As a mathematician, I see friends comment on these things and I'm compelled to go in and explain how these are ill posed and not how we'd write it, and give the actual order of operations. And there's always someone who insists on inventing new rules (e.g. apparently implicit multiplication is stronger than regular multiplication, but I'm sorry, 1/2x=0.5x).
@@spacelemThe implicit multiplication distinction is taught in other countries like India. Positif is used to mean nonnegative in French. In many parts of the world, ]a, b[ is used to denote open intervals.
@@spacelem OK so I'm reasonably well versed in math, but man, when I see "1/2x" my brain sees "1/(2x)" instead of "(1/2)x". I think it's just because the lowercase 'x' is smaller so it looks like how you would write it by hand if you were writing it with 2x as the denominator. I'm with the other guy, I overuse parentheses just to make sure there's no ambiguity.
@@hunted4blood the thing is, I would never type that without the parentheses. If I was writing it with pen and paper, I'd give some hint like writing the 2x a bit closer together and lower (like a very lazy attempt at a full fraction). I'm aware that in some programming languages (e.g. Julia) it will interpret 1/2x as 1/(2x), but... well that's just something you've got to learn, but at least it's unambiguous in the language.
Heavily agreed, it's clickbait math for non-mathematicians. It's a shame because it distracts from REAL and interesting mathematics. It's such a vast and beautiful subject.
Not even non-mathematicians, I’m an Industrial Engineer (basically a business admin degree in my country) and it’s enough to make these internet math problems infuriating
It is not exactly a math problem, but is is a serious problem. Just never codifying the rules to an extent that you can easily and unambiguously write an equation with a keyboard or a calculator is a very serious problem that hurts mathematics because it is pretty important to be able to communicate math and adding unnecessary friction to this is pretty dumb particularly since every other case was handled by order of operations and we just never agreed on this little edge case.
@@wisnoskij Except, you can easily. Just don't write 1/2x, just as you wouldn't write "we saw her duck" without any context if you wanted to be unambiguous. Try 1/(2x) or (1/2)x instead. Boom! Perfectly unambiguous. Not any string of characters has to have an unambiguous meaning to be able to find a string that unambiguously describes the meaning you want.
@@KaiHenningsenI mean, in my math classes my teachers would give us equations like. “y=-x^2” And then say the answer it is -(x^2) because its ACTUALLY -1*x. Ambiguous stuff like that is everywhere sadly, even if it technically makes sense.
@@TJ-hg6op To be fair it's just the conventional way to write that kind of thing (at least in highschool / engineering math), and your teachers should have explained that convention. Anyway get used to it because it's gonna keep happenning, it's also not the only convention they'll forget to teach sadly
for the two people directly above me you are unfortunately dumb and stupid because the real issue that infinite singles would cause is that the cashier will, and probably has to, count them all individually to make sure you're paying in full
The misuse of "Some infinities are bigger than others" combined with a link to the Vsauce video "How to count past infinity" makes my blood boil. If they had actually watched it, they would have been able to determine that those infinities are the same size.
This is the case for Surreal numbers though! There are literally bigger infinities. Also there any algebraic expression involving infinities, for example omega + 1 or omega squared.
Ah I edited the comment giving more explanation, but TH-cam is horribly written so it didn't go through. Basically Surreal numbers are formed from certain positions in two-player perfect information games(such as Chess). You can define the sum of games(positions) to be a new game where players move in either of the two components. You can define the subtraction of two games A - B to be A + (-B), where -B is B but with the moves for each player swapped. If A - B = 0, that is, the player to move to this position has a winning strategy, then the games are said to have equal value. If A - B has winning strategy for the first player, then A > B. If A - B < 0 then the second player has a winning strategy from that position. With this you get all of Cantor's numbers plus any algebraic expression involving them, for example, 1/omega, omega - 1, or sqrt(omega). Also, you get like numbers where adding them is computed by writing each number in binary and computing the bitwise xor of them, and numbers which act like clouds of different numbers depending on the context. Surreal numbers are only regular numbers + infinities and infinitesimals, and I think they form a really amazing basis for analysis that I wish was used more, but the other numbers, generally called Games, capitalized because they form a proper class if you know what that is, are simply amazing and wonderfully alien and I wish everybody would read about them.
Ah I edited the comment giving more explanation, but TH-cam is horribly written so it didn't go through. Also I tried sending this, but again TH-cam is horribly written so it didn't go through. Basically Surreal numbers are formed from certain positions in two-player perfect information games(such as Chess). You can define the sum of games(positions) to be a new game where players move in either of the two components. You can define the subtraction of two games A - B to be A + (-B), where -B is B but with the moves for each player swapped. If A - B = 0, that is, the player to move to this position has a winning strategy, then the games are said to have equal value. If A - B has winning strategy for the first player, then A > B. If A - B < 0 then the second player has a winning strategy from that position. With this you get all of Cantor's numbers plus any algebraic expression involving them, for example, 1/omega, omega - 1, or sqrt(omega). Also, you get like numbers where adding them is computed by writing each number in binary and computing the bitwise xor of them, and numbers which act like clouds of different numbers depending on the context. Surreal numbers are only regular numbers + infinities and infinitesimals, and I think they form a really amazing basis for analysis that I wish was used more, but the other numbers, generally called Games, capitalized because they form a proper class if you know what that is, are simply amazing and wonderfully alien and I wish everybody would read about them.
dog do you know how algorithm works? it just looks at users who have similar content preferences and recommends the same thing they clicked on/looked up. there's no sentient being providing you with niche videos
@@opal_2476 I don't know anything about how the algorithm works.However,I have noticed that up until now,most of the videos that appeared in my recommendations were from creators I already watched/kinda popular channels.I've been getting a lot of videos in my recommendations that are from new and obscure channels lately.Looks like I'm not the only one to notice that change
What I hate about the PEMDAS ones is that it's always the same trick and people seem to just forget that they've had the same exact conversation on repeat every year for the past decade. How are you not bored of this BS?
@@Spellweaver5 You would think so, but I'd also think that someone in these conversations would go "here's a video by [insert popular maths youtuber] from when this happened before explaining it." But no. No one ever mentions that they've done this before in these conversations. It seems ridiculous that it's *all* new people and no one remembers the previous times. So now I don't know what to think.
It's because people only know this topic in math and they are pretty comfy arguing like a math enthusiast. Being human is hard you always have to satisfy that you are educated and convince your brain you are the greatest
@@jamesdurtka2709in fairness, school exams don’t help. In a 5 mark question, you’re guaranteed nearly two marks for writing anything, wrong or right. Ofc, if it’s right you’ll get more. But still.
@@therealgeneralMacArthuri took AP chem this year and when my teacher was explaining acids and bases it was so funny to me. He was basically like, okay so because this is a weak acid, the equilibrium constant is very low. So to calculate the pH, we're just gonna assume x is "close enough" to zero. I was so baffled i thought he was joking at first. I will be going into mechanical engineering in 2 years. Based on what ive heard, i imagine im in for some much funnier approximations.
What always annoys me about that the 0.999...=1 thing is that every time it's brought up you'll get a bunch of people who don't know anything about real or functional analysis trying to "um actually" and "debunk" it. The sheer arrogance to assume you know more than professionals who have dedicated years or decades of their lives to the subject is infuriating
The root of the disconnect, IMO, is that mathematicians understand that maths is a bunch of rules we decide upon (with different rules for different areas of maths, called axioms), and we then see where they lead us. But if your maths education stopped at the end of high school, you'll have probably never seen this and so maths for you isn't about deduction and proofs but memorising times tables and trig identities, so you'll think that all maths is is a collection of titbits of pure knowledge that are true in-and-of-themselves. To the lay person, mathematics is theology, while it's actually playing with lego where each brick is an abstract idea (the analogy works particularly well because so much of maths involves building higher-order objects from simpler ones). And bad internet maths leans into this because it's so much easier to present knowledge from up high without having to go into the thinking behind it, especially when it's what the audience except.
I really think calculus is an important educational step here. Most of math even on a grade school level seems like BS until you understand it, from zero to fractions to irrational numbers to infinitesimals. Forcing people to realize that this staircase of apparent absurdities keeps going higher and higher would probably do a lot to cull this problem. Then again, half of the complaints about how "modern math is useless nonsense" comes from retired engineers, so maybe not...
@@jacksonsmith2955engineers are probably the single field in where the maths are ingrained into their work but they fail to understand why. All they do is approximate everything for their work and hope to god whatever they build doesnt blow up in their labor worker's face.
@@macchiato_1881 that's the most inaccurate depiction of engineering I've ever heard, every engineering course sets boundaries as to what approximations can be made and where and when use a certain model vs another and there is a lot of pure math, it's like 50% of the classes
@@jacksonsmith2955 maths never seemed like nonsense to me. But I was made to prove theorems from 5th grade onwards. That said, I often noticed that I started truly understanding the previous topic only once I started learning the following one. Might be the reason I'm never truly going to understand functional analysis.
Perfectly stated, like there's probably some branch of math where 0.999... can be treated as a distinct entity from 1 and that's fine, you're just operating in a different context with different rules. The statement that 0.999... = 1 is simply commentary on a particular set of rules we've chosen (and commonly use) but if you aren't aware of the way these systems are constructed I guess you'd just have to take it on faith (or, alternatively, argue about it - because written down on paper those two things do NOT look the same)
Unfortunately, the popularity of the phrase "some infinities are bigger than other infinities" was really increased by the book "The Fault in Our Stars". I say unfortunately, because the character saying it is a 16 year old girl who doesn't actually understand the math. But it was a quick, catchy, and incorrect explanation so it's caught on on the internet.
The ordinals take on a natural order (obviously) and measure (less obviously) it just isn't exactly the same as that for strictly cardinal numbers. But there are absolutely infinities of different sizes. For example (technically a proof), when handling ordinal addition, we have that right addition is monotone increasing : α + β = 𝜏 => 𝜏 > α there exists a well-ordered measure on the ordinals. None of those steps are really necessary, as ordinals are definitionally measured, but it helps to illustrate that they are, since their distinction from cardinal numbers can obfuscate that sense of size. There's also the plain case of countable vs uncountable infinities which do have strictly different cardinalities without thinking about ordinals, but that's a lot more vague. I read the book like a decade ago now, though, so I don't know what the explanation given in the book is. It could absolutely be incorrect.
@@tomekk.1889 Well The Fault in Our Stars came out 4 years before the VSauce video but I totally agree, I think that the VSauce video is much more popular and that's definitely where I heard about different sized infinities for the first time. i.e. depends on who you ask
THANK YOU the bedmas ones are so annoying cuz like sometimes they’ll word it or explain the equation awfully and then people get mad if u get it wrong like bro
I appreciate the fact that math has become more accessible lately, but sometimes people forget that you don't actually know a topic after watching a vsauce video (despite it being a good way to fuel curiosity)
Is this about the multiple infinities video? I thought he explained that pretty well, he basically said the exact same thing that was said in this video. What is the part that isn’t accurately conveyed in said video?
The problem with the Vsauce video is that Michael himself demonstrates that an infinite set containing all integers and all even numbers is the same. These people just read the title and didn't watch it.
I still think the worst clickbait is math videos where the title card just states something wrong. And then the video explains that "we were using non-euclidean geometry all along", or whatever excuse they made up.
It's true - the whole point of "you get the right answer because of Bodmas" which is what these things are, is that bodmas is an agreed convention that people use to tie-break ambiguous notation (I mean, sure, it's also the case that you use conventions so that you don't have to draw brackets around things all the time once you establish what kind of things you want to talk about) But the point with conventions is that they are not anything fundamental about the maths - it's like whether people drive on the right or the left is a convention, it's not something fundamental to the nature of the car that driving on one side or the other is better - but it is important that everyone in the same country understands the same convention so you don't crash into oncoming traffic.
I'm always disappointed there aren't more computer scientists in these discussions, since they're the ones who actually care about mathematical formulations of BODMAS/PEMDAS/etc.
This, but i wouldn't even call them conventions. I would call them mnemonics. Their point is to help you _memorize_ the rules (without having to understand them...), but they are not the rules themselves. Apparently that detail is not included in the curriculum since so many people try to interpret them as if they were law.
Dumbest thing is, people treat it like everyone's taught the same BODMAS/PEMDAS but then there's differences internationally where in America (I hear) they prioritize division over multiplication and addition over subtraction, and here in Australia we do division and multiplication at the same time as we step through the function seeing as they're inverse operations of each other, and the same goes for addition and subtraction. It just gets people arguing like everyone who doesn't get one conclusion is stupid without even thinking why they'd get there.
