@GroupNebula563 @TechDesk @404mediaco @emanuelmaiberg @wikipedia @LucasWerkmeister

""you can see right here where it cites two textbooks" - nope. I can see quite clearly they are NOT Maths textbooks, as I said

"explaining exactly why the article is correct" - now go read about limits and/or decimal representations in Maths textbooks and you'll discover why it's wrong. Here's a free head-start explaining why 1/3 isn't actually equal to 0.333... and is only an approximation (now multiply by 3)

@fuchsi @GroupNebula563 @TechDesk @404mediaco @emanuelmaiberg @wikipedia @LucasWerkmeister

"you have to make the line for "period" over the 3 of 0,3" - yes, and is an approximation of 1/3, since it's literally impossible to have an exact decimal representation of 1/3 in base 10, since 3 isn't a factor of 10, as per the textbook

@fuchsi @GroupNebula563 @TechDesk @404mediaco @emanuelmaiberg @wikipedia @LucasWerkmeister

"but it's 3 until infinity. That limes should do that, no?" - I'm not sure what you mean. Even at 3 to infinity, it's still only an approximation of 1/3, so 0.9 to infinity is only an approximation of 1. ALL non-terminating decimals are only approximations, again as per the textbook. Only terminating decimals are exactly equal to a fraction, such as 0.25 is exactly equal to 1/4, as per the textbook

@fuchsi @GroupNebula563 @TechDesk @404mediaco @emanuelmaiberg @wikipedia @LucasWerkmeister

In other words, no matter how many steps you do your long division to, you're still left with remainder 1 - that remainder 1 literally never disappears, even at preceded by infinite zeroes. It's because 3 isn't a factor of 10, and we're doing it in base ten. In base 3 you can exactly represent 1/3 - it's 0.1 - in base ten you can't. The same thing happens in the other direction converting back.

@fuchsi @GroupNebula563 @TechDesk @404mediaco @emanuelmaiberg @wikipedia @LucasWerkmeister

"You don't have the remaining 1. That also gets divided by 3" - which then also gives a remainder of 1, divide by 3 again, another remainder of 1, ad infinitum. That's EXACTLY how infinity works - a never-disappearing remainder of 1, infinitely repeating 3's. Again, every non-terminating decimal is only an approximation, only terminating decimals are exactly equal to fractions, as per Maths textbooks

@fuchsi @GroupNebula563 @TechDesk @404mediaco @emanuelmaiberg @wikipedia @LucasWerkmeister

"I am not sure that's how it works" - That's exactly how it works. Sit down with pen and paper and start doing long (or short) division. 3 doesn't go into 1, add a zero. 3 goes into 10 3 times with 1 remainder, repeat ad infinitum. Again, every non-terminating decimal is only an approximation due to a never-disappearing remainder, due to not being a factor of 10

@fuchsi @GroupNebula563 @TechDesk @404mediaco @emanuelmaiberg @wikipedia @LucasWerkmeister

And the limit is defined as the number it can never reach, hence the name, limit.

@fuchsi @GroupNebula563 @TechDesk @404mediaco @emanuelmaiberg @wikipedia @LucasWerkmeister
Now read all the underlined parts, especially the double-underlined part. You can lead a horse to water...

"That means if you take the infinite term you can make an integer out of it" - no, had you read it, you would understand that means if you take THE LIMIT you can get an integer out of it. The "infinite term" is still less than it, still less than 4, as per the textbook...

@SmartmanApps @GroupNebula563 @TechDesk @404mediaco @emanuelmaiberg @wikipedia @LucasWerkmeister Wait, what? Are you saying I should only read books that are not referenced by Wikipedia?

Isn't that a bit extreme from "I would change one sentence about infinity"?

@fuchsi
He's saying the only textbooks that are correct are the ones that agree with him, of course ;)

God knows how old the one (or maybe it's more than one) he's using there is; the dotted equal symbol isn't used any more. The symbol S_∞ introduced there has no different meaning than S_n (that is, both are said to "approach" the limit) whereas in a modern textbook (A level maths textbooks for example) say S_∞ = 1 (rather than "approaches 1").

