stands.

• If I propose 1=2, then both these counter arguments are invalid:
(1) Since I already know 1=2, this statement is true
(2) Since I already know 1 does not equal 2, this statement is false

The arguments against the validity of the statement should not start by assuming the validity or not of the statement.

If I point out to you that your argument is only valid if 1=2, then has case (1) as its basis and is therefore not valid. My pointing this out to you does not mean that I am assuming (2).

You said: “Therefore, any finite decimal expansion of 0.333… will bear a loss of accuracy as compared to 1/3.”

But if you re-read what I said, you will notice that I am only expanding to the desired level of accuracy (i.e. to a finite number of ‘3’ digits) and then I am adding a ‘remainder’ expression that makes the whole sum equate exactly to 1/3. This maintains mathematical rigour.

You said “in an analogous manner, then you will no longer have the number 0.333…”

But whenever we want to find a square root, find a circumference, or perform a division don’t we use analogous processes?

Do you believe that:
(1) The division process when applied to 1 divided by 3 does not operate in an analogous manner. If so, how else can a division process operate?
(2) The division process does operate in an analogous manner, but the answer it reveals already exists somewhere, and it already has infinite length. If so, where does the square root of 2 exist and how can you retrieve the n-th character from this place?

Getting back to the main point…

Your counter argument to the alleged contradiction is “ if it is true that we have a completed, infinite set of all of the terms in the sum, then it cannot be true that we have infinitely many more terms which are not yet accounted for.”.

Here you appear to be saying we have accounted for all the terms. You have not said how this is possible without having a ‘last term’ and importantly you have not explained how the division process can end, which was the whole point of my original argument. This is not a very convincing counter argument.

• Apologies for my mixing up of ‘analogous’ and ‘algorithmic’ in my previous post!

• But if you re-read what I said, you will notice that I am only expanding to the desired level of accuracy (i.e. to a finite number of ‘3’ digits) and then I am adding a ‘remainder’ expression that makes the whole sum equate exactly to 1/3. This maintains mathematical rigour.

What is the purpose of this decimal expansion?

If the purpose is to create an approximation of 1/3 for use in calculations, then the remainder expression is extraneous, because it will not be used in those calculations. As such, there is certainly a loss in accuracy.

If the purpose is simply to create an expression equal to 1/3, then the whole exercise is simply redundant, as we could simply say “1/3” and be done with it.

Apologies for my mixing up of ‘analogous’ and ‘algorithmic’ in my previous post!

No worries. The point I was making is that Hilbert’s Hotel has no bearing on the discussion because we are not manipulating the digits of 0.333… in any way similar to the manipulation of vacancies in the Grand Hotel.

Here you appear to be saying we have accounted for all the terms. You have not said how this is possible without having a ‘last term’ and importantly you have not explained how the division process can end, which was the whole point of my original argument. This is not a very convincing counter argument.

I see no reason why a last term is necessary. All that is necessary for a number to have meaning is for that number to be well-defined. It is quite easy to define what we mean by the symbol, “0.333…” This definition does not require a last term.

Nor is the Division operation the same as the algorithm by which we evaluate Division. I can offer a number of different algorithms for evaluating Division. The fact that we can iterate a particular algorithm indefinitely does not imply anything about the operation which that algorithm is intended to evaluate.

• You said: “If the purpose is to create an approximation of 1/3 for use in calculations, then the remainder expression is extraneous”

Often a result may be required (by an engineer, say) to make a real-world measurement to a desired level of accuracy. Often such results can feed into other calculations. If the value fed into other calculations is not entirely accurate, then the margin of error increases making later results less accurate and potentially outside of the required level of accuracy.

You said: “It is quite easy to define what we mean by the symbol, “0.333…” This definition does not require a last term.”

But it is not at all obvious because I interpret this in a completely different way than you do. I see this as a process that is endless, whereas you see this as a fixed value that has no more terms after ‘infinitely many’.

You said: “The fact that we can iterate a particular algorithm indefinitely does not imply anything about the operation which that algorithm is intended to evaluate.”

