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

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™„

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...

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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 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)…

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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 you absolutely do not teach the foundations of mathematics in schools. You teach the foundations of mathematical intuition, you teach people to count and multiply. You do not (evidently) teach axiomatic frameworks, the notion of well-foundedness, or the actual methods used to justify our inferences re e.g. limits. There is inherently something intuitional about maths, and sometimes your intuition is wrong. Maths is fundamentally logical implication, not epistemic statements...
Show threadHugh SimpsonApr 14
@SmartmanApps the entirety of the real number line can be defined as isomorphic to set-theoretic constructions, provided one accepts infinite sets, but we sure as hell aren't teaching that in most schools
Show threadHugh SimpsonApr 14
@SmartmanApps the thing that galls me the most is that standard mathematics - the sort that you can actually do stuff with - hugely simplifies the cognitive leap by some incredibly elegant abbreviation of tedious thoughts that let us just say 0.(9)=1 (for example) so as not to leave etas and deltas lying around superfluously, and you characterise your rejection of this as obvious and simple and accepted and it is anything but. You are hugely overcomplicating the whole endeavour to your detriment
Show threadπŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±Apr 14

@silasmariner
"you absolutely do not teach the foundations of mathematics in schools" - that's hilarious dude. 🀣🀣🀣 Arithmetic, algebra,, calculus, trig, sets,...

"you teach people to count" - which is used in things like... finding the cardinality of a set

"e.g. limits" - yep, we teach limits too, and asymptotes, hyperbolas,...

"the entirety of the real number line" - reals don't belong on the number line - it's right there in the name that it's for numbers!

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

@silasmariner
"we sure as hell aren't teaching that in most schools" - because it violates the rules of Maths, which we DO teach in schools πŸ™„

"you characterise your rejection of this as obvious and simple and accepted" - as seen in Maths textbooks

"You are hugely overcomplicating the whole endeavour" - the difference between an infinite sum and the limit of it is quite simple actually - we see it in graphs of a hyperbola for example

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

@silasmariner
"you refuse to accept that (0.9 -0.9) + (0.09 -0.09)... = 0" - what you mean is YOU refuse to accept that the LIMIT is the number that the series literally NEVER REACHES πŸ™„ https://dotnet.social/@SmartmanApps/115365301522477363

"what is it about that that troubles you" - violating the rules of Maths, ignoring literal proofs, need I go on? πŸ™„

If you don't want to accept what's in Maths textbooks about limits I'll just end up blocking you dude - you've wasted enough of my time already - so ball's in your court...

Apr 14 at 8:06pmWeb
Show threadHugh SimpsonApr 15
@SmartmanApps ... No finite truncation of an infinite positive series reaches the limit, true, is that what motivates you? Then you must believe Xeno's paradox to be unresolved. If I halve the distance between Achilles and the tortoise every time, I have an infinite sum. Nonetheless I believe Achilles reaches the tortoise. You, however, must deny this, and consequently believe motion in space to be impossible. It is an absurdity to permit the infinite but deny the implications. Most strange.
Show threadHugh SimpsonApr 15
@SmartmanApps anyway you need to stop appealing to 'the laws of mathematics' without referencing anything specific, because it just makes you look like you don't understand anything you're talking about. Adding infinite series together is clearly permitted by associativity and commutativity. These are axiomatic, and can only be denied in the context of sums by forbidding the infinite to exist; that you do one but not the other leaves you in an incoherent state
Show threadπŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±Apr 15

@silasmariner
"No finite truncation of an infinite positive series reaches the limit, true" - let me fix that for you, no infinite series ever reaches the limit, full-stop

"you must believe Xeno's paradox to be unresolved" - no, it's not a paradox to begin with, since you can't slow time πŸ™„

"I have an infinite sum" - no, you have an impossible situation where you have slowed time

"believe motion in space to be impossible" - no, slowing time is impossible πŸ™„

Show threadπŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±Apr 15

@silasmariner
"you need to stop appealing to 'the laws of mathematics' without referencing anything specific" - I have specifically referred to limits/asymptotes, not allowed to do arithmetic with infinite decimals, cardinality of sets, etc., and you are pretending I haven't, like the Gaslighter that you are

"it just makes you look like you don't understand anything you're talking about." - says person who has repeatedly shown he doesn't know what he's talking about, such as what limits are

Show threadπŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±Apr 15
@silasmariner
"Adding infinite series together is clearly permitted by associativity and commutativity" - adding their LIMITS together is permitted. Again, there's no way to calculate the actual infinite sum, hence why we use their limits, but you refuse to acknowledge the difference between the infinite sum itself and the limit, despite me posting textbooks that explicitly spell out the difference, so as promised, time for you to go now Gaslighter. Bye now