Again he confuses the terms of a sequence with its limit. This is genuinely interesting though, because this textbook uses the ellipsis to mean an unwritten finite number of terms rather than infinitely many, so could really explain why he has this misconception.

@fuchsi Regarding recurring decimals, yes the language used in that book is very much not standard. Actually we should be clear that, in the language of that book (Second Algebra by Hawkes, Luby and Touton, published 1911), SmartmanApps is correct; in that book 0.999... is not 1, because 0.999... does not represent a real number at all. From his screenshot of p176: "The numerical value of 0.666... is a **variable** depending on the number of 6s annexed to the right". (My emphasis)

So this is where he's coming from: for him, 0.999... has *no* fixed value but is more like a variable symbol that is restricted to taking values in the sequence of finite decimals 0.9, 0.99, 0.999, etc. In normal mathematical language we would say it refers to the sequence (or perhaps set) of such values, because the idea that a variable can take on multiple different values within the same piece of mathematical discourse is unnecessary and confusing. (Because if x can take on multiple values, you are forced to accept the statements "x = 0.9" and "x = 0.99" in the same argument, because they might represent different values of x at different times. Can we therefore conclude that x - x = 0.09? No! You must at all times pay attention to whether your statements about a variable were only true contingently. Normal mathematical argument would enclose such sub-scenarios in conditional language, making this explicit.)

I'm not convinced that the wording here "changed" in the sense it being normal in 1911 and rare now. Euler spoke of recurring decimals as definite, not variable quantities in the 1700s, and the textbook Introduction to Algebra by G. Chrystal first printed in 1898 clearly does so too (first screenshot)

Regarding S_∞ actually I think the language hasn't changed, but that book he's quoting is a little sloppily written here. The second screenshot is the modern development, including the "S_∞ = a/(1-r)", from the A-level textbook he's using.

The third screenshot is from the bottom of page 172 of Second Algebra which he clipped, and the top of 173. Notice in fact that the authors abandon the language of "S_∞ approaches..." in favour of giving it a definite value. Their treatment on p173 is in fact quite clear and in contradiction with the Smart Man's conclusion: "when we speak of the sum of the series, we **mean the limit**". From what he says, I don't think he is quite capable of understanding what a mathematician is saying when they say, "by these words we mean this thing", else he wouldn't have underlined all those words which contradict him :P

Further to this, on p175 there is the fourth screenshot, which says that (when x < 1), "the sum of the series 1 + x + x² + x³ + ... is a definite number". "Definite numbers" are just themselves; they don't change and cannot "approach" anything.

I am ashamed that I skipped over his screenshot of p176, because that finally actually clarifies what he thinks recurring decimal notation means in a way that his posts dating back to 2024 failed to. Presumably he also thinks infinite summation is similarly a "variable" waiting for you to decide how many terms to add up.

Sadly he doesn't have the ability to explain this: he doesn't write, "0.999... ≠ 1 because the notation on the left does not denote a specific number, but is a placeholder for one of 0.9, 0.99, 0.999, ... and 1 is not any one of those". Instead he ridicules and dodges questions. And sadly, while he'll probably never respond to this way of putting things, even if he did he would not be able to understand or admit that his preferred language, from one textbook written over 100 years ago, is not standard and that in the language used by everyone else, 0.999... denotes a single, definite number, and that number is 1.

I think this way of writing maths invites such imprecise thinking though - do you agree? If we see a symbol used *as a number* the way recurring decimals universally are, then I think it will always lead to people treating them as a particular number.

@FishFace I think you both should kiss and make up.

I think you are both more passionate about mathematical problems than I ever was. (Even though I competed in math olympiads as a child.)

