ah ha, but there in is the issue. 0.9999... is not a limit. We also are under the delusion that if we have a limit that we can actually reach that limit such that we can actually find the slope of a tangent to a curve at any point. The fact is we can never actually achieve that limit or we are dividing by zero. Even 0/0 is undefined so to have the actual tangent is technically not possible.
Here are my approaches:- Approach1: 1 = 99/99 = (99.99... - 0.99...)/ (100-1) Notice that the numerator and the denominator need to be equal, so that we get 1 as answer. Hence by comparison of numerator and denominator, we get 99.99... = 100 and 0.99... = 1. Approach2: Let X=0.9999... 100X = 99.99.... On substrating them, 100X - X = 99.99... - 0.99... 99X = 99 X= 99/99 X=1. Therefore, 0.9999... = 1.
@@anuragguptamr.i.i.t.2329 still wrong. there is the idea of infinitesimals. 0.999... is such an infinitesimal and you will never reach 1. The idea that a limit is equal to the assigned number is false. f(x)=1/x, as the limit of x approaching 0 is undefined. You will never reach 0. same is true for ln x , as you will never reach 0 nor can x be negative.
@@kennethgee2004 No, you're wrong. A limit is, by definition, that value. There is no question about "reaching" anything. The difference between 0.999... and 1 is zero. The stuff about diving by zero is a red herring. You are confusing yourself by thinking that 0/0 is undefined. It isn't. It's merely indefinite, and when it is the limit of the quotient of two functions, it often has a definite value. Think about sin(x)/x and work out whether its value when x=0 can be anything other than 1.
@@RexxSchneider You really need to revisit how the limit is born. It is the secant line that we are moving ever closer towards the tangent line. The secant line becomes length 0 at the point of tangent, but then 0/0 is undefined. We never actually reach that point. We just see that it is going towards that point and write an equal sign. The value is not actually equal.
which, IF those defending the intuition the 0.9999… ≠ 1 are correct, will be infinitesimally more than zero. Bottom line, no one is really defending that _according with the formal definitions of standard mathematics_ 0.9999… ≠ 1, which would be trivially wrong. People are rather arguing that, given that there’s such a strong intuition conflicting with the standard _convention,_ who is to say that the standard formalization and axioms are for sure correct? Now THAT is a philosophical question, and it’s a legitimate one. Surely there can’t be a _formal indisputable proof_ that the very formal system being called into question is correct. One can argue that “objective truth” in mathematics makes no sense _outside of a rigorously defined formal system,_ which I guess most mathematicians would agree with. But that is a philosophical position, to be argued philosophically. It is NOT an indisputable theorem inside a formal system. Philosophically, it’s perfectly legitimate to disagree with this, and to defend, for example, that there’s an objective structure to abstract quantities - so that any given formal system can be right or wrong in capturing such structure. And if that’s true, it’s no wonder that elusive questions about infinity are places where the standard math could be wrong - after all, it wouldn’t necessarily make any practical or theoretical difference to be wrong in such extreme cases.
@@SelbyClaudeif the axioms are incorrect, then logic is false. 1 does not equal 1, 1+1 does not equal 2. Multiplication isn't real. Nothing is real. Go look at the formal proof, it is basic bitch algebra and it proves definitively that 0.999... is equal to 1. They are just different ways of writing 1. It isn't intuitively wrong. Your intuition is wrong if you think it is wrong. Yes, you can have an intuition is wrong. Contrary to the popular saying about opinions, opinions can in fact be wrong too. Because opinions that claim false things are true, are certifiably wrong.
@@meateaw no it is you who is wrong. Axioms aren't correct or incorrect. They just are. They are the rules by which the game is played. Other rules could have been chosen, and often other rules are chosen. By the most common set of rules 0.999...=1, but you can absolutely pick a consistent different set of rules where 0.999... is not 1
The way I like to shut these people down is asking them what number lies between 0.999… and 1. If they’re not the same number, there must be a number between them because the real numbers are a dense set
And you can’t really disagree with that unless you want to go into hyperreal numbers which is already very advanced and non-standard analysis which is also quite unintuitive. So if you want to stay in the regular real numbers we are familiar with then you have to accept it.
That requires them accepting that there must exist such a number, and .9999.... can't simply be the largest number less than 1. Which is a much easier thing to believe. And is also not entirely incorrect... real numbers might not have such a thing, but number systems like hyperreals freely accept that .9999... is exactly that.
@@vib80 If they say that 0.9999... is the largest number less than 1, then ask them what 1 - 0.9999... yields. ( Ok, if they then insist that the result of that calculation is 0.0000.........1, then you probably have to give up, because obviously they don't understand what "infinite" means. :D )
@@user-ee5kl6oi7t Yes: 1/9 is 0.111... So you take 1/9*9=9/9=1 right? Now lets write the same formula in decimals: 0.111...*9=0.999...=1. 0,999... is 9/9 in fractions.
@@user-ee5kl6oi7t I think the confusion here is that we're all taught the long division algorithm, but there's no way to naturally get 0.999... through the standard long division algorithm. Once you start working in different bases, or even looking into repeated fractions and basic number theory, and understanding what's actually going on under the hood in long division, the misconceptions about 0.999... might start to go away edit: I understand the proofs for why it's true. I'm trying to explain from a pedagogical standpoint, why most people might have a bad first reaction to 0.999... = 1. For example, in bprp's latest video on this topic, th-cam.com/video/C_1AqiKJRhA/w-d-xo.html he uses the same whiteboard explanation for 0.999... = 9/9 to show that 0.444... = 4/9. But tell somebody that 4/9 = 0.444..., and they readily believe it. Why? They've done it by hand in grade school, and can use long division to work out the details. You don't need to whip out alternative proofs, which, while true, aren't how most people internalize repeated fractions as a whole. The intuition is solidified already for most people because they can just divide the fraction using the long division algorithm, and see the repeated decimal pop out. [on a side note, the manipulation of infinite decimals is exactly how you manipulate infinite fractions, so perhaps that would lead to more intuition about what's going on? I really liked his followup video.]
@@coolskeletondude5902 Lots of things you learned in the 3rd great were simplifications designed to make complicated concepts digestible by third graders. If you 3rd grade teacher didn't inculcate this in you they did you a disservice.
@@EvanG529 Yes. A lot of people don't seem to understand that rational numbers live in a completely different conceptual universe from natural numbers.
I think the biggest reason people fail to accept this is because they’re trying to imagine something like this in real life. Take Zeno’s Paradox which is vey similar. No matter how much finite time passes, you’ll never reach the end point. But we are dealing with infinite time which is not a real number itself.
if you consider motion as a supertask, where every infinitely small distance takes proportionately little time to cross at a steady velocity, you can easily pass an infinite number of points in finite time to go 1 meter in 1 second you have to go 1/2 a meter in 1/2 a second, 1/4 meter in 1/4 second, etc. This series converges on 1 meter per second
@Neckhawker No, they don't. 0.999... repeating does not converge at 1, it is equal to 1. It is simply a different way of writing 1, like writing it as 3/3, which is also equal to 1 (and equal to 0.999... repeating, since they are the same).
@@zagreus5773 No, 0.999... defined as 9 repeating to the infinite isn't equal to 1. It is its limit that converge to 1. And no, 3/3 isn't equal to 0.999..., idem, 1/3 isn't equal to 0.3333..., but 0.333... converge to 1/3.
I think a lot of people get hung up on this because they interpret 0.999... as a decimal followed by "an arbitrarily large string of 9s" or something like that. And any very long string of 9s, like a billion or a google-plex 9s after the decimal, IS less than one, so this becomes more of a semantic misunderstanding about what we're representing with 0.999...
I know what infinite means. Try draw 0.999... on a graph. It will never touch 1, so therefore it cant be 1. Yes its infinite, but you can travel down that 0.999... line for as loooooooong as you like, and yet no matter how far you go its always just below the 1 line above it. Its approaching it for infinity, yep agree, but it can and will never actually be the same as 1. The only people who would argue that these are the same thing are the same people who like to say "but actually"
@@bjornfeuerbacher5514 It's kind of the opposite. 'Infinity' isn't really a reasonable mathematical concept, and people try to reason about 0.999... using their conception of 'infinity'. I think the problem is that many people don't understand what a limit is (really it's too bad that the concept of a limit isn't taught before Calculus). If you're thinking in terms of infinity, .9 is finite, and so is .99, and .9999999, and so if you take a finite string of 9s and add another 9 on the end, you will always get a finite number. There's no end to this process, no point where one more nine causes the number to roll over to 'exactly one', even if you keep adding 9s ad infinitum. This line of reasoning does countenance infinity, sort of, but to rigorously understand what's going on you need to look at it in terms of the limit, not of nebulously defined 'infinity'.
@@networkguy153if you want me to prove 0.999… is equal to 1 in an elementary way here u go: 1/3 as you know is equal to 0.333… Now if we double 1/3 we get 0.666…. So far so good, But what if we triple it? Well we should get 0.9999 right? But tripling is the same as multiplying by 3, so 1/3 x 3 =? Oh it’s 1? 😮
@@networkguy153And therefore you immediately say that you do not in fact know what “infinite” means. Because you immediately followed it up by saying “no matter how far you travel down”, which are always *finite* amounts. You cannot draw infinity. Infinity is an abstract concept, not a number, not a length, not ratio, and most certainly not intuitive. Infinity follows certain mathematical rules, and the only way to get a correct result regarding infinity, is to apply those mathematical rules, like in the video.
My son had this problem with his teacher at school. I gave him the simpler algebraic proof that if x = 0.999…, 10x = 9.999..,, subtract first equation from second and get 9x = 9, then divide by 9 and you’re done. He managed to get the teacher to admit that this was right.
Infinity or not, in a base10 system multiplying anything with 10 just shifts the comma to the right by one step.. Sure you could probably come up with a definitive proof, but I doubt that anybody is gonna ask you for one
this exact proof was involved in a famous question that appeared in an entrance exam for one of the biggest universities here in brazil. every math teacher used to talk about this question and to show this proof here. pretty nostalgic
Just ask what is 1/(1-0.999...) and if it's larger than all natural numbers, and if they're willing to accept (real) numbers larger than anything together representable by digits.
Look, they may be wrong in this case, but I don't think you know what philosophy is. Mathematics is a subset of formal logic, formal logic is a subset of logic, logic is a subset of philosophy. We're all trying to do the same thing: understand things and learn HOW to understand things correctly.@@exscape
Certain people get so angry about this, every mathematician ever will tell them 0.9999... = 1 and they will drone on and on about how false it is, refusing to engage with actual proofs, making up terms without properly defining them, etc. It's bewildering how dedicated people are to this.
In highschool, the first time I saw .9 repeating equals 1 it felt very wrong to me. Essentially, it felt like there must be some infinitesimal difference between them. Luckily I had a very good math teacher who took the time to talk me through it, and didn't make me feel like an idiot for questioning it, and expressing my intuition. And now I'm the guy on the internet arguing with people who think .9 repeating isn't 1, lol.
I got one counter to you. When you subtract 0.9999999999 from 1 do you get zero? Absolutely not. No matter how many nines you add at the end, the last number on the 1 is going to be 0, and you'll have to do 10-9 at some point. Therefore, there is a difference. Where people mess up is in misinterpreting 'repeating'. To compare two numbers, you will ALWAYS have to stop at a specific number. And in that moment, there is still a difference between the two.
The way I like to explain it, to help people grasp it intuitively, is that 1/3 is 0.3333... 3 times 1/3 is 1, right? Which means 3 times 0.3333..., which is 0.9999... must also be 1.
The way I explained it to both of my (at that time) third-graders, was to have them compute 1/9 and notice that they get 0.111…. That was probably the first time they had seen a repeating number and given it much thought. Then repeat with 2/9, 3/9 (aka 1/3), 4/9, ... and so on. They'll quickly recognize the pattern. Finally, I asked them what they would get for 9/9. Of course, with this lead-up, the answer from them is going to be 0.999…. And that's the moment when I point out that I always thought that 9/9 was actually 1. After a brief moment of surprise, both of them admitted that my explanation made sense. 0.999… does in fact appear to be just another way of writing 1. This is of course not a rigorous mathematical proof, unless you first show why n/9 produces a repeating number. But that's too difficult for a third-grader. Nonetheless, I found it to be a good way to get an intuitive grasp of what's happening here without having to go much beyond any of the math that a third-grader is familiar with. And for anybody sufficiently curious, it is possible to develop this approach into a full proof. Maybe, they'll do that when they are older and remember this lesson.
@@gutschke "was to have them compute 1/9 and notice that they get 0.111…. That was probably the first time they had seen a repeating number and given it much thought." - That doesn't actually prove anything. That's just a quirk of doing division of a number which is not evenly divisible by your base. In base 9 instead of base 10, 1/9 is exactly .1. And 1/10 becomes the weird repeating thing.
If you went the exact opposite direction the moon was currently in as far as the moon is away from earth on average, _you still_ would have missed by less than million miles.
"you wouldnt even miss the moon by a million miles if you just roughly went in the direction of the moon" You would miss the moon "a million miles" (well sort of as you were already told) if you went in the direction of the moon. Orbital meachanics are weird someway, but at the point in time you are at the moon, the moon is at a completely different place than a few days before. So no, to reach the moon you do not go in the direction of the moon. You can't.
@@pat9353 "If you went the exact opposite direction the moon was currently" The funny thing is: Because how orbital mechanics work, it may be you arrive at the moon by doing this. But then it will be a long trip. ;)
The root cause of this confusion is that most laymen who have not had formal rigorous mathematical education past first year university don't understand that mathematics is based on precise definitions and instead argue based off of their "intuition". They don't understand that every symbol or collection of symbols in math has a definition that is agreed upon, and all statements made about those objects must first begin with the definition. There is no room for "interpretation" or "intuition" or anything about "you." Starting from the definition, each step follows logically and necessarily, and we arrive at the conclusion. People who argue 0.999... is not equal to 1 do so mostly using their intuition about what 0.9999... "should" mean or "ought to" mean. These arguments from intuition leave a ton of wiggle room for hand-wavy logic which isn't air tight. The reason why this equality is not contentious in the mathematical community is because everyone who knows proper math knows the exact procedure you must follow to demonstrate the equality. The collection of symbols "0.9999..." has a very precise definition. BY DEFINITION, it is the limit of the infinite sum of 9/10^n, n going from 1 to infinity. The "infinite sum" also has a precise definition, defined using epsilon and deltas. When you apply the definition of the infinite sum, you find that it evaluates to 1 exactly. Therefore 0.9999... = 1, end of story. The equality follows directly from the definitions of what those symbols mean. There is only ambiguity if you are using a different definition for those symbols from what is agreed upon, or you failed to apply those definitions in your argument, or you are under the misunderstanding that the same number cannot be expressed in two different ways.
The person's fundamental error is that, while he more or less correctly recognizes that the 1 is the *limit* of the sequence of partial series implied by adding nines to 0.9999 (at least, I think that is what he's getting at with "the definition of an asymptote"), he doesn't realize that that is how decimal representations of real numbers work: the number represented by a non-terminating decimal *is* the limit.
This is what I try to explain to people as well. If there are infinite 9s, then 0.9 recurring is the number at infinity, and the number at infinity is 1.
Exactly thought about the same thing. Best intuitive explanation for people who cannot cope with the mathematical formal explanation, it's like Zenon and his arrow and we know that in the end it reaches its target. However, it's possible that both are true. It simultaneously is and isn't equal, it does and doesn't reach the target. In reality we know physics and we know that the arrow's atoms will reach the target's atoms close enough to interact, but they don't really "touch" in the physical sense. Therefore it never reaches the target and it always reaches the target. For us humans it's equal to one. If you somehow could perceive infinity as a finite object, you'd say it doesn't equal.
