[ 3 / biz / cgl / ck / diy / fa / g / ic / jp / lit / sci / tg / vr ] [ index / top / reports / report a bug ] [ 4plebs / archived.moe / rbt ]

# /sci/ - Science & Math

[ Toggle deleted replies ]
File: 95 KB, 656x843, 1425612191013.jpg [View same] [iqdb] [saucenao] [google] [report]

0.999... is NOT 1

No number other than 1 is equal to 1.

9 is not 10 doesnt matter if there are .999999999999... numbers. This whole problem just proves that infinity isnt real and just mental wankery

 >> Anonymous Mon Aug 24 23:14:36 2020 No.12045479 >he doesn't understand what 1 means from a mathematical standpoint
 >> Anonymous Mon Aug 24 23:21:48 2020 No.12045496 >>12045479You're not wrong, but the true interesting observation is that people posting these threads reliably never seem to understand what 0.999... means from a mathematical standpoint, and yet for some reason still feel knowledgeable enough to judge what it does and doesn't equal.
 >> Anonymous Mon Aug 24 23:25:33 2020 No.12045505 >>12044835>just mental wankeryall of math and science is that. there's nothing real about math or logic. rationalizing experimental data is just schizoprenia.
 >> Anonymous Mon Aug 24 23:29:23 2020 No.12045512 >I'll smugly and confidently assert something, that will ensure those anonymous people on the internet know I'm right!
 >> Anonymous Mon Aug 24 23:33:39 2020 No.12045525 This is true in the hyperreal numbers, but I strongly doubt that the anons who start $0.999...$ threads even understand what the hyperreals are.
 >> Anonymous Mon Aug 24 23:34:54 2020 No.12045528 >>120448358/9+1/9= 9/90.8...+0.1...=?
 >> Anonymous Mon Aug 24 23:36:58 2020 No.12045533 >>12045525>This is true in the hyperreal numbers,no it isnt1 - .999... isnt 1/wits just 0limits are a geometric concept, and infinitesimals are an algebraic conceptthey never mix
 >> Anonymous Mon Aug 24 23:40:30 2020 No.12045546 >>12045533>no it isntYes it is
 >> Anonymous Mon Aug 24 23:44:20 2020 No.12045560 1/inf=0
 >> Anonymous Mon Aug 24 23:45:53 2020 No.12045563 1/3 = .3333....2/3 = .6666....3/3 = 11/3 + 2/3 = 1.333.... + .666.... = .999.... = 1OP dumb!also what does lim x->1 x equal?1
 >> Anonymous Mon Aug 24 23:46:16 2020 No.12045565 >>120448350.999... is by definition lim n->oo sum i=1 to n of 9/10^n, which is lim n->oo 1-1/10^n=1. It is a simple calculation from the very definition of what 0.999... is. It isn't hard. If they aren't equal, what number is in between them, lol?
 >> Anonymous Mon Aug 24 23:59:36 2020 No.12045591 >>12045546Why do you retards do this shit? Do you know what the transfer principle is? .9...=1 in the reals so it is true in the hyperreals.
 >> Anonymous Tue Aug 25 00:03:36 2020 No.12045602 >>12045591Bruh you can explicitly construct infinitesimals in an ultraproduct of the reals.
 >> Anonymous Tue Aug 25 00:06:06 2020 No.12045608 >>120448350.999.. is not a number in the formal sense. It is a shorthand hand way of expressing an infinite series.
 >> Anonymous Tue Aug 25 00:07:09 2020 No.12045611 >>12045602Infinitesimals don't split equal numbers you mongoloid.
 >> Anonymous Tue Aug 25 00:07:15 2020 No.12045612 >>12045602Okay and...?
 >> Anonymous Tue Aug 25 00:10:19 2020 No.12045618 >>12044835based
 >> Anonymous Tue Aug 25 00:11:03 2020 No.12045621 >>12045611>>12045612You can construct $\varepsilon$ such that $0 < \varepsilon < 1 - 0.999...$.
 >> Anonymous Tue Aug 25 00:11:41 2020 No.12045624 I swore to stop visiting /sci/ until mods started deleting these threads. I just checked into see if something had changed. Back to redit.
 >> Anonymous Tue Aug 25 00:14:45 2020 No.12045636 File: 3.16 MB, 424x498, laugh.gif [View same] [iqdb] [saucenao] [google] [report] >>12045621He thinks 0<0 in the hyperreals.
 >> Anonymous Tue Aug 25 00:21:46 2020 No.12045648 >>12045621Says it can be constructedDoesn't even try to construct it
 >> Anonymous Tue Aug 25 00:29:36 2020 No.12045667 >>12045621No, you can't. You can find a hyperreal h such that 0
 >> Anonymous Tue Aug 25 00:33:17 2020 No.12045672 >>12045667>o, you can't. You can find a hyperreal h such that 0
 >> Anonymous Tue Aug 25 00:35:52 2020 No.12045675 >>12045648I'm phoneposting, so I won't type out all the latex, I'll just bullet point it:-take direct product of countably many copies of $\mathbb{R}$ indexed by the natural numbers-form ultrafilter of all cofinite sets of $\mathbb{N}$ -form ultraproduct using this ultrafilter (note that by identifying any real number with its constant tuple you can see this contains $\mathbb{R}$)-set $\varepsilon = (\frac{1}{n})_n$-$1$ is no longer the limit of $0.9,$ $0.99,$ $0.999$ etc. because the difference always exceeds $\varepsilon$ (obviously $<$ is defined as per the ultrafilter.>>12045667 read the last point bud
 >> Anonymous Tue Aug 25 00:46:32 2020 No.12045692 >>12045563Limit is not a number. It will never reach 1.0.9... =/= 1
 >> Anonymous Tue Aug 25 00:47:00 2020 No.12045694 >>120448351 > .9 + .09 + .009 + .0009 +...
