I'm Settling This Math Debate Forever (.99 repeating = 1)
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- Опубликовано: 6 сен 2024
- Does .99 repeating equal 1? Spoiler Alert: .99 repeating =1 and it always will.
It's time to formally prove .99999 equals 1.
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#math #brithemathguy #.99=1
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Dude i have never seen a pinned comment with 0 or 1 like
Here is the problem. In base 6, 0.5555555555555555555........................equals 1, in base 7, 0.666666666666666666666666666666.................equals one, and so on. But if you convert numbers between bases, they are not equal. When I try to convert 0.99999999999999999999..............into other bases, the resulting numbers don't equal one in those bases. None of that makes any sense.
@@InternetDarkLord You: "When I try to convert 0.99999999999999999999..............into other bases, the resulting numbers don't equal one in those bases"
Then you aren't doing the base conversion correctly.
@@Chris-5318 OK, how do you covert .9999999999999....................into base 5?
@@InternetDarkLord Use your loaf, it doesn't matter how you do it, the result is 0.444... = 1.
I always liked the demonstration like this:
x = .999...
10x = 9.999...
10x -x = 9.999... - .999...
9x = 9
x = 1 = .999...
I like it too!
YEAAA
I always felt 0.9999=1, but this was the demonstration I first found to make me 100% sure.
The exact same thing is taught in 9th grade schools here (in India) and this method is used to convert any repeating, non terminating decimal number into a rational number.
Wow
"girl you denser than the set of real numbers."
noone has found a reply to this comment
haha
"But I'm a very complex entity that isn't defined by my representation; I have many forms that are all me"
Her response:
"Bro, is this even a compliment? Get real"
Her response: "Atleast I am not denser than a Neutron star like you."
But it has so many cool rich nines how can it be the same as just a lousy poor one? That's right, it cant! (Proof by segregation)
Sadly, that doesn't work in mathematics
@@p_square Oh but now it does, I proved this way of proving using proof by intimidation /jk
@@ShinySwalot haha
@@ShinySwalot Ugh, I hate numberists like you. All numbers are equal! /ˢ
@@brandonchan5387 millions years of evolution and we finally come to numberism
I remember as a kid i came up with a similar solution when my dad told me that 0,999... and 1 are the same! :D I just thought by myself "what would the difference between these two numbers be?" and my answer was 0,000.... with infinite zeroes and a 1 attached to it, but the 1 is literally nowhere since the 0s go on forever! This is way more sound tho hahahah
Actually what you say is more intuitively satisfying.
I prefer your explaination, lol
Saved
I think this is called Epsilon in maths
@@kebab_boi7720 Not necessarily. You're probably thinking of epsilon proofs of limits. The point of those proofs is that you can literally choose any value>0 for epsilon, and the absolute value of the difference between the function value and the limit will eventually be smaller than epsilon. Obviously, this means epsilon can be arbitrarily small, but it doesn't have to be.
Plot twist: 1 is actualy equal to 0.999...
Yooo 😄😂😂
This is funny but it also proves the absurdity of the argument that .999... equals 1. As if either number holds supremecy over the other in anyway. We should teach .999... repeating to children when they count.
And then proof it
i dont think you know what equal means lmao
this isnt a plot twist, its true
I'm still a believer that most of the difficulty arises from the concept of what is a real number.
Yeah it is easy to prove something, if you define a rule to give what you want.
en.wikipedia.org/wiki/Cauchy_sequence#Examples
a real number is any number that doesn’t has i (imaginary number) for example 2i is an imaginary number 5i+10 is a complex number (has an imaginary part and a real part)
@@teshinthai2541 I think what he means is how can we know the usual mathematical rules apply to an infinitely long number? Infinite numbers have many exceptions to the rules, and while .9 repeating isn't infinitely large, it is infinitely LONG, so does the laws apply, then?
Of course you could say it's a human construct and you could decide if it is without it probably having any effect on any real application of mathematics but then why even bother proving it's = 1... I'm sure if I tried I could come up with sound logic for why it's not = 1
@@blamethefranchise1473 agreed .9....... approaches 1 but never reaches one because it is infinitely long it is a form of infinity so it should be treated more like infinity or pi since both are infinitely long and don't have an end
Wikipedia mentions the debate and some reasons why people can't grasp it.