I’m not sure I understand what you are getting at. Does this mean although we try to evaluate pi, or a square root, or a division via processes, in a purer sense these are just transformations of static values into a different form. These values already exist; we are merely using processes to look at the values in different ways. This belief that all numbers/values already exist is known as mathematical platonism with a small ‘p’ (apologies for this sounding very silly if it is not what you mean).

I am still no closer to understanding your counter argument.

• Also…

Symbolic representations of endless iteration and/or endless recursion (often combined with phrases like ‘inductive proof’) are often purported to demonstrate that something is ‘infinite’ and thus contains ‘infinitely many’ elements.

But however many digits are produced by a division algorithm (or any other form of repetition/recursion), it can only ever be a finite number. It does not matter if we think of the algorithm as happening instantaneously; the count still cannot change from being finite to being infinite after some really big number, just like if you start counting the natural numbers you cannot reach ‘infinity’.

Therefore a cyclic algorithm that performs division cannot achieve a result containing ‘infinitely many’ digits. If the ‘division operation’ is something different to the algorithms used to perform division, then the division algorithms are invalid because they are not true to the definition of ‘division’. They should be scrapped and replaced by valid algorithms.

What is the true definition of ‘division’ if it is different to the algorithms? I would like this to be clarified so that I can understand how a result with ‘infinitely many’ digits can be achieved.

It appears we cannot produce ‘infinitely many’ digits and we cannot add ‘infinitely many’ digits and we cannot add up ‘infinitely many’ digits. So the only way we can attempt to ‘prove’ what infinitely many digits add up to is by arguments such as the 10x – x so-called ‘proof’ (which actually proves 0.999… does NOT equal 1 when done with rigour – as in the ‘Karma Peny’ video) and the epsilon-delta argument (which starts out by assuming that 0.999… can exist in ‘entirety’ and does have a static value – it effectively assumes what it sets out to prove).

With no valid mathematical definition for ‘infinitely many’ and with no valid proof, I see no reason to accept the concept.

• There is nothing real about the real numbers. Check out a good book on analysis and see that the words infinite and infinity are only used as a shorthand, and no claim for existence is made.
I can set up a sequence of partial sums of the sequence 0.5, 0.25, 0.125,… or 1/2, 1/4, 1/8, 1/16, … which has the same limit as the 0.9, 0.99, 0.999,…, which is 1.
The infinite, or never ending decimal expansion is a very non mathematical thing. It’s a shame that kids are obliged to deal with it using this description.
It is the limits of sequences that are the real numbers, and the only reason the word ‘number” is used is that you can show that they obey the same rules of arithmetic as the rational numbers.

• Often a result may be required (by an engineer, say) to make a real-world measurement to a desired level of accuracy. Often such results can feed into other calculations. If the value fed into other calculations is not entirely accurate, then the margin of error increases making later results less accurate and potentially outside of the required level of accuracy

I’m not sure how this is meant to address my point. If you split the number into a non-repeating decimal approximation and an error correction, you are still performing your calculations upon the approximation. Carrying along the error correction doesn’t change this, and each new calculation made requires an amendment to that error correction if it’s going to ensure arbitrary accuracy, down the line. You might as well just keep track of the exact calculation, alone, rather than an approximation plus an exact error calculation.

But it is not at all obvious because I interpret this in a completely different way than you do. I see this as a process that is endless, whereas you see this as a fixed value that has no more terms after ‘infinitely many’.

And therein lies the problem. When mathematicians discuss the number 0.333… they are not referring to an iterative algorithm. They are referring to a number just as static as 1 or 4 or 7,362. If you insist on discussing a number as if it was an algorithm, then you are knocking down a Straw Man. If you want to show that the way mathematicians treat a particular number leads to a paradox or a contradiction or an absurdity, you need to address the actual way that mathematicians treat that number. You cannot substitute a different understanding than they utilize and expect it to be truly meaningful.

I’m not sure I understand what you are getting at. Does this mean although we try to evaluate pi, or a square root, or a division via processes, in a purer sense these are just transformations of static values into a different form.

They are not transformations. They are approximations. An approximation takes a number which does not have a simple form, when using a particular symbology, and substitutes another number which is arbitrarily close to the original, but which can be expressed in a simple form using that symbology.