@@jacksonsmith2955 yeah, we know it depends on what order of operations you use instead of there being innate truth and only one solution. It's pretty sad how order of operations is taught as something that makes sure people interperet the same equation the same way for real applications that need consistency, but people aren't all taught the same order of operations... it just makes people fight over who was taught correctly. It's like making seed-based RNG but giving everyone different ways the seeds get converted into random numbers, so the seeds become meaningless except to give the same person the same result every time.
@@volbla BODMAS/PEMDAS are mnemonics to help you memorize rules, the rules themselves are conventions. Doing multiplication before addition isn't some inherent property of math, we just made it up because it made writing some equations easier.
I think the fundamental friction for both the gotcha equations and 0.999... = 1 is that in some sense people think that the notation system _is_ the maths, rather than merely a way of representing it. People get excited about these counterintuitive equations as if it's a demonstration of mastery of mathematics, when in fact there's nothing of mathematical interest in such garden path constructions; the mathematics is the object of the communication, not the communication itself. If the communication doesn't effectively convey that meaning, it's just bad communication. Likewise, if the notation _is_ the maths, 0.999... = 1 is disturbing, because something is two things at once. But these "things" are just names. Getting hung up on it is a bit like my nephew - when he was maybe 4 or 5 - getting hung up on one of his toys being both a T-rex _and_ a dinosaur. I do think it can be helpful to clarify that 0.999... isn't a special case: every finite decimal representation has a counterpart that ends with an infinite sequence of 9s (and likewise for other bases, of course). And you can prepend and append zeros to your heart's content, which doesn't seem to bother people.
as a programmer, an interesting way I've seen the 0.999... = 1 thing interpreted is that it can be thought of that both 0.(9) and 1 are expressions that evaluate to the same value, nobody finds 3-2=1 weird, despite it essentially being the same thing which relates to what you said about the difference between notation and mathematical truth
@@uhrguhrguhrgit’s more amusing as a programmer that floating point error destroys the rigor of almost any statement involving rational numbers, depending on the language and processor you’re using. 3.0 - 2.0 often DOESN’T equal 1.0. There’s a hilarious series of rather cursed examples of Boolean evaluations in JavaScript because of the ways it tries to rectify this (and just ends up making things more absurd).
@@willmungas8964 A more apt example would have been 0.3 - 0.2. Small integers can be exactly represented in floating-point, but decimals cause trouble because we use base ten, which has a messy factor of five in it that computers can't easily divide by.
I really like the way you gave the diagonalization argument as I find that most explanations out there say to "take the digit at that position and add 1" and maybe also "if it happens to be a 9 [bla bla bla I don't want to bore you to death]", I find that it distracts from the actual point of the proof which is that the number you're generating is DIFFERENT in at least one position, thus it cannot have been included in the list.
I may be stupid, but doesn't the proof of rational numbers having the same cardinality of the natural numbers shown in this video NOT match every rational number 1 to 1 to a natural? Because each of the boxes after the 1st definitely contains more than one number Is that not the rigorous proof, or is proving that reals can't be matched one to one with naturals not enough to prove they have a different cardinality (?? Edit: I just googled it and they all use methods to match every rational 1-to-1 to naturals so yea lmao this one probably wasn't the best proof
@@CraftIPeach of the boxes after the one contain more than one number but each box always contains a finite amount of numbers therefore you can match them 1 to 1 with the naturals
@@goshawk6153 it wasn't immediate to me how one implies the other, but yea it definitely makes it so you can match each finite amount incrementally, but that was one more step than my mind made LOL
What I like about “some infinities are bigger than others” is the rich history behind it. The concept of infinity caused a feud in mathematics. Not everybody accepted Cantor’s ideas. Gödel used the diagonal argument to prove his incompleteness theorems. Effectively Gödel created the first compiler in his proof. Turing created the Turing machines to use diagonal argument for the halting problem. So “some infinities are bigger than others” was a big part of the foundation of theoretical computer science. It’s a very important piece in the history of computation. But yeah, this is not what people mean when they bring this up.
Yeah, logical theory, computation, and mathematics (proof theory, type theory, and category theory) are all very densely connected, it's honestly a sad education if you never get to learn these connections
I think in general people mean "some infinities are bigger than others" in exactly the same sense, they just don't necessarily have the right intuition as to why.
You know that are an infinity amount of integer numbers, but you also know that every number can be divided by any other number, so, you have infinite integers and infinity non-integers, but one has more numbers.
But the problem is that you can construct a bijection from the integers to the rationals, so they must be the same size. So even though we have this property where any number can be divided by any number (that isn't zero), we don't necessarily get a larger set. So it has to be something else that makes the set of real numbers have a higher cardinality than the integers.
Fellow math nerd here, mathematics seems to be uniquely misunderstood as a discipline. Much of the work I spend teaching college algebra is in breaking down the student’s stereotypes on what math actually is. Something that still surprises me is how deep the disconnect is between scholars of mathematics and everyone else (particularly foundations of mathematics ie logicians and everyone else). It is fascinating to me how much people wish to imagine mathematics working a certain way versus how it actually works.
everything abt this video is so great. you have such a nice voice and a clear ability to explain (what seem like to me, a non-mathhead) complex concepts. and the animations!! i adore them! great work man. i hope you keep making videos explaining things youre passionate about!
Excellent video! A big part of what makes me sad about these things is that a lot of these topics are interesting, but the Internet just loves turning them into little fun facts or, worse, pissing contests. Take the 4/2(3-1) example. That is a great way to introduce the topics of parsing algorithms, whether they are unique or not, and so on. That's a genuinely interesting subject to talk about! But, no, the Internet is more interested in trying to one-up the other person.
Yes, I have tried to explain the problem using parsing before as well. Thinking about how grammar is a choice made by humans (and arises by natural languages processes), even in a mathematical context, is really important too.
Honestly, it doesn't really matter what the answer to that is, nor does it really matter that it's math. The issue is that people don't care about accuracy and unambiguity because a lot of people scuff at you when you point out the incorrect formulation of the question. It's a lot like grammar honestly. A lot of people out there mix up words like "you're" and "your", or "their" and "they're", and so on, the list is essentially endless. However, even if you try to correct them in as nice of a manner as you can, a lot of people will react with "okay whatever smartass nobody cares" attitude. The same really goes for this; people just don't care about being correct, they just want to be part of a larger group. The lack of knowledge and care to acquire knowledge is honestly one of the most dangerous things to society I believe. I'm not saying that everyone needs to have a PhD in mathematics, but this type of question shouldn't really be a problem. People are so focused on finding the answer to the question that they never stop and think about why people arrive at different answers in the first place. They just don't care about knowing it. And it's really sad because if people just cared about learning things, and asking questions about things, then I genuinely believe that society would be so much better rather than being stuck in this echo-chambery-styled society that we have going on right now.
Yeah for real. I always hated that, and I especially hate how people think the division sign (the minus sign with a dot above and below) is actually ever used in any serious field (engineering, for example, the one I have experience in). That sign is always too ambiguous, and you could always come up with several different "correct" answers. No engineer uses notations like that. You could always add more parentheses or vertical lines to something like that to make it impossible to misinterpret. The only people that unironically use that division sign are 3rd graders learning that sign (for some reason?) and internet math guys who don't actually do any math outside of 15+12=27. And the shocking thing is, those people are always the most confident in their answer because "you do multiplication and division from left to right" or some random standard notation that isn't really used in the real world.
I actually no longer have any memories of using the division sign in grade school, but I am certain we used it. This must mean that once we students understood what division was, they immediately taught us better notation. And clearly it was better, since I evidently never ever ever used the division sign again and just forgot about it.
I paused to comment around the 11 minute mark and almost restated the entire conclusion. For lots of technical subjects the interesting facts that hook people in are oversimplified or of minimal importance to the field as a whole (eg. the idea of a five sided square got me interested in topology). Sometimes curiosity reaches a bit beyond what we're ready for, and the best solution is the one keeping us interested enough to learn it's wrong.
love this video, explains everything I've hated about people who don't know much about math trying to sound smart or saying that I am wrong about things I spend my time looking into because I find it interesting.
YES omg the number of times I've seen that fractions thing pop up on social media again and again only for people to either insist on a particular solution as the only correct solution, or otherwise go down the route of smugly declaring that math isn't that accurate after all. Like dawg just throw away your device then since it's math that lets you talk shit on socials 😂 it's infuriating how "use your damn parentheses" is somehow always the least popular answer :/
I think the big problem with 0.999...=1 is more so a misunderstanding of what the real numbers are. The way i would think of the real numbers is: N is axiomatically a thing Z is N with subtraction Q is Z with division R is Q with suprimum and infimum but that's not how most people think of these. numbers are just numbers, you write them down and they equal other things. so, you learn at school that 1/4=0.25 why? because 2/10+5/100=1/4 this dosen't work for all fractions. take 1/3 this dosen't work. however, the infinite sum 3/10+3/100+... does equal 1/3. so this fact is hidden from you and you just learn that 1/3=0.3333... so, numbers can be infinitely long. Now, look at 1-0.9999... this equals 0.0000...1 which is not 0! so they're not equal! of course, you and I know that 0.0000...1 notation just is meaningless, but most people just don't know what this is (to add to this, in the hyperreal numbers, a field extension of R, there *is* a number smaller than all positive fractions yet larger than 0). my favourite way to show it isn't a thing is to tell people that ...99999=-1 add them digit by digit and you get 0! (the devil is in the details. three dots can lie) while this is true in some sense (p-adic) these are not the real numbers. This obviously like, dosen't matter, school shouldn't teach 9 year olds what a suprimum or what a cauchy series is, but i think this is the fundamental problem with lay people's understanding of math. they don't know what the rules of the game are, so they're sometimes cheated.
Aw man. You just made me remeber the moment from elementary shool when I was shown 0.(9) = 1 for the first time. Where Im from the "()" is the symbol of repeating digits btw. The equality didn't make sense to me. I quickly thought about 1 - 0.(9) and concluded that it must equal 0.(0)1. This made sense for me. I imagined infinity to be some point in the sequence. It's such a point that can never be reached - any finite number of steps would still be not far enough. But I still imagined that it is a fisical point somewhere infinitely far in the sequence. I don't know whether we are teaching what the infinity is in the wrong way or understanding it is a limit that needs to be broken at some point. Now I understand that infinity doesn't really exist. It would more accurately be called "unboundedness". For example let's take natural numbers. There really isn't an infinite number of them. In theory there are, of course, but in practise nearly all of them won't ever exist, be used. I can in principle write down any one natural number and there os no upper bound on how big it could be, but that's it. The natural numbers are only infinite in the sense that we can never name the largest one. A person can only name a finite number of numbers in any given time. It's fascinating to think that there exist the largest natural number that a person have imagined. It is probably massive. I can think of BB(BB(420)) for example. But there always exist Biggest Imagined Number and after it an "infinity" of useless numbers that have never been used
I think math education in schools should be split into calculating class and maths. Preferrably maths as an optional course in the final grades where you start from axioms and basic logic and work your way to some of the basics so that interested students actually get to know what maths is actually about.
I think the actual misunderstanding is of what a decimal representation is I think if you explain to someone what 0.333... = 1/3 actually means conceptually they maybe understand what's going on even not rigorously, you can explain it using the concept of a measurement, when you measure something with a ruler, you're doing a "ok, how many units does this fit into? after the maximum, how many 1/10 units? after that, how many 1/100?..." that's what the decimal representation is supposed to represent so there are two ways of measuring 1, you can just fit it in one unit or keep going to the second smallest unit you can each time
Just found you. Working on my master in mathematics, I must admit I found much of the explanations trivial. Yet your narration is oddly captivating, equally so is your style of animation. I love this! ... and the boss baby Jesus
I think your problem with math edutainment is still true in actual school education. Here in Hong Kong, finding sums of infinite geometric series is taught in high school, and we were never taught the convergence of limits. Either the formula is stated and required to be recited, or the popular "infinite terms cancelling each other" trick is used without considering convergence. In that sense, the reasoning in actual school education is also simplified. But I think there are times when reasoning and presentation do require simplifications and glossing over important details, in order to kick start an audience's understanding. We can't teach set theoretic construction of integers to 6 year olds, we just teach them basic arithmetic directly.