One of them is people imagine 1 as an asymptote, where 0.9... approaches it but never crosses. As if it was a process, not a number. Like imagining someone writing a series of 9s, on and on and on, never truly reaching 1.
However, 0.9... isn't something approaching 1. It's a constant value. The number of 9s isn't increasing, they all already exist.
I like that approach. I guess it means all numbers already exist, even those that no-one and no computer got to write out yet. I was thinking about this in relation to the infinite Hilbert hotel where any new guest can be accommodated by shifting all the other guests to the next room along. My possible solution is to build a new room every time a new guest shows up, but if that means the room doesn't exist till you build it, then it's like saying a number doesn't exist till you formulate it.
What is 1- epsilon?
@@EmmaSquire-ks9nu Unless you’re working in the hyperreals, it’s less than 0.99…
I just used my own "rule of ninths":
1/9 = 0.111...
2/9 = 0.222...
etc., so then
9/9 = 0.999...
and pretty clearly 9/9 = 1.
Slept like a baby ever since.
Your own? But this is taught at school in the 6th grade:
0,xxx....=0,(x)=x/9
0,(xy)=xy/99
0,x(y)=xy-x/90
Here is the thing fractions like 1/9th are represented as repeating decimals because they can't be divided equally .111111 is an imperfect representation of 1/9th, which is why in a lot of cases, like in algebra, the fraction is used and multiplied out instead of converting it, to a decimal and solving it that way
so I guess you could say .111... approaches 1/9th same as .999... approaches 1 they are close enough that we can use them like that, but they aren't inherently equal
Another way to say it is that .111... is 1/9th of .999..... not 1
@@KingArkon He probably meant he came up with it himself, not that he was the first one to come up with it.
@@thecraftycreeper3167 *Calculus: Am I a joke to you?*
@@thecraftycreeper3167 Proof:
Let's assume that 0.11111... is 1/9th of 0.999999... as per your claim. But
0.111111111... = 1/9
If you disagree, type 1/9 on a calculator and you will get 0.111111111... , so far so good. Now:
0.1111.... = 1/9
0.1111.... × 9 = 1/9 × 9
0.9999.... = 9/9
0.9999... = 1
BOOM!
When I first saw this equality, I knew it was true as I can never think of a number between 0.99... and 1.
Correct!
(1+0.99 repeating)/2
@@thecheesegenius3817 sir excuse me as in logic 0.9 sub infinity and what can come then or
1.999.../2=0.99999 as in logic remainder on dividing by 2 is 1 so no number exist
@@BriTheMathGuy
Lol, that doesn't even mean anything. What does it mean to think of a number between 1 and an object you haven't even defined?
Once you define it there is no controvery.
It's just people thinking that the symbols "0.9..." should have some inherent meaning. That's exactly the mistake the two of you are making.
@@MrCmon113 dude how is it undefined we know all the digits of it but it just won’t end
I feel this proof wouldn't change anyone's mind about the problem. If one believes 0.999... is not equal to 1, the same person doesn't believe 1/(10^n) converges to 0
Good point but I think not quite right. I can believe that 1/(10^n) converges to zero without believing that 0.999... equals 1. Because convergence and equality are not the same thing. I believe that 0.999... represents a sequence which converges or approaches the limit of one, but it is not "equal" to 1
@@matthewhowey6564 yes you are correct and I should explain what I mean more clearly,
At 3:20 when he changes 1/10^n to a 0, that's when someone who think a number like 0.000....1 exists would think 0.0...1 is what shows up instead of zero. And then we're kinda back at where we started.
@@eliaslaihorinne yep exactly, fully agree, thanks for clarification
@@matthewhowey6564 no.
the sequence, 0.9(1/10^n) has a limit of 0. the infinite series, which is the sum of that sequence when it has infinite terms, is equal to 1. theres literally a formula for figuring out what the sum of an infinite series is, and its a/(1-r). since a is 0.9, and r is 0.1, that makes the sum of the infinite series, and therefore the 0.99 repeating that its equal to, equal to 0.9/(1-0.1), which is equal to 0.9/0.9, which is equal to 1.
