πŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±

1/9
#MathsMonday #Mathematics
This rubbish article https://www.scientificamerican.com/article/mathematicians-cant-agree-on-whether-0-999-equals-1/ popped up in my feed a few times, and I've already debunked the various points, but will cover it with specific links for each (non-)point.

"Mathematicians can’t agree on whether 0.999... equals 1" - yes they can, it's not, as per division, limits, infinite decimals, and other #Maths topics, all found in #Math textbooks

"by Manon Bischoff" - "is a theoretical Physicist". Maybe just stay in your lane dude... πŸ™„

Apr 13 at 6:07amWeb
Show threadπŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±Apr 13

2/9
"Countless debates in classrooms" - there are no debates in classrooms. Students are taught all of the topics which prove they're not equal.

"Teachers, professors and math-savvy Internet users repeatedly affirm that it does" - nope! Teachers most definitely do NOT say they are equal. Dude has clearly not ASKED any teachers, and is outright just making this up!

"many others still refuse to believe them" - people who can show you actual proofs and textbooks... πŸ™„

Show threadπŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±Apr 13

3/9
"the decimal representations of some fractions are infinite, such as 1⁄3" - which we know are APPROXIMATIONS https://dotnet.social/@SmartmanApps/116316765092652161

"one therefore chooses a symbol because a decimal notation would only approximate the actual value" - that's true of ALL infinite decimals. You so nearly had it!

"1⁄3 corresponds to the decimal number 0.333...." - corresponds, but is only an approximation...

πŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“± (@[email protected])

Attached: 1 image 1/5 #MathsMonday #Mathematics #Math This week I'm coming back to the topic of 0.(3) only being approximately equal to 1/3, which I discussed previously at https://dotnet.social/@SmartmanApps/115207065189190900 At the time, I knew this was true simply from knowing doing the division always leaves a remainder of 1. Since then I've now seen 2 #Maths textbooks which explicitly spell this out, that all non-terminating decimals are only approximations, and that only terminating decimals are exactly equal to fractions...

dotnet.social
Show threadπŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±Apr 13

4/9
"multiply it by 3 to get 0.999...." - which goes against the rules of Maths, hence using symbols for them instead, such as pi https://dotnet.social/@SmartmanApps/115603725411319763

"They reason that because 1⁄3Γ—3=1" - and that reason is wrong. See above

"there are a few other proofs" - that WASN'T a proof. You violated the rules of Maths, meaning it's not a proof https://dotnet.social/@SmartmanApps/115603723212470958

πŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“± (@[email protected])

3/6 Let x=0.(9) (multiply both sides by 10) 10x=9.(9) (substitute 0.(9)=x) 10x=9+x (subtract x from both sides) 9x=9 (divide both sides by 9) x=1 1. in the first place, we can't do arithmetic with infinitely recurring numbers. This is why, as I discussed at https://dotnet.social/@SmartmanApps/115365301522477363, the sum of the infinite series is DEFINED as being the limit of the series, because the limit is finite, and thus we can do arithmetic with it, but it's only an approximation of the infinite sum...

dotnet.social
Show threadπŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±Apr 13

5/9
"you have what’s called a geometric series, something mathematicians have known how to solve for several hundred years" - what we ACTUALLY solve is the LIMIT of the series, given the infinite sum can't actually be calculated https://dotnet.social/@SmartmanApps/115365301522477363

"the term 1⁄10n becomes zero" - no it doesn't. It can literally NEVER be zero - that's exactly what makes it a limit! See previous link. 0/∞=0, 1/∞ doesn't. A non-zero numerator means we have a non-zero fraction. This is so not-complicated!...

πŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“± (@[email protected])

Attached: 1 image 1/5 #MathsMonday #Mathematics #Maths #Math So we've seen that 0.(3) is merely a decimal APPROXIMATION of 1/3, due to the limitations of using Base 10. It's also true, for the same reason, that 1 is merely an integer APPROXIMATION of 0.(9), since 9 is infinitely recurring. If we think of 0.(9) as being 0.9+0.09+0.009+... this leads some to make a false equivalence argument, which is wrong in 2 ways...

dotnet.social
Show threadπŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±Apr 13

6/9
"This example is just one of many proofs" - that ALSO wasn't a proof, for the reasons I just gave.