@@jessepowellr4srry guess my history books were old... It's the one where an object supposedly in motion going along a path has to first travel half distance to destination, then half of the half, then half of that half at nauseam. Because it has to travel infinite halfs it will never reach it's destination in finite time. Which is demonstrably false. Second it's a paradox, what do you want me to do with it? Make it make sense?
@@Washeek chillax, man, you were half right. Zeno postulated at least nine paradoxes that we know of, with three of them being the most famous, namely: the dichotomy paradox; Achilles and the tortoise paradox; the arrow paradox. You were stating the premise of the dichotomy paradox, which divides space into infinite halves of smaller and smaller segments, but calling it the arrow paradox, which divides time into infinite instants where no motion occurs
It's possible to have systems where two numbers can differ by an infinitesimal. But the real numbers are defined in such a way that each number is equal to the limit of any sequence that approaches it.
What made the most sense to me was when one afternoon I decided to try finding what 0.999... was in base 3. In base 10, the way most people would find 0.999... is 1/3=0.333... then 0.333*3=0.999.... But interestingly, this only happens because our base 10 system doesn't like to divide 3s. We can see this type of phenomenon when we try to divide 1/2 in base 3. In base 10, the answer is obviously 0.5, but in base 3, if you go through the long division process, you find that 1/2=0.111.... So if we were to multiply that by 2, we would get 0.222..., which seems like an asymptote again, but the expression we just did is identical to the expression 0.5*2, just in a different numerical system. So what if we try to go through this process to find what happens when you try to reach 0.999... in base 3? Well, 3 is written as 10 in base 3, so the expression is 1/10, and we find that in base 3, 1/10 equals 0.1, because 3 essentially takes the numerical role in base 3 that 10 does in base 10, so their properties are similar. So, base 3's 0.1 is equal to base 10's 0.333.... But if we multiply 0.1 by 3 in base three, we get 0.1*10=1, which in base 10 would look like 0.333...*3=1. I like this explanation the most because it gets at the heart of what the actual reason we have this phenomenon is, rather than just trying to prove its truth. The only reason you can't divide one by 3 evenly with integers is because you can't divide 10 by 3 evenly with integers. Humans semi-aebitrarily chose a base 10 system of counting, but numbers don't exist only in base 10. You can't make sweeping mathematical judgements based on an artifact of our system of computing.
Fun fact, if you write a Python script to print 0.1 to 50 decimal places, you'll get a number slightly greater than 0.1. This is because 0.1 is a repeating decimal in base 2, aka binary. This will work in any programming language that represents decimals as floating point numbers, and it's why you should never use floats for things requiring exact precision.
This was the best explanation I have heard of this topic and it completely switched my incorrect opinion. None of the other common explanations worked for me but this did. Thank you!
Hmm yeah this logic checks out for me… I think for me the problem is context. 0.3333… being 1/3 is just because we can’t write the fraction any better as a number, so we know what it means. But if you just stick 0.9999… in front me and refuse to elaborate that really what ur saying is 3x1/3, then that’s when it becomes questionable, like why not just write “1” to begin with lol. coming from a non math guy btw
@@SN.LurkinG Okay, I don't know if you mind reading a bit, but I tried explaining in a way that would be approachable to a non math guy. In short, you really should write 1, unless you have a good reason not to. It's just that there are multiple ways to write some numbers. The decimal numbers are a representation that uses a geometric series with some chosen base. We mostly use 10, but you will encounter base 2 and 16 in programming and occasionally some others. This basically means that if I write for example 1.29, it means 1*10^0+2*10^(-1)+9*10^(-2). When you have a number with infinite number of decimals, it represents an infinite sum. And those are somewhat weird. Because those sums represent numbers, if two sums add up to the same number, the associated decimals represent the same number. It just so happens the sums 1*10^1 and 0*10^1+9*10^(-1)+9*10^(-2)... both sum up to 1 (as shown in the video). It will actually happen any time you end on infinite decimal places that contain a 9, you will get two possible representations. This doesn't work for 0.333... though, there's only one such representation. The reason is basically that 0.999... is in some sense the "last" number between 0 and 1, you can kinda approach it with the sum both from above and below in base 10. Then again, there is no last number between 0 and 1, if you name some, I can name one closer to 1, so 0.999... can't actually be smaller than 1. By the way, there's another intuitive way to see they're the same number. It can be shown that numbers are the same if their distance on the number line is zero, in other words if you can't find any number that would lie between them on the number line, it's the same number. Well, what number is supposed to be between 0.9999... and 1? There is none! This works because if the distance between two points is 0, the two points have to be a single one. I hope this helps. Sorry if it got too long or complicated, I am a math guy and I can be terrible at explaining stuff to non-mathematicians.
@@wiener_process Wow thanks for the reply, can’t say I understood all of it. What I meant to say originally, is that your simple 0.333… explanation did the trick for me, so thanks for that! This “number line” philosophy is very interesting tho, crazy how abstract and detached from reality math can get, I think that’s where I and most other people struggle with - the fact that these numbers only exists in the abstract “representation” sense.
I never considered .333... to equal 1/3rd, I believed that 1/3rd can't be expressed in a base 10 floating point system. Is 2/3rds .666... with a 7 at the "end"? That's more equal to 2/3rds than .666... is it not? Is .1 in base 3 between .333... and .333...4? I'm asking... I don't know
My sister is a maths teacher and recently had a parent meeting, because a father did not believe that his son was wrong by saying that 0,999... is not equal to 1.
Ha ha. At what grade level? Here in Denmark I have experienced students in my math class at the gymnasium who got to know in elementary school that 7^0 is 0. The explanation I got from the student was that their elementary school teacher said that if you calculate 7^2 you write 7 two times in the product but if you write 7^0, then it occurs zero times in the product (i.e. you have nothing). So that the answer should be zero. To this explanation I said to my student, "that was an excellent way to memorize the wrong answer, I am sorry that you learned incorrectly back then".
7^2 means 1*7*7. Is how we were taught. The powers shows how many times the number is multiplied to 1. So 0^0 is considered 1 but 0^n where n is not 0 Is considered 0.
@@patricklincoln5942 It's a bad way to memorize the wrong answer, really. An empty set of numbers, all multiplied together, is 1, because 1 is the multiplicative identity. Just like how zero Boolean values all ANDed together is true, and zero Boolean values all ORed together is false.
You get 0.0 followed by an infinite number of 0s with a 1 at the end. If there is an infinite number of 0s followed by 1, that is the same as 0 because the nature of infinity means you will never be able to reach that 1
@@danielhobbyistUnfortunately that means that particular explanation is not helpful or convincing to the audience (unless you have a very small audience)
@@danielhobbyist that’s literally what I said. 1-0.999… would supposedly have a 1 after an infinite amount of 0s, but infinity has no end, so it would be the same as 0. Someone clearly didn’t take basic English classes, or calc 2
"a 1 after an infinite amount of 0s, but infinity has no end, so it would be the same as 0" no, it would be undefined. It does happen to equal zero, but not for the reason you say.
I think what makes this difficult for non-mathematicians (including me) to grasp is that it asks a layperson to reassess their assumptions about how certain things work, like exactly 'what' infinity means in a mathematical context, and what repeating decimals are actually meant to represent. Actually 'proving' that 0.999... = 1 mathematically requires referencing higher level mathematics, and without more intuitive examples available, it comes off to a layperson as a mathematician saying "Just trust me", and I think people have more trouble 'just trusting' mathematicians compared to other experts because everyone has their own understanding of how math is 'supposed' to work. I know I personally needed the 1/3 = 0.333... examples from the comments to really be able to conceptualize it and reconcile it with my current understanding of math because unfortunately geometric sequences aren't something I've ever even approached in my life. That being said, I'm not a fan of how people in the comments or on the internet are smugly approaching this like people who don't understand are stupid or unteachable. Sometimes when someone pushes back against teaching, it's because a part of them doesn't want to accept it or learn, but sometimes it's just a part of trying to reconcile things that don't immediately make sense. If you make someone feel directly attacked or insulted, then chances are it doesn't matter how sound your logic is, you will not get through to them. If you 'do', then you've made learning an unnecessarily unpleasant experience (which makes people even more hesitant to learn in the future).
For what it's worth, I wouldn't take what people say too seriously. Especially since there exist models of math where 0.9999.... =/=1. Anyway, I'm assuming that you gst 1/3=.3333... from long division of 1 by 3, infinitely many times? It's possible to reformulate 1/3 as a geometric series in the same way as the video if you want to try a geometric series example.
Its just some dude decided 0.999... is 1, to make it easier to calculate stuff. That's all. This guy and others in youtube comments spouting same stuff that they memorized from their classes, almos like brainwashed.
@@jersefrenzer1265 Yeah, that's how I got 0.333... I don't doubt you could use a geometric series, but like I said, those are out of my grasp, I never learned about them. Though at this point I am considering trying, if for no other reason than to keep learning.
My take on this is that the root of the misunderstanding is that 0.9999... is considered to be the representation of a number, and thus this number cannot be 1, because as they say, only 1=1. But here's the thing, "2-1" is also equal to 1. But "2-1" is strictly not a number, it's an expression that has to be evaluated to get a number. So maybe, one way to help people understand the issue is that "0.999..." is an *expression* that evaluates to exactly 1, just as "2-1", or "lim x when x -> 1".
You're not wrong. I think it's definitely important to stress to people that an expression that evaluates to a number is still just a number. I differ with you slightly because 2 - 1 is still just a number, strictly a number even. Like, don't even bring evaluation into it. Introduce people to more cases where the same number can be written in "strange" ways. But I do think you're right, that people get the idea that there's a difference between an arithmetic expression (that evaluates to a number) like 2 - 1 and a "bare" number, like 0.9 repeating. I think we just need to stress harder that they are the same thing, even though it seems to be counterintuitive to many people. 2 - 1 and 1 are exactly the same thing. Cheers.
But that's the thing, everything you write down is an expression. A number is an abstract concept, 1 is not actually the number 1, it's a vertical line with a little flag on top. We use it as a symbol for the number 1. Once you realise that representation and semantics are separate concepts, this becomes a whole lot less confusing.
@@hannessteffenhagen61 You are right of course, "1" is also an expression. But I still think this approach of stressing that there is a need to evaluate the expression "0.999..." to get to "1" provides a way to reach the correct conclusion, namely that these two expressions are both equal to the number 1.
It's a chose your poison situation. And the rules you decide to play by. Whether you decide It's equal to 1 or not, leads you to different "Games" of mathematics. You shouldn't convince yourself that it being equal to 1 is *Always* the case.
Yes. All numbers are expressions. "123.4" is, by definition, (1•10² + 2•10¹ + 3•10⁰ + 4•10⁻¹). They teach this in second grade, with ones and teens and hundreds. Base 10 and all other bases are just a representation of the abstract number, not the number itself.
If they were not equal, then there should exist another real number strictly between them. Now we could rigourously show this, but it’s also pretty clear and intuitive that there’s no number you could find that’s strictly between the two. The “…” in the decimals really signifies that we are taking the limit of the sequence with finite decimals, i.e. 0.9, 0.99, 0.999, … And the answer is precisely the limit, which is 1. And if you want to think of it as a sum like you did, we also say that even though the partial sums only approach the limit, we say the entire infinite sum is *equal* to the limit as long as it exists.
@@marksandsmith6778 infinity is a concept/idea. It is not something you can count/measure like a number. And just like any other concept or idea, we have rigorous definitions and rules on how infinity is to be handled and intepreted. And for our particular context it is in the form of a limit. It is unintuitive for sure because we live in a finite world, but we have proper mathematical tools on how to properly use and understand it. So merely saying “infinity does not exist” is so meaningless and worthless because it clearly shows you have very little understanding of the topic.
Good point on the "..." implying a limit, the "0.999...=1" is so confusing because the argument about 0≠1 on the first position seems to seal the deal that the numbers can't be equal, but this is a limit so you can't just compare positions, as this works only between a number and a number
@@Firefly256 that is ONLY if you want to get into the hyperreal/surreal numbers, which is very advanced, non-standard analysis, which is both unintuitive and not at all for the average math person like you and me. If you want to stay in the regular real numbers that we all are familiar and comfortable with, then infinitesimals cannot exist.
I think the confusion stems from the fact that pure math isn't always indicative of anything you can find in the real world. You can't have 0.999... of an apple, for instance. Whatever particles you use to measure how much of an apple you have are necessarily finite, and so you'll eventually hit a last number.
This is actually the same thing as the universe expanding but, you're not really supposed to talk about that during Zeno's paradox because they did not know those things yet. So it's not a part of the real thought experiment.
But you can have 0.999... of an apple. It's the entire apple. Because 0.999... is defined as having NO LAST NUMBER. You think you'll "eventually hit a last number" but that's literally not what 0.999... means. You'll never hit a last number, because it's = 1.
My understanding basically that numbers are idealized forms that exist within the mind. We can only imagine everything as singles objects, just as we can only idealize a perfect circle or absolutely straight line. An infinite series is an irrational number within the real number set of objects. They tend to be an abstraction of real objects based on our ability to deduce subject matter. Anyways, that's my understanding of the problem.
The only thing I know for sure after seeing this is that the comments section of a TH-cam video about math is a postapocalyptic war zone populated with homicidal nerds.
I was expecting to find hundreds of comments talking about how hilarious it was that the guy denying 0.9999..=1 opened his first comment in the thread with "dear united States public school victims" Instead I found a load of people siding with him, a minute ago I read someone deny that if you divide 1 by 3 you get 0.33333... TH-cam is weird
@@matta5749 three points for ya doofus 1. My interest in maths is purely a casual one, I'm not trying for a doctorate 2. I dont know if you noticed but of course my comment was mathematically non rigorous, it was a social comment, not a maths one And 3. You're just actually wrong, someone pointed out the whole 1/3×3=0.999...=1 thing, and the other guy said "this statement begs the question that 1/3 is 0.333...", that's legitimately him denying that 0.333... Is 1/3 Take yourself less seriously ya goofball, we're all having fun here
i love the videos but yeah the comments section can be really toxic. even though i enjoy math and science in themselves, the community kinda scares me away from the fields lol
Tell people that 1/2, 2/4, 3/6, 0.5, and 50% are the same number and no one bats an eye. Tell them that 0.999(repeating) and 1 are the same number and everyone loses their minds.
@@MichaelMoore99 1 - 0.999(repeating) = 0.000(repeating)1. Which is still not 0. Red = blue, 1 = 2, 0.999...999 = 1, taxation isn't theft, same acrobatic reasoning. No amount of ball twisting mathematics will ever magically turn 0.999...999 into 1.
My understanding is that asymptote, by definition, is a line that continually gets closer to a certain distance (in this case 1) without ever meeting at a FINITE distance. But infinitely recurring is not finite. That asymptote, taken to infinity, therefore is 1.
This is correct. Infinity is the next step after forever. So asymptotes are not reached with just natural numbers that take forever but infinity in a sequence is ω the next sequence after the forever sequence of natural numbers. So you reach the asymptote after forever but without the one additional step (forever+1) you never reach the asymptote.
@@ferretyluv Yes and we are talking about infinite sequence of terms so the correct infinity to use is the ordinals. 0.999... does not have the size of aleph0 but it has ω digits.