@@matthewhowey6564 0.999... is the limit of the sequence 0.9, 0.99, 0.999, ..... So is 1. In fact
0.999... := lim n->oo 0.999...9 (n 9s) = lim n->oo 1 - 1/10^n = 1
I think the main reason why people believe 0.9999... is not equal to 1 is because they lack the sense of what it means to pull the row of nines to infinity. They always see it as finite, however long it is. And sure enough, for any finite row of nines in 0.9999..., it is less than 1.
Thy also lack an understanding of how limits work, and they are unable to distinguish conceptually between "sequence", "number", "process", and "limit".
This misconception is furthered by people generally representing decimals in the no-.9... form for brevity. 0.4 = 0.39999..., but it's infinitely easier (literally) to write it as 0.4.
An infinite "row of nines" isn't equal to one. It's not even a number. It's a row of nines. The problem is people like you thinking that a set of symbols has some inherent meaning prior to any definition. "0.999..." isn't a word in common use. It's a mathematical expression that requires a definition.
Moreover the actual "0.999..." doesn't require an "infinite row" or anything like that. Just infinite sets.
@@angelmendez-rivera351
That describes Kashua.
@@MrCmon113 "The problem is people like you thinking that a set of symbols has some inherent meaning prior to any definition. "0.999..." isn't a word in common use. It's a mathematical expression that requires a definition."
Symbols have no inherent meaning, true. Neither has a language. That's all valid, and there is certainly a place for that discussion, however, you could say that about everything. In particular, you could say that 3+7=6 might be true, it's just symbols. And if I were to work in Z4 or Z2, that would indeed be the case. But there is a common notion of what 3+7 means, it isn't 4, even for those who never properly defined those symbols ever. I think so far, you agree.
And I would argue, though that might certainly be subjective, that what people commonly understand by "0.999..." pretty much aligns with what the conventional mathematical definition says, so I think we can cut someone some slack for saying "an infinite row of 9s" in the context of what they're trying to say.
This video really squeezes my theorem.
Among mathematicians, this has never been a debate. The fact that 0.9999... = 1, as well as 17.563999999... = 17.564 and 1.29999999... = 1.3 is just a redundancy of the decimal notation. We just decided to stick with the form that ends with an infinite string of 0's, instead of the one that ends with an infinite string of 9's.
More formally, every real number is defined to correspond to a certain equivalence class of cauchy successions of rational numbers, where two rational cauchy successions {a_n} and {b_n} are defined to be equivalent if the succession {a_n-b_n} converges to zero. This definition obviously includes all the pre-defined subsets of real numbers; e.g., if a class of cauchy successions converges to a rational number in Q, then define that class of cauchy successions to represent that number.
Of course, given an arbitrary real number, there exist infinite cauchy successions that represent it; we can pick the one we prefer the most from its equivalence class when we want to make maths with it. The decimal notation is just choosing the succession to be in the same form for every real number: the number a⁰.a¹a²a³... is the one represented by the class of rational successions that, among the other infinite ones, includes the succession:
{a⁰, a⁰ + a¹/10, a⁰ + a¹/10 + a²/100,
a⁰ + a¹/10 + a²/100 + a³/1000, ...}
In easier-to-grasp terms we say that a⁰.a¹a²a³... = a⁰ + 0.a¹ + 0.0a² + 0.00a³ + ...
Now let's take a rational number p/q; it can be shown that its decimal expantion is either periodic or finite, depending if the prime factorization of the denominator q (after simplifying every common factor between p and q) has only 2's and 5's or not. Let's stick with the 'problematic' case, when the decimal expantion is finite.
Let's make it even simpler, lemme just take the star of the video, the number 1. How can we write it as 1 = a⁰ + a¹/10 + a²/100 + ... for a certain set of digits a⁰,a¹,...? Well, of course choosing a⁰ = 1 and a¹=a²=...=0 works fine, so the decimal expantion of 1 is 1.000000...