 >> Anonymous Tue Aug 25 00:58:31 2020 No.12045707   >>12045692>Limit is not a numberwith infinite series, it it>lrn2math
 >> Anonymous Tue Aug 25 00:59:34 2020 No.12045709 >>12045692 >Limit is not a number with infinite series, it is >lrn2math
 >> Anonymous Tue Aug 25 01:08:15 2020 No.12045724 >>12045692Wait, so what do you think the limit of1+ 1/2 + 1/4 + 1/8 ..... is?
 >> Anonymous Tue Aug 25 01:10:28 2020 No.12045733 >>120457242^1/2, still the sum of numbers in that sequence will never be equal to the limit, it's a limit.
 >> Anonymous Tue Aug 25 01:11:23 2020 No.12045734 >>12045675>-set ε=(1n)nthats 0its not an infinitesimalyou dont form infinitesimals through a limiting processquoting my first post>limits are a geometric concept, and infinitesimals are an algebraic concept>they never mix
 >> Anonymous Tue Aug 25 01:12:44 2020 No.12045739 >>12045733>2^1/2how?
 >> Anonymous Tue Aug 25 01:13:28 2020 No.12045741 >>12045675Well, we've done the ultraproduct construction so far so fine, but then the issue is near the end where there are some implicit assumptions.Okay, great-we've created the standard model of 'the' hyperreals. So far, it appears not that useful if we don't clearly define what is '0.999...' and furthermore what is 'limit' in hyperreals. 0.999... is generally notation for real numbers, representing the limit of the sequence sum 9/10^n, which exists. How are we extending this notion of 'limit'-after all, the derivative 'realifies' the hyperreal quotient, does the limit do so as well? How about 0.999..., do we intend this instead to be based on a notion of decimal expansion in hyperreals, in which case it will mean something different then above.0.999... as itself being a real number, being found to be 1 in the reals, is, by extension, also 1 in the hyperreals. If you don't start with 0.999... as being a prior defined number but instead defined by some process you are extending into hyperreals, then it will of course be possibly different from 1, but you have to define what this process is. 0.999... is notation I've only seen for real numbers, in turn, it should be associated with this prior real number, and then embedded in the hyperreal.
 >> Anonymous Tue Aug 25 01:13:39 2020 No.12045742 >>12045733>2^1/2kek
 >> Anonymous Tue Aug 25 01:21:51 2020 No.12045757 >>12045742kek. Now I understand how treacherous it was for the Wolfshekl Committee to sieve through the vast number of alleged proofs of Fermat's Last Theorem by Amateur Mathematicians.
 >> Anonymous Tue Aug 25 01:26:05 2020 No.12045764 >>12045602You don’t know what the fuck you’re talking about, bud. Stop using words you don’t understand.
 >> Anonymous Tue Aug 25 01:31:36 2020 No.12045771 >>12045525I've heard about surreal numbers, but not hyperreal nummbers, is there a practical difference?
 >> Anonymous Tue Aug 25 01:31:56 2020 No.12045773 >>12045734>thats 0How is 0 if not a single coordinate of the tuple is 0?>>12045741Come on man, I know it was just a bullet point explanation, but surely you could fill in the gaps? Limit is defined exactly the way it is in the reals. $0.999...$ is defined at the limit of $0.9,$ $0.99,/math] [math]0.999$, ...
 >> Anonymous Tue Aug 25 01:32:35 2020 No.12045774 >>12045621>>12045675In the hyperreals, the limit extends into the hypernatural.
 >> Anonymous Tue Aug 25 01:33:46 2020 No.12045777 >>12045773Yes, and the ... notation extends beyond the naturals in the hyperreal context. See the transfer principle.
 >> Anonymous Tue Aug 25 01:33:51 2020 No.12045778 >>12045773Wow I should've previewed the latex in that post. It's always the simple posts where you make errors...
 >> Anonymous Tue Aug 25 01:35:28 2020 No.12045781 >>12045773who cares if you have some sequence in the hyperrealswe're talking about .999... in the reals, which are embedable into the hyperrealsthis entire tangent is unrelated to the original topic
 >> Anonymous Tue Aug 25 01:36:48 2020 No.12045785 >>12045771In truth, there is no '1' hyperreal field. A hyperreal field is a field [formalization of 'number'] that otherwise satisfy the axioms [or rather, first-order implementations] of the real numbers [called real closed fields] and contain elements that can be considered infinitesimal and infinities. When you hear about 'the' hyperreals, they are generally referring to what is known as the ultraproduct construction. Surreals are when one considers how large one can go with hyperreals, which uh, contains all ordered fields and all transfinite ordinals and yeah [I'm no expert though].
 >> Anonymous Tue Aug 25 01:37:57 2020 No.12045787 >>120457780.999... still equals 1 in the hyperreals because ... is meant to extend over the *entire* sequence which is now hypernatural in length. The 9s don’t stop at the naturals.
 >> Anonymous Tue Aug 25 01:40:23 2020 No.12045793 >>12045781You are correct, but it was to refute >>12045533>>12045787This post is utter handwavy garbage
 >> Anonymous Tue Aug 25 01:43:13 2020 No.12045800 >>12045773Oh I am not the one making the bullet point. He is referring to some more advanced mathematical constructs, which are in truth quite necessary for a concrete construction of hyperreals. I am responding that while it looks all technical and rigorous, implicit reference to 0.999... or 'limit' without a clear meaning of what either of these mean in the hyperreals is the error. 0.999... is notation quite unique to the real numbers and so you would be correct in that being the 'real' limit. Defined as a real number, it is then contained in the hyperreal as a real. However, the person I am responding to seems to be indicating 0.999... as not being a real number defined by real means by as some type of notation extendable into the hyperreals, which will give an entirely different number. I am challenging of what in the world this means, since as you mentioned, 0.999... is purely real number notation.