@@hadrienlondon4990 wait, really? how tf does that even make sense? you learn to compute them before you even understand what they are or how to derive them? that's absurd
@@jacksonsmith2955It's the same in Australia. You learn derivatives from first principles in a very hand-wavy "just consider as x goes to a" without really learning what a limit is. In first year uni, I learnt more about limits and convergence, but it was still a little hand-wavy. It wasn't until second year that the epsilon-delta formulation of convergence was ever taught and that's when limits actually became something properly defined.
@@echo.1209 okay, so you do talk about the concept of limits, just informally. i think i'm fine with that tbh, the intuition is probably more important than the theory for 95% of students anyways.
@@jacksonsmith2955 They tell us "uuuhhh you write the limit dont worry abt it" and then they say "see when theres no h at the denominator we can put h=0 to get the limit"
Also, 0.(9) may not even be a defined number. We only assume it is a real number because it acts kind of like an infinite monotone bounded sequence. It also depends on how the operator is defined.
From Wiki :"Certain procedures for constructing the decimal expansion of *_x_* will avoid the problem of trailing 9's. For instance, the following algorithmic procedure will give the *_standard_* decimal representation" (description follows).
7:00 The word 'infinity' and the term 'infinite set' are not synonymous. The former refers to particular points in topological spaces, while the latter refers to sets that are not finite.
You're right that "infinity" is used inappropriately in the phrase "some infinities are bigger than others" because what is really meant is "infinite sets." I'm not sure I've heard of referring to "infinity" as a point in a topological space. I would consider it more a concept or way to think about certain ideas. Rigorous definitions in mathematics never talk about points at "infinity." The definition of a limit as a variable tends towards infinity doesn't involve infinity at all. It involves talking about arbitrarily small values representing allowance for error and sufficiently large values representing how large the variable must be in order to ensure the function stays within that error.
@@bartholomewhalliburton9854 'I'm not sure I've heard of referring to "infinity" as a point in a topological space' Literally done all the time as early as in introductory courses and textbooks on calculus (although, at that point, the students are not yet aware of the fact that they are dealing with such spaces). When working with limits of sequences, you are already dealing with a subspace of Alexandrov compactification or a similar space. 'I would consider it more a concept or way to think about certain ideas' A limit of a sequence in some space is some point L of the relevant space, such that for every neighbourhood U(L) there exists a punctured neighbourhood of U(inf)\{inf} in Alexandrov compactification, such that for every natural n from U(inf)\{inf} the nth element of the sequence is in U(L). Similarly, when dealing with limits of functions, people very often deal with Alexandrov compactification or extended real line (i.e. the standard two-point compactification of the space of real numbers). Pretty sure that the projective compactification is taught almost immediately in complex analysis courses. This stuff is perfectly rigorous. 'Rigorous definitions in mathematics never talk about points at "infinity."' Yeah, they do. 'The definition of a limit as a variable tends towards infinity doesn't involve infinity at all' Hmm? Are you sure about that? How do you interpret the expression 'lim(1/x) as x->inf' when dealing with the definition of limits through filter bases? Because the expression 'x->inf' does not make sense unless you define what the point 'inf' is. 'It involves talking about arbitrarily small values representing allowance for error and sufficiently large values representing how large the variable must be in order to ensure the function stays within that error' This is what lacks rigour. This explanation of limits does not work in any sort of non-metric space, let alone in any non-metrisable space.
Seems like TH-cam hid my previous reply, so I will be brief this time. Google 'Alexandrov compactification', 'extended real line', 'Riemann sphere'. Your understanding of limits seems to be very shallow, and you seem to have not paid enough attention in introductory calculus classes, as students start rigorously working with such points very early in calculus - it's needed for limits of sequences.
Seems like TH-cam hid my previous reply, so I will be brief this time. So, here's the short of it: This stuff is, contrary to your claim, rigorous, and students start working with it within the first few weeks of their introduction to calculus.
@@thetaomegatheta Can you inform me what particular points in euclidean space (i.e. the space students work with in calculus) are "infinite"? You said "infinity" refers to particular points in a topological space. I'm just confused what you mean by this since I've never heard of this.
I like the visual proof for matching fractions to natural numbers. Draw all the fractions in a grid with the numerator on one axis and the denominator on the other. Count down the diagonal so start at coordinates 1,1 then 2,1 then 1,2 then 3,1 … skip all equivalently repeated fractions. This also proves that there are that same number of 1 dimensional and two dimensional spatial vectors assuming they can only contain natural numbers.
The other major thing people don’t get with infinities is that they aren’t going to intuitively make sense. People like saying .99…. = 1 is absurd and doesn’t make sense, when the premise of infinite 9s is already absurd and hard to grasp. The answer is confusing because the premise is equally confusing
As the video shows, there’s some context where it does make sense to say that some infinities are bigger than others. We just have to be clear on what we mean by “infinity” and “bigger”.
Weird statements like these are mainly thought of by mathematicians, as a way to showcase how irrational some of these numbers are. They aren't meant to be solved, they're kinda just like games.
nah bud internet math is just restating already discovered math in a simplified/more approachable manner. you don't really think 100% of these youtubers really invented what they said in their videos do you?
That was very graciously said, you have a wonderfully caring tone and I appreciate that you point out how the plight of disinterested people(not uninterested) is applicable in all subjects and topics. We should all be more discerning with what we learn and how we teach it while being thoughtful of what people might misunderstand without judging them harshly.
@@BlackLegVinesmokeSanji idk why, we take concepts that would likely never see the light of day and use them to create marvelous things, meanwhile compilers miscount 1 as .9999999 causing inaccurate results messing up everything we try and do
As someone who is just learning discrete math in university this video was super interesting and I still learned a lot. I've always hated the "some infinities are bigger than others" saying because it just felt wrong. This conceptually really helped me wrap my mind around infinity.
@@mms7146 Why do you think "denser" is a better word? I have heard a lot of people (but not professional mathematicians) suggest "denser" is a better word, but I have never understood why. So if you could explain it, I would really appreciate it!
@@MuffinsAPlentyI don't have an opinion re. bigger vs denser, but the reason denser might make more sense is because it doesn't feed into people's misconceptions about different sizes of infinity. For example, on the number line, you cannot reach a "bigger" infinity than that of the naturals just by going farther to the right than a smaller infinity. Instead you must, for example, take the entire interval between 0 and 1.
@@eel9 Thank you! I appreciate your answer! So, it seems to me that perhaps people think about the very specific example of naturals/integers/rationals vs reals, rather than thinking about cardinality as a general concept which applies to _any_ set, and where _any_ two sets can be "compared" cardinality-wise.
@@MuffinsAPlenty @eel9 gave quite a nice answer. It is more of an intuitive way of seeing it rather than a formal one, as an infinity cannot be *literally* bigger than another infinity. But one may be denser than the other as, for example, there is a greater number of infinities per unit of rational numbers for every unit of natural number. In any case, I have no formal background in mathematics, so you shouldn't take my explanation for granted.
You're really gifted with how you're able to talk in such an approachable and appealing way. Keep using this ability. I don't have a background in maths, but this video is entertaining.
I am in love with your style of animation, your voice and delivery. I've seen some videos before about 0.(9)=1, never with this simple yet reasonable explanation. I tend to use knowledge I learned from math videos when teaching kids/young students math and I like to throw them some more interesting math facts that require a little more skill to solve or to think about to maybe interest them in the subject. I really love what we may do with math and how everything in it is so connected, one to another. I hope I'll see more videos from you, having subscribed I certainly hope so. Also, wouldn't it be an enough explanation for not being able to compare the sizes of two infinities saying that they're just not numbers? As far as I know, infinity is not a number, so therefore we shouldn't be able to tell if it's smaller/larger. Let me know if I think logically!
yup, we cant compare the sizes (in the usual sense of the word) of infinities so that why we use cardinality which is a different though analogous concept. So when people say "some infinities are bigger than others" they are usually referring to cardinality but are often using it wrong
Infinities are numbers though, at least in some precise sense. I think the problem arises from a deep conceptual misunderstanding about mathematics. That it is just whatever we want it to be. Axioms and definitions are arbitrary. They need not serve a physical purpose. After all, we cannot physically experience infinity. As an example: Is there a cardinality between the naturals and the reals? Yes and no, depending on the axiomatic framework. And let's not forget that all of this is confused with limits as well. Additionally, 0.9999... need not be equal to 1, it is not in some frameworks of analysis.
@@horstheinemann2132 Mathematicians cant even agree on what a number is so I would argue infinities are a number "in some precise sense" to only a subgroup of mathematicians.
@@horstheinemann2132 In any other framework (that still has a useful notion of convergence ala the geometric series), it is equal to some 'monad' of 1, so it essentially means the same thing.
@@MagicGonads Proving that 0.999... = 1 relies on the trichotomy of numbers (either < or > or =). This relies on the law of excluded middle. So in intuitionist logic, there might be a number in between. In a classical framwork (classical logic + ZFC set theory) we can define the hyperreal numbers in the field of non-standard analysis. They are the reals plus infinitely small (smaller than all 1/n) and infinitely large (larger than all natural N) numbers. The standard part of 0.9999... and 1 will still agree though.
The worst one to me is "1+2+3+... = -1/12" because it's so obviously false. People get tricked due to two proofs that seems right, but have nothing rigorous. This just shows that if you do not follow the established rules, you get absurd results like this one. You might find a correlation, but it's crazy to see that so many people are persuaded that a sum of positive numbers is negative.
you are criticizing people not using enough rigor when believing things said at them (sum of positive ints = -1/12) but then your argument is based on pure intuition "how can a sum of positive numbers be negative?". There are many surprising facts in math, and basic intuition shouldn't stop you from seeing them. In this case, it's true that the sum of all ints doesn't equal -1/12, but it does lead to interesting places (tho the og numberphile way of getting there is just plain wrong, and could be used to get even more meaningless results if maniupulated), related to complex analysis/riemann's zeta function. Point is you can't just apply raw intuition to something like infinite sums. The sum of 1 -1 +1 -1 +1 -1 .... is undefined traditionally (diverges), but using Cesàro's sum you get the result 1/2, which would also be nonsensical if you apply just intuition to it (how can you get a fraction out of a sum of integers?).
Or I would rather say that it's implied by its definition. You're still making a good point, however triviality isn't intuition. If I'm not wrong, Cesàro's sum doesn't give you an equality, which would be absurd (unless you give specific definitions ?)
@@ulyssebeauchamp4629 not an equality ig, usually you write it with an = sign with a letter on top (i think C for Cèsaro), "equal" in a Cèsaro sum context. Also for convergent sums (like 1+1/2+1/4...) you use = to say the series converges to that number, is that necessarily equality? From my knowledge, we just didn't have a definition for what [inf. series (convergent)] = number meant so we just defined it to be the number it converges to
@@ArbitraryCodeExecution Well, in my lesson we define a series with a limit of a sum (for example, 1 + 1/2 + 1/4... = sum of 1/2^n to infinity = limit of (sum of 1/2^n to N) when N reaches infinity). So to me it's equal in the same meaning as a limit... I'm actually not sure now tho, if it's rigorous to say it's an exact equality
Thank you so much! I get so frustrated sometimes by the "pop" mathematics here on the internet and yeah i guess it just has something to do with that disconnect between what we can say from rigorous mathematics and those who have never been exposed to mathematical rigor
the video was great, the animations, the music. i never was big into math but this video was very insightful, as i had heard about these type of conclusions on the internet
I got downvoted for saying this: "Division requires 2 inputs. How you input these should be well defined. In the case of ÷ or / symbol it is not well defined as there is no end to how much the first and second number or expression extends to. In this case 4÷2(3-1) the second number in the division might extend to 2(3-1) or just 2. As far as I know there is no clear definition to solve this syntax issue. If you input this to a computer they won't agree either. Most of your modern languages and calculators will ignore everything and do the division with the closest number left to right but some calculators will say otherwise."
That’s correct, and it’s well documented that historically there’s always been ambiguity. A nice article that includes documents from the early XX century is the article “Ambiguous PEMDAS”, by Oliver Knill at Harvard
it's not a syntax issue, as you can define unambiguous grammars for this problem the issue is in *choosing a grammar that matches human language* as we disagree culturally on what the answer would be.
I just had this on my recommended page. The video is calming, interesting and informative at the same time. I'm glad I found this channel (I will now binge the other videos). Thank you for creating this masterpiece, I'm excited for the future :)
I have often been infuriated with those kind of expressions you mention in the beginning. My take has been to explain that maths notation isn't maths itself. It's a way of communicating, and we need to communicate in a precise, useful way. On the flip side, many professional mathematicians will, through what we call "abuse of notation" write stuff that is strictly speaking meaningless or even wrong, but that communicates the idea well, trusting that the reader can make sense of it. (One pet peeve of mine: Cryptographers will often talk about sampling a vector from Z^n. That's meaningless, but extremely intuitive, so we're happy to go along.)