"if we switch to binary notation, which consists only of 0’s and 1’s, the same problem arises" - the problem being your lack of familiarity with Maths πŸ™„

"Even though mathematics is a subject in which you can derive correlations exactly, with minimal room for interpretation" - and yet here you are making up your own interpretations... πŸ™„

Show threadπŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±Apr 13

7/9
"if you look at the number line and pick any two numbers, there are always infinitely many more between them" - no there isn't https://dotnet.social/@SmartmanApps/115682175610484093

"You have found a break in the number line" - no you haven't. Numbers are discrete, there are no gaps between them. See previous link

"As soon as you calculate a sum, you have to round up" - no you don't. Calculators do https://dotnet.social/@SmartmanApps/115207066209800987, but there's no such rule of Maths. Again you're making things up...

πŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“± (@[email protected])

Attached: 1 image 1/7 #Maths #Math This #MathsMonday Why there isn't an #infinite amount of #numbers between 0 and 1 on the #numberline AKA The number-line isn't a TARDIS! Firstly, one thing to point out is that people often conflate numbers with #numerals. This is important because #scalars also use numerals. Even though both numbers and scalars use numerals, they aren't the same thing! In #Mathematics, especially #arithmetic, we use numbers to count things. They enumerate how many things there are...

dotnet.social
Show threadπŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±Apr 13
8/9
"This rounding up also applies to multiplication, such that 0.999...Γ—1=1, which means a basic rule of mathematics, that anything multiplied by 1 is itself, no longer applies" - it's an Identity in Maths, that ALWAYS applies. 0.(9)x1=0.(9). πŸ™„ This dude sure does like making up imaginary rules of Maths - here the dude uses an imaginary "rule" of Maths to claim that an actual rule of Maths doesn't apply!...
Show threadπŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±Apr 13

9/9
"nonstandard analysis, which allows for so-called infinitesimals" - so does high school Maths, used in all the actual proofs that they aren't equal

"if they differ by one infinitesimal" - they do! πŸ˜‚

"no more so than conventional calculus" - which ALSO says they aren't equal

"there is still a debate whether 0.999... = 1" - no there isn't. It's a fact of Maths that they aren't equal

"working with the numbers and calculation familiar to most of us" - but apparently not familiar to you πŸ™„

Show threadHugh SimpsonApr 13
@SmartmanApps a field that includes real numbers, but excludes infinitesimals, is entirely consistent...? It's not like this is some hard-hitting truth to mathematics? Weird to me you accept infinitesimals but previously have denied the arithmetic of infinite series, essentially claiming that S(0,inf,9*10^(-n)) - S(1,inf,9*10^(-n)) should be illegal. I'm not sure how you can retain interesting or useful properties in such a system. Anyway 0.(9) is clearly the sum of the latter series. Which is ?
Show threadFish FaceApr 13
@silasmariner how much have you read from this guy? Do you know what you're getting yourself into?
Show threadHugh SimpsonApr 13
@FishFace I had not read much. Was nice to remind myself of the relevant maths whilst thinking the claim through (and its implications in e.g breaking field axioms for the implied indivisible infinitesimal iota). I haven't thought about this sort of claim since back at uni, but I now have dug deeper and realise I should probably step back lol.
Show threadFish FaceApr 13
@silasmariner yeah I am on the same journey! My plan is to eventually write everything up because I agree it's very interesting to work out. It's tricky to get definitive statements out of him (like, I don't think he really knows himself whether he thinks 0.999... is a specific real number or whether it's a way of denoting all possible finite decimals consisting of 0. followed by repeated nines) so it's tough to balance covering all bases with writing something a) succinct and b) that is about maths rather than drama.
Show threadHugh SimpsonApr 13
@FishFace I'm gonna be honest, I'm genuinely kinda interested in the implications of a model that's essentially just the reals plus some indivisible infinitesimals. So instead of a field you might think you get a ring, because you don't get multiplicative inverses.. But then iota is defined as the difference between a limit and a series, so its not well defined enough to even multiply.. Iota is like a dirty bit? Fascinating.
Show threadFish FaceApr 13

@silasmariner yeah, so this is where it gets unclear in which direction to go. First off, you could adjoin a single infinitesimal value i = 1 - 0.999..., and study the field you get by taking the closure under the field operations. This is clearly not what he's talking about, because 0.999... and 1 are then separated by infinitely many values: 1 - i/2, 1 - i/3, etc, and you need infinitely many actual infinities (1/i, 2/i, for example) in your structure. The structure resembles the hyperreals but I remember reading that it's not isomorphic.