@fuchsi @[email protected]
"passionate about mathematical" - I'm passionate about #debunking #Gaslighters and their #bullying. He's just passionate about Gaslighting/bullying people. Note in the following how often he edited out the references which contradict him...

"The numerical value of 0.666... is a **variable** depending on the number of 6s annexed to the right". (My emphasis)" - also your EDIT, where you deliberately omitted V is the Variable being described, hence your lack of a screenshot

@fuchsi @[email protected]
"Notice in fact that the authors abandon the language of "S_∞ approaches..." in favour of giving it a definite value" - notice that your screenshot in fact deliberately omits the very next paragraph, which says that the sum of the terms is approaching the limit 🙄

"when we speak of the sum of the series, we **mean the limit**"" - says person who can't provide a screenshot of them saying that anywhere, given they make the quite clear distinction between them repeatedly

@fuchsi @[email protected]
""the sum of the series 1 + x + x² + x³ + ... is a definite number". "Definite numbers" are just themselves; they don't change and cannot "approach" anything" - says person who, yet again, left the very next paragraph out of their screenshot, which quite clearly states "the sum of whose terms approach a limit", the limit being the "definite number" being discussed 🙄

"Instead he ridicules and dodges questions" - #EveryAccusationIsAConfession 🙄

@fuchsi @[email protected]
"he'll probably never respond" - why would you think I'd stop debunking your Gaslighting?? 😂

"people treating them as a particular number" - they treat the limit as being the result of the infinite sum because that's what textbooks have been teaching them for more than 100 years, 🙄 whilst making the clear distinction between the two (which you keep editing out of your replies)

"denied the existence of the "Key Point 6.12"" - nope! I never denied the existence of the limit 🙄

@fuchsi @[email protected]
From your other reply "despite my helpful purple annotations ^_^" - I'm pointing out all the times "approximate" and "approximation" are used that you're trying to draw people away from noticing Gaslighter 🙄

P.S. you also need to learn the difference between equals and identically equal

"he hasn't read it" - and you haven't quoted it, so??

"Elsewhere I suggested he doesn't have a maths degree" - because you can't back anything you say up and thus make up lying ad hominems

@fuchsi Do you not feel the power of my typing compelling you? Damn, I must have forgotten to switch my internet to "hypnotise"...

I am fascinated though. Apparently, "the numerical value of" a number can be different from "the numerical value V of" the same number? At least, that's how I interpret his complaint that I did not include the "V" in that quote (and it's true, I didn't. *And* it's true that I didn't include the screenshot that we had already seen :) ).

This IMO would be a fundamental lack of understanding of one of the most basic aspects of mathematics, that the value of a variable symbol is, exactly and always, the value of the object it denotes. I don't see how a graduate of a mathematics degree comes away thinking that there is a difference here.

Either way it's bizarre, because that was the one chance he had to actually be saying something true (albeit wrongly expressed, albeit contrary to all ordinary mathematical language, but nevertheless true on its own terms), and he's denying it! Maybe next time he'll enlighten us as to whether 0.666... is:

* a number
* greater than 0.6, 0.66, 0.666, and every decimal number 0.66..66 with a finite number of sixes
* at most two thirds

but since he ignored this question the last time I asked it (about 0.999...) I doubt it :( Will he say whether "the numerical value of 0.666..." is actually different from "the numerical value V of 0.666..." and if so, how? Still less likely. He doesn't like making precise assertions.

It would be fascinating to sit down with him and see exactly how his brain works. Apparently the mere presence of the word "approximate" on a page means that the equality symbol does not have its usual meaning, so the statement "⅓ = 0.333..." can be ignored. Can anyone really believe that "this gives us two approximations to x" implies that "the infinite decimal expansion of x", then denoted with the "=" symbol, is approximate? I assume this is what he's trying to say by drawing lines between the word "approximations" there and my highlighting of the exact equality below.

Yes, Donald, we all agree that the finite terms 0.3, 0.33, 0.333, etc are approximations to ⅓! And so does Wikipedia, where this discussion started. No-one believes this extends to the recurring decimal, except you.

