@GroupNebula563 @TechDesk @404mediaco @emanuelmaiberg @wikipedia
"for every troll who edits an article" - Professor Rick Norwood isn't a troll https://www.researchgate.net/profile/Rick-Norwood and yet his Maths corrections keep getting backed out by admins. Welcome to why I post my Maths facts on Mastodon, where no-one can back them out https://dotnet.social/@SmartmanApps/110968910722113903
@GroupNebula563 @TechDesk @404mediaco @emanuelmaiberg @wikipedia
"theyβre constantly being fact-checked by tons of experts" - and getting backed out again by admins. See previous comment. None of the Maths pages ever cite any Maths textbooks, despite the fact there are many available for free on the Internet Archive
"many policies bypass all that" - including bypassing fact-checking π so either the policies don't work, or aren't followed. Either way Wikipedia has a facts problem
@GroupNebula563 @TechDesk @404mediaco @emanuelmaiberg @wikipedia @LucasWerkmeister
"the policies do work, and are followed" - clearly not, given pages like https://en.wikipedia.org/wiki/0.999... exist
"IMO tend to be far more authoritative" - I see you haven't read any of their blog posts then, where they can't even get order of operations right (spoiler alert: they don't teach it at university, it's taught in high school, which they've long since left - high school textbooks are the references to use)
@SmartmanApps @TechDesk @404mediaco @emanuelmaiberg @wikipedia @LucasWerkmeister ...you can see right here where it cites two textbooks explaining exactly why the article is correct
https://books.google.com/books?id=jWgPAQAAMAAJ
https://archive.org/details/mathematicalanal02edtomm/page/n3/mode/2up
Algebra is abstract mathematics - let us make no bones about it - yet it is also applied mathematics in its best and purest form. It is not abstraction for its own sake, but abstraction for the sake of efficiency, power and insight. Algebra emerged from the struggle to solve concrete, physical problems in geometry, and succeeded after 2000 years of failure by other forms of mathematics. It did this by exposing the mathematical structure of geometry, and by providing the tools to analyse it. This is typical of the way algebra is applied; it is the best and purest form of application because it reveals the simplest and most universal mathematical structures. The present book aims to foster a proper appreciation of algebra by showing abstraction at work on concrete problems, the classical problems of construction by straightedge and compass. These problems originated in the time of Euclid, when geometry and number theory were paramount, and were not solved until th the 19 century, with the advent of abstract algebra. As we now know, alge bra brings about a unification of geometry, number theory and indeed most branches of mathematics. This is not really surprising when one has a historical understanding of the subject, which I also hope to impart.
@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...
Attached: 1 image 4/5 We can see in this textbook, it uses a dot above an equals sign to show "x approaches the limit a", and says we can also write it as "lim x=a". I have also seen textbooks say x:=a, where := means "is defined as". Note that in all 3 cases, the textbook has quite conspicuously never said x=a, only variations on the limit of x is defined as a, which we do so that we have a finite number that we can perform further calculations with...
@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_β 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_β = 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_β 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_β = 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_β 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 + xΒ² + xΒ³ + ... 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... β 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_β 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 π
"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 + xΒ² + xΒ³ + ... 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 π
"Instead he ridicules and dodges questions" - #EveryAccusationIsAConfession π
@fuchsi @[email protected]
"he'll probably never respond" - why would you think I'd stop debunking your Gaslighting?? π
"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, π 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 π
@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 π
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 @[email protected]
"I don't see how a graduate of a mathematics degree comes away thinking that there is a difference here" - says person who deliberately omitted it in their "quote" π
"since he ignored this question the last time I asked it" - nope, you ignored that I answered it, same as every time, because you have this running false narrative that I can't answer your questions, despite the fact I always do π
@fuchsi @[email protected]
"means that the equality symbol does not have its usual meaning" - I already said that it's usual meaning is DIFFERENT from the meaning of IDENTICALLY EQUAL, and here you are yet again ignoring that I pointed that out to you Gaslighter
"And so does Wikipedia" and so does the textbook, explicitly stating the number of appended 6's is "unlimited"
"No-one believes this extends to the recurring decimal, except you" - and Maths textbooks, and other Maths teachers π
@fuchsi @[email protected]
"It was nice of him to link another textbook" - says Gaslighter outright making stuff up, given I did nothing of the sort π
"in that textbook (p38, screenshot)" - he says, without a screenshot of this mysterious page and book
"(p42, same screenshot)" - there are no screenshots of mine which include 5 whole pages. Pull the other one dude - it's got bells on it
"Now's a good time to make a table of sources in support of writing 0.999... = 1" - in which you lied, as usual
@fuchsi @[email protected]
And in response to this rubbish comment in the screenshot... here's another part of the EXACT SAME TEXTBOOK, saying that the limit APPROXIMATES the infinite sum π
"not a response to what I wrote" - that was in the other 6 posts, you know, the ones you ignored
"That looks pretty consistent to me" - consistent lies
"it's confusing that 0,9999... = 1" - it isn't, as per multiple textbooks, not to mention you can prove it yourself by doing long division of 1/3 and other methods
@SmartmanApps "the infinite sum is always less than the limit, but we use the limit in it's place when doing arithmetic. "
Isn't that also what Wikipedia and our friend said?
Just in more words?