So let's not close under the field operations and go for a ring instead. We are then not forced to have many values between 0.999... and 1, but we do have to have distinct values 1 + iΓ—n for each integer n.

If we work this out as a set of decimal sequences with strict equality, I guess we represent i as the sequence given by i(n) = 0 for each natural number n, but permit sequences of length omega+1, so that i(omega) = 1. 0.999... is then truly the "last real number before 1". I don't know what 1 - 2Γ—i should be, though. Any ideas? It's possible this isn't even a ring.

I feel like you can get something resembling this also by taking ℝ×{0,1} where operations in the ℝ part are standard, and operations in the binary bit are modulo 2! You wouldn't get a ring if you had saturating arithmetic in the binary part, right? (Like a dirty bit).

Show threadHugh SimpsonApr 13
@FishFace actually shit we don't even have a group, do we, because we're past the Rubicon and you're not allowed to manipulated the sums of infinite series. So it's a magma I think? But it does have two operations that under most (??) scenarios we should expect to be allowed to be commutative, associative and distributive. I don't know the name for that.
Show threadHugh SimpsonApr 13
@FishFace probably we're just in the realms of some highly-constrained constructivist mathematics and this is more like a Turing machine than an algebra
Show threadFish FaceApr 13

@silasmariner with saturating arithmetic yeah, because you don't have the cancellation law!

Not sure what you mean by it being like a Turing machine but I have wondered if there's a constructivist background here because he talks about practical limits from time to time.

Show threadHugh SimpsonApr 13
@FishFace oh I only mentioned turing machine in the sense that the rules don't really have the neat fit of maths. Everything's all a bit too bitsy and case-by-case. It was a metaphor at best. I tap out about here I suspect, having exhausted my intellectual curiosity. Was nice to chat!
Show threadFish FaceApr 13
@silasmariner nice talking to ya!
Show threadπŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±Apr 13

@silasmariner
"real numbers" - Reals aren't numbers, they're scalars https://dotnet.social/@SmartmanApps/115840140785226744

"have denied the arithmetic of infinite series" - no I haven't. We calculate the limit

"S(0,inf,9*10^(-n)) - S(1,inf,9*10^(-n)) should be illegal" - not should be, is, that's why we calculate the limit in it's place for arithmetic

"Anyway 0.(9) is clearly the sum of the latter series. Which is" - not able to be calculated, so we calculate the limit instead. Not sure how many times I need to say that

πŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“± (@[email protected])

Attached: 1 image 1/5 This #MathsMonday I'm expanding on #scalars in #Mathematics. Previously we discussed that integers in #Maths are discrete, and used for counting things. In #Math we also have the Real Numbers, which is a bit of a misnomer since we don't usually use them for counting! It's more a case of people conflating them with numbers, because both use numerals. Rather, the Reals are used as multipliers, to provide a scale (though sometimes used as a fraction's decimal equivalent)…

dotnet.social
Show threadHugh SimpsonApr 14

@SmartmanApps your usage of 'scalar' vs 'number' is not one I've ever seen before, I wonder where you picked it up from. What is the distinction, to you? For me I have only seen the use of scalar vs domain element in the realms of linear algebra (scalar vs vector) -- never before have I heard the claim that a scalar is something different to a number...

Anyway we don't calculate the limit in its place -- the limit of a sum of an infinite series is its value, proof by reductio ad absurdum.

Show threadHugh SimpsonApr 14
@SmartmanApps not that you need it in the instance of that particular sum, because the elements literally cancel out; that's what makes it so easy to determine the limits of the original sum for such series. This is how people do maths, it is _ubiquitous_.
Show threadHugh SimpsonApr 14
@SmartmanApps are you, perhaps, of the opinion that there should be no such thing as infinite collections in mathematics? That one cannot talk freely of, say, the set of all integers? Would you dispute the Cantor diagonalisation argument and proclaim that there is the same cardinality of reals as rationals because anything over finite is just unmanageable? That transfinite induction is an illegal technique? Genuinely curious.
Show threadHugh SimpsonApr 14
@SmartmanApps ah I've followed your link now. I see that you do indeed deny the existence of the infinite. Finite constructivism or something I think that school's called? It's pretty niche. Odd that you're so adamant on what you must know is a minority position, but nonetheless fascinating. Alas, most of my favourite mathematics (complex analysis! Set theory!) all relies rather heavily on the manipulation of the infinite, so I suspect we'll find little to agree on.
Show threadHugh SimpsonApr 14
@SmartmanApps I will add, finally, that I do however harbour a soft spot for model theory, so your model is interesting. Philosophically, however, I think it's wrongheaded to insist on some models being Platonic truths and others being nonsensical -- internal consistency is sufficient IMO, albeit obviously unprovable within a system (_thanks_ GΓΆdel, you ruined a perfectly lovely dream). We can apply the maths we dream of with infinity in eg electrical engineering so clearly it has merit.
Show threadπŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±Apr 14