I like this handwavey explanation I use: 1/3 = 0.333… 2/3 = 0.666… 1 = 1/3 + 2/3 = 3/3 = 0.333… + 0.666… = 0.999… Fun fact! Any repeating decimal can be represented as the digits of the repetition divided by an equivalent number of 9s. 0.111… = 1/9 0.151515… = 15/99 0.123123123… = 123/999 And of course: 0.999… = 9/9 = 1
I had a hand-wavy method to show things like this in 6th grade, I recall my teacher asking me what 9.9999… was equal to and my method said it’d have to equal 10. He laughed and said therefore my method was wrong. About a week later he met with me and apologized, and explained the flaw in his thinking and was kind enough to show me the rigorous method for proving this (which, to be honest, I didn’t understand at the time but it’s the thought that counts). 😅
This is the result of two definitions: 1. 0.9999... is defined as the series 0.9 + 0.09 + 0.009 + ... 2. A series of real numbers, a_0 + a_1 + ... is said to equal a real number L iff the limit of the partial sums converges to L. This is one intuitive translation of what it means to add a number "infinitely" many times. However, it is not the only one. I think it can be more helpful to teach non-mathematicians that this is why mathematical definitions are important, rather than, what I sometimes see, 'uhmm akshully' X is defined this way. For example, the idea that a series "equals" rather than "converges" to something is honestly probably just a matter of convenience with how we express maths. This is merely a definitional thing -- it doesn't *have* to be that way. Although, it'd be tedious to write out many theorems. In a similar sense, in the above definitions we make use of real numbers (usually implicit), yet again, one could easily define equality via some other number system (perhaps one that contains infinitesimals). In the hyperreals, the partial sum of 0.9999... is *never* infinitesimally close to 1, hence in some definitions we say it is not convergent to 1.
Thank you, it's relieving to read such a sober comment among so many comments trying to "dunk" on people. Yes, 0.999 recurring = 1, but it's ultimately a matter of *convention*. Once it's defined as a series, then it becomes 1. It's not 1 because "you're a dumb high school dropout who can't do math".
My other take: The sequence 0.9, 0.99, 0.999 is an assimptote. But we are not talking about the sequence, we are talking about the number 0.999... that is the limit of the sequence.
The only thing I don't understand about this is that if the limit of the geometric series approaches 1, how can the geometric series itself be equal to 1.
Saying that the geometric series is equal to 1 is just a colloquial way of saying that the sum of this series is equal to one (i.e. the limit of its partial sums).
You're right, he equivocated those. The real answer is mathematicians decided 150y ago to treat infinitesimals as 0 because it works in the real world and makes math 1000x easier.
'You're right, he equivocated those' Incorrect. It is proven that 0.999... = 1. 'The real answer is mathematicians decided 150y ago to treat infinitesimals as 0' Again, incorrect. There are literally no infinitesimals in the space of real numbers. A positive infinitesimal is a number that is less than any real number, but greater than 0. For any positive real number to be an infinitesimal it would have to be less than itself, but the relation '
The way I would explain it is that 1 - 0.999… is 0.000… followed by a one. However you would never get to the 1 so the value is 0 meaning the difference is zero, or in other words, there is literally no difference.
Something that helped me understand this: When you write a sequence of decimal digits to describe a number (infinite or not), the number you are describing is not that sequence. It is the **limit** of that sequence. Numbers are not sequences and they do not "approach" anything. The sequence of decimal digits that represented by 0.999... **approaches** 1, but if you want to interpret it as a singular **number** then the number can only be 1.
I like this argument! You just use the uniqueness of limits :) Similarly, you can just show that if 0.999… is the limit of the sequence 0.9, 0.99, 0.999, etc. then 1 is also a valid limit point of the sequence, but again, the limit is unique so they must be the same number But also, I wouldn’t say that 0.999… is a sequence in itself. I think your second interpretation is a better way to see this number, it really is the limit of the sequence.
No it is 1. The easiest way to think about the number line. If you go by digits of 0.99... you see that for each digit it gets closer to 1. But every time you think there is a distinction between 1 and 0.99... the next digits will remove it. since it goes for infinity that means that there will never be anything between 1 and 0.99... Therefore both numbers occupy the same place in the line. Thus are the same number
"Numbers are not sequences" Weeeell... There are 2 popular ways of constructing the real numbers. One is using Dedekind cuts. These are quite abstract, but probably more useful when proving some properties of real numbers. The other way is using sequences. If a sequence of rational numbers (where here, those rational numbers are not stirctily the subset of the reals) converges to some value, we "assign" that value to that sequence. Or rather, if 2 sequences have the same limit, we say these sequences are equivalent and we set the real number x to be the equivalence class of those sequences
Interesting, havent seen Dedekind cuts yet. But I fully agree with you, my point was just that the number is not the sequence itself, the number is the limit of the sequence(s) that converge to it (unless im missing something relevant)@@methatis3013
I've debated this with a (humanities) professor before. He thought it was some trick, like the "proof" that 1 = 0. He started from the assumption that the equality is false, and then talked about how interesting it was that you can use math to prove something that isn't true.
Haha but math knows this. Logic tells you that from a false statement you can derive false conclusions. So he started with a false statement that equality is false, and derived another false statement that math lets you prove false conclusions. It's logical implication 101. (False => false) is true.
The reason I think many people are so convinced that 0.999... has to be less than 1 is that they have misunderstood what infinity means and confuse it with arbitrarily large, but finite. It is obvious that any finite number of 9's after the decimal would represent a number ever so slightly smaller than 1 and it leads to phrases like "asymptote", "arbitrarily close but not equal" and "at the end of the 9's" being common in such arguments. With an infinite number of 9's however there is no last digit or point where the supposed approximation stops as infinite literally means never ending and the result is exactly 1.
Exactly. I think an easier way to think about it is by subtracting 1 from both sides of the equation to get 0 = 0.000…1 There are an infinite number of 0s after the decimal point followed by a 1. Because of the nature of infinity, it is impossible to reach that 1 because the number of 0s before it are, well, infinite. Therefore, the 1 doesn’t really exist so 0.000…1 it is just equal to 0
My interpretation would be that there is a reason someone writes 0.999... instead of just writing 1. What is trying to be communicated when someone writes 0.999... other than just 1. It's plainly obvious that 0.999... and 1 are literally different symbols.
This exact thing is what I am trying to explain to doubters too. The whole thing boils down to the infinite or not. Also trying to ask what number comes between 0.999… and 1. It must be the difference of the two numbers, that does not equal to 0, but the difference does equal 0, so the numbers have to be exactly the same. 1-0.999…=0.00000… And if you are tempted to put a ‘1’ at the end somewhere, you just stopped the infinite process of putting zeroes, making it a finite number, no longer infinite. So the difference is 0. 1 is exactly equal to 0.999…
there are several easier ways of showing intuitively that they are the same. 1. if x and y are different, then there should be some number that could come between them. but 1-0.999 repeating = 0 2. 0.333 repeating = 1/3. multiply both sides by 3 and we get 0.999 repeating = 1 3. x*10-x = 9 works for 0.9999 repeating and 1.
There are two types of people. One will figure: OK, that makes sense. The other will, from that point on, refuse to accept that 0.333... is the same as 1/3.
My favorite way that my professor showed and proved this (not serious proof but just a quick thing) was asking what 1/3 equals. 0.333….and then he asked us what if we were to multiple0.333 times 3. We would get 0.999… but if you take 1/3 and multiply it by three, you get one!
@@johnlabonte-ch5ul What a steaming great muppet you are. Belief is not required. 0.999... = 1 precisely for all purposes (except appearance), 1 - 0.999... = 0 on the nose. You do not know what the word "equivalent" means in mathematics (or probably everyday English either). Yet you persist in using it all the time. Are you ever going to try to prove any of your claims. Are you ever gong to try to find a fault on any of the following three arguments: 1) 0.999... := lim n->oo 0.999...9 (n 9s) = lim n->oo 1 - 1/10^n = 1 2) 10 * 0.999... = 9.999... => 9 * 0.999... + 0.999... = 9 + 0.999... => 9 * 0.999... = 9 => 0.999... = 9/9 = 1 3) If, as you "believe", 1 - 0.999... = d > 0 then 0.999... only has floor(log10(1/d)) consecutive 9s. To get the required infinitely many 9s, we need d = 0. Come on, combine the full power of both of your braincells, and show everyone how smart you aren't by disposing of those three arguments. I'm still WAAAAAAAIIIIIIIITINGTINGTINGTING... LOL
@@johnlabonte-ch5ul KarenTheBonehead, you haven't addressed any of them. OK if 3) is circular that would disqualify it as a proof, and I'd have to scrap it. So prove that it is circular. You will fail because it isn't circular. I can't begin to imagine how even a humongous muppet like you thinks it's circular. You: "In order to believe d>0 you need d=0???" You have lost the plot again. How on Earth did you come to that "conclusion". (I guess that's what it's supposed to be!!!). I said if d > 0 then there are only finitely many 9s, but if d = 0 we'd get the required infinitely many 9s. You reading comprehension sucks as bad as your math comprehension. I guess that tumour is taking it's toll. You forgot to take your meds again, didn't you.
The way I was taught this was "Divide 1 by 3, you get 0.333... which you then multiply by 3 to get 0.999... but similarly if you do 1:3×3 in one go you get 1. Therefore 1 and 0.999... are the same."
@@Felipe-sw8wp You can express this as forall j in N: exists k >= j: x_{−k} B-1 where x = sigma B^n sum_{i=1}^{infty} x_{-i} B^{-i} for base B >= 2 and sign sigma in {-1,1}
I learnt this in 8th grade ( India) x=0.999999.... -(equation i) 10x=9.999999.... -(equation ii) SUBTRACT equation i from equation ii We have, 10x-x = (9.999....) - (0.9999....) So by subtracting 9x = 9 (divide by 9) x=1 Therefore, 0.999.... = x = 1 . . . is it correct sir??? iam in grade 10 btw, just completed my sst boards and studying for maths 😅😅😅
Yes this is correct. A counter argument I heard was that 9 recurring is -1 (or something along the lines of that), but the difference between 999... and 0.999... is that recurring decimals are convergent. In other words, your proof works because 0.9 recurring is convergent.
1 divided by 3 = 0.333... one third times 3 = 1 0.3333.... * 3 = 0.999999.... If 0.9999.... does not equal 1, arithmetic falls apart. There are several proofs (why is the plural of proof not proves? /actually asking) for this. This is the one I prefer because it's easy to remember and simple to convey
I have seen situations with fractals where it's significant that 0.999... is not equal to 1, but it's important to realise that they're not using the standard real numbers but are using infinitesimals. So perhaps the quarrel is simply a mismatch between number systems.
Nah. I've had people argue these kinds of things with me that have masters degrees in engineering. You barely need to understand a limit, to practically apply a limit.
@@ralphinoful engineers have funny understanding of math, they just understand it enough to use it in life but not to the core. That makes them potentially the most delusional kind. I am an engineer btw.
simple proof that 0.9 recurring = 1 (i learnt this in this year, grade 9 ) let x = 0.99999….. 10x = 9.999999…… subtracting both equations 9x = 9.0 which is 9 x = 9/9 x = 1 hence proved
The problem here is that people don't quite understand the details about limits. Any increasing sequence that has a supremum, always has an upper limit, which equals to that supremum. However, the supremum itself is not part of the sequence. In other words, whatever number of 9s you write, the number in question shall always be less than 1. That's why we can only deal with limits here, not with the sequence numbers themselves. From a computational standpoint, this question is nonsensical, because a Turing machine trying to compute an infinite number of 9s will never stop, therefore, never yielding an answer to the question. So basically, there is a convention that when we write infinite sums (improper integrals too) by that we really mean limits, not that the sum of those numbers actually yields anything specific.
The super deep answer is that there's nothing stopping you from insisting that 0.999... ≠ 1. Math is just a system of rules and conventions made up by people. There's no ground truth. Mathematicians will often consider math with different rules, sometimes leading to interesting developments. However, if you want to insist that 0.999... ≠ 1, you need to acknowledge that you're not using the same math as everyone else, and it's very likely your system of math is both internally inconsistent and lacking any practical value.
I mean there’s nothing stopping you from insisting 1 = 0 or that the sky is red. Simply insisting something doesn’t inherently make that something useful or mean anyone has to give it credence.
@@alephnull3535 Yes, exactly. My broader point is that people seem to think that math has inherent truth, and you cannot possibly argue against it, but actually the foundations are "arbitrary", albeit the foundations most commonly used are broadly useful and nearly universally accepted.
on the other hand, sometimes defying common assumptions can also lead to new valid and interesting things, as was the case with Euclid's 5th postulate.
Not all maths agree that .99... is 1. I am not a math, my vote doesn't count, but from the maths I know and see they are outnumbered but not an infinitesimal.
@@johnlabonte-ch5ul 'Not all maths agree that .99... is 1. I am not a math' This is just your assumption - you can't point to any who don't agree with that.
In pure mathematics, it ALWAYS matters. There's no such thing as "close enough". Either =, < or >. There's is no other option. We're not taking a measurement or approximation. We have the perfect exact precise answer.
Its like a bell curve on mathematical knowledge. People with little knowledge can easily see this they dont require complicated proof to accept reality. Then we have people who think themselves genius just because they learned some advanced math operations, things proven by a simple google search are now questioned again, all those people who looked into it before must have been false. And then we have people who are on the other side of the bell curve. "Haha, look i made a Charizard-like curve"
Regardless of your understanding of this concept, screaming at someone who doesn’t understand a theoretical mathematical concept shows either a lack of your own patience or an inability to teach in a way that is comprehensible to others. That is why the educator referenced in the original post should be fired, not because the OP was mistaken about the concept itself.
It's an axiomatic thing, but yes in usual axioms .999... equals 1. That's a weakness of the decimal representation system, there are two equally valid representations. I a nutshell, .999... is not a sequence, it's the limit of the sequence, and that limit is 1.
I think the best way to help non-mathematicians, or anyone with a Jr. High or lower level of mathematics, is to simply demonstrate how this "problem" is just an illusion and arises when you start dividing by primes that aren't factors in your base. Show 1/3 and 1/5 in base-6, for example, or how 1/3 creates a repeating decimal in not just base-10 but also base-2. In my experience they eventually have an "ah-ha!" moment after just a few examples showing that ultimately the problem isn't with mathematics, but their imagination / mental representation of values like 1/3 when they see a decimal representation.
I think a problem with writing 0.999…=1 is that we assume the reader to intuitively understand the left side as the limit of a series. As mathematicians we know that the equal sign in this case is a convention that is well defined and may fail to understand why someone unfamiliar with the formal definition of limits wouldn’t accept that.
'As mathematicians we know that the equal sign in this case is a convention' Count me out, because I know that it's not a mere convention. 0.999... = 1. And why do you say 'as mathematicians, we...' when you are not a mathematician (otherwise you would know that 0.999... = 1)?
@@thetaomegatheta I think your disagreement with OP is misdirected. There is certainly convention involved, you just don't place it in the equals sign. But there's undeniably convention involved. 0.999... is meaningless. Until you have some convention to explain what it means. Like, for example, the limit of a sequence of partial sums i.e. the sum of a convergent series i.e. a decimal expansion. Correct me if I'm wrong of course. I do know there are more subtleties at play, as before you can play around with real numbers you do need to define them in the first place, but I'm fairly strict on the fact that the limit of partial sums definition is just that, a definition.
I had this argument once... with my high school math teacher. Eventually when I brought in a mathematical proof it reduced her to spluttering about how if she chopped a bit of a penny off it would be less than one penny, demonstrating her proof with a penny on her desk and her hands in a decisive chopping motion. When I tried bringing it back to math after that, she would just get upset and threaten to send me to detention if I didn't stop.
I do understand the initial scepticism when you first hear about this. It does take a bit of processing to fully come to terms with this concept, but gagging the student won’t help much.