Edit: and I didn't leave the rest out. I am sorry if it looked that way. I just try to find common ground and go on from there. I don't think it's useful to always write a whole page if we don't find the starting point first.
@SmartmanApps your textbook also says they are equal "The simplest answer is because it works".
The text also says it's strange and "mathematicians and philosophers have struggled with the idea". So you are not alone.
And they conclude that 0,333... = 1/3 (is equal)
They don't say 0,333... < 1/3 (is smaller) or 0,333... ~ 1/3 (is rounded). They explain that they use the symbol "equal" because in infinity there is no difference between both.
And again: the text also sees this contradiction.
@fuchsi
"your textbook also says they are equal" - no it doesn't. "the answer is always less than 4"
"So you are not alone" - I'm not struggling with the idea at all. This has been well-established for a very long time
"And they conclude that 0,333... = 1/3 (is equal)" - no they don't. They conclude we use it in it's place "because it works", just like we can use 22/7 in place of pi if we choose
"They don't say 0,333... < 1/3 (is smaller)" - yes they do. "the answer is always less than 4"...
@fuchsi
You're also ignoring the textbook which explicitly stated that 0.(6) is less than 2/3. Now divide by 2
"Another way to say it is: we defined it that way" - nope. We've defined infinite sums one way, the limit another way
"mathematics is a sort of language for numbers" - no it isn't - it's a tool for calculating things, and sometimes we do things to make the calculations easier, like using 22/7 for pi
"And we defined both to be equal" - no we didn't, just like pi isn't defined as 22/7
@SmartmanApps
"You're also ignoring the textbook which explicitly stated that 0.(6) is less than 2/3. Now divide by 2"
I am not exactly sure what you mean by that.
Yes 0,6 < 2/3
But we are talking about sums that go into infinity.
So you can say
6/10+6/100+6/1000+... = 0,6666... = 2/3
(Not < 2/3 because we don't stop the sum. We take all of it. Until infinity. That's why there are 3 dots "..." behind the term. Because it goes on and on.
Funfact:In Germany you would write a line above the 6.)
@SmartmanApps
Okay. Now I am struggling to understand you.
>"And they conclude that 0,333... = 1/3 (is equal)" - no they don't.
But they literally do. It's in the text. Third line from the bottom.
> They conclude we use it in it's place "because it works"
Yes. And if we can use it in it's place, it's equal. (Note that they use the word "equal" several times.)
@SmartmanApps
> "They don't say 0,333... < 1/3 (is smaller)" - yes they do. "the answer is always less than 4"...
Mathematical symbols are important. They use = and not <
And again: they also confirmed that contradiction. And they still concluded that
"The sums are getting as close to 1/3 as we want, so we say that the sum of the infinite series is equal to 1/3, and we write 0,333... = 1/3"
"We can easily write any recurring decimal as an exact fraction."
@fuchsi
"I am struggling to understand you" - that's only because you're unwilling to let go of your preconceived idea that they are equal. Students don't have any trouble with this
"But they literally do. It's in the text." - no, they literally don't. "These terms are getting AS CLOSE AS WE LIKE to 1/3, so WE SAY THAT the sum of the infinite series is equal to 1/3...". They also say at the top that the limit only APPROXIMATES the infinite sum
@fuchsi
"if we can use it in it's place, it's equal." - nope. Pi isn't equal to 22/7, the infinite sum isn't equal to the limit
"Note that they use the word "equal" several times" - and less than and approximates even more times π
"Mathematical symbols are important. They use = and not <" - yep, and also not ==. You're doing same thing as the Gaslighter insisting equals and identically equal mean the same thing - they don't. See screenshot for what equals means in this context
@fuchsi
"they also confirmed that contradiction" - I don't know what you're even talking about now. There's no contradictions
"We can easily write any recurring decimal as an exact fraction" - only by fudging it and using 99 in the denominator instead of 100, and the reason we have to fudge it is because they're not equal π https://dotnet.social/@SmartmanApps/116316775610449251
"0,6 < 2/3" - and 0.(6)<2/3, the textbook explicitly tells you that
"talking about sums that go into infinity" - yep <2/3 all the way to infinity
@fuchsi
"you can say 6/10+6/100+6/1000+... = 0,6666... = 2/3" - yep, because we allow substituting the limit for the actual infinite sum when you need to do arithmetic with it. Doesn't mean they're actually equal, just like pi isn't actually equal to 22/7
"Not < 2/3 because we don't stop the sum" - still <2/3 all the way to infinity - they just told you that π
"We take all of it. Until infinity" - and it's still less than 2/3, until infinity, hence why we use the limit in it's place
@SmartmanApps
"you can say 6/10+6/100+6/1000+... = 0,6666... = 2/3" - yep, because we allow substituting the limit for the actual infinite sum when you need to do arithmetic with it."
See. We are on the same page. You just use a weird formulation.
I told you it's a wording issue.
@SmartmanApps == is a logical operator for Informatics to return a Boolean.
That's not math.
Don't mix that up.
Also don't mix up the mathematical formulas and the describing texts. They sometimes use inaccurate words. Because of the contradiction
@SmartmanApps yes. They say both in the text. They acknowledge this seemingly contradiction and say that people have been struggling with it for hundreds of years.
But this is just a wording issue. The formulas are clearly stating equality with an = symbol.
The approximation was only for the picture. And the picture is an analogy. A metaphor for the infinite sum.
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