@silasmariner
"your usage of 'scalar' vs 'number' is not one I've ever seen before" - we teach it in high school πŸ™„

"I wonder where you picked it up from" - school, textbooks

"we don't calculate the limit in its place" - yes we do

"the limit of a sum of an infinite series is its" - limit

"not that you need it in the instance of that particular sum" - you need it for every infinite sum

"so easy to determine the limits of the original sum for such series" - which is exactly why we do it

Show threadπŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±Apr 14

@silasmariner
"This is how people do maths, it is _ubiquitous_" - that's right. Calculating the limit is ubiquitous. You haven't even come up with any way that one could actually calculate the infinite sum

"the set of all integers? "- we don't. It's just the Integers. Sets are finite, by definition.

"Would you dispute the Cantor diagonalisation argument" - it's easily proven wrong. That's why Mathematicians said he was a madman πŸ™„ https://dotnet.social/@SmartmanApps/116276431640577565

πŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“± (@[email protected])

Attached: 1 image 1/6 Further extending this #MathsMonday on the laughable Cantor #Mathematics claim that the "infinite sets" of the Naturals and the Evens are "the same size", let's accept that #Math argument for a moment, and prove by contradiction that it can't be true... If sets can be infinite, that means not only do they come with the #Maths bracket form of set notation, and cardinality, but also set operations, like Union and Intersection, and ways to depict those sets/operations... with Venn diagrams...

dotnet.social
Show threadπŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±Apr 14

@silasmariner
"I see that you do indeed deny the existence of the infinite" - says person failing to point out anywhere where I did any such thing. πŸ™„ We use it in calculating limits for starters

"Odd that you're so adamant on what you must know is a minority position" - odd that you call a majority position "minority". πŸ™„

"internal consistency is sufficient IMO" - and Cantor's "proof" is inconsistent with the rules of Maths, hence why Mathematicians called him a madman πŸ™„

Show threadHugh SimpsonApr 14
@SmartmanApps dude the axiom of choice in ZFC - the 'standard model', if you will - necessitates understanding of and accord with the uncountability of the reals. Cantor is not like some weird kook, he's foundational
Show threadπŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±Apr 14

@silasmariner
"uncountability of the reals" - because they're continuous, not discrete like numbers, as taught in high school, as I already said. πŸ˜‚ This was all well-known well before Cantor came along to try and abuse it https://dotnet.social/@SmartmanApps/115840140785226744

"he's foundational" - yet somehow manages to never get mentioned at any time in school, where we teach the foundations of Maths, including the rules used with sets πŸ˜‚

"not like some weird kook" - Mathematicians said otherwise

πŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“± (@[email protected])

Attached: 1 image 1/5 This #MathsMonday I'm expanding on #scalars in #Mathematics. Previously we discussed that integers in #Maths are discrete, and used for counting things. In #Math we also have the Real Numbers, which is a bit of a misnomer since we don't usually use them for counting! It's more a case of people conflating them with numbers, because both use numerals. Rather, the Reals are used as multipliers, to provide a scale (though sometimes used as a fraction's decimal equivalent)…

dotnet.social
Show threadHugh SimpsonApr 14
@SmartmanApps As far as denying infinity... well, like, you refuse to accept that (0.9 -0.9) + (0.09 -0.09)... = 0. So if not infinity, what is it about that that troubles you?
Show threadFish FaceApr 14

@silasmariner I suspect his thinking is not so refined as that, because he is a high school teacher and generally teaches to high school syllabi. So I would guess (though you'd have to check) that he thinks it's perfectly ok to talk about "the identity function on the integers", and hence you get infinite sets in by the back door.

There was, however, a conversation below one of his threads where he didn't seem to think of a function on integers as being a complete object. My suspicion is that this doesn't come from a considered finitistic position, because he never uses the term "finitist" to describe his views and never explicitly rejects infinitary reasoning, only its results, and because someone who's thought about this properly would be aware that most mathematicians by far allow infinitary reasoning.