 >> Anonymous Tue Aug 25 01:46:14 2020 No.12045806 File: 5 KB, 454x88, Oh no no no.png [View same] [iqdb] [saucenao] [google] [report]
 >> Anonymous Tue Aug 25 01:49:09 2020 No.12045814 >>12045800Clarification: by $0.999...$ in the reals we mean the limit of the sequence $0.9$, $0.99$, $0.999$ and so on. Now map these real numbers to their constant tuples in the hyperreals. Then $0.999...$ in the hyperreals is the limit of this new sequence. This limit is NOT $1$>>12045806So you kicked off the whole tangent then
 >> Anonymous Tue Aug 25 01:51:29 2020 No.12045818 >>12045814>So you kicked off the whole tangent thenno you didyou were the first person to mention hyperrealshence the fact that i fucking quoted you talking random shit about the hyperreals
 >> Anonymous Tue Aug 25 01:54:23 2020 No.12045826 >>12045793*Your* post is utter hand wavy garbage, retard.https://en.wikipedia.org/wiki/Transfer_principle
 >> Anonymous Tue Aug 25 01:54:34 2020 No.12045827 >>12045818I mentioned hyperreals as an interesting aside (because OP's statement is true in the hyperreals) and that would have been that. Instead you tried to insist that it is not the case, so I had to post a bunch of stuff explaining it to you
 >> Anonymous Tue Aug 25 01:58:20 2020 No.12045837   >>12045826>>12045787>over the *entire* sequence which is now hypernatural in lengthThis is what's garbage. The sequence is *exactly* the same length as it is in $\mathbb{R}$ because it's the same sequence with constant tuples instead of real numbers
 >> Anonymous Tue Aug 25 02:03:55 2020 No.12045845 File: 1.82 MB, 4032x3024, A86B6CF9-35C5-4C6A-9A6A-2BA9478EEC75.jpg [View same] [iqdb] [saucenao] [google] [report] Where does the logic in pic related fail in the hyperreals? Math newbie here.
 >> Anonymous Tue Aug 25 02:04:08 2020 No.12045846 >>12045814Okay, I figured that much out (though 0.999... could have referred to some hyperreal decimal expansion with more 9's). Now how is limit being defined.[I don't even think the limit in a naive sense would exist here, which I mean typical epsilon but using hyperreals. Let {s_n} be the sequence {0.9,0.99,...} Let us try epsilon limit definition. Then, L=1+h for some infinitesimal hyperreal h and 1 both satisfy the epsilon condition [when the epsilon is considered real], so the limit isn't unique. If we try hyperreal epsilon, then we need to find N such that n>N implies |L-s_n|N which is impossible since it has to hold when e is less then every other real. So, by naively extending the notation of ... into hyperreals like this, have made it unusable.
 >> Anonymous Tue Aug 25 02:06:44 2020 No.12045850 >>12045826I think we're talking about different numbers. See my clarification here >>12045814. I'm talking about the $0.999...$ of the reals when embedded in the hyperreals. Seems like we've been at each others throats over a miscommunication. Apologies
 >> Anonymous Tue Aug 25 02:11:45 2020 No.12045859 File: 57 KB, 500x369, 1337850440812.jpg [View same] [iqdb] [saucenao] [google] [report] >>12045505I mean, there is some fundamental reality to math and logic, because we can use mathematical formulas to predict real-world physical interactions down to the atoms.Physics is just reality behaving according to mathematics that we've discovered.
 >> Anonymous Tue Aug 25 02:14:37 2020 No.12045866 >>12045859That's like saying the Vatican and Italy are one and the same. Math is infinitely larger than physics.
 >> Anonymous Tue Aug 25 02:18:45 2020 No.12045875 Anon, you haven't answered why the sum of 1 + 1/2 + 1/4 + 1/8 .... = 2^1/2
 >> Anonymous Tue Aug 25 02:20:13 2020 No.12045876 >>12045866And all of it is reflected in reality, even if we can't observe some of it.
 >> Anonymous Tue Aug 25 02:20:45 2020 No.12045878 >>12045827>because OP's statement is true in the hyperrealsno its not.999... doesnt stop meaning the real number once we go the hyperrealscall f the embedding of R into the hyperrealsthen 1=f(1)=f(.99...)=.999....99... is not a hyperreal sequence, its a real sequenceyou cant just redefine agreed upon notions and then say "ackshually in the hyperreals this isnt true"youre no longer talking about .999..., but youre still calling it .999...you mean to say that (.9...9)_n as a sequence in the hyperreals is not (1)_n in the hyperreals, which is different from (.999...)_nthere is a distinction here you do not understandreal numbers are classes of cauchy sequences of rationals.999... is the sequence of .9 .99 .999 etcelements of the hyperreals are sequences of realstaking the sequence .9 .99 .999 etc as a sequence of reals to define a hyperreal is not the same as taking the class of .9 .99 .999 etc as a sequence of rationalsthey live in totally different spaces
 >> Anonymous Tue Aug 25 02:21:38 2020 No.12045883 >>12045876>all of itlolno
 >> Anonymous Tue Aug 25 02:24:06 2020 No.12045891 >>12045878Not him, but do you have a PhD in mathematics?
 >> Anonymous Tue Aug 25 02:24:16 2020 No.12045893 >>12045883What branches of math have absolutely no basis in the physical reality in which we live?
 >> Anonymous Tue Aug 25 02:27:12 2020 No.12045896 >>12045878>taking the sequence .9 .99 .999 etc as a sequence of reals to define a hyperrealThat is NOT what I've been doing. The $0.9$s and $0.99$s etc. are all shorhand for the hyperreals that those real numbers map to when we embed $\mathbb{R}$. Those are sequences *of* hyperreals, NOT sequences defining hyperreals
 >> Anonymous Tue Aug 25 02:28:53 2020 No.12045899 >>12045896I even said this in my clarification >>12045814:>Now map these real numbers to their constant tuples in the hyperreals
 >> Anonymous Tue Aug 25 03:06:27 2020 No.12045967 >>12045893any of it that doesn't use the same axioms as physics
 >> Anonymous Tue Aug 25 03:08:52 2020 No.12045971 >>12045967Physics generally doesn't have 'axioms'.