This belief that all numbers/values already exist is known as mathematical platonism with a small ‘p’ (apologies for this sounding very silly if it is not what you mean).

I am not a mathematical platonist. I am a Nominalist, through and through. When I use the phrase, “a number exists,” I do not mean it in the same way as I mean, “a horse exists.” Rather, I simply mean that the number in question can be well-defined.

What is the true definition of ‘division’ if it is different to the algorithms? I would like this to be clarified so that I can understand how a result with ‘infinitely many’ digits can be achieved.

A Division is the ratio of two numbers. An algorithm can help us to evaluate this ratio, but the algorithm is not, itself, the ratio.

I am still no closer to understanding your counter argument.

Then I will narrow the scope. I see no reason to think that your assertion from (5) is true. Specifically, I see no reason to think that there could be any remainder after an infinite expansion of 0.333…

• First my reply to howardat58…

If a series has endless non-zero terms then it is incorrect to pretend it equals ‘its limit’. An endless series is a perfectly valid object to use in mathematics just by correctly representing its property of endlessness. It is wrong to pretend it is something that it is not.

You can define what the concept of a ‘limit’ means, and you can associate a limit with an endless series. But the only reason anyone ever does this is to then pretend that in some respect the endless series equals the limit, it does not. This is self-delusion.
Furthermore, the concept of a ‘limit’ can be misleading and it leads to inconsistencies.

An endless series can have alternating positive and negative terms, where the sum to the n-th term alternates above and below a certain value, but which gets ever closer to the value as the series extends. The n-th sum is not ‘limited’ by the so-called limit in this case.

(2) An example of ‘inconsistency’
Some endless series are said to have a limit and other are not, which is inconsistent. To determine if a series like 1 + 1/2 + 1/3 + 1/4 +… has a limit we have to perform something called a ‘convergence test’, which is not elegant.
It also causes inconsistencies on graphs. For example, the famous Zeta function is defined as being an endless series but when we do a graph for it, we actually plot the ‘limits’ and we pretend this represents the actual value of the function at the plotted x-axis values.
This produces the misnomer that the Zeta function returns fixed values for input values greater than 1, but it “blows up to infinity” for values of 1 and under (hence the need for ‘analytic continuations’).

But if we stop deluding ourselves that we are plotting the actual value of the function and we realise what we are actually plotting, we can plot any value without the need for analytic continuations.

For any given input value, the Zeta function produces an endless series. We can form an expression for the sum to the n-th term for this series. This expression has a fixed part and a variable part. If we plot the fixed part of the expression for the sum to the n-th term, then we can plot a value for any given input, and when the value is greater than 1 this will equal the so-called ‘limit’. We have no inconsistency here.

And my reply to Boxing Pythagoras…

You said: “When mathematicians discuss the number 0.333… they are not referring to an iterative algorithm. They are referring to a number just as static as 1 or 4 or 7,362.”

Exactly, it is the assertion that such a number, with ‘infinitely many’ digits is a valid concept. I assumed it was and found a contradiction. With no valid mathematical definition for ‘infinitely many’ and with no valid proof, I see no reason to accept the concept.

You said: “ I simply mean that the number in question can be well-defined.”

But as I said in my last point, a definition based on repetition/recursion does not define ‘infinitely many’ because however many loops you count, it will always be a finite value. If you start counting the natural numbers you cannot reach ‘infinity’ you can only reach finite values. If your definition is based on inductive repetition then it is not well-defined.

You said “A Division is the ratio of two numbers. An algorithm can help us to evaluate this ratio, but the algorithm is not, itself, the ratio.”

If this is correct then to do a division we just need to show the two numbers and a symbol to indicate ‘ratio’. All the algorithms that try to get a single value are incorrect and false.

• If a series has endless non-zero terms then it is incorrect to pretend it equals ‘its limit’.

Again, I see no reason why such a series wouldn’t equal its limit. The mathematics is consistent and provides reliable results. You haven’t yet demonstrated a reason to think otherwise.

An endless series can have alternating positive and negative terms, where the sum to the n-th term alternates above and below a certain value, but which gets ever closer to the value as the series extends. The n-th sum is not ‘limited’ by the so-called limit in this case.