@@tedzards509 Technically, a vector space needs to be over a field. The analogue of vector spaces for rings is called a module, so Z^n would be a Z-module. However I think that since you can embed Z^n in R^n, calling elements of Z^n vectors is not meaningless.
@@hadrienlondon4990it's also worth noting that there's no corresponding word for an element of a module; this is probably because mathematicians were more interested in individual vectors when they were developing linear algebra, instead of the global algebraic structure when they were developing module theory. so the question is: what else do you call an element of a module? i guess you can call it just an element, but that gets cumbersome quickly; it's probably easier to just borrow terminology, even if it's a bit nonsensical.
Every STEM major is an expert at math until the first time they have to contend with real analysis. The grounded computations with familiar physical intuition and easy to recognize patterns disappear. Seeing something like, "the Cantor set is of measure 0, but also uncountable," is a lot to swallow and prove but a lot more interesting than "what is 4/2(3-1)".
I didn’t click on this video for a long time because the thumbnail and title made it sound like it was going to be an angry guy ranting about math. But it was just a guy even calmer than 3blue1brown explaining math stuff
What one learns as a child one knows as an adult. People take the early education as something almost god given. Multiplication by juxtaposition taught years later and then then just blindly apply what they learned early on to it.
@okaro6595 exactly, it's people treating math like they did in 4th grade not understanding that back then it was taught to them that way only so that they could wrap their tween mind around the concepts
@@godlyvex5543it clarifies ambiguity but it isn’t effective when people don’t use basic and proper notation. it isn’t a different method, it’s just a small addition to pemdas
from one door to another. Great video :) i liked all your explanations! and honestly i am equally frustraited by ppl who dont understand what they are talking about using some random quote they heard so this definitly resonated with me
Awesome video! The argument people make as to why 0.999... =/= 1 because "erm actually 1 is just the limit as the number goes on" is particularity annoying because, yes, it is just the limit of a geometric series, it just so happens that every single mathematician agrees on a real number being equal to the complete sum of the series of its infinite digits. Another peeve of mine is when principal square roots are assumed to stand for both the possitive and negative roots, even though by convention it's just the positive root
0:00 I feel like you put in more gentle way the anger i had at this equasion, people posting it and even more so at school system I'm not mad that people "solve it wrong" and are "bad at math" because of that. I'm vert VERY MAD that school made people think, and (in turn since they have been teached that)that people think that this is what math is about, and what being good or bad at math is about. Which is sad because while this stupid equasion is put on pedestal, sierpiński triangle femains hidden in Tri Force, Mandalbrot set is set to the side, and how not all randoms aren't equal, and how something random can still show paterns (like diferent dystrybutons or above mention Sierpiński triangle which can be made with chaos theory) needs to be discovered on TH-cam and not in school, (well at least not in Polish school, all we had in terms of probability were boring ass probability trees based on draw of balls of other shit from boxes) So when another time this shit is posted, knife in my pocket opens and i want to scream into monitor "CAN WE TALK ABOUT LITERALLY ANYTHING ELSE !?" PS. i know my english be bad, at this point i'm tried of this incosistent speling and not bothering that much.
On a completely unrelated note, my favorite argument regarding the infinite 1 vs 20 dollar bills is that you get more spending value with the 20s, not because you have more cash, but because you have more time Once you have unlimited currency, your capacity to spend it comes down to a practical finite limit- that of time. If I wanted to spend 1 million dollars, assuming it takes around half a second to pull a 1 dollar bill from my infinite wallet, I would have to spend almost 6 days straight just gathering the bills. Meanwhile, if I have 20s, I can get that done in just 7 hours. Therefore, over the course of my life, it becomes possible to spend more of my infinite money with bigger bills
This is fully a failure on the part of the viewer, and absolute arrogance on their part. If you watch videos by Vsauce, Veritasium, PBS, StandUpMaths, Numberphile, Vihart, etc... they ALL provide clear explanations of infinity in ways that couldnt possibly lead someone to say an infinite number of 20s is a bigger Infinity than the infinite number of 1s. Literally all of them take pains to point out that this isnt about the "mass density" of the infinity, literally all of them use the evens vs naturals example which is 1:1 what the $20 to the 1$ bills is about. It takes a particular kind of person, clearly an unfortunately common kind, to watch a video like that, NOT understand, and then somewhere between minutes and years later have the confidence to CORRECT someone with their misunderstanding. They're cruising through life on headlines and catchy statements. If you asked them if they can outrun a tortoise then they'd think back to "the toroise and the hare", consider that they can't outrun a hare, and conclude that the tortoise would indeed beat them in a race.
"∞" is not strictly a number. It's a limit of an endless evaluation process, where evaluation result grows without a seeming end. This limit by itself only points to one of the edges of the numeric space. When people talk about comparing infinities, what they really talk about, is comparing growth rates of process above. Which property, could arguably, be used by limits themselves, by treating them as variables of unknown magnitudes, but known magnitude proportions.
I hate the "some infinities are bigger than others" thing so much. In my undergrad, cardinality was NEVER talked about like that. The terms we used, and which I massively prefer, are "countable" and "uncountable" infinities. The way you prove that the reals are uncountably infinite just happens to be by showing that whenever you try to count them, you can always create a new one, which lends itself to the idea that its "bigger", but the idea of size doesnt really even have much meaning in terms of infinity, so we don't talk about it like that. All we're trying to show is that you could assign count the naturals, integers, rationals, etc since you can find a way to map them to each other, which is the very essence of "counting", making them "countably infinite" and that when you try to do the same thing with the reals, it is impossible, making them "uncountably infinite". No where in that is it necessary to describe one as "larger" than the other, because even though the practical effect is that "I can always have more reals in the set", the end result of sets being infinite is that they are infinite, and their apparent relative scale is useless to consider in favor of the difference in the way they can be formed/organized.
I have been taught the idea of larger and smaller infinities in the context of N vs R sets - yes, the set of natural numbers is infinite, but even a subset of R bounded by a rational x to x+1 is infinite, so one could think of R as being made up of infinite infinities which, intuitively, has a higher cardinality. Is that a wrong way of looking at things? Probably...
it has more relevance when talking about measure theory (in continuous measures like 'length' or 'area', usually only uncountable subsets have a non-zero measure if they have a measure at all (except for distributions with impulses)) or the arithmetic hierarchy (computational complexity that goes beyond intractable and into uncomputable, and then levels of oracles)
It's very natural to talk about them like that in order theory. Cardinality is an equivalence relation, and with the axiom of choice, the equivalence classes are the cardinals (a subset of the ordinals) which are well ordered. Very directly, they are bigger or smaller than each other in even any naive understanding of that statement. Calling them countable/uncountable only looks at the subscript of the aleph and asks "Is it 0 or not 0?". The only reason we do that is because most of math is very focused on the real numbers, and it's enough detail to ask "Does it look like N?". If we ever develop beyond the reals, it might be more useful to distinguish between the number 1 and the number 2. The thing I don't like is when cardinality is equated with the entire concept of size.
@@SlipperyTeeth As I understand it, 'the cardinality' (the cardinal corresponding to a set) is not an equivalence relation, but rather the equivalence class under the equivalence relation of bijection over sets. (but it's just naming conventions I guess, or let me know if I confused something)
@@SlipperyTeeth the cardinality of the reals is not necessarily aleph 1 (so, thanks for saying 'not zero', but then you said distinguishing 1 and 2), it's consistent to say any number of alephs can exist between the naturals and the reals. But beth 1 is the cardinality of the reals, so we usually mean the beth numbers just because we don't generally care if the continuum hypothesis holds or in what way it doesn't.
This video was nice, I’ve never really been a maths guy, the only things I ever took away from class is to always show your working and that poisson is French for fish so your sentiments at 12:45 really spoke to me. Thanks for that 😊
Another way that I have seen it shown that .999999 repeating = 1 is with algebra. Let x = .999999 repeating then… 10x = 9.99999 repeating x = .99999 repeating -------------- (subtract bottom from top) 9x = 9 x = 1
This feels like a video from vsauce, if vsauce was more concise and wholesome. Awesome all round! Edit: 10 hours ago this video was a day old and had 150 views. Now at 12k! Epic!!!
@@godofmath1039 For some reason, the word “wholesome” has started being wildly overused. It seems to be applied to anything remotely in the semantic vicinity of “pleasant / gentle / comforting / traditional.” This person probably just means that this video is quieter and slower-paced than vsauce.
This is why we should teach real analysis to children, so they can have a rigourous foundation of maths and not say "err acksually some infinities are more than others"
“We saw her duck” actually has a hidden, third meaning, in which you kill her duck with a saw.
Interesting...
Don't kill her drafting ducks!
well, I guess it's more likely her duck is already dead and you're butchering it with a saw (very foolish)
answer: these equations aren't written correctly
no serious mathematician uses inline division
i would use it only when putting both sides of it in parentheses and the fraction itself in parentheses
As a mathematician, I see friends comment on these things and I'm compelled to go in and explain how these are ill posed and not how we'd write it, and give the actual order of operations. And there's always someone who insists on inventing new rules (e.g. apparently implicit multiplication is stronger than regular multiplication, but I'm sorry, 1/2x=0.5x).
@@spacelemThe implicit multiplication distinction is taught in other countries like India. Positif is used to mean nonnegative in French. In many parts of the world, ]a, b[ is used to denote open intervals.
@@spacelem OK so I'm reasonably well versed in math, but man, when I see "1/2x" my brain sees "1/(2x)" instead of "(1/2)x". I think it's just because the lowercase 'x' is smaller so it looks like how you would write it by hand if you were writing it with 2x as the denominator. I'm with the other guy, I overuse parentheses just to make sure there's no ambiguity.
@@hunted4blood the thing is, I would never type that without the parentheses. If I was writing it with pen and paper, I'd give some hint like writing the 2x a bit closer together and lower (like a very lazy attempt at a full fraction).
I'm aware that in some programming languages (e.g. Julia) it will interpret 1/2x as 1/(2x), but... well that's just something you've got to learn, but at least it's unambiguous in the language.
Heavily agreed, it's clickbait math for non-mathematicians. It's a shame because it distracts from REAL and interesting mathematics. It's such a vast and beautiful subject.
Not even non-mathematicians, I’m an Industrial Engineer (basically a business admin degree in my country) and it’s enough to make these internet math problems infuriating
It is not exactly a math problem, but is is a serious problem. Just never codifying the rules to an extent that you can easily and unambiguously write an equation with a keyboard or a calculator is a very serious problem that hurts mathematics because it is pretty important to be able to communicate math and adding unnecessary friction to this is pretty dumb particularly since every other case was handled by order of operations and we just never agreed on this little edge case.
@@wisnoskij Except, you can easily. Just don't write 1/2x, just as you wouldn't write "we saw her duck" without any context if you wanted to be unambiguous. Try 1/(2x) or (1/2)x instead. Boom! Perfectly unambiguous.
Not any string of characters has to have an unambiguous meaning to be able to find a string that unambiguously describes the meaning you want.
@@KaiHenningsenI mean, in my math classes my teachers would give us equations like. “y=-x^2” And then say the answer it is -(x^2) because its ACTUALLY -1*x. Ambiguous stuff like that is everywhere sadly, even if it technically makes sense.
@@TJ-hg6op To be fair it's just the conventional way to write that kind of thing (at least in highschool / engineering math), and your teachers should have explained that convention.
Anyway get used to it because it's gonna keep happenning, it's also not the only convention they'll forget to teach sadly
The true reason to take Infinite $20s over Infinite $1s is to avoid pissing off cashiers when you buy something expensive.
counterpoint: infinite singles are funnier
Counterpoint 20$ bill for buying 5.5% the tell them to cheap the change (would not piss them off)
for the two people directly above me you are unfortunately dumb and stupid because the real issue that infinite singles would cause is that the cashier will, and probably has to, count them all individually to make sure you're paying in full
The misuse of "Some infinities are bigger than others" combined with a link to the Vsauce video "How to count past infinity" makes my blood boil. If they had actually watched it, they would have been able to determine that those infinities are the same size.
Such are Vsauce fans
Yeah, some people understand videos the way they understand news headlines.
This is the case for Surreal numbers though! There are literally bigger infinities. Also there any algebraic expression involving infinities, for example omega + 1 or omega squared.