 >> Anonymous Tue Aug 25 03:10:08 2020 No.12045973 File: 112 KB, 953x613, 1595913191512.jpg [View same] [iqdb] [saucenao] [google] [report]
 >> Anonymous Tue Aug 25 03:10:25 2020 No.12045974 >>12045971lol ok sweetie
 >> Anonymous Tue Aug 25 03:14:32 2020 No.12045980 >>12045973Thanks anon, do you think you could make that image a bit harder to read though? There's not enough jpeg artifacting, I can still barely make out the text. Also, the colors need to contrast more, I only have a mild headache from trying to read those boxes.
 >> Anonymous Tue Aug 25 03:15:24 2020 No.12045982 >>12045974Please, show me an instance in physics where we go 'Ax. 1 blah blah, Ax. 2 bluh bluh', and after some derivation, we get blub. No, physicists have a certain intuitive connection between physical world and corresponding mathematics, physicists then may use some less rigorous mathematics [say, working with dirac-delta as function] to get answers that can be interpreted physically. Not axioms of an abstract formal system.
 >> Anonymous Tue Aug 25 03:17:24 2020 No.12045984 >>12045982>'Ax. 1 blah blah, Ax. 2 bluh bluh', and after some derivation, we get blub.Impressive, did you quote that from your PhD?
 >> Anonymous Tue Aug 25 03:19:23 2020 No.12045987 File: 3 KB, 125x114, 1589316322831s.jpg [View same] [iqdb] [saucenao] [google] [report]
 >> Anonymous Tue Aug 25 03:22:12 2020 No.12045989 >>12045984As you can clearly tell, since I don't think physics actually uses 'axioms', I can't actually insert such a sentence that would make a convincing example (since I don't think they exist) I have to term it in terms of 'blah' 'bluh' and 'blub' as placeholder terms for whatever axioms YOU think physics is based on.
 >> Anonymous Tue Aug 25 03:23:14 2020 No.12045990 >>12045989bro he ethered you, it's over just close the tab and do something else with your night
 >> Anonymous Tue Aug 25 03:25:22 2020 No.12045994
 >> Anonymous Tue Aug 25 03:26:24 2020 No.12045995 >>12045994Okay now I officially choked rip https://www.youtube.com/watch?v=PDNZX2nql2Y
 >> Anonymous Tue Aug 25 03:27:04 2020 No.12045997 >>12045994not an argument. kek. You've lost. you're an embarrassment to human kind. Return to the wild and live among the chimps.
 >> Anonymous Tue Aug 25 03:31:24 2020 No.12046003 >>12045997Too much of a degen to do that. Now STFU and just give the axioms of physics or whatever.
 >> Anonymous Tue Aug 25 03:35:01 2020 No.12046015 >>12045896>That is NOT what I've been doing.sick, if thats not what youre doing then .999... is unequivocally 1>The 0.90.9s and 0.990.99s etc. are all shorhand for the hyperreals that those real numbers map to when we embed R.call f the embedding1 = .999... in the reals implies f(1) = f(.999...) in the hyperreals by welldefinednesswe just follow convention and call f(1)=1 and f(.999...)=.999..., so 1=.999... as the embedded hyperreals.999... is not 1 - 1/w in the hyperreals, it is still always 1
 >> Anonymous Tue Aug 25 03:36:22 2020 No.12046018 >>12045982>working with the dirac delta as a functionIt's a distribution and if you still don't understand that then you're rarted. Colloquially calling it a function does not change how you use it (aka like a distribution).
 >> Anonymous Tue Aug 25 03:37:01 2020 No.12046019 >>12044835The universe is discrete and finite and so are numbers. Anything else is the work of the Devil.Fuck this. I am sick of all these arguments. Its time we put our faith to the test and settled this issue with SWORDS. ON THE HOLY FIELD OF BATTLE!I have no doubt that we shall prevail, for GOD is on our side. We shall march upon the CURSED SODOMITES who have defiled our world with their heinous lies and infinities and infinitesimals and complex number planes and and and shit! They shall perish at our hands and then THEY WILL discretely BURN IN HELL!...for a finite amour of time.Brothers and Sister of the ONE TRUE FINITE UNIVERSE, say it with me...DEUS VULT!DEUS VULT!BY GOD! DEUS VULT!
 >> Anonymous Tue Aug 25 03:37:11 2020 No.12046020 >>12046003>Axioms play a key role not only in mathematics, but also in other sciences, notably in theoretical physics. In particular, the monumental work of Isaac Newton is essentially based on Euclid's axioms, augmented by a postulate on the non-relation of spacetime and the physics taking place in it at any moment.>In 1905, Newton's axioms were replaced by those of Albert Einstein's special relativity, and later on by those of general relativity.>Another paper of Albert Einstein and coworkers (see EPR paradox), almost immediately contradicted by Niels Bohr, concerned the interpretation of quantum mechanics. This was in 1935. According to Bohr, this new theory should be probabilistic, whereas according to Einstein it should be deterministic. Notably, the underlying quantum mechanical theory, i.e. the set of "theorems" derived by it, seemed to be identical. Einstein even assumed that it would be sufficient to add to quantum mechanics "hidden variables" to enforce determinism. However, thirty years later, in 1964, John Bell found a theorem, involving complicated optical correlations (see Bell inequalities), which yielded measurably different results using Einstein's axioms compared to using Bohr's axioms. And it took roughly another twenty years until an experiment of Alain Aspect got results in favour of Bohr's axioms, not Einstein's. (Bohr's axioms are simply: The theory should be probabilistic in the sense of the Copenhagen interpretation.)>As a consequence, it is not necessary to explicitly cite Einstein's axioms, the more so since they concern subtle points on the "reality" and "locality" of experiments.