Can you give an example, and explain why you do not feel that the proposed limit is the actual limit of the series?

(2) An example of ‘inconsistency’
Some endless series are said to have a limit and other are not, which is inconsistent. To determine if a series like 1 + 1/2 + 1/3 + 1/4 +… has a limit we have to perform something called a ‘convergence test’, which is not elegant.

The fact that a concept is defined only for a limited set of parameters does not imply that it is inconsistent. If the concept produced differing results for the same set of parameters, then it would be inconsistent. Can you provide any examples of such?

Nor do I find anything inelegant about the concept of a convergence test. Why do you find this inelegant?

Exactly, it is the assertion that such a number, with ‘infinitely many’ digits is a valid concept. I assumed it was and found a contradiction.

As I’ve stated several times, I do not agree that you have demonstrated any contradiction. You’ve baldly asserted that one exists, but I’ve seen no demonstration of this claim, as yet.

But as I said in my last point, a definition based on repetition/recursion does not define ‘infinitely many’ because however many loops you count, it will always be a finite value.

At no point did I say that the definition is based upon repetition or recursion. I simply said that the number needs to be well-defined.

That said, I disagree with your statement, even as regards infinite iterative definitions. Once again, it is fallacious to pretend that you can draw conclusions about an infinite number of iterations based upon the properties of finite numbers of iterations.

If this is correct then to do a division we just need to show the two numbers and a symbol to indicate ‘ratio’. All the algorithms that try to get a single value are incorrect and false.

Yes, when we want to perform an operation, we can symbolize this. That does not, in any way, imply that algorithms which are utilized in helping to evaluate such operations are “incorrect and false.” Algorithms are tools which we utilize to help perform an operation. They are not the operation, itself.

A hammer and nails are tools which are used to attach things together. A hammer and nails are not the concept of attachment. Similarly, the algorithms which we utilize to evaluate a division are tools. These algorithms are not, themselves, division.

• You said: “I see no reason why such a series wouldn’t equal its limit. The mathematics is consistent ”

Either an endless series is:

(1) Finite. This is clearly false as the series would have a last term and would not be endless.

(2) Infinite. If we assume this to be true and then try to prove that the sum of the series 0.999… equates to 1 then we fail, in fact we end up proving that it cannot equal 1 (As is shown from 06:00 to 08:45 in this video: https://www.youtube.com/watch?v=–HdatJwbQY ).

(3) Endless. This third option is usually not even contemplated, but mathematics is more than capable of modelling a series as an endless process with no fixed value.

Choosing option (2) leads to paradoxes (google ‘infinity paradox’), contradictions (most obvious of which is how can something that is endless somehow end), and lots of counter-intuitive results (such as the Ramanujan sums for diverging series).

Choosing (3) leads to provable results, no paradoxes, and no counter-intuitive results.

You said: “give an example, and explain why you do not feel that the proposed limit is the actual limit of the series”

My claim is that the word ‘limit’ is misleading. A limit is a value beyond which something does not pass, but in the case of an alternating series it passes this ‘limit’ after each subsequent term is included.

If you want an example, just take any endless series consisting of all positive terms and which is said to ‘converge’. Next multiply each term by [(-1) to the power (n-1)].

A big downside of the way the word ‘limit’ is interpreted is that when we try to plot values for the Zeta function we fail for input values of 1 and below. A better term in my opinion would be something like ‘fixed part’, which would be short for the fixed part of the expression for the n-th sum. Then we could plot the ‘fixed part’ of the Zeta function for any input value and we would no longer be limited by the word limit.

You said: “If the concept produced differing results for the same set of parameters, then it would be inconsistent. Can you provide any examples”

My understanding of the word inconsistent is “not staying the same throughout”. This is why I stand by my example. Another example is that if we accept that in decimal form, real numbers have ‘infinitely many’ digits to the right of the decimal point, then our notation system allows three ways to represent numbers that are allowed all trailing zeros/nines (e.g. 1, 0.999… and 1.000…) but it allows just one way to represent other numbers (e.g. 0.333…).