But is it the *only* possible choice? Let's try to change some of the digits, starting from a⁰. We clearly can't plug in it a 2 or a 3: since the succession a⁰, a⁰+a¹/10, ... Is strictly increasing, we can't hope to make it converge to 1 if already the first term is bigger than 1. So we can try putting in a⁰ = 0.
What about a¹? Well, we know that a²/100 + a³/1000 + ..., for *every* choice of digits a²,a³,... Is surely smaller than (or at most equal to) 0.1. if we hope to build a succession that converges to 1, we have to set a¹ = 9, otherwise the limit would be smaller than (or at most equal to) 0.(a¹+1). The same goes for all the other digits: if we hope to build a succession that converges to 1, we have to set them all to 9.
So we understood that 0.9999... is the only second possible candidate to represent 1, together with 1.00000... but do they represent the same number? That is, does the difference between the succession
{1, 1+0, 1+0+0, ...}
And the succession
{0, 0+0.9, 0+0.9+0.09,...}
Converge to zero? Well, of course it does: the difference gives the succession:
{1, 0.1, 0.01, ...}
Which clearly approaches zero.
Tldr the definition of decimal expantion is ambiguous for the numbers with a finite decimal expantion, as the definition is compatible both with a finite sum of terms and with a neverending sum of 9's. But everyone who knows what we decided the meaning of "real number" to be has no doubts about the fact that 0.999.... = 1.
Maths always comes down to definitions. Many people think that there's a sense of 'trueness' behind some mathematical identities, that some things are truer than others and that there's room for debate. But it all always boils down to the definitions we gave for the stuff we work with. "0.9999... = 1" because of what mathematicians mean with "1", of what they mean with "0.99999..." and of what they mean with "=". There's no mystical and inconceivable truth waiting for us at the end of that infinite string of 9's.
I want to copy your comment and write it all over my school walls
This is gold btw
I agree ! This is really gold ! Thanks for this comment, I never thought of this problem with this angle of view !
Yes but is defining the limit of a sequence of partial sums to be equal to whatever it converges to a reasonable definition? Imo most mathematicians just take the construction of the reals for granted, which lacks intellectual integrity. But I guess the grants they have aren't for such philosophizing, so it's understandable but disappointing
@@tommyhopkins6431 Maths isn't reasonable or unreasonable. Maths is just a pile of definitions built over a small set of propositions assumed to be true. The idea of a "reasonable" or an "unreasonable" definition, or set of axioms, is simply ill-defined and with little meaning.
...But I know what you mean with that. "Heeey, wait a sec! Sure, Maths itself is nothing but a chain of tauthologies with no _a priori_ importance, but humanity didn't pick the most commonly used axioms and definition blindly from the infinite world of formal logic! Before we formalized real numbers, we measured lenghts! Before we discovered/invented probability theory, we knew how likely it was for a dice to roll on a 4! Before introducing natural numbers and addition with an elaborate construction, we counted the sheeps in our farm! Maths must be deeply rooted in the natural world, or at least part of it is, and it's our goal to choose our axioms and definitions such that our mathematics perfectly mimics the maths that exists in nature. The rest of logically possible constructions may be interesting to study, but are clearly less important than the ones on top of which nature thrives."
To which I answer... I dunno? This is straightup philosophy and I'm definetely not competent enough to answer these questions (I mean, to be honest, no one is and no one ever will - that's the cool and bad thing about philosophy). Is maths something more fundamental than empty abstract tauthologies? Does the mysterious bond between maths and nature suggest that some mathematical concepts effectively _exist in nature_ in some way (whatever it means), or it's just because we built maths to be that way? And assuming a deep connection between maths and reality, which mathematical concepts have a physical meaning and which ones are the "bad apples"? The concept of natural number seems very fundamental, but what about, e.g., imaginary numbers? Are they nothing but a clever way to manipulate more fundamental concepts, or do they _exist in nature_ in some way too (whatever that means)? And what about real numbers? And negative numbers?