Ah I edited the comment giving more explanation, but TH-cam is horribly written so it didn't go through. Basically Surreal numbers are formed from certain positions in two-player perfect information games(such as Chess). You can define the sum of games(positions) to be a new game where players move in either of the two components. You can define the subtraction of two games A - B to be A + (-B), where -B is B but with the moves for each player swapped. If A - B = 0, that is, the player to move to this position has a winning strategy, then the games are said to have equal value. If A - B has winning strategy for the first player, then A > B. If A - B < 0 then the second player has a winning strategy from that position. With this you get all of Cantor's numbers plus any algebraic expression involving them, for example, 1/omega, omega - 1, or sqrt(omega). Also, you get like numbers where adding them is computed by writing each number in binary and computing the bitwise xor of them, and numbers which act like clouds of different numbers depending on the context. Surreal numbers are only regular numbers + infinities and infinitesimals, and I think they form a really amazing basis for analysis that I wish was used more, but the other numbers, generally called Games, capitalized because they form a proper class if you know what that is, are simply amazing and wonderfully alien and I wish everybody would read about them.
Ah I edited the comment giving more explanation, but TH-cam is horribly written so it didn't go through. Also I tried sending this, but again TH-cam is horribly written so it didn't go through. Basically Surreal numbers are formed from certain positions in two-player perfect information games(such as Chess). You can define the sum of games(positions) to be a new game where players move in either of the two components. You can define the subtraction of two games A - B to be A + (-B), where -B is B but with the moves for each player swapped. If A - B = 0, that is, the player to move to this position has a winning strategy, then the games are said to have equal value. If A - B has winning strategy for the first player, then A > B. If A - B < 0 then the second player has a winning strategy from that position. With this you get all of Cantor's numbers plus any algebraic expression involving them, for example, 1/omega, omega - 1, or sqrt(omega). Also, you get like numbers where adding them is computed by writing each number in binary and computing the bitwise xor of them, and numbers which act like clouds of different numbers depending on the context. Surreal numbers are only regular numbers + infinities and infinitesimals, and I think they form a really amazing basis for analysis that I wish was used more, but the other numbers, generally called Games, capitalized because they form a proper class if you know what that is, are simply amazing and wonderfully alien and I wish everybody would read about them.
algorithm lately has been crazy with recommending low subscriber creators and I love it
Same
Likewise
@1stnorthernfrontiercool animations always win over the crowd.
dog do you know how algorithm works? it just looks at users who have similar content preferences and recommends the same thing they clicked on/looked up. there's no sentient being providing you with niche videos
@@opal_2476 I don't know anything about how the algorithm works.However,I have noticed that up until now,most of the videos that appeared in my recommendations were from creators I already watched/kinda popular channels.I've been getting a lot of videos in my recommendations that are from new and obscure channels lately.Looks like I'm not the only one to notice that change
What I hate about the PEMDAS ones is that it's always the same trick and people seem to just forget that they've had the same exact conversation on repeat every year for the past decade.
How are you not bored of this BS?
Seriously. It's the same thing over and over and over again. Maybe it's just bots arguing with each other.
Exactly! And everytime they are thinking that they are solving a new problem!
I thought it was different people getting exposed to the same problem every year...
@@Spellweaver5
You would think so, but I'd also think that someone in these conversations would go "here's a video by [insert popular maths youtuber] from when this happened before explaining it." But no. No one ever mentions that they've done this before in these conversations. It seems ridiculous that it's *all* new people and no one remembers the previous times.
So now I don't know what to think.
It's because people only know this topic in math and they are pretty comfy arguing like a math enthusiast. Being human is hard you always have to satisfy that you are educated and convince your brain you are the greatest
my maths teacher years ago said "a little bit of knowledge is a dangerous thing" and it is extremely true.
Your teacher taught you the Dunning Kruger effect lol
Also, it ain't what you don't know that'll get you into trouble, it's what you know that just isn't so
@@jamesdurtka2709in fairness, school exams don’t help.
In a 5 mark question, you’re guaranteed nearly two marks for writing anything, wrong or right. Ofc, if it’s right you’ll get more. But still.
@@tinthatisfullofbeansexactly what i was thinking, low knowledge should be more self concieveable
As an engineering student, 0.999... = 1 isn't too hard to accept compared to π = 3, but I still really liked this explanation.
Lol approximation & estimation. "Close enough".
@@EpicMiniMeatwad "fuck it, good enough" is how most math is done in engineering tbh.
@@therealgeneralMacArthuri took AP chem this year and when my teacher was explaining acids and bases it was so funny to me. He was basically like, okay so because this is a weak acid, the equilibrium constant is very low. So to calculate the pH, we're just gonna assume x is "close enough" to zero. I was so baffled i thought he was joking at first. I will be going into mechanical engineering in 2 years. Based on what ive heard, i imagine im in for some much funnier approximations.
Cuz i mean like it makes SENSE right, 1.0 • 10^-20 is BASICALLY zero and your margin of error is low, but that was pretty funny regardless.
I mean sure 0.999... = 1, but then again 0.9999 = 1
What always annoys me about that the 0.999...=1 thing is that every time it's brought up you'll get a bunch of people who don't know anything about real or functional analysis trying to "um actually" and "debunk" it. The sheer arrogance to assume you know more than professionals who have dedicated years or decades of their lives to the subject is infuriating
the level of disconnection between the quality and the subscribers and views this channel has is insane. Loved the video.
Woah, I just noticed that. Less than 300 subs? Nah, this dude deserves at least 300k.
thanks for saying this, wanted to but in a mean way
I deadass thought this channel had over 100k subs or even a million. Lol
The root of the disconnect, IMO, is that mathematicians understand that maths is a bunch of rules we decide upon (with different rules for different areas of maths, called axioms), and we then see where they lead us. But if your maths education stopped at the end of high school, you'll have probably never seen this and so maths for you isn't about deduction and proofs but memorising times tables and trig identities, so you'll think that all maths is is a collection of titbits of pure knowledge that are true in-and-of-themselves.
To the lay person, mathematics is theology, while it's actually playing with lego where each brick is an abstract idea (the analogy works particularly well because so much of maths involves building higher-order objects from simpler ones). And bad internet maths leans into this because it's so much easier to present knowledge from up high without having to go into the thinking behind it, especially when it's what the audience except.
I really think calculus is an important educational step here. Most of math even on a grade school level seems like BS until you understand it, from zero to fractions to irrational numbers to infinitesimals. Forcing people to realize that this staircase of apparent absurdities keeps going higher and higher would probably do a lot to cull this problem.
Then again, half of the complaints about how "modern math is useless nonsense" comes from retired engineers, so maybe not...
@@jacksonsmith2955engineers are probably the single field in where the maths are ingrained into their work but they fail to understand why. All they do is approximate everything for their work and hope to god whatever they build doesnt blow up in their labor worker's face.
@@macchiato_1881 that's the most inaccurate depiction of engineering I've ever heard, every engineering course sets boundaries as to what approximations can be made and where and when use a certain model vs another and there is a lot of pure math, it's like 50% of the classes
@@jacksonsmith2955 maths never seemed like nonsense to me. But I was made to prove theorems from 5th grade onwards.
That said, I often noticed that I started truly understanding the previous topic only once I started learning the following one. Might be the reason I'm never truly going to understand functional analysis.
Perfectly stated, like there's probably some branch of math where 0.999... can be treated as a distinct entity from 1 and that's fine, you're just operating in a different context with different rules. The statement that 0.999... = 1 is simply commentary on a particular set of rules we've chosen (and commonly use) but if you aren't aware of the way these systems are constructed I guess you'd just have to take it on faith (or, alternatively, argue about it - because written down on paper those two things do NOT look the same)
Unfortunately, the popularity of the phrase "some infinities are bigger than other infinities" was really increased by the book "The Fault in Our Stars". I say unfortunately, because the character saying it is a 16 year old girl who doesn't actually understand the math. But it was a quick, catchy, and incorrect explanation so it's caught on on the internet.
The ordinals take on a natural order (obviously) and measure (less obviously) it just isn't exactly the same as that for strictly cardinal numbers. But there are absolutely infinities of different sizes. For example (technically a proof), when handling ordinal addition, we have that right addition is monotone increasing : α + β = 𝜏 => 𝜏 > α there exists a well-ordered measure on the ordinals. None of those steps are really necessary, as ordinals are definitionally measured, but it helps to illustrate that they are, since their distinction from cardinal numbers can obfuscate that sense of size.
There's also the plain case of countable vs uncountable infinities which do have strictly different cardinalities without thinking about ordinals, but that's a lot more vague.
I read the book like a decade ago now, though, so I don't know what the explanation given in the book is. It could absolutely be incorrect.
@@ChancePhilbin the girl in the book says that the "Infinity" of numbers between 0 and 2 is bigger than between 0 and 1, even though it's really not.
@@furryhunter110 Oh, yeah, that’s definitely wrong.
Wasn't it vsauce who popularized it?
@@tomekk.1889 Well The Fault in Our Stars came out 4 years before the VSauce video but I totally agree, I think that the VSauce video is much more popular and that's definitely where I heard about different sized infinities for the first time. i.e. depends on who you ask
THANK YOU the bedmas ones are so annoying cuz like sometimes they’ll word it or explain the equation awfully and then people get mad if u get it wrong like bro
I appreciate the fact that math has become more accessible lately, but sometimes people forget that you don't actually know a topic after watching a vsauce video (despite it being a good way to fuel curiosity)
Is this about the multiple infinities video? I thought he explained that pretty well, he basically said the exact same thing that was said in this video. What is the part that isn’t accurately conveyed in said video?
@@tacokitten nah, more in general, and not towards him alone
@@tacokitten even if it was explained correctly that doesn't mean the person will fully understand or remember after only watching the video once
The problem with the Vsauce video is that Michael himself demonstrates that an infinite set containing all integers and all even numbers is the same. These people just read the title and didn't watch it.
I still think the worst clickbait is math videos where the title card just states something wrong. And then the video explains that "we were using non-euclidean geometry all along", or whatever excuse they made up.
It's true - the whole point of "you get the right answer because of Bodmas" which is what these things are, is that bodmas is an agreed convention that people use to tie-break ambiguous notation (I mean, sure, it's also the case that you use conventions so that you don't have to draw brackets around things all the time once you establish what kind of things you want to talk about)
But the point with conventions is that they are not anything fundamental about the maths - it's like whether people drive on the right or the left is a convention, it's not something fundamental to the nature of the car that driving on one side or the other is better - but it is important that everyone in the same country understands the same convention so you don't crash into oncoming traffic.
I'm always disappointed there aren't more computer scientists in these discussions, since they're the ones who actually care about mathematical formulations of BODMAS/PEMDAS/etc.
This, but i wouldn't even call them conventions. I would call them mnemonics. Their point is to help you _memorize_ the rules (without having to understand them...), but they are not the rules themselves. Apparently that detail is not included in the curriculum since so many people try to interpret them as if they were law.
Dumbest thing is, people treat it like everyone's taught the same BODMAS/PEMDAS but then there's differences internationally where in America (I hear) they prioritize division over multiplication and addition over subtraction, and here in Australia we do division and multiplication at the same time as we step through the function seeing as they're inverse operations of each other, and the same goes for addition and subtraction. It just gets people arguing like everyone who doesn't get one conclusion is stupid without even thinking why they'd get there.
@@jacksonsmith2955 yeah, we know it depends on what order of operations you use instead of there being innate truth and only one solution. It's pretty sad how order of operations is taught as something that makes sure people interperet the same equation the same way for real applications that need consistency, but people aren't all taught the same order of operations... it just makes people fight over who was taught correctly. It's like making seed-based RNG but giving everyone different ways the seeds get converted into random numbers, so the seeds become meaningless except to give the same person the same result every time.
@@volbla BODMAS/PEMDAS are mnemonics to help you memorize rules, the rules themselves are conventions. Doing multiplication before addition isn't some inherent property of math, we just made it up because it made writing some equations easier.
I think the fundamental friction for both the gotcha equations and 0.999... = 1 is that in some sense people think that the notation system _is_ the maths, rather than merely a way of representing it. People get excited about these counterintuitive equations as if it's a demonstration of mastery of mathematics, when in fact there's nothing of mathematical interest in such garden path constructions; the mathematics is the object of the communication, not the communication itself. If the communication doesn't effectively convey that meaning, it's just bad communication.
Likewise, if the notation _is_ the maths, 0.999... = 1 is disturbing, because something is two things at once. But these "things" are just names. Getting hung up on it is a bit like my nephew - when he was maybe 4 or 5 - getting hung up on one of his toys being both a T-rex _and_ a dinosaur.