You said “Nor do I find anything inelegant about the concept of a convergence test. Why do you find this inelegant?”

Because I see an endless series as one type of object and I believe it should be treated in the same way in mathematics regardless of if the n-th sum is increasing, decreasing or alternating as n increases.

Performing a convergence test and then asserting that if a series converges it can be said to equate to a real number, but other endless series don’t, does not appear elegant to me. But this is a futile argument because beauty is in the eye of the beholder.

You said “ I do not agree that you have demonstrated any contradiction. ”

Put simply, the contradiction is this:
(a) After any amount of terms, there will still be more positive non-zero terms remaining.
(b) After ‘infinitely many’ terms there will be no more terms

You may assume ‘infinitely many’ is valid and devise your own logic for infinite objects, but unless you can demonstrably prove that there is some overlap with finite and infinite objects you have no basis to assert that (b) is right.

You cannot demonstrate how the division process can end, and you cannot prove mathematically that ‘infinitely many’ digits will make 0.333… equate to 1/3. Therefore you cannot assert that an object described in finite terms, like 1/3, has anything at all to do with your infinite object 0.333…

You said “At no point did I say that the definition is based upon repetition or recursion. I simply said that the number needs to be well-defined”

If you are going to connect things consisting of finite terms (like 1/3 and 1), to things supposedly consisting of ‘infinitely many’ terms, (like how you define 0.333… and 0.999…), then you need to clearly define how finite terms and infinite terms relate to one another.

If your definition leads to a clear understanding of how an infinite object can be constructed, then I will concede it is well-defined. Otherwise it is not and you cannot assert a relationship that you cannot prove exists.

You said “Algorithms are tools which we utilize to help perform an operation. They are not the operation, itself.”

Words are very important in mathematics. Complicated formulas have no intrinsic meaning without precise descriptions of what the symbols mean. At the lowest level, mathematics is all about clarity of understanding using words.

Unfortunately words can have multiple meanings and this is where a lack of clarity is allowed to muddy the waters and create a lack of clarity.

One trick is the claim that ‘infinite’ means the same as ‘endless’ and then to claim that this means we must have ‘infinitely many’ of something.

Another trick is to say we can associate the limit of an endless series with the series itself, and in this way we can equate the series to a fixed value.

Why not call a ratio “a ratio” and call the process of division “division”? Either way, you know what I am talking about when I ask “how can the division process end?” To claim that it is not division is to avoid answering the question.

• (2) Infinite. If we assume this to be true and then try to prove that the sum of the series 0.999… equates to 1 then we fail, in fact we end up proving that it cannot equal 1 (As is shown from 06:00 to 08:45 in this video

Again, I do not see that “we end up proving that it cannot equal 1,” even accounting for Mr. Peny’s video.

Choosing option (2) leads to paradoxes (google ‘infinity paradox’), contradictions (most obvious of which is how can something that is endless somehow end), and lots of counter-intuitive results

I completely agree that there are counterintuitive results. However, beyond your bald assertions that they exist, you have yet to demonstrate that there are any legitimate paradoxes or contradictions implicit in the notion of infinity.

My claim is that the word ‘limit’ is misleading. A limit is a value beyond which something does not pass

This is not what the word “limit” means in mathematics. A brief discussion of the mathematical definition for “limit” can be found, here: http://www.wolframalpha.com/input/?i=mathematical+definition+of+limit

If you are basing your objections to mathematical limits based upon the definition you gave, you are committing an equivocation fallacy.

My understanding of the word inconsistent is “not staying the same throughout”.

Again, this is not what mathematicians mean by “inconsistent.” So, again, using such a definition to object to mathematics is to commit an equivocation fallacy.

Put simply, the contradiction is this:
(a) After any amount of terms, there will still be more positive non-zero terms remaining.
(b) After ‘infinitely many’ terms there will be no more terms

The problem is that you have not demonstrated (a) to be true. I will agree that after any finite amount of terms, there will still be more terms remaining. You have not demonstrated that this extends to any infinite amount of terms, as well.

You cannot demonstrate how the division process can end

Once again, division is not a process. The algorithms which we utilize to evaluate division are processes. Division is a mathematical operation. Mathematical operations are descriptions of relationships between quantities.