But hey, at the very least the conversation shifted from "0.9999... = 1 is wrong" to "0.9999... = 1 is unreasonable", in the sense that the set of definitions that imply 0.999... = 1 doesn't correctly represent reality. To be fair though, the canonical construction of the real numbers is based on very natural assumptions (the Cauchy one is a bit cumbersome, but formalizes the very intuitive idea of "looking at the limit of a string of numbers"), so I don't think that you can find a simpler/more intuitive/more "natural" (whatever that means) number system in which 0.9999... somehow formalizes the idea of chaining an infinite string of 9's, but ends up being something different from 1.
What are other math topics that are true but hard to accept?!
i dont accept this
I day ago? How?
@UCjjJd2ysyKg1F8h7sM99M6Q no! It Equals -13/12 my dad said
Here are some math topics that are true but hard to accept:
1-1+1-1+1-1+.....= 1/2
1-2+3-4+5-6+.....= 1/4
1+2+3+4+5+.....= -1/12
And ever heard of Kaprekar's constant.
1. Numbers exist.
2. Numbers don't exist.
I don't understand how/why there is so much disagreement in the comments. 2 real numbers are different if and only if there is an uncountably infinite number of real numbers between them.
Just because the way we write .9999.. doesn't look like the symbol '1' it does not mean it isn't the same value. Stop relying too heavily on notation.
it all depends on what rules you want to base yourself on. We can define the 0.99(...) as the number right before 1. This rule of "2 real numbers are different if there is an infinite amount of numbers between them" is just a logical rule and not an actual measurable rule
@@andrejosue98 you cannot define define .(9) as the number right before one because I assume you're taking that it has an exact and unambiguous definition as a sum.
@@andrejosue98 Also you're wrong that it's not a measurable rule. If there is an infinite amount of numbers between any two distinct real numbers a and b, then I am able to tell you a number in between a and b that is not equal to a or b. I cannot do that with .(9) and 1 because they are the same number.
@@andrejosue98 the only other point to make on this I've heard is that .(9) itself is an abuse of notation. I somewhat agree and always will use .(9) as lim x->infty 9*sum_1^x 1/10^x. If you're using a different meaning of .(9) you're going to have to explain what *exactly* that is.
Its not about the symbols looking different. Its about the logical meaning behind the symbols. You probably know this stuff, but I'll write it for the sake of being explicit and providing context. 0.2 corresponds to 1/5th. This is a finite decimal because 10 is divisible by 5 and we have a base 10 number system. If you go through the process of computing a division step by step, the computation will never end if the number systems base is not divisible by the number you are dividing by. 10 is not divisible by 3, so we end up getting .3 with a remained, then .33 with a remainder, then .333 with a remainder, and so on.
So. There is never any point at which the decimal you get from dividing 1 by 3 in a base 10 system actually becomes equal to 1/3rd. It can only approach it. Our education system is inconsistent with how it treats this issue compared to other similar issues. Saying the limit of a function approaches a value as the input approaches a given value and saying that the function equals a value at a given value are two different things. You can't just get rid of the limit and it still be the same logical statement. The function could be undefined at that point. But then that's exactly what they do when it comes to .9999... = 1. The limit of this division process of diving 1 by 3 in a base 10 system is 1/3rd. And if you completed the division, you'd have 1/3rd. But .3 hasn't complted it. .33 hasn't completed it. .333 hasn't completed it. And saying that the 3's go on forever doesn't show that you have completed it. Sure, you can say that the decimal is infinite and that the division process is infinite. But what is their cardinality? Aleph null? Aleph 1? It is undefined. You can't make a conclusion about it.