I do think it can be helpful to clarify that 0.999... isn't a special case: every finite decimal representation has a counterpart that ends with an infinite sequence of 9s (and likewise for other bases, of course). And you can prepend and append zeros to your heart's content, which doesn't seem to bother people.
as a programmer, an interesting way I've seen the 0.999... = 1 thing interpreted is that it can be thought of that both 0.(9) and 1 are expressions that evaluate to the same value, nobody finds 3-2=1 weird, despite it essentially being the same thing
which relates to what you said about the difference between notation and mathematical truth
@@uhrguhrguhrgit’s more amusing as a programmer that floating point error destroys the rigor of almost any statement involving rational numbers, depending on the language and processor you’re using. 3.0 - 2.0 often DOESN’T equal 1.0.
There’s a hilarious series of rather cursed examples of Boolean evaluations in JavaScript because of the ways it tries to rectify this (and just ends up making things more absurd).
@@willmungas8964 A more apt example would have been 0.3 - 0.2. Small integers can be exactly represented in floating-point, but decimals cause trouble because we use base ten, which has a messy factor of five in it that computers can't easily divide by.
I really like the way you gave the diagonalization argument as I find that most explanations out there say to "take the digit at that position and add 1" and maybe also "if it happens to be a 9 [bla bla bla I don't want to bore you to death]", I find that it distracts from the actual point of the proof which is that the number you're generating is DIFFERENT in at least one position, thus it cannot have been included in the list.
I may be stupid, but doesn't the proof of rational numbers having the same cardinality of the natural numbers shown in this video NOT match every rational number 1 to 1 to a natural? Because each of the boxes after the 1st definitely contains more than one number
Is that not the rigorous proof, or is proving that reals can't be matched one to one with naturals not enough to prove they have a different cardinality (??
Edit: I just googled it and they all use methods to match every rational 1-to-1 to naturals so yea lmao this one probably wasn't the best proof
@@CraftIPeach of the boxes after the one contain more than one number but each box always contains a finite amount of numbers therefore you can match them 1 to 1 with the naturals
@@goshawk6153 it wasn't immediate to me how one implies the other, but yea it definitely makes it so you can match each finite amount incrementally, but that was one more step than my mind made LOL
"People act like they know shit they have no clue about" Vol. ∞+1
Thank you for making this, I feel so validated in my frustrations about mainstream math discourse.
Excellent video! I like how you basically snuck in a real analysis proof at the start there :p
we stay silly
Yes
Also when explaining comparison of cardinalities
@@doorwaydude Collab with MAKit
this video is so cozy and sweet, I don't know how to describe it
Personally I would describe it as cozy and sweet
brother you just described it wdym
It's the music
Please learn then
You just described it though
The music in the background made this so enjoyable to watch, great video!
you know the name of it?
I don't know how you landed on my YT recommendation, but I am GLAD you did, thank you
What I like about “some infinities are bigger than others” is the rich history behind it. The concept of infinity caused a feud in mathematics. Not everybody accepted Cantor’s ideas.
Gödel used the diagonal argument to prove his incompleteness theorems. Effectively Gödel created the first compiler in his proof. Turing created the Turing machines to use diagonal argument for the halting problem.
So “some infinities are bigger than others” was a big part of the foundation of theoretical computer science. It’s a very important piece in the history of computation.
But yeah, this is not what people mean when they bring this up.
Yeah, logical theory, computation, and mathematics (proof theory, type theory, and category theory) are all very densely connected, it's honestly a sad education if you never get to learn these connections
I think in general people mean "some infinities are bigger than others" in exactly the same sense, they just don't necessarily have the right intuition as to why.
You know that are an infinity amount of integer numbers, but you also know that every number can be divided by any other number, so, you have infinite integers and infinity non-integers, but one has more numbers.
That's a good find
But the problem is that you can construct a bijection from the integers to the rationals, so they must be the same size. So even though we have this property where any number can be divided by any number (that isn't zero), we don't necessarily get a larger set.
So it has to be something else that makes the set of real numbers have a higher cardinality than the integers.
“Some infinities are bigger”. Brotha name one convergence test…
Fellow math nerd here, mathematics seems to be uniquely misunderstood as a discipline. Much of the work I spend teaching college algebra is in breaking down the student’s stereotypes on what math actually is. Something that still surprises me is how deep the disconnect is between scholars of mathematics and everyone else (particularly foundations of mathematics ie logicians and everyone else). It is fascinating to me how much people wish to imagine mathematics working a certain way versus how it actually works.
your voice, pacing, humor, animation style, etc etc etc are all amazing. so thankful this ended up on my homepage :) subbed!!
0:47 soo, I' the only one who imagined a group of psychos sawing some poor girls pet duck in half? 😅
Yes
That’s some poor grammar 😂
@@blew319 it's correct if 'saw' is a verb in 'present imperfect tense'
@@MagicGonadsI think you are confabulating it with spanish. It's just present tense
It could be simply them cutting her dinner in an unconventional manner.
But no, you're not the only one.
everything abt this video is so great. you have such a nice voice and a clear ability to explain (what seem like to me, a non-mathhead) complex concepts. and the animations!! i adore them! great work man. i hope you keep making videos explaining things youre passionate about!
Excellent video!
A big part of what makes me sad about these things is that a lot of these topics are interesting, but the Internet just loves turning them into little fun facts or, worse, pissing contests. Take the 4/2(3-1) example. That is a great way to introduce the topics of parsing algorithms, whether they are unique or not, and so on. That's a genuinely interesting subject to talk about! But, no, the Internet is more interested in trying to one-up the other person.
Yes, I have tried to explain the problem using parsing before as well.
Thinking about how grammar is a choice made by humans (and arises by natural languages processes), even in a mathematical context, is really important too.
Honestly, it doesn't really matter what the answer to that is, nor does it really matter that it's math. The issue is that people don't care about accuracy and unambiguity because a lot of people scuff at you when you point out the incorrect formulation of the question. It's a lot like grammar honestly. A lot of people out there mix up words like "you're" and "your", or "their" and "they're", and so on, the list is essentially endless. However, even if you try to correct them in as nice of a manner as you can, a lot of people will react with "okay whatever smartass nobody cares" attitude. The same really goes for this; people just don't care about being correct, they just want to be part of a larger group.
The lack of knowledge and care to acquire knowledge is honestly one of the most dangerous things to society I believe. I'm not saying that everyone needs to have a PhD in mathematics, but this type of question shouldn't really be a problem. People are so focused on finding the answer to the question that they never stop and think about why people arrive at different answers in the first place. They just don't care about knowing it. And it's really sad because if people just cared about learning things, and asking questions about things, then I genuinely believe that society would be so much better rather than being stuck in this echo-chambery-styled society that we have going on right now.
Yeah for real. I always hated that, and I especially hate how people think the division sign (the minus sign with a dot above and below) is actually ever used in any serious field (engineering, for example, the one I have experience in).
That sign is always too ambiguous, and you could always come up with several different "correct" answers. No engineer uses notations like that. You could always add more parentheses or vertical lines to something like that to make it impossible to misinterpret.
The only people that unironically use that division sign are 3rd graders learning that sign (for some reason?) and internet math guys who don't actually do any math outside of 15+12=27. And the shocking thing is, those people are always the most confident in their answer because "you do multiplication and division from left to right" or some random standard notation that isn't really used in the real world.
I actually no longer have any memories of using the division sign in grade school, but I am certain we used it. This must mean that once we students understood what division was, they immediately taught us better notation. And clearly it was better, since I evidently never ever ever used the division sign again and just forgot about it.
I paused to comment around the 11 minute mark and almost restated the entire conclusion. For lots of technical subjects the interesting facts that hook people in are oversimplified or of minimal importance to the field as a whole (eg. the idea of a five sided square got me interested in topology). Sometimes curiosity reaches a bit beyond what we're ready for, and the best solution is the one keeping us interested enough to learn it's wrong.
love this video, explains everything I've hated about people who don't know much about math trying to sound smart or saying that I am wrong about things I spend my time looking into because I find it interesting.
It’s unreal what a masterpiece this video is. Thank you
These math "problems" really lay out the fundamental problem with infix notation.
The people that write equations like this have never used a Ti-84.
YES omg the number of times I've seen that fractions thing pop up on social media again and again only for people to either insist on a particular solution as the only correct solution, or otherwise go down the route of smugly declaring that math isn't that accurate after all. Like dawg just throw away your device then since it's math that lets you talk shit on socials 😂 it's infuriating how "use your damn parentheses" is somehow always the least popular answer :/
I think the big problem with 0.999...=1 is more so a misunderstanding of what the real numbers are.
The way i would think of the real numbers is:
N is axiomatically a thing
Z is N with subtraction
Q is Z with division
R is Q with suprimum and infimum
but that's not how most people think of these.
numbers are just numbers, you write them down and they equal other things.
so, you learn at school that 1/4=0.25
why? because 2/10+5/100=1/4
this dosen't work for all fractions.
take
1/3
this dosen't work.
however, the infinite sum 3/10+3/100+...
does equal 1/3. so this fact is hidden from you and you just learn that
1/3=0.3333...
so, numbers can be infinitely long.
Now, look at
1-0.9999...
this equals
0.0000...1
which is not 0! so they're not equal!
of course, you and I know that 0.0000...1 notation just is meaningless, but most people just don't know what this is (to add to this, in the hyperreal numbers, a field extension of R, there *is* a number smaller than all positive fractions yet larger than 0). my favourite way to show it isn't a thing is to tell people that
...99999=-1
add them digit by digit and you get 0! (the devil is in the details. three dots can lie) while this is true in some sense (p-adic) these are not the real numbers.
This obviously like, dosen't matter, school shouldn't teach 9 year olds what a suprimum or what a cauchy series is, but i think this is the fundamental problem with lay people's understanding of math. they don't know what the rules of the game are, so they're sometimes cheated.
Aw man. You just made me remeber the moment from elementary shool when I was shown 0.(9) = 1 for the first time. Where Im from the "()" is the symbol of repeating digits btw. The equality didn't make sense to me. I quickly thought about 1 - 0.(9) and concluded that it must equal 0.(0)1. This made sense for me. I imagined infinity to be some point in the sequence. It's such a point that can never be reached - any finite number of steps would still be not far enough. But I still imagined that it is a fisical point somewhere infinitely far in the sequence.
I don't know whether we are teaching what the infinity is in the wrong way or understanding it is a limit that needs to be broken at some point. Now I understand that infinity doesn't really exist. It would more accurately be called "unboundedness". For example let's take natural numbers. There really isn't an infinite number of them. In theory there are, of course, but in practise nearly all of them won't ever exist, be used. I can in principle write down any one natural number and there os no upper bound on how big it could be, but that's it. The natural numbers are only infinite in the sense that we can never name the largest one.
A person can only name a finite number of numbers in any given time. It's fascinating to think that there exist the largest natural number that a person have imagined. It is probably massive. I can think of BB(BB(420)) for example. But there always exist Biggest Imagined Number and after it an "infinity" of useless numbers that have never been used
I think math education in schools should be split into calculating class and maths.
Preferrably maths as an optional course in the final grades where you start from axioms and basic logic and work your way to some of the basics so that interested students actually get to know what maths is actually about.
@@feliksporeba5851 i also for some time in middle school believed you can do things like 0.(0)1, and just put a number after repeating digits
I think the actual misunderstanding is of what a decimal representation is
I think if you explain to someone what 0.333... = 1/3 actually means conceptually they maybe understand what's going on
even not rigorously, you can explain it using the concept of a measurement, when you measure something with a ruler, you're doing a "ok, how many units does this fit into? after the maximum, how many 1/10 units? after that, how many 1/100?..." that's what the decimal representation is supposed to represent
so there are two ways of measuring 1, you can just fit it in one unit or keep going to the second smallest unit you can each time
@DarkPortall *supremum
Just found you. Working on my master in mathematics, I must admit I found much of the explanations trivial. Yet your narration is oddly captivating, equally so is your style of animation. I love this!
... and the boss baby Jesus
cantor's diagonal argument is actually the same as the halting problem proof, if you're willing to abstract enough
Get in line! I teach pre-algebra (among many things) and my students are routinely misled by this trash.
You really tricked everyone who hates math into clicking on a math video
And some of them proceed to not watch it, as expected
I think your problem with math edutainment is still true in actual school education. Here in Hong Kong, finding sums of infinite geometric series is taught in high school, and we were never taught the convergence of limits. Either the formula is stated and required to be recited, or the popular "infinite terms cancelling each other" trick is used without considering convergence. In that sense, the reasoning in actual school education is also simplified.