 >> Anonymous Tue Aug 25 03:39:34 2020 No.12046025 >>12046019superpositions are anything but discrete
 >> Anonymous Tue Aug 25 03:40:39 2020 No.12046027 >>12046025Say that to my Sword, HERETIC!
 >> Anonymous Tue Aug 25 03:40:44 2020 No.12046028 is 0.8888888..... the same as 0.99999999999.....?
 >> Anonymous Tue Aug 25 03:41:07 2020 No.12046029 >>12046028no.
 >> Anonymous Tue Aug 25 03:45:52 2020 No.12046043   >>120460288/9
 >> Anonymous Tue Aug 25 03:46:54 2020 No.12046047 >>120460288/9 and 9/9
 >> Anonymous Tue Aug 25 03:48:32 2020 No.12046051 >>12046018I'm aware. I'm looking at the 'sloppy mathematics' seen in physics.
 >> Anonymous Tue Aug 25 03:49:20 2020 No.12046052 >>12045876>>12045893Where do you think transfinite set theory, e.g. large cardinals and the continuum hypothesis, are reflected in reality? Genuinely curious.
 >> Anonymous Tue Aug 25 04:08:24 2020 No.12046117 >>12046020Copy pasta from wiki is fine and all but they are hardly what considered 'axioms' in the mathematical sense [which, is what is being referred to]. Generally speaking, for Newton they are referred to as laws, for Einstein as postulates. Only Bell's theorem, I would say, really truly counts as a theorem of certain 'axioms', perhaps.Axioms are formal statements used to derive other ones by matter of derivation. When I consider something like Einstein's postulates, namely that the speed of light is invariant and that the laws of physics are the same in all inertial frames, when we derive the Lorentz transformations, we are making implicit assumptions of what in the hell a change of frames is and what counts as 'laws of physics' as 'being the same'. These are based on an intuitive understanding of the connection between math and physics and not from rigorous proof. So, the original person I was responding to was complaining that mathematics concepts not based on physics axioms had no basis in physical reality is 100% correct because your notion of 'axiom' [meaning 'assumed truth', in the loosest sense] is entirely distinct from the mathematical one.
 >> Anonymous Tue Aug 25 04:11:38 2020 No.12046127 >>12044835Everyone is making fun of OP, but he's right. I can say that 1 = 2 + 2 if I'm working over modulo 3. The point is, without which equivalence relation is being used, the only equality that matters is literal equality, of which these symbols are different and can represent different things. However, using the equivalence rule that a ≈ b iff |a - b| < 1/n for any positive integer, we find that 0.9999... ≈ 1.
 >> Anonymous Tue Aug 25 04:19:59 2020 No.12046152 >>12046117>Newton they are referred to as lawsBut didn't we build the whole of classical mechanics from just 3 laws (this may be an understatement but the general idea still holds)?
 >> Anonymous Tue Aug 25 05:16:40 2020 No.12046269 >>12045546why do you think the definition of 0.999... should change when you move from reals to hyperreals ?
 >> Anonymous Tue Aug 25 05:47:30 2020 No.12046308 Why wouldn't 0.000...1Always in between 0.999... and 1?
 >> Anonymous Tue Aug 25 05:52:08 2020 No.12046319 >>12046308>let's pretend finite is infinite
 >> Anonymous Tue Aug 25 05:53:48 2020 No.12046321 >>12046308because 0.000...1 doesn't represent a real number
 >> Anonymous Tue Aug 25 06:33:07 2020 No.12046367 File: 660 KB, 1432x4127, 1585406223787.jpg [View same] [iqdb] [saucenao] [google] [report]
 >> Anonymous Tue Aug 25 11:02:41 2020 No.12046881 >>12046308This is a not at all rigorous way to look at it but in essence, you would get, in this view [if it even makes sense and it doesn't really make sense I think] 0.999... 1. This 1 at the end can't pair with any 9 at the end because there is no 9 at the end. Naturally, when it comes to hyperreals, 0.999... 1 doesn't actually exist and neither does 0.000... 1, though the idea between splitting between real decimal and infinitesimal decimal exists.
 >> Anonymous Tue Aug 25 11:22:19 2020 No.12046928 >>12046152I mean, yes, in a certain sense-though to be clear, what exactly counts as Newtonian mechanics isn't really fixed. Newtonian mechanics could mean Newton's original mechanics which means no galilean relativity-with galilean relativity, we'd also be fine and this is generally the type of mechanics of introductory physics. If we abstract instead into working with real space rather then euclidean geometry, we get a different framework in doing physics [generally the type used for classical mechanics] How about Lagrangian and Hamiltonian and stuff that is considered 'classical mechanics'? Unlike if I were talking about the axioms of real numbers, which tend to define a quite clear system, the laws of Newton don't really do the same. Looking at Newton's book, we see that euclidean geometry is really the vehicle used to prove results, with the physical arguments moving forward by his laws.Perhaps I was being a bit too gung-ho in my initial statement, but I really like to characterize the difference in approaches between physics and mathematics. If I open up a math textbook, it shouldn't take me more then like 10 seconds to find a direct declared 'axiom'. I don't think I can do the same in physics. There are postulates and all that, sure, but I prefer to not imagine physics as being so rigorously based on these as absolutely foundation of physical systems but rather as principles that connect the physics to math, so that whatever math we are using as our vehicle physics, is well, physics-it has physical meaning. For instance, one can prove all kinds of results about vector calculus and one can consider E and M as arbitrary vector fields but this isn't physical-restrain these by Maxwell's equations and we haven't necessarily made the mathematics, the vehicle we carry out proofs, any different-we've just added physical conditions which can be used to actually solve problems [say, the wave equation]
 >> Anonymous Tue Aug 25 11:32:40 2020 No.12046960 >>12046928>verbal diarrhea cope
 >> Anonymous Tue Aug 25 11:36:59 2020 No.12046982 I'm waiting for fags to prove $0.999... \neq 1$ in the real numbers. All anybody has done is reject proofs of $0.999... = 1$, but not prove the negation.After showing ellipses notation to be contradictory and vague, then we can finally get rid of ellipses notation and use superior formal symbols like >>12045845 that makes clear that it was all limits, anyways.