@@AscendantOatthat won’t help, since she‘s just going to imagine “yes, but 0.0000… *is* bigger than 0 - there will be a 1 at the end”! Infinity eludes our imagination
infinitesimal isnt zero, and decimal notation necessarily defines 0.x as less than zero, you tell someone that something that definitionally doesnt equal one does equal is a bold faced lie and they will naturally be upset and it wont ever go away unless you admit that as a decimal it doesnt but if it were a series then it would approach 1 and so is functionally identical
All of the comments I've seen are good explanations, but rely on some result or intuition; I think ultimately the answer comes down to how real numbers are defined in the first place. There are several ways to do so in Analysis, but the most helpful method here is defining them as equivalence classes on Cauchy sequences (sequences whose terms get arbitrarily close together. As it turns out, this is equivalent to the least upper bound property commonly taught). So take the sequence of just infinitely many 1's (which surely has a limit of 1) and the sequence 0.9, 0.99, 0.999, and so on, which is Cauchy. If you take the limit of the difference of these two sums, we see that clearly the distance between the terms is shrinking to an arbitrarily small amount, and so converges to zero. Even though it technically never "reaches" 0 in a finite number of terms, we just define them as being equal because we can't impose any stopping point where they stop getting closer. So they fall into the same equivalence class, which is just how we define real numbers.
It IS equal to 1 because of the definition of the real numbers. Think of 0.999999... as a SEQUENCE: 0.9, 0.99, 0.999, ... this will converge to 1. The real number 1 is defined as the set of Cauchy sequences of rational numbers that converge to 1, an d (0.9, 0.99, 0.999...) is an element of this set. Hence it is equal to 1.
The important thing to ask is 'in what context'. There is no context other than abstract mathematics where 0.999...=/=1. In computing, two numbers are equal if the difference between them is small enough, so a computer would say they are equal. In any real world scenario, the difference is undetectable to measurement devices, so they are equal.
That's a bad way to explain. Equal would imply no difference. Saying "the difference is undetectable" means there is a difference, you just need a better measurement device. Because your current application can treat them as equal, does not mean that will hold true in the future with more advanced technologies.
@@vicc6790Saying that things are equal literally means that the difference is undetectable. As I said at first, abstract mathematics is the only time that 0.999...=/=1
I've always been taught that 0.99999... Is 1 using a variable. Let's say x = 0.99999... Which means 10x = 9.99999.... Subtract 10x to x, 10x-x = 9.99999... - 0.99999.... Then you get 9x = 9, and x=1 Which means 0.99999=1 It also works with 0.6666 and 3.333 where it's equal to 2/3 and 1/3 respectively.
@@jm329 basically. If you remember the infinite hotel paradox (i think ted ed explained it in a video), there's always an infinite number of rooms and infinite numbers of people, in which the hotel will always have enough room and every person will get a room. Every time you add a person, the person inside would just move to their room number + 1, then theres always a vacant spot in room number 1. Just like 0.999... (infinite amount of 9) Times 10 is just 9.9999... (Still infinite amount of 9) And then subtract infinite amount of 9 with an infinite amount of 9 and you get 0
One minus an infinitesimal is not equal to one. That’s why conditionals like “approach” and “converge” are used. If you want to make a context where it can be true, then sure.
And there is one more point to consider. Infinitesimal numbers do not exist in the standard real number system (they exist in other number systems, such as the surreal number system and the hyperreal number system). 0.(9) - is an ordinary real number, a constant, which by definition is equal to the limit of the sequence 0.9, 0.99, 0.999, ..., which is 1.
'One minus an infinitesimal is not equal to one' Cool. And irrelevant, as 0.999... is not 'one minus an infinitesimal'. Like, what infinitesimal would that be, and not any other one if there even were any infinitesimals in this context? 'That’s why conditionals like “approach” and “converge” are used' Lol no. Learn what the words you use mean instead of making mindless guesses about their meaning and telling a mathematician about them supposedly not knowing what those words mean.
What infinitesimal would that be? Why ask that question when the context of the conversation makes it obvious it’s the one closest to zero without being zero. See, context establishes a valid framework for discussion, which goes to my point above. If you’re not comfortable working with infinitesimals and simply need to think of everything in terms of the real number system, that’s its own context, too.
@@edwardpotereiko "eed to think of everything in terms of the real number system" Because here we are talking about the real numbers. "ee, context establishes a valid framework for discussion" And the context here are real numbers.
@@edwardpotereiko 'What infinitesimal would that be? Why ask that question when the context of the conversation makes it obvious it’s the one closest to zero without being zero' Now, ask yourself, is there such an infinitesimal in hyperreal numbers, let alone real numbers - the context of the video? 'See, context establishes a valid framework for discussion, which goes to my point above' The context is the space of real numbers. There are no infinitesimals in the space of real numbers. 'If you’re not comfortable working with infinitesimals' You don't even know what they are. 'and simply need to think of everything in terms of the real number system, that’s its own context, too' When people say '1+1 = 2', they mean it in the context of real numbers and standard notation using Indo-Arabic numerals, unless another context is specified. You may whine that it's not the case, because there are contexts where that might be not true, but most people will understand what the actual context is.
From a science perspective, we can only gauge a value by a limited amount of accuracy. So at some point its no longer possible to know if the next value is nine so that value will be rounded up resulting in the value of an .999x being just 1 for all usable purposes outside of purely hypothetical or mathematical usage.
The adult is arguing that the real number represented by the string „0.999..“ is the same as the real number represented by the string „1“. The strings are different. You point this out at 0:40, different formulas but same values.
Numbers aren’t formulas or strings, they’re numbers, they’re completely defined by their values. 2/2, 7-6, and 1 are all different “strings” but that doesn’t change the fact that they’re all still 1. The same goes for 0.999…
@@alephnull3535 I did not write that numbers are formulas or strings. There is a clear distinction between the representation (symbols consisting of glyphs or black and white pixels, strings = tuples of symbols) and the associated numbers.
I feel like it's important to note that it is equal to 1 because of the properties of geometric series or limits whenever it's written down. Like the "+c" after an indefinite integral. It's a little nuance that can cause chaos when not formally acknowledged.
For as much as people are attached to the definition of real numbers as infinite decimals, let's be honest: no one uses it for doing arithmetic. Adding 1/3 + 2/3 as 0.333...+0.666...=0.999... is cute, but it's useless. Adding fractions is much easier. It's way faster to do 1/17+1/19 in fraction form in your head than do it with the help of a calculator but using only the infinite decimal representation of those numbers. If you exchange the + sign with -, * or obelus, the problem gets easier and easier do to with fractions, and gets harder and harder to do with infinite decimals. In conclusion: it doesn't matter what 0.999... is, it won't make any difference.
@@alephnull3535 I think their ellipses in the first number are meant to imply "many" and not "infinite"... but the use of the same ellipses in the second number to mean infinite does make things a bit confusing...
@@alephnull3535 Other way around. 0.999...9 is a number. 0.999... is an infinite series. To be a bit more pedantically correct, 0.999...9 represents all numbers where the count of the nines is arbitrarily large, yet finite. Each of those is a number less than 1. And 0.999... represents the infinite series where the series of 9s is infinite. The value of the infinite series is 1.
The much simpler way to explain it is just if 1/3=0.333… then 3*(1/3)=3*(0.333…)=1=0.999… That might be circular reasoning but it’s much easier to accept that 1/3 = 0.333… and explain it on that basis.
This is the route I always take when I just want to get the conversation over with. It's not as rigorous as hopping back to the geometric series that the "..." represents, but it'll do.
I've had people ask me this and I usually say: What is 1/3 as a decimal? 0.333..... What is 1/3 + 1/3 + 1/3 as a decimal? 0.999... What is that as a fraction? 3/3 What is that? 1
Real numbers are defined as the equivalence class of all sequences of rational numbers whose sequence values get closer to each other and who's sequence differences converge to zero (Cauchy sequences). All real number math can be defined through this construction. So, 0.9 repeating is a member of the equivalence class with 1.
It’s probably a 1:1 ratio. I went to a catholic school so my math teachers believed the earth and its biosphere was created as is in a week 6000 years ago and even they understood 0.999…=1.
I remember learning 1=0.9999999999 in calculus 101. I also know that 1=0 via: a = b a - b = b - b (a - b)/(a - b) = (b - b)/(a - b) Such that b - b = 0 as a number reduced by itself is 0 (a - b)/(a - b) = 0/(a - b) Thus: 1 = 0 Therefor all numbers equal 0 as all numbers are groups of 1s added together. Therefor all numbers are lies and dont exist.
'I also know that 1=0 via: a = b a - b = b - b (a - b)/(a - b) = (b - b)/(a - b) Such that b - b = 0 as a number reduced by itself is 0 (a - b)/(a - b) = 0/(a - b) Thus: 1 = 0' You don't, because, by the initial condition, a = b, and you also divided by zero there. But sure, cute attempt at refuting a basic fact that 0.999... = 1.
Ominous I knew a man I told not to argue with a disabled person and instead he berated and humiliated him explaining that he vainly insisted he was correct..thank you for verbally explaining this to others
Geometric sequence and series formulas that you need to know: 👉th-cam.com/video/Pvee2Zbh_Zk/w-d-xo.htmlsi=-0oeVuI4RZ5ABNd_
ah ha, but there in is the issue. 0.9999... is not a limit. We also are under the delusion that if we have a limit that we can actually reach that limit such that we can actually find the slope of a tangent to a curve at any point. The fact is we can never actually achieve that limit or we are dividing by zero. Even 0/0 is undefined so to have the actual tangent is technically not possible.
Here are my approaches:-
Approach1:
1 = 99/99 = (99.99... - 0.99...)/ (100-1)
Notice that the numerator and the denominator need to be equal, so that we get 1 as answer.
Hence by comparison of numerator and denominator, we get 99.99... = 100 and 0.99... = 1.
Approach2:
Let X=0.9999...
100X = 99.99....
On substrating them,
100X - X = 99.99... - 0.99...
99X = 99
X= 99/99
X=1.
Therefore, 0.9999... = 1.
@@anuragguptamr.i.i.t.2329 still wrong. there is the idea of infinitesimals. 0.999... is such an infinitesimal and you will never reach 1. The idea that a limit is equal to the assigned number is false. f(x)=1/x, as the limit of x approaching 0 is undefined. You will never reach 0. same is true for ln x , as you will never reach 0 nor can x be negative.
@@kennethgee2004 No, you're wrong. A limit is, by definition, that value. There is no question about "reaching" anything. The difference between 0.999... and 1 is zero.
The stuff about diving by zero is a red herring. You are confusing yourself by thinking that 0/0 is undefined. It isn't. It's merely indefinite, and when it is the limit of the quotient of two functions, it often has a definite value. Think about sin(x)/x and work out whether its value when x=0 can be anything other than 1.
@@RexxSchneider You really need to revisit how the limit is born. It is the secant line that we are moving ever closer towards the tangent line. The secant line becomes length 0 at the point of tangent, but then 0/0 is undefined. We never actually reach that point. We just see that it is going towards that point and write an equal sign. The value is not actually equal.
"You would would miss the moon by a million miles." No, I'm pretty sure you would miss the moon by 0.000... miles
which, IF those defending the intuition the 0.9999… ≠ 1 are correct, will be infinitesimally more than zero.
Bottom line, no one is really defending that _according with the formal definitions of standard mathematics_ 0.9999… ≠ 1, which would be trivially wrong. People are rather arguing that, given that there’s such a strong intuition conflicting with the standard _convention,_ who is to say that the standard formalization and axioms are for sure correct? Now THAT is a philosophical question, and it’s a legitimate one. Surely there can’t be a _formal indisputable proof_ that the very formal system being called into question is correct.
One can argue that “objective truth” in mathematics makes no sense _outside of a rigorously defined formal system,_ which I guess most mathematicians would agree with. But that is a philosophical position, to be argued philosophically. It is NOT an indisputable theorem inside a formal system. Philosophically, it’s perfectly legitimate to disagree with this, and to defend, for example, that there’s an objective structure to abstract quantities - so that any given formal system can be right or wrong in capturing such structure. And if that’s true, it’s no wonder that elusive questions about infinity are places where the standard math could be wrong - after all, it wouldn’t necessarily make any practical or theoretical difference to be wrong in such extreme cases.
@@SelbyClaude I ain't reading all that.
How much of infinity is useful?
@@SelbyClaudeif the axioms are incorrect, then logic is false.
1 does not equal 1, 1+1 does not equal 2.
Multiplication isn't real.
Nothing is real.
Go look at the formal proof, it is basic bitch algebra and it proves definitively that 0.999... is equal to 1.
They are just different ways of writing 1.
It isn't intuitively wrong. Your intuition is wrong if you think it is wrong.
Yes, you can have an intuition is wrong. Contrary to the popular saying about opinions, opinions can in fact be wrong too. Because opinions that claim false things are true, are certifiably wrong.
@@meateaw no it is you who is wrong. Axioms aren't correct or incorrect. They just are. They are the rules by which the game is played. Other rules could have been chosen, and often other rules are chosen. By the most common set of rules 0.999...=1, but you can absolutely pick a consistent different set of rules where 0.999... is not 1
The way I like to shut these people down is asking them what number lies between 0.999… and 1. If they’re not the same number, there must be a number between them because the real numbers are a dense set
That’s an excellent demonstration. Density is so much fun.
And you can’t really disagree with that unless you want to go into hyperreal numbers which is already very advanced and non-standard analysis which is also quite unintuitive.
So if you want to stay in the regular real numbers we are familiar with then you have to accept it.
That requires them accepting that there must exist such a number, and .9999.... can't simply be the largest number less than 1. Which is a much easier thing to believe. And is also not entirely incorrect... real numbers might not have such a thing, but number systems like hyperreals freely accept that .9999... is exactly that.
@@vib80 If they say that 0.9999... is the largest number less than 1, then ask them what 1 - 0.9999... yields.
( Ok, if they then insist that the result of that calculation is 0.0000.........1, then you probably have to give up, because obviously they don't understand what "infinite" means. :D )
That's my move! I never get an answer to that though. How about you?
0.999... is not an asymptote in this case, it's a flaw in our ability to write certain numbers without using fractional form
but u cant use fractions to write .9999…
@@user-ee5kl6oi7t Yes: 1/9 is 0.111... So you take 1/9*9=9/9=1 right? Now lets write the same formula in decimals: 0.111...*9=0.999...=1. 0,999... is 9/9 in fractions.
@@user-ee5kl6oi7tYou can because 1 is a rational number.
@@user-ee5kl6oi7t I think the confusion here is that we're all taught the long division algorithm, but there's no way to naturally get 0.999... through the standard long division algorithm. Once you start working in different bases, or even looking into repeated fractions and basic number theory, and understanding what's actually going on under the hood in long division, the misconceptions about 0.999... might start to go away
edit: I understand the proofs for why it's true. I'm trying to explain from a pedagogical standpoint, why most people might have a bad first reaction to 0.999... = 1.
For example, in bprp's latest video on this topic, th-cam.com/video/C_1AqiKJRhA/w-d-xo.html he uses the same whiteboard explanation for 0.999... = 9/9 to show that 0.444... = 4/9. But tell somebody that 4/9 = 0.444..., and they readily believe it. Why? They've done it by hand in grade school, and can use long division to work out the details. You don't need to whip out alternative proofs, which, while true, aren't how most people internalize repeated fractions as a whole. The intuition is solidified already for most people because they can just divide the fraction using the long division algorithm, and see the repeated decimal pop out.
[on a side note, the manipulation of infinite decimals is exactly how you manipulate infinite fractions, so perhaps that would lead to more intuition about what's going on? I really liked his followup video.]