It was nice of him to link another textbook, but the part about graphs, rather than about recurring decimals. Still, in that textbook (p38, screenshot), fractions *are* equated, using the equality symbol, to recurring decimals. But this book does one better, because it strictly uses a different symbol for "approximately equal" (p42, same screenshot), which is nowhere to be found on p38. Surely *this* will convince him that recurring decimals are exact representations of rational numbers! Now's a good time to make a table of sources in support of writing 0.999... = 1. I think maybe the time has come for the Smart Man to find some entries which can fill in the final column, or to admit that Wikipedia is entirely justified in writing 0.999... = 1 :) Surely if he believes 0.999... ≠ 1, he can find at least one source which says so in clear, unambiguous symbols. *Surely!*

In partial answer to your question, I believe his use of the term "gaslighter" may have started when someone else said that a "+=" key on a calculator was not an equals key. (It combines the functions of both + and =, and featured on early electronic calculators and on mechanical adding machines. This is a whole other story!). Shortly after he directed it at me for unclear reasons, and now he throws it out at anyone who disagrees with him.

@FishFace @SmartmanApps

That looks pretty consistent to me.

And yes, it's confusing that 0,9999... = 1 .
Infinity is strange.

I also had some mind blockers in university that were similar. I needed some time to get over them.

@fuchsi @[email protected]

> Nope, you ignored that I answered it

If I misplaced a comment of yours in this long (and hard to read, due to blocking - happy to reverse that at any point) thread, I do apologise. It's not intentional, and you can swiftly rectify the situation by linking to the comment I missed.

I asked the question here: https://ioc.exchange/@FishFace/116317586606785481 and as far as I'm aware you haven't replied. Please do tell us whether recurring decimals like 0.999... are real numbers - feel free to link to somewhere you said this if you already did. But to be clear, it's a yes-or-no question: a mathematical object either is or is not a real number. Whatever your answer is should be obviously an answer, when we read it.

And of course we still await your explanation (always sad to be proven correct) of the difference between the sentences "the numerical value V of 0.666..." and "the numerical value of 0.666...". If there is no such explanation, you might wish to consider retracting your outrage at my omission :)

> "And so does Wikipedia" and so does the textbook

So you're happy that Wikipedia and the textbook both use the notation 0.999... = 1 viz. 0.333... = ⅓. Can you now explain why the Wikipedia article is wrong to write such an equation, but the textbooks are not?

The screenshot you couldn't find is the first one of my comment https://ioc.exchange/@FishFace/116335108529026456, which is from p38 of Mathematics Extension 1 Year 11, and which you referred to here: https://dotnet.social/@SmartmanApps/116323534108291577

It's not especially important; it's in the table already.

> "Now's a good time to make a table of sources in support of writing 0.999... = 1" - in which you lied, as usual

The full titles, authors and page/other references are all there for you to dispute any cases where I've misquoted them - or indeed to find a reference for where it says something like ⅓ ≠ 0.333...

> "not a response to what I wrote" - that was in the other 6 posts, you know, the ones you ignored

Then I'll have to request that you organise your thoughts better, so that I can see what is supposed to be a response to what. Thanks in advance.

I've added some missing texts to the tally, and created an unanswered questions log to help us keep track.

Whenever you're ready to add some sources that actually explicitly say 0.999... ≠ 1 (other recurring decimals are available) please do link them.

> (which is a direct rejection of Cantor's "proof", who tried to treat it as a number)

Oh dear, that topic. Cantor treated infinity as a type (actually two different types) of equivalence class of sets. Let's not get distracted though.

> NO MATTER HOW MANY TERMS WE ADD UP, THE ANSWER IS ALWAYS LESS THAN 4

Do you think this is supposed to mean that, "if we add up 'infinitely many' terms, the answer is still less than 4"? Because that would be doing exactly what you just said not to do, and treating infinity as a number. It means "for all natural numbers n, the partial sum of n terms is less than 4". Infinity is not a natural number. It would be incorrect to say "the sum to infinity is less than 4", which is why you have seen many textbooks say "the sum to infinity is ..." and then the limit.