 >> Anonymous Tue Aug 25 11:40:55 2020 No.12047002 >>12046982>ellipses notationit's fine, autist
 >> Anonymous Tue Aug 25 11:43:45 2020 No.12047008 >>12045505I said infinity isnt realWhat number is before 0.999...? Nothing
 >> Anonymous Tue Aug 25 11:46:15 2020 No.12047012 >>12044835900.900900900... = 1000Cope harder faggot
 >> Anonymous Tue Aug 25 11:46:57 2020 No.12047014 >>12047008-13
 >> Anonymous Tue Aug 25 11:49:38 2020 No.12047026 >>12046982It's literally decimal notation. There is a correspondence between real numbers r and sequence of integers. For a real number r, choose the greatest integer a_0 such that r-a_0>0. Then, pick the largest integer a_1 such that r-a_0-a_1/10>0. Pick next largest integer a_2 such that r-a_0-a_1/10-a_2/100>0, and so forth. This process gives us a well defined sequence {a_0,...,a_n,...} associated to the real number. In reverse, consider a sequence like this and consider the sum from i=0 to n of a_i/10^i. The limit of this sequence exists and is considered the real number determined by the decimal expansion. The number associated to this decimal expansion {a_0,a_1, a_2...} is sometimes notated as a_0.a_1a_2... where the sequence is understood before hand. So, play this number with the sequence {0,9,9,9,...} We call the real number associated 0.999... as a matter of notation, and by the definition, it evaluates to 1.
 >> Anonymous Tue Aug 25 11:50:03 2020 No.12047028 >>120470121000x=900900.900...x=900.900...999x=900000x=900000/999=100000/111
 >> Anonymous Tue Aug 25 11:50:40 2020 No.12047031 >>12046960something something 4 second attention span
 >> Anonymous Tue Aug 25 11:52:05 2020 No.12047037 >>12047031something something liquid shit
 >> Anonymous Tue Aug 25 11:56:59 2020 No.12047057 >>12047037na man this is spaghetti-o shit
 >> Anonymous Tue Aug 25 12:28:50 2020 No.12047170 >>12046928What about QM? I'm not good at QM, but doesn't it rest on fundamental laws?
 >> Anonymous Tue Aug 25 13:03:10 2020 No.12047305 >>12047170I mean, there are sometimes 'postulates' of quantum mechanics, but I am emphasizing that these are not so hard and fixed-they serve as physical guidelines. Generally, the postulates of QM are something like each quantum state is associated with a vector [actually a ray] in hilbert space, each observable is associated with a hermitian operator with the state of a particle of measured property value a as being an eigenvector of that operator with eigenscalar a, the measurement of an observable collapses it onto an eigenstate, the existence of a probability function of sorts on a quantum state and observable, and the time-dependent Schrodinger equation or some variations of equivalence. In fact, see Dirac-von Neumann axioms for probably the closest rigorization of this I've seen [though I haven't actually studied it this far]But these axioms don't serve to actually elucidate what is physically going on. That connection is entirely unmathematical and based on physical insight.You can't use this to actually 'show' that momentum operator in position space is iħ∇ for instance [or better said, perhaps, this is an arbitrary example of an operator in the formalization-the physical meaning isn't granted by the axioms] Or, for perhaps an even stronger example, of spin. You could imagine QM without spin, but as it happens spin exists and so we introduce it based on physical insight. Trust me, as a mathematically oriented person, I like seeing more and more detailed descriptions of physics. I've seen classical mechanics formalized in quite an interesting way by arnold in his book, quantum mechanics in the above mentioned way-there was a book on relativity by Reichenbach making an axiomatization of relativity. I got interested in this as 'einstein synchronization' isn't the only method [see Reichenbach synchronization] and all of this is fine but I really don't think much mind is payed attention to how rigorous our axiomatization of physics is.
 >> Anonymous Tue Aug 25 13:09:47 2020 No.12047333 >>12044835Which is bigger, 0/1, or 0/2?
 >> Anonymous Tue Aug 25 13:11:46 2020 No.12047338
 >> Anonymous Tue Aug 25 13:17:45 2020 No.12047361 >>12047338so uhm yeah fun fact physics requires talking about physics
 >> Anonymous Tue Aug 25 13:20:41 2020 No.12047373 >>12047361[HAND WAVING INTENSIFIES]
 >> Anonymous Tue Aug 25 13:21:19 2020 No.12047374 >>12047373just another day in the works of a physicist
 >> Anonymous Tue Aug 25 13:21:55 2020 No.12047377 >>12047305Ok, so there's no true axiomatic understanding of physics, rather pseudo axiomatic could be a better term I guess. >>12047305>I really don't think much mind is payed attention to how rigorous our axiomatization of physics is.Do you think physics could ever be truly axiomatized? Or is the nature of physics itself such that there may not exist true axioms upon which GR, SR, CM and QM are built?
 >> Anonymous Tue Aug 25 13:26:33 2020 No.12047396 >>12045814The limit is 1. The set however, doesn't include 1.
 >> Anonymous Tue Aug 25 13:29:47 2020 No.12047408 >>12047396But the notation is defined to be the limit [see what I wrote about the decimal expansion {a_0, a_1,...}.