I dunno if this is helpful; I try to offer such nuggets in the hope that you can ask questions that reveal what I don't know about his beliefs... so I can then write about them XD

Show threadsedApr 13
@SmartmanApps just to understand your logic, you don't accept that 0.(9) * 10 = 9 + 0.(9), right? For you this is an operation we can't do/write, am I correct?
Show threadFish FaceApr 13

@sed good luck.

From his dump before this one, it's apparent he conceives of 0.(3) as "(3)/1(0)". But I don't think he means this notation to literally denote transfinite numbers because his mathematical understanding is not sophisticated enough for that; instead judging by his wording "for any given number of digits we can rewrite 1(0) as (9)+1", he doesn't have a well-defined understanding of recurring decimals at all; to him they are, I think, a way of talking about decimals with an *undefined* rather than *infinite* number of decimal places. So he talks a lot like he thinks 0.(9) is 0. followed by an unspecified number of nines.

You're unlikely to get any confirmation of this; he seems to be aware this isn't how everyone else treats the notation so won't sign up to it (and he's blocked me so I won't get an answer; I'm still here because I find it all so fascinating).

But from that point of view, it *does* make a kind of sense that you "can't do" 0.(9) Γ— 10 because the former doesn't refer to a specific number. Buuut, if you do the obvious thing and treat 0.(9) Γ— 10 as "9. followed by an unspecified number of nines" the normal argument goes through.

Not sure how he'd reconcile that with the textbooks he likes to bring up though - this is literally examinable material (example attached)!

Enjoy.

Show threadsedApr 14

RE: https://tribe.net/@ahau/116400200666533607

@FishFace he/she gave a reply to my question. I think I won't go farther than that. He/She rejects some notations/operations for I don't know what reason, I think it's a waste of time to interact more.

I just now saw this: https://hostux.social/@[email protected]/116400200644462704

Either a joke or... something else.

Show threadFish FaceApr 14
@sed that at least looks like something designed with enough rational thought to appear coherent, even if wrong!
Show threadsedApr 30

@FishFace I just read this https://www.quantamagazine.org/what-can-we-gain-by-losing-infinity-20260429/

it sheds some light on people (working in math and physics) fighting (or trying to get rid of) the existence of infinity

interesting reading

What Can We Gain by Losing Infinity? | Quanta Magazine

Ultrafinitism, a philosophy that rejects the infinite, has long been dismissed as mathematical heresy. But it is also producing new insights in math and beyond.

Quanta Magazine
Show threadFish FaceApr 30

@sed I haven't read it right to the end yet, but it is very interesting. It has been suggested that SmartmanApps is a finitist (not an ultrafinitist - I have seen him give a proof by induction!), but he rejected the label.

Nevertheless, he displays some characteristics of one: he seems to think of infinite decimals as a process rather than actual numbers, and in one of his flawed arguments against the infinitude of intervals of rationals, he goes on a tangent about measurement accuracy and atoms, suggesting he can't, or has difficulty, conceiving of numbers separate from something the number measures.

But his arguments about the finite nature of intervals are just wrong (he states, and does not prove, that there are finitely many rationals in an interval over a given denominator. He states as if it follows, that there are finitely many rationals in the interval.) A proper finitist would have an argument here that isn't completely bogus: she'd say, "we can conceive of a process for generating new rationals in the interval which will never run out, but it is fallacious to conceive of all of them at once as a completed whole because we can never in practice generate them all". (An ultrafinitist would say that there is some actual limit on how fine we can divide the interval, but wouldn't say what the limit is.)

It's kind of annoying that he touches on these very interesting topics but doesn't have the understanding to discuss them properly!

Show threadπŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±Apr 13
@sed
"just to understand your logic, you don't accept that 0.(9) * 10 = 9 + 0.(9), right?" - the rules of Maths don't, yes. That's why we use symbols like pi in their place for calculations, and then only substitute in a value for it when all arithmetic has been done https://dotnet.social/@SmartmanApps/115603723212470958
πŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“± (@[email protected])

2/6 Something can be a proof if, and ONLY if, the rules of Maths have been obeyed at every step. For example, I could claim to prove that 2+3x4=20, by going... 2+3x4 =5x4 =20 But this violates the order of operations rules, in the second step by doing the addition first, and therefore is NOT a proof. The "proof" that 0.(9)=1 similarly violates, not just one, but multiple rules of Maths, and is thus not a proof at all. A reminder, this (fake) proof goes as follows...

dotnet.social