@@user-ee5kl6oi7t 9/9
The way that made sense to me was if 0.999... / 3 = .333... And 1 / 3 = .333... then by the transitive property, 0.999... must be equal to 1.
1/3 is not equal to 0.333... but 0.333 converges to 1/3. That's circular logic.
@@EvanG529 what? 1 divided by three is . 3 repeating we learned that in like 3-4th grade
@@coolskeletondude5902 1 is not divisible by 3, so we can only approach it with a repeating decimal. It's the same argument, just rehashed.
@@coolskeletondude5902 Lots of things you learned in the 3rd great were simplifications designed to make complicated concepts digestible by third graders. If you 3rd grade teacher didn't inculcate this in you they did you a disservice.
@@EvanG529 Yes. A lot of people don't seem to understand that rational numbers live in a completely different conceptual universe from natural numbers.
I think the biggest reason people fail to accept this is because they’re trying to imagine something like this in real life. Take Zeno’s Paradox which is vey similar. No matter how much finite time passes, you’ll never reach the end point. But we are dealing with infinite time which is not a real number itself.
if you consider motion as a supertask, where every infinitely small distance takes proportionately little time to cross at a steady velocity, you can easily pass an infinite number of points in finite time
to go 1 meter in 1 second you have to go 1/2 a meter in 1/2 a second, 1/4 meter in 1/4 second, etc. This series converges on 1 meter per second
Yeah, the 0.9999... is not being constructed in some temporal process. The whole thing exists right now.
People are just confusing "being equal to", and "converging to".
@Neckhawker No, they don't. 0.999... repeating does not converge at 1, it is equal to 1. It is simply a different way of writing 1, like writing it as 3/3, which is also equal to 1 (and equal to 0.999... repeating, since they are the same).
@@zagreus5773 No, 0.999... defined as 9 repeating to the infinite isn't equal to 1. It is its limit that converge to 1.
And no, 3/3 isn't equal to 0.999..., idem, 1/3 isn't equal to 0.3333..., but 0.333... converge to 1/3.
I think a lot of people get hung up on this because they interpret 0.999... as a decimal followed by "an arbitrarily large string of 9s" or something like that. And any very long string of 9s, like a billion or a google-plex 9s after the decimal, IS less than one, so this becomes more of a semantic misunderstanding about what we're representing with 0.999...
In other words, they don't understand what "infinite" actually means.
I know what infinite means. Try draw 0.999... on a graph. It will never touch 1, so therefore it cant be 1. Yes its infinite, but you can travel down that 0.999... line for as loooooooong as you like, and yet no matter how far you go its always just below the 1 line above it. Its approaching it for infinity, yep agree, but it can and will never actually be the same as 1. The only people who would argue that these are the same thing are the same people who like to say "but actually"
@@bjornfeuerbacher5514 It's kind of the opposite. 'Infinity' isn't really a reasonable mathematical concept, and people try to reason about 0.999... using their conception of 'infinity'. I think the problem is that many people don't understand what a limit is (really it's too bad that the concept of a limit isn't taught before Calculus). If you're thinking in terms of infinity, .9 is finite, and so is .99, and .9999999, and so if you take a finite string of 9s and add another 9 on the end, you will always get a finite number. There's no end to this process, no point where one more nine causes the number to roll over to 'exactly one', even if you keep adding 9s ad infinitum. This line of reasoning does countenance infinity, sort of, but to rigorously understand what's going on you need to look at it in terms of the limit, not of nebulously defined 'infinity'.
@@networkguy153if you want me to prove 0.999… is equal to 1 in an elementary way here u go:
1/3 as you know is equal to 0.333…
Now if we double 1/3 we get 0.666….
So far so good,
But what if we triple it?
Well we should get 0.9999 right?
But tripling is the same as multiplying by 3, so 1/3 x 3 =?
Oh it’s 1? 😮
@@networkguy153And therefore you immediately say that you do not in fact know what “infinite” means. Because you immediately followed it up by saying “no matter how far you travel down”, which are always *finite* amounts. You cannot draw infinity. Infinity is an abstract concept, not a number, not a length, not ratio, and most certainly not intuitive. Infinity follows certain mathematical rules, and the only way to get a correct result regarding infinity, is to apply those mathematical rules, like in the video.
My son had this problem with his teacher at school. I gave him the simpler algebraic proof that if x = 0.999…, 10x = 9.999..,, subtract first equation from second and get 9x = 9, then divide by 9 and you’re done. He managed to get the teacher to admit that this was right.
This is not a good proof, unless you can show that .999... * 10 = 9.999..., algebraic proofs just don't work well with infinity😊
Infinity or not, in a base10 system multiplying anything with 10 just shifts the comma to the right by one step.. Sure you could probably come up with a definitive proof, but I doubt that anybody is gonna ask you for one
this exact proof was involved in a famous question that appeared in an entrance exam for one of the biggest universities here in brazil. every math teacher used to talk about this question and to show this proof here. pretty nostalgic
Just ask what is 1/(1-0.999...) and if it's larger than all natural numbers, and if they're willing to accept (real) numbers larger than anything together representable by digits.
Doesn’t actually work as proof tbh
The counter-arguments to this seem to be more philosophical than mathematical
They have to be since they're trying to prove something that's mathematically incorrect.
The counter-arguments to this are just incorrect
it's just the sad tendency of human thinking "it's not intuitive to me therefore it must be wrong"
Look, they may be wrong in this case, but I don't think you know what philosophy is. Mathematics is a subset of formal logic, formal logic is a subset of logic, logic is a subset of philosophy. We're all trying to do the same thing: understand things and learn HOW to understand things correctly.@@exscape
@@UmaROMCyes, a *subset*
If you use a tool from different philosophical subsets, it might not work to prove formal mathematics.
Before I watched the video I thought the title was implying you were the one arguing 0.999... wasn't equal to 1 lol
That was what I thought for .999(repeating) of the video 😂
Yes, same thing. And I got really confused because I was sure of the answer.
a successful title, then! :P
I wasn't going to click because of it until I realized it was just the quote from the pic
Certain people get so angry about this, every mathematician ever will tell them 0.9999... = 1 and they will drone on and on about how false it is, refusing to engage with actual proofs, making up terms without properly defining them, etc. It's bewildering how dedicated people are to this.
While at the same time completely accepting that 0.3333... = 1/3
In highschool, the first time I saw .9 repeating equals 1 it felt very wrong to me. Essentially, it felt like there must be some infinitesimal difference between them. Luckily I had a very good math teacher who took the time to talk me through it, and didn't make me feel like an idiot for questioning it, and expressing my intuition.
And now I'm the guy on the internet arguing with people who think .9 repeating isn't 1, lol.
This is why engineering maths is easier, because you simply don't care about trivial things like this
I got one counter to you. When you subtract 0.9999999999 from 1 do you get zero? Absolutely not. No matter how many nines you add at the end, the last number on the 1 is going to be 0, and you'll have to do 10-9 at some point. Therefore, there is a difference. Where people mess up is in misinterpreting 'repeating'. To compare two numbers, you will ALWAYS have to stop at a specific number. And in that moment, there is still a difference between the two.
@@tervalasthat's not 0.999... repeating then, that's a finite number of 9s, the entire point is that's is infinite 9s
The way I like to explain it, to help people grasp it intuitively, is that 1/3 is 0.3333...
3 times 1/3 is 1, right? Which means 3 times 0.3333..., which is 0.9999... must also be 1.
The way I explained it to both of my (at that time) third-graders, was to have them compute 1/9 and notice that they get 0.111…. That was probably the first time they had seen a repeating number and given it much thought. Then repeat with 2/9, 3/9 (aka 1/3), 4/9, ... and so on. They'll quickly recognize the pattern. Finally, I asked them what they would get for 9/9. Of course, with this lead-up, the answer from them is going to be 0.999…. And that's the moment when I point out that I always thought that 9/9 was actually 1.
After a brief moment of surprise, both of them admitted that my explanation made sense. 0.999… does in fact appear to be just another way of writing 1.
This is of course not a rigorous mathematical proof, unless you first show why n/9 produces a repeating number. But that's too difficult for a third-grader. Nonetheless, I found it to be a good way to get an intuitive grasp of what's happening here without having to go much beyond any of the math that a third-grader is familiar with. And for anybody sufficiently curious, it is possible to develop this approach into a full proof. Maybe, they'll do that when they are older and remember this lesson.
Would that person agree that .333333... is exactly 1/3? Dunno.
@@zachansen8293I don't know any other decimal expansion for 1/3
A flat earther would not agree 3/9 is 1/3. Some people are unteachable, aka stupid. :D
@@gutschke "was to have them compute 1/9 and notice that they get 0.111…. That was probably the first time they had seen a repeating number and given it much thought."
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That doesn't actually prove anything. That's just a quirk of doing division of a number which is not evenly divisible by your base. In base 9 instead of base 10, 1/9 is exactly .1. And 1/10 becomes the weird repeating thing.
you wouldnt even miss the moon by a million miles if you just roughly went in the direction of the moon
If you went the exact opposite direction the moon was currently in as far as the moon is away from earth on average, _you still_ would have missed by less than million miles.
@@pat9353 true.. didnt think of that - since the moon isnt half a million miles away
@@La_sagne I am shocked at how much closer the moon is than I thought it was
"you wouldnt even miss the moon by a million miles if you just roughly went in the direction of the moon"
You would miss the moon "a million miles" (well sort of as you were already told) if you went in the direction of the moon. Orbital meachanics are weird someway, but at the point in time you are at the moon, the moon is at a completely different place than a few days before. So no, to reach the moon you do not go in the direction of the moon. You can't.
@@pat9353 "If you went the exact opposite direction the moon was currently"
The funny thing is: Because how orbital mechanics work, it may be you arrive at the moon by doing this. But then it will be a long trip. ;)
The root cause of this confusion is that most laymen who have not had formal rigorous mathematical education past first year university don't understand that mathematics is based on precise definitions and instead argue based off of their "intuition". They don't understand that every symbol or collection of symbols in math has a definition that is agreed upon, and all statements made about those objects must first begin with the definition. There is no room for "interpretation" or "intuition" or anything about "you." Starting from the definition, each step follows logically and necessarily, and we arrive at the conclusion.
People who argue 0.999... is not equal to 1 do so mostly using their intuition about what 0.9999... "should" mean or "ought to" mean. These arguments from intuition leave a ton of wiggle room for hand-wavy logic which isn't air tight. The reason why this equality is not contentious in the mathematical community is because everyone who knows proper math knows the exact procedure you must follow to demonstrate the equality. The collection of symbols "0.9999..." has a very precise definition. BY DEFINITION, it is the limit of the infinite sum of 9/10^n, n going from 1 to infinity. The "infinite sum" also has a precise definition, defined using epsilon and deltas. When you apply the definition of the infinite sum, you find that it evaluates to 1 exactly. Therefore 0.9999... = 1, end of story. The equality follows directly from the definitions of what those symbols mean. There is only ambiguity if you are using a different definition for those symbols from what is agreed upon, or you failed to apply those definitions in your argument, or you are under the misunderstanding that the same number cannot be expressed in two different ways.
The person's fundamental error is that, while he more or less correctly recognizes that the 1 is the *limit* of the sequence of partial series implied by adding nines to 0.9999 (at least, I think that is what he's getting at with "the definition of an asymptote"), he doesn't realize that that is how decimal representations of real numbers work: the number represented by a non-terminating decimal *is* the limit.
This is what I try to explain to people as well. If there are infinite 9s, then 0.9 recurring is the number at infinity, and the number at infinity is 1.
Zeno's Paradox, except he is going 90% of the distance to the finish line from each waypoint.
Exactly thought about the same thing. Best intuitive explanation for people who cannot cope with the mathematical formal explanation, it's like Zenon and his arrow and we know that in the end it reaches its target.
However, it's possible that both are true. It simultaneously is and isn't equal, it does and doesn't reach the target. In reality we know physics and we know that the arrow's atoms will reach the target's atoms close enough to interact, but they don't really "touch" in the physical sense. Therefore it never reaches the target and it always reaches the target. For us humans it's equal to one. If you somehow could perceive infinity as a finite object, you'd say it doesn't equal.
Similar, a snapshot of an arrow in flight, even at infinite shutter speed, looks motionless. Don't stand in front of the arrow, it is always moving.
@@jessepowellr4srry guess my history books were old... It's the one where an object supposedly in motion going along a path has to first travel half distance to destination, then half of the half, then half of that half at nauseam. Because it has to travel infinite halfs it will never reach it's destination in finite time. Which is demonstrably false.
Second it's a paradox, what do you want me to do with it? Make it make sense?
@@Washeek chillax, man, you were half right. Zeno postulated at least nine paradoxes that we know of, with three of them being the most famous, namely: the dichotomy paradox; Achilles and the tortoise paradox; the arrow paradox.
You were stating the premise of the dichotomy paradox, which divides space into infinite halves of smaller and smaller segments, but calling it the arrow paradox, which divides time into infinite instants where no motion occurs
Zeno's paradox would be in binary, where 0.111...=1.
If an adult is yelling at you and trying to get you fired over this problem, they lost the argument.
It's possible to have systems where two numbers can differ by an infinitesimal. But the real numbers are defined in such a way that each number is equal to the limit of any sequence that approaches it.
Yeah, Cauchy sequences.
Finally someone who knows what hes talking about.
succinct and well said 🎉
Each number is constructed as the equivalence class of all Cauchy sequences converging together.
What made the most sense to me was when one afternoon I decided to try finding what 0.999... was in base 3. In base 10, the way most people would find 0.999... is 1/3=0.333... then 0.333*3=0.999.... But interestingly, this only happens because our base 10 system doesn't like to divide 3s. We can see this type of phenomenon when we try to divide 1/2 in base 3. In base 10, the answer is obviously 0.5, but in base 3, if you go through the long division process, you find that 1/2=0.111.... So if we were to multiply that by 2, we would get 0.222..., which seems like an asymptote again, but the expression we just did is identical to the expression 0.5*2, just in a different numerical system. So what if we try to go through this process to find what happens when you try to reach 0.999... in base 3? Well, 3 is written as 10 in base 3, so the expression is 1/10, and we find that in base 3, 1/10 equals 0.1, because 3 essentially takes the numerical role in base 3 that 10 does in base 10, so their properties are similar. So, base 3's 0.1 is equal to base 10's 0.333.... But if we multiply 0.1 by 3 in base three, we get 0.1*10=1, which in base 10 would look like 0.333...*3=1.
I like this explanation the most because it gets at the heart of what the actual reason we have this phenomenon is, rather than just trying to prove its truth. The only reason you can't divide one by 3 evenly with integers is because you can't divide 10 by 3 evenly with integers. Humans semi-aebitrarily chose a base 10 system of counting, but numbers don't exist only in base 10. You can't make sweeping mathematical judgements based on an artifact of our system of computing.
Fun fact, if you write a Python script to print 0.1 to 50 decimal places, you'll get a number slightly greater than 0.1. This is because 0.1 is a repeating decimal in base 2, aka binary. This will work in any programming language that represents decimals as floating point numbers, and it's why you should never use floats for things requiring exact precision.
This was the best explanation I have heard of this topic and it completely switched my incorrect opinion.
None of the other common explanations worked for me but this did. Thank you!
@@theJACKHAMMER13 You have no idea how happy it makes my math nerd brain when I successfully explain something to someone. Glad I could help out :)
What's up with people, they have no problem with 0.3333...=1/3, but multiply it by three and suddenly they can't wrap their head around it.
They can’t understand rational numbers with a repeating digit.