 >> Anonymous Tue Aug 25 13:31:49 2020 No.12047416 Go post this on reddit or something, you are not telling/informing shit in here
 >> Anonymous Tue Aug 25 13:58:34 2020 No.12047527 >>12047377The core issue is that the connections established between abstract math and the physical world isn't really one suitable to formalization. For instance, if I want to say F=ma, I would first need to establish a primitive concept of 'force' and 'mass' associated to a given particle and then a mapping of 'force' to a vector, mass to a real number, as well as space and time to define acceleration. Then, our axioms would be this correspondence in addition to F=ma. Establishing such abstract connections based on very real physical things under hot philosophical and scientific debate, based on the inductive methods of science rather then 'axiom based' is why I view this as being not being done. That being said, there is an interesting paper I saw that formalized the principle of relativity by way of set theory and 'results' of an experiment, I think this was it: https://www.researchgate.net/publication/51937070_Formal_statement_of_the_special_principle_of_relativity [published in Springer] I'm sure it isn't impossible to do more things like this but if I am working in physics, it's kind of more like 'what's the point'. In physics, all our formal models don't matter if they fail experimentation. We could have made the most robust version of Newtonian mechanics by axioms and it wouldn't account for the most basics of modern physics discovered. Not even electromagnetism.
 >> Anonymous Tue Aug 25 14:35:39 2020 No.12047704 >>12045479other than that we can make 1=2?
 >> Anonymous Tue Aug 25 15:09:20 2020 No.12047856 >>12046015OK, but how would you define the limit of the sequence in >>12045814 in the hyperreals*? It definitely isn't 1, by the argument in >>12045675*i.e. $f(0.9),$ $f(0.99),$ $f(0.999),$ and so forth, where $f$ is the embedding.
 >> Anonymous Tue Aug 25 15:27:58 2020 No.12047941 >>120455281/9>0.111...
 >> Anonymous Tue Aug 25 15:37:58 2020 No.12047984 If you claim that the infinite sequence of numbers makes sense, then you already postulate that infinity is possible. Then you shouldn't complain about infinitesimals.
 >> Anonymous Tue Aug 25 15:44:41 2020 No.12048018 >he doesn't knowkek
 >> Anonymous Tue Aug 25 16:40:59 2020 No.12048228 >>12047984"real numbers don't include infinitesimals" is not complaining, it's a fact
 >> Anonymous Tue Aug 25 17:03:32 2020 No.12048325 >>12047984>then you already postulate that infinity is possiblethis is unrelated to the stuff inside of infinite sets, the real numbers have a definition that doesnt include infinitesimals>>12047856it is oneyou still arent realizing the difference between the kinds of sequences hereyou are taking a sequence of hyperreals and limiting to another hyperrealyou arent constructing a hyperreal as a sequence of realsyoure taking .9 + 0/w.99 + 0/w.999 + 0/w...at the end of the limit, the infinitesimal part is still 0with that being said, you technically cant even define limits in the hyperreals, since they dont form a metricbut the obvious way, taking limits on real parts and infinitesimal parts separately would be one of the better waysbut even easier is just noticing that f(.9) are all real numbers themselves, so we can just use the transfer principle
 >> Anonymous Tue Aug 25 17:31:15 2020 No.12048479 >>12047527thank you anon
 >> Anonymous Tue Aug 25 17:37:27 2020 No.12048504 >>12048228Why should we prefer your "real numbers" to numbers containing infinitesimals? Are you claiming that they describe the actual universe better?
 >> Anonymous Tue Aug 25 17:44:46 2020 No.12048527 >>12048504no, why would you think that
 >> Anonymous Tue Aug 25 17:47:16 2020 No.12048544 File: 58 KB, 704x414, hyperreals.png [View same] [iqdb] [saucenao] [google] [report] >>12048325I think I've found out why we are at loggerheads; see pic related (yeah I know it's r*ddit, but the people on r/math know their stuff). The limit of the sequence I defined is not 1, because it doesn't have a limit at all. We are each talking about different sequences and thus we are both right about our own points. Neither of us is wrong (about our own sequences), but neither of us is 100% correct.
 >> Anonymous Tue Aug 25 18:02:21 2020 No.12048605 >>12048504I mean, literally every physics book I have ever seen uses calculus and real numbers so... yes.
 >> Anonymous Tue Aug 25 18:05:44 2020 No.12048617 >>12048544[Different person giving my though] Personally, I think hyperreals are above most people to even argue about. 0.999... is quite strictly terminology used in the reals, assigned to a real. This real is embedded in hyperreals. Only if one wants to define a different hyperreal by the notation 0.999... [in the context of .9,.99 and so forth being considered in the hyperreals and reals and considering limiting questions that why] can one get a different answer, but this requires a higher level of tech. Personally, all I've actually seen about the hyperreals are in their use as real closed fields in mathematical logic. The terminology is not well defined for hyperreals is the ultimate answer.
 >> Anonymous Tue Aug 25 18:14:43 2020 No.12048658 File: 120 KB, 400x333, 1584418870693.png [View same] [iqdb] [saucenao] [google] [report] >>12048617>The terminology is not well defined for hyperreals is the ultimate answerYeah $...$ is a vague and imprecise way of representing a hyperreal number. A massive chunk of this thread is a debate which arose from 2 different interpretations of what those dots mean.
 >> Anonymous Tue Aug 25 19:01:34 2020 No.12048831 >>12048605Will the physics in it break if there will be numbers between 0.(9) and 1?
 >> Anonymous Tue Aug 25 19:12:35 2020 No.12048873 >>12048831Well, math itself would break, so let's say yes.
 >> Anonymous Tue Aug 25 19:55:21 2020 No.12049009 >>12048325You don't need a metric in order to define a limit.