> So yes, the textbook is correct - the infinite sum is always less than the limit, but we use the limit in it's place when doing arithmetic

Sorry, you said "the textbook is correct", and then some words that were not in the textbook. It did not say "the infinite sum is always less than the limit". It said "the sum of the infinite geometric series ... is 4".

Perhaps you would like to try answering Steffi's question again? It was a simple yes-or-no affair, and didn't need the application of your imagination ;)

@FishFace seems that I am now a gaslighter, too. And your alt account.

That's interesting.

@fuchsi

Yep, how does it feel? Do you feel our identities blending together? XD Any excuse for all the pushback he gets, I suppose.
Did you enjoy yourself? I did but I think these kinds of interactions are nothing but frustrating to most people. (I find it frustrating as well, but also fascinating).

My next question to him was going to directly be what he would add to Wikipedia to make it correct - given that he accepts the equation 0.999... = 1, there must some verbiage that he would like to add and be happy. Sadly, because he persistently misquotes textbooks, inserting his own incorrectly-synthesised understanding of two separate domains (finite and infinite) I suspect that he would answer with a sentence that is not in fact present in any of the textbooks.

Never mind that the Euler and Robertson sources have no such precise discussion of limits because the concept didn't exist yet.

Never mind that, just as with getting that answer to what exactly 0.666... *is*, he'd probably ignore the question :)

The thing about "using" an approximation in a calculation is an interesting one. I don't know about you, but I was always taught that, when using an approximation like 3.142 for pi, to write ≈ instead of = throughout. Or else, to write "to 3 decimal places" after each such statement. Given that the textbook by Pender he used specifies a specific approximation symbol, it's clear that in such texts they do not mean an approximation.

In such a discussion I am always trying to find the internal consistencies as well as the inconsistencies. I think there is a way you could get muddled into believing this "using" idea if you for some reason really strongly believed that recurring decimals are not equal to their equivalent fractions, but how do you get to that point? No matter what he says, it's *not* how it's taught in schools!

I think the fundamental inconsistency that he will never get over can be summed up with his statement, "We've defined infinite sums one way, the limit another way" - compare this to "it's literally impossible to calculate an actual infinite sum". So are they "defined" or "impossible to calculate"? What are they defined as? We'll never know - so sad!

But what I'm most interested in is whether you think he's trolling or serious, and whether he actually ever studied maths at university (as we mean it, not as part of a teaching qualification or something)? I don't think it's possible to believe that defining two things as equal is akin to an approximation if you got a maths degree. I am not sure whether he's trolling though - he's pretty subtle if so, and moreover he *only* has these few special topics he spreads misinformation about, and he has done so on the topic of Order of Operations *persistently* on many websites, since at least 2023. I feel like a troll would be more wide-ranging, but I would love to know if you agree.

Oh btw, the "identically equal" thing is a weird one that I never did pick up on. If you look this up in a school textbook, it's the difference between an identity like sin² x + cos² x = 1 (true for all x) and an equation like x² - 4 = 0 (only true if x is ±2). In school we were inconsistently presented with identities using the ≡ symbol. Yet another tell-tale, I think: you don't use this in university maths courses at all because you learn quantifiers. But this distinction is obviously only relevant for equations with variable symbols in, so I think it speaks to an even worse misunderstanding.

@FishFace looks like the dunning kruger effect for me.

But I am too long out of it to know it better. So I can only parrot the book.

@fuchsi honestly parroting the book is all it takes if you understand English (or the language of the book) (maybe you need a tiny understanding of mathematical language?)

You probably remember more than you give yourself credit for - I surprised myself by being able to prove a limit/supremum from first principles when talking about it before 😁