1/3 + 2/3 = 1
0.333(3)… + 0.666(6)… = 1
0.999(9)… = 1
Hmm yeah this logic checks out for me… I think for me the problem is context. 0.3333… being 1/3 is just because we can’t write the fraction any better as a number, so we know what it means. But if you just stick 0.9999… in front me and refuse to elaborate that really what ur saying is 3x1/3, then that’s when it becomes questionable, like why not just write “1” to begin with lol. coming from a non math guy btw
@@SN.LurkinG Okay, I don't know if you mind reading a bit, but I tried explaining in a way that would be approachable to a non math guy.
In short, you really should write 1, unless you have a good reason not to. It's just that there are multiple ways to write some numbers. The decimal numbers are a representation that uses a geometric series with some chosen base. We mostly use 10, but you will encounter base 2 and 16 in programming and occasionally some others. This basically means that if I write for example 1.29, it means 1*10^0+2*10^(-1)+9*10^(-2). When you have a number with infinite number of decimals, it represents an infinite sum. And those are somewhat weird. Because those sums represent numbers, if two sums add up to the same number, the associated decimals represent the same number. It just so happens the sums 1*10^1 and 0*10^1+9*10^(-1)+9*10^(-2)... both sum up to 1 (as shown in the video). It will actually happen any time you end on infinite decimal places that contain a 9, you will get two possible representations. This doesn't work for 0.333... though, there's only one such representation. The reason is basically that 0.999... is in some sense the "last" number between 0 and 1, you can kinda approach it with the sum both from above and below in base 10. Then again, there is no last number between 0 and 1, if you name some, I can name one closer to 1, so 0.999... can't actually be smaller than 1.
By the way, there's another intuitive way to see they're the same number. It can be shown that numbers are the same if their distance on the number line is zero, in other words if you can't find any number that would lie between them on the number line, it's the same number. Well, what number is supposed to be between 0.9999... and 1? There is none! This works because if the distance between two points is 0, the two points have to be a single one.
I hope this helps. Sorry if it got too long or complicated, I am a math guy and I can be terrible at explaining stuff to non-mathematicians.
@@wiener_process Wow thanks for the reply, can’t say I understood all of it. What I meant to say originally, is that your simple 0.333… explanation did the trick for me, so thanks for that! This “number line” philosophy is very interesting tho, crazy how abstract and detached from reality math can get, I think that’s where I and most other people struggle with - the fact that these numbers only exists in the abstract “representation” sense.
I never considered .333... to equal 1/3rd, I believed that 1/3rd can't be expressed in a base 10 floating point system.
Is 2/3rds .666... with a 7 at the "end"? That's more equal to 2/3rds than .666... is it not?
Is .1 in base 3 between .333... and .333...4?
I'm asking... I don't know
My sister is a maths teacher and recently had a parent meeting, because a father did not believe that his son was wrong by saying that 0,999... is not equal to 1.
Ha ha. At what grade level? Here in Denmark I have experienced students in my math class at the gymnasium who got to know in elementary school that 7^0 is 0. The explanation I got from the student was that their elementary school teacher said that if you calculate 7^2 you write 7 two times in the product but if you write 7^0, then it occurs zero times in the product (i.e. you have nothing). So that the answer should be zero. To this explanation I said to my student, "that was an excellent way to memorize the wrong answer, I am sorry that you learned incorrectly back then".
7^2 means 1*7*7. Is how we were taught. The powers shows how many times the number is multiplied to 1. So 0^0 is considered 1 but 0^n where n is not 0 Is considered 0.
@@patricklincoln5942 I had an elementary school English teacher who couldn’t actually speak English - and I still remember some error she taught! 😅
@@KiyoGam1 0^0 is undefined
@@patricklincoln5942 It's a bad way to memorize the wrong answer, really. An empty set of numbers, all multiplied together, is 1, because 1 is the multiplicative identity. Just like how zero Boolean values all ANDed together is true, and zero Boolean values all ORed together is false.
If you subtract 1 - 0.999... you get 0.000..., which is more intuitively understandable to be equal to 0
You get 0.0 followed by an infinite number of 0s with a 1 at the end. If there is an infinite number of 0s followed by 1, that is the same as 0 because the nature of infinity means you will never be able to reach that 1
@@dadonkas5541 if it is infinite there is no end. Someone clearly didn't take calc 2
@@danielhobbyistUnfortunately that means that particular explanation is not helpful or convincing to the audience (unless you have a very small audience)
@@danielhobbyist that’s literally what I said. 1-0.999… would supposedly have a 1 after an infinite amount of 0s, but infinity has no end, so it would be the same as 0. Someone clearly didn’t take basic English classes, or calc 2
"a 1 after an infinite amount of 0s, but infinity has no end, so it would be the same as 0" no, it would be undefined. It does happen to equal zero, but not for the reason you say.
I think what makes this difficult for non-mathematicians (including me) to grasp is that it asks a layperson to reassess their assumptions about how certain things work, like exactly 'what' infinity means in a mathematical context, and what repeating decimals are actually meant to represent. Actually 'proving' that 0.999... = 1 mathematically requires referencing higher level mathematics, and without more intuitive examples available, it comes off to a layperson as a mathematician saying "Just trust me", and I think people have more trouble 'just trusting' mathematicians compared to other experts because everyone has their own understanding of how math is 'supposed' to work. I know I personally needed the 1/3 = 0.333... examples from the comments to really be able to conceptualize it and reconcile it with my current understanding of math because unfortunately geometric sequences aren't something I've ever even approached in my life.
That being said, I'm not a fan of how people in the comments or on the internet are smugly approaching this like people who don't understand are stupid or unteachable. Sometimes when someone pushes back against teaching, it's because a part of them doesn't want to accept it or learn, but sometimes it's just a part of trying to reconcile things that don't immediately make sense. If you make someone feel directly attacked or insulted, then chances are it doesn't matter how sound your logic is, you will not get through to them. If you 'do', then you've made learning an unnecessarily unpleasant experience (which makes people even more hesitant to learn in the future).
For what it's worth, I wouldn't take what people say too seriously. Especially since there exist models of math where 0.9999.... =/=1.
Anyway, I'm assuming that you gst 1/3=.3333... from long division of 1 by 3, infinitely many times? It's possible to reformulate 1/3 as a geometric series in the same way as the video if you want to try a geometric series example.
Its just some dude decided 0.999... is 1, to make it easier to calculate stuff. That's all. This guy and others in youtube comments spouting same stuff that they memorized from their classes, almos like brainwashed.
@adamfarmer7665 Exactly and people using the 1/3 argument drive me insane. It’s like defining a word by using the same word.
@@jersefrenzer1265 Yeah, that's how I got 0.333... I don't doubt you could use a geometric series, but like I said, those are out of my grasp, I never learned about them. Though at this point I am considering trying, if for no other reason than to keep learning.
Finally someone sane
My take on this is that the root of the misunderstanding is that 0.9999... is considered to be the representation of a number, and thus this number cannot be 1, because as they say, only 1=1.
But here's the thing, "2-1" is also equal to 1. But "2-1" is strictly not a number, it's an expression that has to be evaluated to get a number.
So maybe, one way to help people understand the issue is that "0.999..." is an *expression* that evaluates to exactly 1, just as "2-1", or "lim x when x -> 1".
You're not wrong. I think it's definitely important to stress to people that an expression that evaluates to a number is still just a number. I differ with you slightly because 2 - 1 is still just a number, strictly a number even. Like, don't even bring evaluation into it. Introduce people to more cases where the same number can be written in "strange" ways. But I do think you're right, that people get the idea that there's a difference between an arithmetic expression (that evaluates to a number) like 2 - 1 and a "bare" number, like 0.9 repeating. I think we just need to stress harder that they are the same thing, even though it seems to be counterintuitive to many people. 2 - 1 and 1 are exactly the same thing. Cheers.
But that's the thing, everything you write down is an expression. A number is an abstract concept, 1 is not actually the number 1, it's a vertical line with a little flag on top. We use it as a symbol for the number 1. Once you realise that representation and semantics are separate concepts, this becomes a whole lot less confusing.
@@hannessteffenhagen61 You are right of course, "1" is also an expression. But I still think this approach of stressing that there is a need to evaluate the expression "0.999..." to get to "1" provides a way to reach the correct conclusion, namely that these two expressions are both equal to the number 1.
It's a chose your poison situation. And the rules you decide to play by. Whether you decide It's equal to 1 or not, leads you to different "Games" of mathematics. You shouldn't convince yourself that it being equal to 1 is *Always* the case.
Yes. All numbers are expressions. "123.4" is, by definition, (1•10² + 2•10¹ + 3•10⁰ + 4•10⁻¹). They teach this in second grade, with ones and teens and hundreds.
Base 10 and all other bases are just a representation of the abstract number, not the number itself.
Gota love non-mathmaticians arguing about math.
Edit: guys shut up its not that deep
Is it anything like mathematicians arguing about the Humanities, in which case they often make utter fools of themselves?
@@Bulhakaslol I've seen someone give a lecture on why the Mandelbrot set is the "atheist's worst nightmare"
just like religious people arguing about science
If us religious people didn't understand science, it wouldn't be here today for you atheists to build upon
@@CreepyMemes Or atheists forging false evidence.
If they were not equal, then there should exist another real number strictly between them. Now we could rigourously show this, but it’s also pretty clear and intuitive that there’s no number you could find that’s strictly between the two.
The “…” in the decimals really signifies that we are taking the limit of the sequence with finite decimals, i.e. 0.9, 0.99, 0.999, …
And the answer is precisely the limit, which is 1.
And if you want to think of it as a sum like you did, we also say that even though the partial sums only approach the limit, we say the entire infinite sum is *equal* to the limit as long as it exists.
Ok but infinity isnt real... sorry.
@@marksandsmith6778 infinity is a concept/idea. It is not something you can count/measure like a number.
And just like any other concept or idea, we have rigorous definitions and rules on how infinity is to be handled and intepreted. And for our particular context it is in the form of a limit. It is unintuitive for sure because we live in a finite world, but we have proper mathematical tools on how to properly use and understand it.
So merely saying “infinity does not exist” is so meaningless and worthless because it clearly shows you have very little understanding of the topic.
Good point on the "..." implying a limit, the "0.999...=1" is so confusing because the argument about 0≠1 on the first position seems to seal the deal that the numbers can't be equal, but this is a limit so you can't just compare positions, as this works only between a number and a number
In before someone says (1 - epsilon)
@@Firefly256 that is ONLY if you want to get into the hyperreal/surreal numbers, which is very advanced, non-standard analysis, which is both unintuitive and not at all for the average math person like you and me.
If you want to stay in the regular real numbers that we all are familiar and comfortable with, then infinitesimals cannot exist.
I think the confusion stems from the fact that pure math isn't always indicative of anything you can find in the real world.
You can't have 0.999... of an apple, for instance. Whatever particles you use to measure how much of an apple you have are necessarily finite, and so you'll eventually hit a last number.
This is actually the same thing as the universe expanding but, you're not really supposed to talk about that during Zeno's paradox because they did not know those things yet. So it's not a part of the real thought experiment.
But you can have 0.999... of an apple. It's the entire apple. Because 0.999... is defined as having NO LAST NUMBER. You think you'll "eventually hit a last number" but that's literally not what 0.999... means. You'll never hit a last number, because it's = 1.
My understanding basically that numbers are idealized forms that exist within the mind. We can only imagine everything as singles objects, just as we can only idealize a perfect circle or absolutely straight line. An infinite series is an irrational number within the real number set of objects. They tend to be an abstraction of real objects based on our ability to deduce subject matter.
Anyways, that's my understanding of the problem.
@@thane9 _"Because 0.999... is defined as having NO LAST NUMBER."_
0.222... also has no last number.
The only thing I know for sure after seeing this is that the comments section of a TH-cam video about math is a postapocalyptic war zone populated with homicidal nerds.
Correction: an approximation of a "postapocalyptic war zone populated with homicidal berserkers", but it's off by an infinitesimally small amount.
Survival of the Nerdiest out here
I was expecting to find hundreds of comments talking about how hilarious it was that the guy denying 0.9999..=1 opened his first comment in the thread with "dear united States public school victims"
Instead I found a load of people siding with him, a minute ago I read someone deny that if you divide 1 by 3 you get 0.33333...
TH-cam is weird
@@matta5749 three points for ya doofus
1. My interest in maths is purely a casual one, I'm not trying for a doctorate
2. I dont know if you noticed but of course my comment was mathematically non rigorous, it was a social comment, not a maths one
And 3. You're just actually wrong, someone pointed out the whole 1/3×3=0.999...=1 thing, and the other guy said "this statement begs the question that 1/3 is 0.333...", that's legitimately him denying that 0.333... Is 1/3
Take yourself less seriously ya goofball, we're all having fun here
i love the videos but yeah the comments section can be really toxic. even though i enjoy math and science in themselves, the community kinda scares me away from the fields lol
Tell people that 1/2, 2/4, 3/6, 0.5, and 50% are the same number and no one bats an eye. Tell them that 0.999(repeating) and 1 are the same number and everyone loses their minds.
Lol because they are not the same number
I find your (and other mathematicians') ability to claim that red = blue very concerning.
@@Jwellsuhhuh If they're not the same number, then tell me what 1 - 0.999(repeating) equals that's not 0.
@@leamael00 Red = blue? What? What are you saying?
@@MichaelMoore99 1 - 0.999(repeating) = 0.000(repeating)1. Which is still not 0.
Red = blue, 1 = 2, 0.999...999 = 1, taxation isn't theft, same acrobatic reasoning. No amount of ball twisting mathematics will ever magically turn 0.999...999 into 1.
My understanding is that asymptote, by definition, is a line that continually gets closer to a certain distance (in this case 1) without ever meeting at a FINITE distance.
But infinitely recurring is not finite. That asymptote, taken to infinity, therefore is 1.
This is correct. Infinity is the next step after forever. So asymptotes are not reached with just natural numbers that take forever but infinity in a sequence is ω the next sequence after the forever sequence of natural numbers. So you reach the asymptote after forever but without the one additional step (forever+1) you never reach the asymptote.
@@kazedcatOmega is for ordinal numbers. Cardinal numbers are aleph null.
@@ferretyluv Yes and we are talking about infinite sequence of terms so the correct infinity to use is the ordinals. 0.999... does not have the size of aleph0 but it has ω digits.
You could also just ask them to find a number between the two, and if you can't they need to be equal.
@@DKNguyen3.1415 it doesn't work, they always insist it's 1-0.999... is 0.00000000......1
a+b/2 will always result in a new number (c) where a
I like this handwavey explanation I use:
1/3 = 0.333…
2/3 = 0.666…
1 = 1/3 + 2/3 = 3/3 = 0.333… + 0.666… = 0.999…
Fun fact! Any repeating decimal can be represented as the digits of the repetition divided by an equivalent number of 9s.
0.111… = 1/9
0.151515… = 15/99
0.123123123… = 123/999
And of course: 0.999… = 9/9 = 1
No, that's circular logic, because 0.333... only approaches 1/3 because 1 is not divisible by 3.
@@EvanG529 You are wrong
@@lengors7327 fantastic point, don't think I'm recovering from that one
@@EvanG529 I wasn't debating you, only making a factual statement. So, I don't really care about making "good points".
@@lengors7327 Gee, I wonder why discussions of this topic don't stay civil. Please, if you have nothing to say, don't say it.
I had a hand-wavy method to show things like this in 6th grade, I recall my teacher asking me what 9.9999… was equal to and my method said it’d have to equal 10. He laughed and said therefore my method was wrong. About a week later he met with me and apologized, and explained the flaw in his thinking and was kind enough to show me the rigorous method for proving this (which, to be honest, I didn’t understand at the time but it’s the thought that counts). 😅
How nice to have a curious teacher.