But I think there are times when reasoning and presentation do require simplifications and glossing over important details, in order to kick start an audience's understanding. We can't teach set theoretic construction of integers to 6 year olds, we just teach them basic arithmetic directly.
In highschool in France, we learn derivatives before limits...
@@hadrienlondon4990 wait, really? how tf does that even make sense? you learn to compute them before you even understand what they are or how to derive them? that's absurd
@@jacksonsmith2955It's the same in Australia. You learn derivatives from first principles in a very hand-wavy "just consider as x goes to a" without really learning what a limit is. In first year uni, I learnt more about limits and convergence, but it was still a little hand-wavy. It wasn't until second year that the epsilon-delta formulation of convergence was ever taught and that's when limits actually became something properly defined.
@@echo.1209 okay, so you do talk about the concept of limits, just informally. i think i'm fine with that tbh, the intuition is probably more important than the theory for 95% of students anyways.
@@jacksonsmith2955 They tell us "uuuhhh you write the limit dont worry abt it" and then they say "see when theres no h at the denominator we can put h=0 to get the limit"
Also, 0.(9) may not even be a defined number. We only assume it is a real number because it acts kind of like an infinite monotone bounded sequence. It also depends on how the operator is defined.
Right. You can never get it "back" by dividing 9/9 (unlike 1/3 = 0.(3)), so it's kinda "not valid".
From Wiki :"Certain procedures for constructing the decimal expansion of *_x_* will avoid the problem of trailing 9's. For instance, the following algorithmic procedure will give the *_standard_* decimal representation" (description follows).
So it's rather _non-standard,_ and you may easily get rid of it.
7:00 The word 'infinity' and the term 'infinite set' are not synonymous. The former refers to particular points in topological spaces, while the latter refers to sets that are not finite.
You're right that "infinity" is used inappropriately in the phrase "some infinities are bigger than others" because what is really meant is "infinite sets." I'm not sure I've heard of referring to "infinity" as a point in a topological space. I would consider it more a concept or way to think about certain ideas. Rigorous definitions in mathematics never talk about points at "infinity." The definition of a limit as a variable tends towards infinity doesn't involve infinity at all. It involves talking about arbitrarily small values representing allowance for error and sufficiently large values representing how large the variable must be in order to ensure the function stays within that error.
@@bartholomewhalliburton9854
'I'm not sure I've heard of referring to "infinity" as a point in a topological space'
Literally done all the time as early as in introductory courses and textbooks on calculus (although, at that point, the students are not yet aware of the fact that they are dealing with such spaces).
When working with limits of sequences, you are already dealing with a subspace of Alexandrov compactification or a similar space.
'I would consider it more a concept or way to think about certain ideas'
A limit of a sequence in some space is some point L of the relevant space, such that for every neighbourhood U(L) there exists a punctured neighbourhood of U(inf)\{inf} in Alexandrov compactification, such that for every natural n from U(inf)\{inf} the nth element of the sequence is in U(L).
Similarly, when dealing with limits of functions, people very often deal with Alexandrov compactification or extended real line (i.e. the standard two-point compactification of the space of real numbers).
Pretty sure that the projective compactification is taught almost immediately in complex analysis courses.
This stuff is perfectly rigorous.
'Rigorous definitions in mathematics never talk about points at "infinity."'
Yeah, they do.
'The definition of a limit as a variable tends towards infinity doesn't involve infinity at all'
Hmm? Are you sure about that?
How do you interpret the expression 'lim(1/x) as x->inf' when dealing with the definition of limits through filter bases?
Because the expression 'x->inf' does not make sense unless you define what the point 'inf' is.
'It involves talking about arbitrarily small values representing allowance for error and sufficiently large values representing how large the variable must be in order to ensure the function stays within that error'
This is what lacks rigour. This explanation of limits does not work in any sort of non-metric space, let alone in any non-metrisable space.
Seems like TH-cam hid my previous reply, so I will be brief this time.
Google 'Alexandrov compactification', 'extended real line', 'Riemann sphere'.
Your understanding of limits seems to be very shallow, and you seem to have not paid enough attention in introductory calculus classes, as students start rigorously working with such points very early in calculus - it's needed for limits of sequences.
Seems like TH-cam hid my previous reply, so I will be brief this time.
So, here's the short of it:
This stuff is, contrary to your claim, rigorous, and students start working with it within the first few weeks of their introduction to calculus.
@@thetaomegatheta Can you inform me what particular points in euclidean space (i.e. the space students work with in calculus) are "infinite"? You said "infinity" refers to particular points in a topological space. I'm just confused what you mean by this since I've never heard of this.
I like the visual proof for matching fractions to natural numbers. Draw all the fractions in a grid with the numerator on one axis and the denominator on the other. Count down the diagonal so start at coordinates 1,1 then 2,1 then 1,2 then 3,1 … skip all equivalently repeated fractions. This also proves that there are that same number of 1 dimensional and two dimensional spatial vectors assuming they can only contain natural numbers.
The other major thing people don’t get with infinities is that they aren’t going to intuitively make sense.
People like saying .99…. = 1 is absurd and doesn’t make sense, when the premise of infinite 9s is already absurd and hard to grasp. The answer is confusing because the premise is equally confusing
The statement that some infinities are bigger than other infinities is like saying some zeroes are bigger than other zeroes
As the video shows, there’s some context where it does make sense to say that some infinities are bigger than others. We just have to be clear on what we mean by “infinity” and “bigger”.
I think that this internet math is good, because it breaks the limits of math to make more math
More math ❌
More meth ✅
Weird statements like these are mainly thought of by mathematicians, as a way to showcase how irrational some of these numbers are. They aren't meant to be solved, they're kinda just like games.
nah bud internet math is just restating already discovered math in a simplified/more approachable manner. you don't really think 100% of these youtubers really invented what they said in their videos do you?
That was very graciously said, you have a wonderfully caring tone and I appreciate that you point out how the plight of disinterested people(not uninterested) is applicable in all subjects and topics. We should all be more discerning with what we learn and how we teach it while being thoughtful of what people might misunderstand without judging them harshly.
Did I just find my new favorite channel? I think I just found my new favorite channel.
0.99999999999 = 1 is a nightmare for computer programmers
And
Math people hate programmers anyways so I'm happy with how that is annoying to programmers
Programmers have
1+x=x or 1+1=10
@@BlackLegVinesmokeSanji idk why, we take concepts that would likely never see the light of day and use them to create marvelous things, meanwhile compilers miscount 1 as .9999999 causing inaccurate results messing up everything we try and do
As someone who is just learning discrete math in university this video was super interesting and I still learned a lot. I've always hated the "some infinities are bigger than others" saying because it just felt wrong. This conceptually really helped me wrap my mind around infinity.
i feel like it’s better to think of it as “some infinities are denser than others”
@@mms7146 Why do you think "denser" is a better word? I have heard a lot of people (but not professional mathematicians) suggest "denser" is a better word, but I have never understood why. So if you could explain it, I would really appreciate it!
@@MuffinsAPlentyI don't have an opinion re. bigger vs denser, but the reason denser might make more sense is because it doesn't feed into people's misconceptions about different sizes of infinity.
For example, on the number line, you cannot reach a "bigger" infinity than that of the naturals just by going farther to the right than a smaller infinity. Instead you must, for example, take the entire interval between 0 and 1.
@@eel9 Thank you! I appreciate your answer!
So, it seems to me that perhaps people think about the very specific example of naturals/integers/rationals vs reals, rather than thinking about cardinality as a general concept which applies to _any_ set, and where _any_ two sets can be "compared" cardinality-wise.
@@MuffinsAPlenty @eel9 gave quite a nice answer.
It is more of an intuitive way of seeing it rather than a formal one, as an infinity cannot be *literally* bigger than another infinity. But one may be denser than the other as, for example, there is a greater number of infinities per unit of rational numbers for every unit of natural number.
In any case, I have no formal background in mathematics, so you shouldn't take my explanation for granted.
You're really gifted with how you're able to talk in such an approachable and appealing way. Keep using this ability.
I don't have a background in maths, but this video is entertaining.
I am in love with your style of animation, your voice and delivery. I've seen some videos before about 0.(9)=1, never with this simple yet reasonable explanation. I tend to use knowledge I learned from math videos when teaching kids/young students math and I like to throw them some more interesting math facts that require a little more skill to solve or to think about to maybe interest them in the subject. I really love what we may do with math and how everything in it is so connected, one to another. I hope I'll see more videos from you, having subscribed I certainly hope so.
Also, wouldn't it be an enough explanation for not being able to compare the sizes of two infinities saying that they're just not numbers? As far as I know, infinity is not a number, so therefore we shouldn't be able to tell if it's smaller/larger. Let me know if I think logically!
yup, we cant compare the sizes (in the usual sense of the word) of infinities so that why we use cardinality which is a different though analogous concept. So when people say "some infinities are bigger than others" they are usually referring to cardinality but are often using it wrong
Infinities are numbers though, at least in some precise sense. I think the problem arises from a deep conceptual misunderstanding about mathematics. That it is just whatever we want it to be. Axioms and definitions are arbitrary. They need not serve a physical purpose. After all, we cannot physically experience infinity. As an example: Is there a cardinality between the naturals and the reals? Yes and no, depending on the axiomatic framework. And let's not forget that all of this is confused with limits as well. Additionally, 0.9999... need not be equal to 1, it is not in some frameworks of analysis.
@@horstheinemann2132 Mathematicians cant even agree on what a number is so I would argue infinities are a number "in some precise sense" to only a subgroup of mathematicians.
@@horstheinemann2132 In any other framework (that still has a useful notion of convergence ala the geometric series), it is equal to some 'monad' of 1, so it essentially means the same thing.
@@MagicGonads Proving that 0.999... = 1 relies on the trichotomy of numbers (either < or > or =). This relies on the law of excluded middle. So in intuitionist logic, there might be a number in between. In a classical framwork (classical logic + ZFC set theory) we can define the hyperreal numbers in the field of non-standard analysis. They are the reals plus infinitely small (smaller than all 1/n) and infinitely large (larger than all natural N) numbers. The standard part of 0.9999... and 1 will still agree though.
Those kinds of questions aren't meant for people who are good at math, they're meant for people who _think_ they're good at math (twitter users)
The worst one to me is "1+2+3+... = -1/12" because it's so obviously false.
People get tricked due to two proofs that seems right, but have nothing rigorous. This just shows that if you do not follow the established rules, you get absurd results like this one.
You might find a correlation, but it's crazy to see that so many people are persuaded that a sum of positive numbers is negative.
you are criticizing people not using enough rigor when believing things said at them (sum of positive ints = -1/12) but then your argument is based on pure intuition "how can a sum of positive numbers be negative?". There are many surprising facts in math, and basic intuition shouldn't stop you from seeing them. In this case, it's true that the sum of all ints doesn't equal -1/12, but it does lead to interesting places (tho the og numberphile way of getting there is just plain wrong, and could be used to get even more meaningless results if maniupulated), related to complex analysis/riemann's zeta function. Point is you can't just apply raw intuition to something like infinite sums. The sum of 1 -1 +1 -1 +1 -1 .... is undefined traditionally (diverges), but using Cesàro's sum you get the result 1/2, which would also be nonsensical if you apply just intuition to it (how can you get a fraction out of a sum of integers?).
"A sum of natural numbers (positive integers) is positive" is a proved property, not intuition
Or I would rather say that it's implied by its definition.
You're still making a good point, however triviality isn't intuition.
If I'm not wrong, Cesàro's sum doesn't give you an equality, which would be absurd (unless you give specific definitions ?)
@@ulyssebeauchamp4629 not an equality ig, usually you write it with an = sign with a letter on top (i think C for Cèsaro), "equal" in a Cèsaro sum context. Also for convergent sums (like 1+1/2+1/4...) you use = to say the series converges to that number, is that necessarily equality? From my knowledge, we just didn't have a definition for what [inf. series (convergent)] = number meant so we just defined it to be the number it converges to
@@ArbitraryCodeExecution Well, in my lesson we define a series with a limit of a sum (for example, 1 + 1/2 + 1/4... = sum of 1/2^n to infinity = limit of (sum of 1/2^n to N) when N reaches infinity).