Words are very important in mathematics.

I completely agree! This is why your equivocation fallacies are particularly egregious.

One trick is the claim that ‘infinite’ means the same as ‘endless’ and then to claim that this means we must have ‘infinitely many’ of something.

Can you cite a single Set Theorist or textbook which has ever claimed “that ‘infinite’ means the same as ‘endless’ and [therefore] this means we must have ‘infinitely many’ of something?” Because I’ve never seen such a claim.

It is true that the concept of “infinity” has an unfortunate bit of ambiguity attached to it. It can refer to an indefinitely repeating iterative process, as you mention. However, it can also refer to a property of numbers. These are two entirely different concepts. When we talk about the quantity of digits in 0.999…, we are referring to a property of numbers; we are not referring to an iterative process. Objecting to the numeric property based upon an understanding of iterative processes is yet another equivocation fallacy.

Why not call a ratio “a ratio” and call the process of division “division”?

Because division is not a process.

Either way, you know what I am talking about when I ask “how can the division process end?” To claim that it is not division is to avoid answering the question.

It’s certainly not avoiding answering the question. It’s pointing out that the question is malformed and irrelevant. The fact that a particular algorithm for evaluating division may iterate indefinitely does not imply anything about the division, itself.

• You cannot prove that objects that can be expressing in a finite number of terms (like 1/3 or 1) are in any way related to your objects you claim to contain ‘infinitely many’ terms (in this case, namely the decimals 0.333… and 0.999…).

The well known proof of 10x – x proves the reverse. You cannot show how an algorithm can get from one to the other, and all you are left with are assertions and assumptions.

• However, it has been very enjoyable discussing this with you, so thank you very much.

As we have got to the point where we are repeating ourselves, I think this has run its course.

Once again many thanks, Bye.

• You cannot prove that objects that can be expressing in a finite number of terms (like 1/3 or 1) are in any way related to your objects you claim to contain ‘infinitely many’ terms (in this case, namely the decimals 0.333… and 0.999…).

Once again, with the exception of only a tiny few, every mathematician on the planet disagrees with you. It’s fairly simple to prove that 0.999…=1.

The well known proof of 10x – x proves the reverse. You cannot show how an algorithm can get from one to the other, and all you are left with are assertions and assumptions.

It most certainly does not “prove the reverse.” And I can quite easily show how an algorithm “can get from one to the other.” Some specific, iterative algorithms can be indefinitely repeated, but that does not imply that they cannot be shown to produce this particular result; and it certainly doesn’t imply that there are no algorithms which can demonstrate this particular result.

However, it has been very enjoyable discussing this with you, so thank you very much.
As we have got to the point where we are repeating ourselves, I think this has run its course.
Once again many thanks, Bye.

I have also enjoyed this discussion, and thank you very much for it.

• “(9) The ‘epsilon-delta’ argument says that 0.999… must equal 1 because there is no number between the two.”
No! The beauty of the epsilon-delta argument is that it is totally finite.

4. I understand the instinctive response that 0.999… shouldn’t equal 1. It’s hard to shake the feeling one gets from a finite expansion of 0.999…999. But it should at least be intellectually convincing when you try working out how big the difference between the infinite series and 1 should be.

5. I am not admitting that 0.3 recurring equals ‘infinitely many’ terms, which is what you are saying with that statements that includes “from n= 1 to infinity”.

What I did was I assumed ‘infinitely many’ was possible, and then I showed that this leads to a contradiction. Therefore statements such as ‘for n=1 to infinity’ and ‘infinitely many’ are invalid.

If my logic is correct and there is a contradiction, then Stevin’s real numbers and Cantor’s infinite sets are invalid and cannot be used as arguments against the original contradiction (i.e. ‘infinite sets’ and Aleph Null are inadmissible).

This is a fundamental problem and it needs to be addressed using fundamental core logic. .

But I think the meaning of your reply can still be gleamed. Your counter argument appears the be that after ‘infinitely many’ terms there are no more terms. This is the same as the first assertion in point (5) of my original argument.

Therefore my original argument still stands.

6. I give up !