This is the result of two definitions:
1. 0.9999... is defined as the series 0.9 + 0.09 + 0.009 + ...
2. A series of real numbers, a_0 + a_1 + ... is said to equal a real number L iff the limit of the partial sums converges to L.
This is one intuitive translation of what it means to add a number "infinitely" many times. However, it is not the only one. I think it can be more helpful to teach non-mathematicians that this is why mathematical definitions are important, rather than, what I sometimes see, 'uhmm akshully' X is defined this way.
For example, the idea that a series "equals" rather than "converges" to something is honestly probably just a matter of convenience with how we express maths. This is merely a definitional thing -- it doesn't *have* to be that way. Although, it'd be tedious to write out many theorems. In a similar sense, in the above definitions we make use of real numbers (usually implicit), yet again, one could easily define equality via some other number system (perhaps one that contains infinitesimals). In the hyperreals, the partial sum of 0.9999... is *never* infinitesimally close to 1, hence in some definitions we say it is not convergent to 1.
I agree, and I think the difference is important. To say that converges is the same as equals, is the same as to say that lim f(x) = f(x)
Thank you, this is the point I was looking for someone to make.
Thank you, it's relieving to read such a sober comment among so many comments trying to "dunk" on people.
Yes, 0.999 recurring = 1, but it's ultimately a matter of *convention*. Once it's defined as a series, then it becomes 1.
It's not 1 because "you're a dumb high school dropout who can't do math".
My other take:
The sequence 0.9, 0.99, 0.999 is an assimptote.
But we are not talking about the sequence, we are talking about the number 0.999... that is the limit of the sequence.
The only thing I don't understand about this is that if the limit of the geometric series approaches 1, how can the geometric series itself be equal to 1.
The limit does not approach anything. It is 1.
Saying that the geometric series is equal to 1 is just a colloquial way of saying that the sum of this series is equal to one (i.e. the limit of its partial sums).
You're right, he equivocated those. The real answer is mathematicians decided 150y ago to treat infinitesimals as 0 because it works in the real world and makes math 1000x easier.
'You're right, he equivocated those'
Incorrect. It is proven that 0.999... = 1.
'The real answer is mathematicians decided 150y ago to treat infinitesimals as 0'
Again, incorrect. There are literally no infinitesimals in the space of real numbers.
A positive infinitesimal is a number that is less than any real number, but greater than 0. For any positive real number to be an infinitesimal it would have to be less than itself, but the relation '
A summation series doesn't "approach" a number. It "converges" to it. There's no limit involved in these situations. You can use an equals sign.
The way I would explain it is that 1 - 0.999… is 0.000… followed by a one. However you would never get to the 1 so the value is 0 meaning the difference is zero, or in other words, there is literally no difference.
Something that helped me understand this: When you write a sequence of decimal digits to describe a number (infinite or not), the number you are describing is not that sequence. It is the **limit** of that sequence.
Numbers are not sequences and they do not "approach" anything. The sequence of decimal digits that represented by 0.999... **approaches** 1, but if you want to interpret it as a singular **number** then the number can only be 1.
I like this argument! You just use the uniqueness of limits :)
Similarly, you can just show that if 0.999… is the limit of the sequence 0.9, 0.99, 0.999, etc. then 1 is also a valid limit point of the sequence, but again, the limit is unique so they must be the same number
But also, I wouldn’t say that 0.999… is a sequence in itself. I think your second interpretation is a better way to see this number, it really is the limit of the sequence.
No it is 1.
The easiest way to think about the number line.
If you go by digits of 0.99... you see that for each digit it gets closer to 1.
But every time you think there is a distinction between 1 and 0.99... the next digits will remove it.
since it goes for infinity that means that there will never be anything between 1 and 0.99...
Therefore both numbers occupy the same place in the line. Thus are the same number
"Numbers are not sequences"
Weeeell...
There are 2 popular ways of constructing the real numbers. One is using Dedekind cuts. These are quite abstract, but probably more useful when proving some properties of real numbers.
The other way is using sequences. If a sequence of rational numbers (where here, those rational numbers are not stirctily the subset of the reals) converges to some value, we "assign" that value to that sequence. Or rather, if 2 sequences have the same limit, we say these sequences are equivalent and we set the real number x to be the equivalence class of those sequences
Interesting, havent seen Dedekind cuts yet. But I fully agree with you, my point was just that the number is not the sequence itself, the number is the limit of the sequence(s) that converge to it (unless im missing something relevant)@@methatis3013
@@affegpus4195 But inversely it also shows that it will never reach one.
I've debated this with a (humanities) professor before. He thought it was some trick, like the "proof" that 1 = 0. He started from the assumption that the equality is false, and then talked about how interesting it was that you can use math to prove something that isn't true.
“Professor”.
Haha but math knows this. Logic tells you that from a false statement you can derive false conclusions. So he started with a false statement that equality is false, and derived another false statement that math lets you prove false conclusions.
It's logical implication 101. (False => false) is true.
The reason I think many people are so convinced that 0.999... has to be less than 1 is that they have misunderstood what infinity means and confuse it with arbitrarily large, but finite. It is obvious that any finite number of 9's after the decimal would represent a number ever so slightly smaller than 1 and it leads to phrases like "asymptote", "arbitrarily close but not equal" and "at the end of the 9's" being common in such arguments. With an infinite number of 9's however there is no last digit or point where the supposed approximation stops as infinite literally means never ending and the result is exactly 1.
Exactly. I think an easier way to think about it is by subtracting 1 from both sides of the equation to get 0 = 0.000…1
There are an infinite number of 0s after the decimal point followed by a 1. Because of the nature of infinity, it is impossible to reach that 1 because the number of 0s before it are, well, infinite. Therefore, the 1 doesn’t really exist so 0.000…1 it is just equal to 0
My interpretation would be that there is a reason someone writes 0.999... instead of just writing 1. What is trying to be communicated when someone writes 0.999... other than just 1. It's plainly obvious that 0.999... and 1 are literally different symbols.
@@Babyblasphemy So what meaning should the expression 0.999... have instead while remaining consistent with other decimal notation?
This exact thing is what I am trying to explain to doubters too. The whole thing boils down to the infinite or not.
Also trying to ask what number comes between 0.999… and 1.
It must be the difference of the two numbers, that does not equal to 0, but the difference does equal 0, so the numbers have to be exactly the same.
1-0.999…=0.00000…
And if you are tempted to put a ‘1’ at the end somewhere, you just stopped the infinite process of putting zeroes, making it a finite number, no longer infinite.
So the difference is 0. 1 is exactly equal to 0.999…
@@Babyblasphemy so 1/3 = 0.333… are litterally different symbols?
there are several easier ways of showing intuitively that they are the same.
1. if x and y are different, then there should be some number that could come between them. but 1-0.999 repeating = 0
2. 0.333 repeating = 1/3. multiply both sides by 3 and we get 0.999 repeating = 1
3. x*10-x = 9 works for 0.9999 repeating and 1.
Divide by 3. Multiply by 3. Done.
There are two types of people. One will figure: OK, that makes sense. The other will, from that point on, refuse to accept that 0.333... is the same as 1/3.
My cal 2 teacher liked to explain it like this. If we all agree 1/3 = .333... and 2/3 = .666... then 3/3 = .999... or 1.
My favorite way that my professor showed and proved this (not serious proof but just a quick thing) was asking what 1/3 equals. 0.333….and then he asked us what if we were to multiple0.333 times 3. We would get 0.999… but if you take 1/3 and multiply it by three, you get one!
Yes, this is the easiest way to show that 0.(9) is indeed = 1
If you believe 1/3 is exactly = to 1, with no remainder.
".99..." is equivalent to 1 for any practicle purpose.
@@johnlabonte-ch5ul What a steaming great muppet you are. Belief is not required. 0.999... = 1 precisely for all purposes (except appearance), 1 - 0.999... = 0 on the nose. You do not know what the word "equivalent" means in mathematics (or probably everyday English either). Yet you persist in using it all the time.
Are you ever going to try to prove any of your claims. Are you ever gong to try to find a fault on any of the following three arguments:
1) 0.999... := lim n->oo 0.999...9 (n 9s) = lim n->oo 1 - 1/10^n = 1
2) 10 * 0.999... = 9.999...
=> 9 * 0.999... + 0.999... = 9 + 0.999...
=> 9 * 0.999... = 9
=> 0.999... = 9/9 = 1
3) If, as you "believe", 1 - 0.999... = d > 0 then 0.999... only has floor(log10(1/d)) consecutive 9s. To get the required infinitely many 9s, we need d = 0.
Come on, combine the full power of both of your braincells, and show everyone how smart you aren't by disposing of those three arguments. I'm still WAAAAAAAIIIIIIIITINGTINGTINGTING... LOL
KC I have not addressed 3) because it is circular nonsense.
(Baffle'm with El Toro poo poo)
In order to believe d>0 you need d=0???
@@johnlabonte-ch5ul KarenTheBonehead, you haven't addressed any of them.
OK if 3) is circular that would disqualify it as a proof, and I'd have to scrap it. So prove that it is circular. You will fail because it isn't circular. I can't begin to imagine how even a humongous muppet like you thinks it's circular.
You: "In order to believe d>0 you need d=0???"
You have lost the plot again. How on Earth did you come to that "conclusion". (I guess that's what it's supposed to be!!!).
I said if d > 0 then there are only finitely many 9s, but if d = 0 we'd get the required infinitely many 9s. You reading comprehension sucks as bad as your math comprehension. I guess that tumour is taking it's toll.
You forgot to take your meds again, didn't you.
The way I was taught this was "Divide 1 by 3, you get 0.333... which you then multiply by 3 to get 0.999... but similarly if you do 1:3×3 in one go you get 1. Therefore 1 and 0.999... are the same."
The representation of a real number as decimal expansion is not always unique, as this example shows.
It could be if we straight out banned 9 repeating decimals.
@@Felipe-sw8wp Yes. This can be expressed as
∀j∈N: ∃k≥ j: x_{-k} ≠ B−1
where
x = σ B^n ∑_{i=1}^{\infty} x_{−i} B^{−i}
and B≥ 2 is the base, σ the sign
@@Felipe-sw8wp You can express this as
∀j∈N: ∃k≥ j: x_{−k} B-1
where
x = σ B^n ∑_{i=1}^{\infty} x_{-i} B^{-i}
for base B≥2 and sign σ
@@Felipe-sw8wp You can express this as
∀j∈N: ∃k≥ j: x_{−k} B-1
where
x = σ B^n ∑_{i=1}^{\infty} x_{-i} B^{-i}
for base B≥2 and sign σ from {-1, 1}
@@Felipe-sw8wp You can express this as
forall j in N: exists k >= j: x_{−k} B-1
where
x = sigma B^n sum_{i=1}^{infty} x_{-i} B^{-i}
for base B >= 2 and sign sigma in {-1,1}
I learnt this in 8th grade ( India)
x=0.999999.... -(equation i)
10x=9.999999.... -(equation ii)
SUBTRACT equation i from equation ii
We have,
10x-x = (9.999....) - (0.9999....)
So by subtracting
9x = 9 (divide by 9)
x=1
Therefore,
0.999.... = x = 1
.
.
.
is it correct sir???
iam in grade 10 btw, just completed my sst boards and studying for maths
😅😅😅
Yes this is correct. A counter argument I heard was that 9 recurring is -1 (or something along the lines of that), but the difference between 999... and 0.999... is that recurring decimals are convergent. In other words, your proof works because 0.9 recurring is convergent.
The intuitive explanations always made me feel like I was being tricked, thank you for doing things a little more rigorously
The real explanation is that mathematicians chose to drop/ignore infinitesimals.
@@PhyloGenesis There are no infinitesimals involved in this discussion. None dropped, none ignored. They were never there to begin with.
1 divided by 3 = 0.333...
one third times 3 = 1
0.3333.... * 3 = 0.999999....
If 0.9999.... does not equal 1, arithmetic falls apart.
There are several proofs (why is the plural of proof not proves? /actually asking) for this. This is the one I prefer because it's easy to remember and simple to convey
I have seen situations with fractals where it's significant that 0.999... is not equal to 1, but it's important to realise that they're not using the standard real numbers but are using infinitesimals. So perhaps the quarrel is simply a mismatch between number systems.
yes, thats why advance 0.999...=1 deniers always bring up hyperreal number
I bet $0.99999999999(repeating) the guy who argued with you also goes around telling people 2+2=5
Some of y’all in the comments section seem to be still stuck on zeno’s paradox 💀
They are called adults and have a government backed profession as a teacher.
When you skip "rational" day in math class... but show up on irrational day... of course you're gonna yell at the teacher. logic!
Mathematicians: "0.9999... = 1"
Engineers: "0.9999... = Pi = 3"
Astrophysicists:
When folk who maybe barely passed Algebra 1 argue with Math majors.
Nah. I've had people argue these kinds of things with me that have masters degrees in engineering. You barely need to understand a limit, to practically apply a limit.
Not understanding how 0.99...=1 does not make a person dumb. It is not naturally intuitive to how most people understand numbers.
@@eseguy2 True enough, although not germane, I made no reference to intelligence, only competence.
@@ralphinoful engineers have funny understanding of math, they just understand it enough to use it in life but not to the core. That makes them potentially the most delusional kind. I am an engineer btw.
@@eseguy2but arguing without listening to reason does make you a dummy.
simple proof that 0.9 recurring = 1
(i learnt this in this year, grade 9 )
let x = 0.99999…..
10x = 9.999999……
subtracting both equations
9x = 9.0 which is 9
x = 9/9
x = 1 hence proved
The problem here is that people don't quite understand the details about limits. Any increasing sequence that has a supremum, always has an upper limit, which equals to that supremum. However, the supremum itself is not part of the sequence. In other words, whatever number of 9s you write, the number in question shall always be less than 1. That's why we can only deal with limits here, not with the sequence numbers themselves. From a computational standpoint, this question is nonsensical, because a Turing machine trying to compute an infinite number of 9s will never stop, therefore, never yielding an answer to the question. So basically, there is a convention that when we write infinite sums (improper integrals too) by that we really mean limits, not that the sum of those numbers actually yields anything specific.
Judging by those boxes it looks like he's corning the world market of red and black pens 🤣
Reddit is such a cesspool of the most pseudo-intellectual self-righteous keyboard warriors to ever exist.
Yet its better than twitter
As if YT comments are better.
reddit is way better than yt comments sections anyways
@@MythicWiz Fanboy makes his presence known. 🤣
@@razi_man Fanboy makes his presence known. 🤣
The super deep answer is that there's nothing stopping you from insisting that 0.999... ≠ 1. Math is just a system of rules and conventions made up by people. There's no ground truth. Mathematicians will often consider math with different rules, sometimes leading to interesting developments. However, if you want to insist that 0.999... ≠ 1, you need to acknowledge that you're not using the same math as everyone else, and it's very likely your system of math is both internally inconsistent and lacking any practical value.
I mean there’s nothing stopping you from insisting 1 = 0 or that the sky is red. Simply insisting something doesn’t inherently make that something useful or mean anyone has to give it credence.
@@alephnull3535 Yes, exactly. My broader point is that people seem to think that math has inherent truth, and you cannot possibly argue against it, but actually the foundations are "arbitrary", albeit the foundations most commonly used are broadly useful and nearly universally accepted.
on the other hand, sometimes defying common assumptions can also lead to new valid and interesting things, as was the case with Euclid's 5th postulate.
I hate this with every fiber of my being. If it is not 1, it is not 1. Don't do this to me.
'If it is not 1, it is not 1'
Well, it is 1.