So to me it's equal in the same meaning as a limit... I'm actually not sure now tho, if it's rigorous to say it's an exact equality
Thank you so much! I get so frustrated sometimes by the "pop" mathematics here on the internet and yeah i guess it just has something to do with that disconnect between what we can say from rigorous mathematics and those who have never been exposed to mathematical rigor
"hey check it out, I found that 1+2=3"
"no it's not, some numbers are bigger than others"
the video was great, the animations, the music. i never was big into math but this video was very insightful, as i had heard about these type of conclusions on the internet
I got downvoted for saying this:
"Division requires 2 inputs. How you input these should be well defined. In the case of ÷ or / symbol it is not well defined as there is no end to how much the first and second number or expression extends to. In this case 4÷2(3-1) the second number in the division might extend to 2(3-1) or just 2. As far as I know there is no clear definition to solve this syntax issue. If you input this to a computer they won't agree either. Most of your modern languages and calculators will ignore everything and do the division with the closest number left to right but some calculators will say otherwise."
That’s correct, and it’s well documented that historically there’s always been ambiguity. A nice article that includes documents from the early XX century is the article “Ambiguous PEMDAS”, by Oliver Knill at Harvard
it's not a syntax issue, as you can define unambiguous grammars for this problem
the issue is in *choosing a grammar that matches human language*
as we disagree culturally on what the answer would be.
I just had this on my recommended page. The video is calming, interesting and informative at the same time. I'm glad I found this channel (I will now binge the other videos). Thank you for creating this masterpiece, I'm excited for the future :)
this is why we write division as a fraction and not like this a/b what a troglodyte way of doing it
👍 you have my full support on this
There’s no reason not to if you’re not being ambiguous, you will see some people use that style of division sometimes.
At least computer scientists have fully accepted a single operator precedence ordering……………… *cries in bitwise operators*
I have often been infuriated with those kind of expressions you mention in the beginning. My take has been to explain that maths notation isn't maths itself. It's a way of communicating, and we need to communicate in a precise, useful way.
On the flip side, many professional mathematicians will, through what we call "abuse of notation" write stuff that is strictly speaking meaningless or even wrong, but that communicates the idea well, trusting that the reader can make sense of it.
(One pet peeve of mine: Cryptographers will often talk about sampling a vector from Z^n. That's meaningless, but extremely intuitive, so we're happy to go along.)
Is Z^n not a Z-Vectorspace? Or is the issue somewhere else in the notation?
@@tedzards509Z is not a field
@@tedzards509 Technically, a vector space needs to be over a field. The analogue of vector spaces for rings is called a module, so Z^n would be a Z-module.
However I think that since you can embed Z^n in R^n, calling elements of Z^n vectors is not meaningless.
@@hadrienlondon4990it's also worth noting that there's no corresponding word for an element of a module; this is probably because mathematicians were more interested in individual vectors when they were developing linear algebra, instead of the global algebraic structure when they were developing module theory. so the question is: what else do you call an element of a module? i guess you can call it just an element, but that gets cumbersome quickly; it's probably easier to just borrow terminology, even if it's a bit nonsensical.
@@bayleev7494 I would use the word 'point' or 'value' probably
Every STEM major is an expert at math until the first time they have to contend with real analysis. The grounded computations with familiar physical intuition and easy to recognize patterns disappear. Seeing something like, "the Cantor set is of measure 0, but also uncountable," is a lot to swallow and prove but a lot more interesting than "what is 4/2(3-1)".
There's a difference between math heads and meth heads
I didn’t click on this video for a long time because the thumbnail and title made it sound like it was going to be an angry guy ranting about math. But it was just a guy even calmer than 3blue1brown explaining math stuff
Yes. People using pemdas as holy and not knowing multiplication by juxtaposition annoys me more than I want to admit.
If you take it like a 4/2x where x=(3-1), then answer is 1 or 4 again
What one learns as a child one knows as an adult. People take the early education as something almost god given. Multiplication by juxtaposition taught years later and then then just blindly apply what they learned early on to it.
@okaro6595 exactly, it's people treating math like they did in 4th grade not understanding that back then it was taught to them that way only so that they could wrap their tween mind around the concepts
Why use a different method when it only introduces ambiguity?
@@godlyvex5543it clarifies ambiguity but it isn’t effective when people don’t use basic and proper notation. it isn’t a different method, it’s just a small addition to pemdas
I've never watched math videos before but it appeared on my home page, and THIS IS THE BEST THING IVE EVER SEEN INSTANTLY SUBBED
Amazin video! Subscribed and waiting for more content from you, hope your channel grows!
from one door to another. Great video :) i liked all your explanations!
and honestly i am equally frustraited by ppl who dont understand what they are talking about using some random quote they heard so this definitly resonated with me
you hit the jackpot with this vid
Awesome video! The argument people make as to why 0.999... =/= 1 because "erm actually 1 is just the limit as the number goes on" is particularity annoying because, yes, it is just the limit of a geometric series, it just so happens that every single mathematician agrees on a real number being equal to the complete sum of the series of its infinite digits. Another peeve of mine is when principal square roots are assumed to stand for both the possitive and negative roots, even though by convention it's just the positive root
It is one of the greatest videos I've seen on TH-cam.
Straight forward art in my opinion!
Please keep up if that is what you want!
0:00 I feel like you put in more gentle way the anger i had at this equasion, people posting it and even more so at school system
I'm not mad that people "solve it wrong" and are "bad at math" because of that.
I'm vert VERY MAD that school made people think, and (in turn since they have been teached that)that people think that this is what math is about, and what being good or bad at math is about.
Which is sad because while this stupid equasion is put on pedestal, sierpiński triangle femains hidden in Tri Force, Mandalbrot set is set to the side, and how not all randoms aren't equal, and how something random can still show paterns (like diferent dystrybutons or above mention Sierpiński triangle which can be made with chaos theory) needs to be discovered on TH-cam and not in school, (well at least not in Polish school, all we had in terms of probability were boring ass probability trees based on draw of balls of other shit from boxes)
So when another time this shit is posted, knife in my pocket opens and i want to scream into monitor "CAN WE TALK ABOUT LITERALLY ANYTHING ELSE !?"
PS. i know my english be bad, at this point i'm tried of this incosistent speling and not bothering that much.
This channel is really underrated, holy
On a completely unrelated note, my favorite argument regarding the infinite 1 vs 20 dollar bills is that you get more spending value with the 20s, not because you have more cash, but because you have more time
Once you have unlimited currency, your capacity to spend it comes down to a practical finite limit- that of time. If I wanted to spend 1 million dollars, assuming it takes around half a second to pull a 1 dollar bill from my infinite wallet, I would have to spend almost 6 days straight just gathering the bills. Meanwhile, if I have 20s, I can get that done in just 7 hours. Therefore, over the course of my life, it becomes possible to spend more of my infinite money with bigger bills
This is fully a failure on the part of the viewer, and absolute arrogance on their part. If you watch videos by Vsauce, Veritasium, PBS, StandUpMaths, Numberphile, Vihart, etc... they ALL provide clear explanations of infinity in ways that couldnt possibly lead someone to say an infinite number of 20s is a bigger Infinity than the infinite number of 1s. Literally all of them take pains to point out that this isnt about the "mass density" of the infinity, literally all of them use the evens vs naturals example which is 1:1 what the $20 to the 1$ bills is about.
It takes a particular kind of person, clearly an unfortunately common kind, to watch a video like that, NOT understand, and then somewhere between minutes and years later have the confidence to CORRECT someone with their misunderstanding. They're cruising through life on headlines and catchy statements. If you asked them if they can outrun a tortoise then they'd think back to "the toroise and the hare", consider that they can't outrun a hare, and conclude that the tortoise would indeed beat them in a race.
I was intrigued by this video, so I decided to check out your channel.
First thing I see: Boss baby and baby Christ
4:50 got me
All of this is the consequence of Mathematical Illiteracy.
"∞" is not strictly a number. It's a limit of an endless evaluation process, where evaluation result grows without a seeming end.
This limit by itself only points to one of the edges of the numeric space.
When people talk about comparing infinities, what they really talk about, is comparing growth rates of process above. Which property, could arguably, be used by limits themselves, by treating them as variables of unknown magnitudes, but known magnitude proportions.
I didn't know Joseph Anderson knew so much math
I hate the "some infinities are bigger than others" thing so much.
In my undergrad, cardinality was NEVER talked about like that. The terms we used, and which I massively prefer, are "countable" and "uncountable" infinities. The way you prove that the reals are uncountably infinite just happens to be by showing that whenever you try to count them, you can always create a new one, which lends itself to the idea that its "bigger", but the idea of size doesnt really even have much meaning in terms of infinity, so we don't talk about it like that. All we're trying to show is that you could assign count the naturals, integers, rationals, etc since you can find a way to map them to each other, which is the very essence of "counting", making them "countably infinite" and that when you try to do the same thing with the reals, it is impossible, making them "uncountably infinite". No where in that is it necessary to describe one as "larger" than the other, because even though the practical effect is that "I can always have more reals in the set", the end result of sets being infinite is that they are infinite, and their apparent relative scale is useless to consider in favor of the difference in the way they can be formed/organized.
I have been taught the idea of larger and smaller infinities in the context of N vs R sets - yes, the set of natural numbers is infinite, but even a subset of R bounded by a rational x to x+1 is infinite, so one could think of R as being made up of infinite infinities which, intuitively, has a higher cardinality.
Is that a wrong way of looking at things? Probably...
it has more relevance when talking about measure theory (in continuous measures like 'length' or 'area', usually only uncountable subsets have a non-zero measure if they have a measure at all (except for distributions with impulses)) or the arithmetic hierarchy (computational complexity that goes beyond intractable and into uncomputable, and then levels of oracles)
It's very natural to talk about them like that in order theory. Cardinality is an equivalence relation, and with the axiom of choice, the equivalence classes are the cardinals (a subset of the ordinals) which are well ordered. Very directly, they are bigger or smaller than each other in even any naive understanding of that statement.
Calling them countable/uncountable only looks at the subscript of the aleph and asks "Is it 0 or not 0?". The only reason we do that is because most of math is very focused on the real numbers, and it's enough detail to ask "Does it look like N?". If we ever develop beyond the reals, it might be more useful to distinguish between the number 1 and the number 2.
The thing I don't like is when cardinality is equated with the entire concept of size.
@@SlipperyTeeth As I understand it, 'the cardinality' (the cardinal corresponding to a set) is not an equivalence relation, but rather the equivalence class under the equivalence relation of bijection over sets. (but it's just naming conventions I guess, or let me know if I confused something)
@@SlipperyTeeth the cardinality of the reals is not necessarily aleph 1 (so, thanks for saying 'not zero', but then you said distinguishing 1 and 2), it's consistent to say any number of alephs can exist between the naturals and the reals. But beth 1 is the cardinality of the reals, so we usually mean the beth numbers just because we don't generally care if the continuum hypothesis holds or in what way it doesn't.
Thanks bro. This is the best explanation of these basic Set Theory problems I‘ve learned so far.
As a math student, thank you.
Also you're so underrated, I would expect a video like this from a channel with 100x your sub count ngl
This video was nice, I’ve never really been a maths guy, the only things I ever took away from class is to always show your working and that poisson is French for fish so your sentiments at 12:45 really spoke to me. Thanks for that 😊
100 subs , Congo
4:55 took me by surprise lmao, 10/10 segment
Another way that I have seen it shown that .999999 repeating = 1 is with algebra.
Let x = .999999 repeating then…
10x = 9.99999 repeating
x = .99999 repeating
-------------- (subtract bottom from top)
9x = 9
x = 1
this presupposes that x exists (we must show unique convergence)
nice video i agree with everything said, the editing and your voice felt pretty chill and cozy too i liked it
This feels like a video from vsauce, if vsauce was more concise and wholesome. Awesome all round!
Edit: 10 hours ago this video was a day old and had 150 views. Now at 12k! Epic!!!
Concise, yes. Wholesome? Vsauce literally never curses in any of his videos.
@@godofmath1039 For some reason, the word “wholesome” has started being wildly overused. It seems to be applied to anything remotely in the semantic vicinity of “pleasant / gentle / comforting / traditional.” This person probably just means that this video is quieter and slower-paced than vsauce.
@@godofmath1039 Yes, I was kinda joking :)
i hate it so much when people do the 6➗2(1+3) thing when THERE ARE LITERAL DOTS on the symbol for where you have to put expressions in 😭😭😭
0:20 What's a "pem dosser"? Can please anybody elnorate on that?
Someone who follows the rules of PEMDAS. (Parentheses, exponents, multiplication, division, addition, subtraction.)
This is why we should teach real analysis to children, so they can have a rigourous foundation of maths and not say "err acksually some infinities are more than others"