 >> Anonymous Tue Aug 25 20:02:08 2020 No.12049025 >>12049009this comment was made by topology gang
 >> Anonymous Tue Aug 25 22:30:08 2020 No.12049485 >>12045973How is bottom-left a proof by induction?
 >> Anonymous Tue Aug 25 22:47:30 2020 No.12049555 >>12044835op is retarded
 >> Anonymous Tue Aug 25 23:37:20 2020 No.12049694 >>120448352 + 2.999... = 5
 >> Anonymous Tue Aug 25 23:42:20 2020 No.12049705 >>120459733*a^n+3*b^n=3*c^n has no natural solutions for natural n>2.Therefore a^n+b^n=c^n has no natural solutions for natural n>2.Here, I proved Fermat's theorem in two lines! What was all the fuss about?
 >> Anonymous Wed Aug 26 01:33:30 2020 No.12049987 File: 31 KB, 665x624, 10400000.png [View same] [iqdb] [saucenao] [google] [report] Cirno already proved this
 >> Anonymous Wed Aug 26 01:37:37 2020 No.12049997 >>12049705this is the 4th time youve posted this, and its still dumb as shitif youve proven the top line, then yes, the second line follows, theres no issue
 >> Anonymous Wed Aug 26 01:41:05 2020 No.12050010 >>12049705That's why there is the box on the right for the truly skeptical bastardfucker that demands a rigorous proof. The rest are more like persuasive arguments.
 >> Anonymous Wed Aug 26 10:16:50 2020 No.12050967 File: 18 KB, 330x165, swayello.png [View same] [iqdb] [saucenao] [google] [report] >>12049987Cool. I don't think I've seen the proof of the formula for the sum to $\infty$ of a geometric series before.
 >> Anonymous Wed Aug 26 10:19:40 2020 No.12050973 >>12049997>if youve proven the top line, then yes, the second line follows, theres no issueBingo. Now you can see why your 0.333... proof is as silly.
 >> Anonymous Wed Aug 26 10:29:05 2020 No.12050990 >>12050010Where do your infinite series and convergence definitions introduce infinitesimals?
 >> This video had me crying Wed Aug 26 11:31:18 2020 No.12051142 >>120479410.3 written in binary is 0.0100110011001...Are you going to say that 0.3 != 3/10?3/10, 0.3, and [0.010011001...]base2 are just different ways of representing the same number.Same is true for 1.9/9, 1, 0.999... Are all the same thing written in different ways.
 >> Anonymous Wed Aug 26 11:33:00 2020 No.12051146 >>120511420.3000... or 0.2999...?
 >> This video had me crying Wed Aug 26 11:40:28 2020 No.12051152 >>12051146[0.3000..]base10 = [0.2999...]base10 = [0.010011001...]base2
 >> Anonymous Wed Aug 26 11:43:57 2020 No.12051169 >>12051152[0.3000..]base10 = [0.2999...]base10 + [0.0000...1]base10
 >> This video had me crying Wed Aug 26 12:03:14 2020 No.12051224 >>12051169>[0.000...1]base10What .00...1??? There is no 1 at the end of infinite series of 0s. If there was, then it wouldn't be infinite. This isn't 0 + epsilon. 0 + epsilon is a number, but it is distinctly different from 0. Similarly, 1 - epsilon is a real number, but it is distinctly different from 1.One of the properties of the real numbers is that if you have two number, x and y, then you can always find another unique number between them. Put simply,x < (x+y)/2 < y for all real numbers where x, y are unique elements of the real numbers.This holds for (1 - epsilon).(1 - epsilon) < ((1 - epsilon) + 1)/2 < 1This doesn't hold for 1 and 0.999...If it did, what number would be in between them?Therefore, 1 and 0.999... must be the same number.
 >> Anonymous Wed Aug 26 12:53:35 2020 No.12051374 >>12044835>No number other than 1 is equal to 1..999... is not a number is a mathematical expression.Are so saying cos(0) = 1 isn't valid?
 >> Anonymous Wed Aug 26 13:29:45 2020 No.12051479 >>120511420.9...=0!!!
 >> Anonymous Wed Aug 26 16:01:15 2020 No.12052014 >>12050990They don't.
 >> This video had me crying Thu Aug 27 00:45:08 2020 No.12053301 >>12051479Well, yeah. 0! = 1 = 0.999...
 >> Anonymous Thu Aug 27 01:22:57 2020 No.12053375 >>12050973you actually arguing against 1/3 = .333... ? you are retard, correct?
 >> Anonymous Thu Aug 27 17:14:45 2020 No.12055655 >>12044835You're right.0.99999... doesn't exist.0.33333... doesn't exist.It's a flaw in the decimal system.We roll with it.
 >> Anonymous Thu Aug 27 17:36:42 2020 No.12055714 File: 301 KB, 172x172, 117015-full.gif [View same] [iqdb] [saucenao] [google] [report] >>12055655>It's a flaw in the decimal systemThere's nothing special about base 10 (ten) in this regard. The same phenomenon occurs in other bases.
 >> Anonymous Thu Aug 27 17:42:12 2020 No.12055732 >>12055714>The same phenomenon occurs in other bases.Not all of them, people usually argue that 12 is objectively the best base to use in but it never stuck because of some bullshit like "we have 10 fingers doe".
 >> Anonymous Thu Aug 27 17:48:02 2020 No.12055751 >>12055732>1x2x3x4>What about x5? You just found out why we have 60 minute hours.
 >> Anonymous Thu Aug 27 17:48:53 2020 No.12055755 >>12055732In base 12 you have $1=0.BBBBB...$
 >> Anonymous Thu Aug 27 17:52:00 2020 No.12055763 >>12053375Is your worldview shattered now?
 >> Anonymous Thu Aug 27 19:32:45 2020 No.12056035 >>12055763no, i already knew morons existed before i met you
>>