0,(9) and 1 are different forms of writing the same number.
Not all maths agree that .99... is 1. I am not a math, my vote doesn't count, but from the maths I know and see they are outnumbered but not an infinitesimal.
@@johnlabonte-ch5ul "Not all maths agree that .99... is 1."
Idiots are everywhere. So it doesn't matter if some agree to nonsense.
@@johnlabonte-ch5ul
'Not all maths agree that .99... is 1. I am not a math'
This is just your assumption - you can't point to any who don't agree with that.
we round 0.9 into 1
This isn't about 0.9 or rounding. It's about 0.999999999.... repeating forever, which is 1.
No, we don't.
insert meme:
"0.999...=1"
"So hot right now"
Thats a whole lotta words to say "close enough not to matter"
You didn't even watch the video, because that's not what was said.
0.999... = 1 exactly, and not approximately.
In pure mathematics, it ALWAYS matters. There's no such thing as "close enough". Either =, < or >. There's is no other option. We're not taking a measurement or approximation. We have the perfect exact precise answer.
By the looks of his shelf, he's sponsored by dry erase markers.
I now feel like I didn’t pay enough attention to maths at school.
Its like a bell curve on mathematical knowledge. People with little knowledge can easily see this they dont require complicated proof to accept reality. Then we have people who think themselves genius just because they learned some advanced math operations, things proven by a simple google search are now questioned again, all those people who looked into it before must have been false. And then we have people who are on the other side of the bell curve. "Haha, look i made a Charizard-like curve"
1-.999… does not equal 0
0/0 undefined
Lim x-> 0 x/x = 1
@@NewesSkiller stuck on the middle of the bell curve I see. 👍
The internet wouldn't be fun if it wasn't for these people.
Regardless of your understanding of this concept, screaming at someone who doesn’t understand a theoretical mathematical concept shows either a lack of your own patience or an inability to teach in a way that is comprehensible to others. That is why the educator referenced in the original post should be fired, not because the OP was mistaken about the concept itself.
It's an axiomatic thing, but yes in usual axioms .999... equals 1. That's a weakness of the decimal representation system, there are two equally valid representations. I a nutshell, .999... is not a sequence, it's the limit of the sequence, and that limit is 1.
This was almost word for word the lecture i had in Calculus when we discussed this very problem.
I think the best way to help non-mathematicians, or anyone with a Jr. High or lower level of mathematics, is to simply demonstrate how this "problem" is just an illusion and arises when you start dividing by primes that aren't factors in your base. Show 1/3 and 1/5 in base-6, for example, or how 1/3 creates a repeating decimal in not just base-10 but also base-2. In my experience they eventually have an "ah-ha!" moment after just a few examples showing that ultimately the problem isn't with mathematics, but their imagination / mental representation of values like 1/3 when they see a decimal representation.
I was actually doing this precise exercise on my math class in this week. Was I suprised - yeah. Was it hard to prove - no.
I think a problem with writing 0.999…=1 is that we assume the reader to intuitively understand the left side as the limit of a series. As mathematicians we know that the equal sign in this case is a convention that is well defined and may fail to understand why someone unfamiliar with the formal definition of limits wouldn’t accept that.
'As mathematicians we know that the equal sign in this case is a convention'
Count me out, because I know that it's not a mere convention. 0.999... = 1.
And why do you say 'as mathematicians, we...' when you are not a mathematician (otherwise you would know that 0.999... = 1)?
Yeah. This.
The equal sign implies an ability to substitute instances of one with the other. This "substitution property of equality" is unique to this symbol.
@@thetaomegatheta I think your disagreement with OP is misdirected. There is certainly convention involved, you just don't place it in the equals sign. But there's undeniably convention involved.
0.999... is meaningless. Until you have some convention to explain what it means. Like, for example, the limit of a sequence of partial sums i.e. the sum of a convergent series i.e. a decimal expansion. Correct me if I'm wrong of course. I do know there are more subtleties at play, as before you can play around with real numbers you do need to define them in the first place, but I'm fairly strict on the fact that the limit of partial sums definition is just that, a definition.
I had this argument once... with my high school math teacher. Eventually when I brought in a mathematical proof it reduced her to spluttering about how if she chopped a bit of a penny off it would be less than one penny, demonstrating her proof with a penny on her desk and her hands in a decisive chopping motion. When I tried bringing it back to math after that, she would just get upset and threaten to send me to detention if I didn't stop.
I do understand the initial scepticism when you first hear about this. It does take a bit of processing to fully come to terms with this concept, but gagging the student won’t help much.
Okay, let's figure out how much of the penny you'd chop off!
1 - 0.999... = 0.000...
So, if 0.999... < 1, then it must also be true that 0 < 0.000...
@@AscendantOatthat won’t help, since she‘s just going to imagine “yes, but 0.0000… *is* bigger than 0 - there will be a 1 at the end”! Infinity eludes our imagination
@@TomJakobW Seems to me there would always be a 5 at the end.😬 Let's get into a huge argument over it!
infinitesimal isnt zero, and decimal notation necessarily defines 0.x as less than zero, you tell someone that something that definitionally doesnt equal one does equal is a bold faced lie and they will naturally be upset and it wont ever go away unless you admit that as a decimal it doesnt but if it were a series then it would approach 1 and so is functionally identical
All of the comments I've seen are good explanations, but rely on some result or intuition; I think ultimately the answer comes down to how real numbers are defined in the first place. There are several ways to do so in Analysis, but the most helpful method here is defining them as equivalence classes on Cauchy sequences (sequences whose terms get arbitrarily close together. As it turns out, this is equivalent to the least upper bound property commonly taught). So take the sequence of just infinitely many 1's (which surely has a limit of 1) and the sequence 0.9, 0.99, 0.999, and so on, which is Cauchy. If you take the limit of the difference of these two sums, we see that clearly the distance between the terms is shrinking to an arbitrarily small amount, and so converges to zero. Even though it technically never "reaches" 0 in a finite number of terms, we just define them as being equal because we can't impose any stopping point where they stop getting closer. So they fall into the same equivalence class, which is just how we define real numbers.
It IS equal to 1 because of the definition of the real numbers. Think of 0.999999... as a SEQUENCE: 0.9, 0.99, 0.999, ... this will converge to 1. The real number 1 is defined as the set of Cauchy sequences of rational numbers that converge to 1, an d (0.9, 0.99, 0.999...) is an element of this set. Hence it is equal to 1.
Confidently incorrect sounds like a fun subreddit.
Just to be clear, 0.999... = 1 is a math fact.
@@Chris_5318 Thanks dude. I'm a mathematician following a TH-cam channel where a guy explains math facts. But I really need to hear that from you 🙄
@@TheFiddleFaddle Ooops, I now realise that you were referring to something said in the video. My bad.
The important thing to ask is 'in what context'. There is no context other than abstract mathematics where 0.999...=/=1.
In computing, two numbers are equal if the difference between them is small enough, so a computer would say they are equal. In any real world scenario, the difference is undetectable to measurement devices, so they are equal.
That's a bad way to explain. Equal would imply no difference. Saying "the difference is undetectable" means there is a difference, you just need a better measurement device. Because your current application can treat them as equal, does not mean that will hold true in the future with more advanced technologies.
@@vicc6790Saying that things are equal literally means that the difference is undetectable. As I said at first, abstract mathematics is the only time that 0.999...=/=1
Is no one going to talk about how immature that adult was? Even though he was correct, screaming at someone should get you fired.
I've always been taught that 0.99999... Is 1 using a variable.
Let's say x = 0.99999...
Which means 10x = 9.99999....
Subtract 10x to x, 10x-x = 9.99999... - 0.99999....
Then you get 9x = 9, and x=1
Which means 0.99999=1
It also works with 0.6666 and 3.333 where it's equal to 2/3 and 1/3 respectively.
okay i get it now man math is stupid
Infinity minus infinity
@@jm329 basically. If you remember the infinite hotel paradox (i think ted ed explained it in a video), there's always an infinite number of rooms and infinite numbers of people, in which the hotel will always have enough room and every person will get a room.
Every time you add a person, the person inside would just move to their room number + 1, then theres always a vacant spot in room number 1.
Just like 0.999... (infinite amount of 9)
Times 10 is just 9.9999... (Still infinite amount of 9)
And then subtract infinite amount of 9 with an infinite amount of 9 and you get 0
One minus an infinitesimal is not equal to one. That’s why conditionals like “approach” and “converge” are used.
If you want to make a context where it can be true, then sure.
And there is one more point to consider. Infinitesimal numbers do not exist in the standard real number system (they exist in other number systems, such as the surreal number system and the hyperreal number system).
0.(9) - is an ordinary real number, a constant, which by definition is equal to the limit of the sequence 0.9, 0.99, 0.999, ..., which is 1.
'One minus an infinitesimal is not equal to one'
Cool. And irrelevant, as 0.999... is not 'one minus an infinitesimal'.
Like, what infinitesimal would that be, and not any other one if there even were any infinitesimals in this context?
'That’s why conditionals like “approach” and “converge” are used'
Lol no.
Learn what the words you use mean instead of making mindless guesses about their meaning and telling a mathematician about them supposedly not knowing what those words mean.
What infinitesimal would that be? Why ask that question when the context of the conversation makes it obvious it’s the one closest to zero without being zero.
See, context establishes a valid framework for discussion, which goes to my point above.
If you’re not comfortable working with infinitesimals and simply need to think of everything in terms of the real number system, that’s its own context, too.
@@edwardpotereiko "eed to think of everything in terms of the real number system"
Because here we are talking about the real numbers.
"ee, context establishes a valid framework for discussion"
And the context here are real numbers.
@@edwardpotereiko
'What infinitesimal would that be? Why ask that question when the context of the conversation makes it obvious it’s the one closest to zero without being zero'
Now, ask yourself, is there such an infinitesimal in hyperreal numbers, let alone real numbers - the context of the video?
'See, context establishes a valid framework for discussion, which goes to my point above'
The context is the space of real numbers. There are no infinitesimals in the space of real numbers.
'If you’re not comfortable working with infinitesimals'
You don't even know what they are.
'and simply need to think of everything in terms of the real number system, that’s its own context, too'
When people say '1+1 = 2', they mean it in the context of real numbers and standard notation using Indo-Arabic numerals, unless another context is specified.
You may whine that it's not the case, because there are contexts where that might be not true, but most people will understand what the actual context is.
From a science perspective, we can only gauge a value by a limited amount of accuracy. So at some point its no longer possible to know if the next value is nine so that value will be rounded up resulting in the value of an .999x being just 1 for all usable purposes outside of purely hypothetical or mathematical usage.
Math is independent of physical limitations. The accuracy is infinte, there is no error at all.
@@Chris-5318In any use case there is a limitation, only specific theoretical or pedagogical uses does a lack of limitation have a place.
@@jcpkill1175 Whatever, 0.999... = 1 precisely.
Explain that to the IRS.
The adult is arguing that the real number represented by the string „0.999..“ is the same as the real number represented by the string „1“. The strings are different.
You point this out at 0:40, different formulas but same values.
Numbers aren’t formulas or strings, they’re numbers, they’re completely defined by their values. 2/2, 7-6, and 1 are all different “strings” but that doesn’t change the fact that they’re all still 1.
The same goes for 0.999…
@@alephnull3535 I did not write that numbers are formulas or strings.
There is a clear distinction between the representation (symbols consisting of glyphs or black and white pixels, strings = tuples of symbols) and the associated numbers.
I feel like it's important to note that it is equal to 1 because of the properties of geometric series or limits whenever it's written down. Like the "+c" after an indefinite integral. It's a little nuance that can cause chaos when not formally acknowledged.
For as much as people are attached to the definition of real numbers as infinite decimals, let's be honest: no one uses it for doing arithmetic.
Adding 1/3 + 2/3 as 0.333...+0.666...=0.999... is cute, but it's useless. Adding fractions is much easier. It's way faster to do 1/17+1/19 in fraction form in your head than do it with the help of a calculator but using only the infinite decimal representation of those numbers. If you exchange the + sign with -, * or obelus, the problem gets easier and easier do to with fractions, and gets harder and harder to do with infinite decimals.
In conclusion: it doesn't matter what 0.999... is, it won't make any difference.
My high school calculus teacher back in the 1980s proved that 0.99... = 1 and it blew my teenage mind. Thanks for bringing back the memories.
People are still confusing 0.99....9 to 0.999... , unbelievable
0.99…9 isn’t a number
@@alephnull3535 I think their ellipses in the first number are meant to imply "many" and not "infinite"... but the use of the same ellipses in the second number to mean infinite does make things a bit confusing...
@@alephnull3535
Other way around. 0.999...9 is a number. 0.999... is an infinite series.
To be a bit more pedantically correct, 0.999...9 represents all numbers where the count of the nines is arbitrarily large, yet finite. Each of those is a number less than 1. And 0.999... represents the infinite series where the series of 9s is infinite. The value of the infinite series is 1.
The much simpler way to explain it is just if 1/3=0.333… then 3*(1/3)=3*(0.333…)=1=0.999…
That might be circular reasoning but it’s much easier to accept that 1/3 = 0.333… and explain it on that basis.
That's the exact way I've always thought of it.
This is the route I always take when I just want to get the conversation over with. It's not as rigorous as hopping back to the geometric series that the "..." represents, but it'll do.
I asked 2 math professors about this, and both said that 9.9(9) = 1, so I can only agree
I've had people ask me this and I usually say:
What is 1/3 as a decimal? 0.333.....
What is 1/3 + 1/3 + 1/3 as a decimal? 0.999...
What is that as a fraction? 3/3
What is that? 1
That's right, but is it worth screaming and threatening to have someone fired over?
Real numbers are defined as the equivalence class of all sequences of rational numbers whose sequence values get closer to each other and who's sequence differences converge to zero (Cauchy sequences). All real number math can be defined through this construction. So, 0.9 repeating is a member of the equivalence class with 1.
The "flaw" is that some people can't grasp that "infinitely close" is the same as "equal". "Exact" is an illusion - it doesn't exist in reality.
0.999... is, in fact, exactly 1.
I was hoping for a discussion on the relationship between limits, asymptotes, and equality
Curves and their asymptotes don't have much to do with this topic.
The decimals '0.999...' and '1' have the same referent.
Mathematicians: Does 0.999... equal 1?
Engineers: Measurement says the plate is 0.87mm thick. Documentation says 1mm. I say it's good enough.
I wonder how exhaustive the intersection between flat-earthers and people who don’t believe 0.99… = 1 is.
It’s probably a 1:1 ratio. I went to a catholic school so my math teachers believed the earth and its biosphere was created as is in a week 6000 years ago and even they understood 0.999…=1.
I remember learning 1=0.9999999999 in calculus 101.
I also know that 1=0 via:
a = b
a - b = b - b
(a - b)/(a - b) = (b - b)/(a - b)
Such that b - b = 0 as a number reduced by itself is 0
(a - b)/(a - b) = 0/(a - b)
Thus: 1 = 0
Therefor all numbers equal 0 as all numbers are groups of 1s added together.
Therefor all numbers are lies and dont exist.
'I also know that 1=0 via:
a = b
a - b = b - b
(a - b)/(a - b) = (b - b)/(a - b)
Such that b - b = 0 as a number reduced by itself is 0
(a - b)/(a - b) = 0/(a - b)
Thus: 1 = 0'
You don't, because, by the initial condition, a = b, and you also divided by zero there.
But sure, cute attempt at refuting a basic fact that 0.999... = 1.
Ominous I knew a man I told not to argue with a disabled person and instead he berated and humiliated him explaining that he vainly insisted he was correct..thank you for verbally explaining this to others