Tactics I have been presented to prove why 0.9 recurring is not 1:
- Invent an impossible non-existent real number: _"0.9 recurring differs from 1 by the smallest real number larger than zero (which certainly exists! Never mind all those mathematicians who claim such a number doesn't exist. Clearly they don't know what they are talking about! I know better than them!)"_
- Use the magic fancy word: _"Infinitesimals! (I have absolutely no idea what they are, but I heard about them in some random Numberphile video, and thus they are the answer!)"_
- Deny the validity of the notation: _"0.9 recurring is not a number at all, because you can't write an infinite number of 9's. It's invalid notation! (Never mind that 0.3 recurring is a valid representation of 1/3. Pay no attention.)"_
- Thinking about 0.9 recurring always in terms of a finite expansion: _"No matter how many 9's you add, it will always be smaller than 1."_
First the system including infinitesimal and infinity as member of “real number”, is called the “hyperreal”. You can do arithmetics with both infinitesimal and infinity, with special rules of course, and they make a lot calculations easier, simply because they discard the unnecessary part (mostly the elements mathematicians added to make the system rigorous) in the process. The result mostly doesn’t differ from the standard system, which means it is an extremely efficient system for calculations. These notions are widely use in both engineering and physics, and maybe more fields, without identifying with the name “hyperreal”.
Second, this maybe sounds ridiculous, but for mathematicians, decimal expression is just an expression after all. It is just a way to express real numbers in a more intuitive way, so that we can have a general idea of the size of the numbers and how they compare. If you ever learn vector and its coordinate expression, you will notice that most of the time, we have “standard basis” as the underlying basis when there’s no further clarification, but you can also use basis say it goes twice far in north than in east. It’s still an valid coordinate system, just with different underlying basis. You can even have basis such one goes north and one goes northeast. It’s also valid. However you will end up with different coordinates with this basis. It’s mostly convenient to use the standard basis, but there’re also a lot situations you would like to utilize other bases, like in a grid made by triangles.
Long story short, “the expression only tells half of the story.” The other half is hidden in the basis. Same story for decimal expression, the definition plays equally important rule in this case as well. The base n expression is defined by the convergent series of powers of n. Different series can converge to the same number. So does the expression. There’s always an expression for a real number, but there can be several expressions for a single number.
However, if you define it another way around, by saying that you have a set, in which each member is a decimal expression. Then these two expressions are different “by definition”. This definition actually has an analogy in ring theory called “formal power series”, which is defined by its infinitely many coefficients, regardless of convergence or anything we usually worry about. It has its own purpose, such as generating functions, which can diverge generally. Similarly, even if two formal power series converge to the same value for every value you plug in, there’re still distinct member of “formal power series”.
However, if you regard these formal power series as functions and they have same value at every point, then they are the same function by definition, since they agree on every value you plug in. Same for the expressions. If you are regarding these two expressions as real numbers, and you can not tell the difference by any possible manipulation, then they are the same number.
Are the expressions the same? No
Are they the same number? Yes
@@gunhasirac The thing about 0.9 recurring is that it's merely an alternative representation of the value 1. It's no different than "1/3" being an alternative representation of "0.1 base 3", which is an alternative representation of "0.3 recurring". They may each use a different notation but they all represent the exact same real value.
In the same vein "1" and "0.9 recurring" are merely two different notations to represent the same real value.
Well, the notion of "infinitely many digits" doesn't actually make any sense and when we write that, we define it as the limit of the sequence upto n digits, which makes obvious why 0.999... Should be 1, as the limit of the sequence 0.9,0.99,0.999 and so on is equal to 1
@@DjVortex-wok yes however that’s the whole point of the hyper reals the fact that now we are infinitely precise in the real numbers and basically any other scenario yes 0.9… is equal to 1 and nobody will argue that but we define this infinitesimal value as not in the real numbers there for 0.9… doesn’t equal 1 in the hyper reals I get what you mean by saying they are the same number but that’s the entire idea of the infinitesimals (I’ll admit I don’t have the greatest understanding of the hyper reals but this should still be an understandable argument)
@@elreturner1227
So, just invent an abstract concept that cannot exist in actual math, which has absolutely no use in math, which cannot be used in any calculations for any actual results, and which actually is just completely equal to 0 (yet is somehow different from it at the same time, which is a contradiction).
I was debating on bprp's channel and now I am here
Another incredible video. I really prefer these types of videos that explore a concept more in-depth (denser information, you could say) and try to address different perspectives. Thanks for sharing and keep ‘em coming!
Glad you enjoyed it!
@@BriTheMathGuy Its a convention, that is being given the status of proof, without an actual proof.
The disproof is easy :
0.999... < 1.000... by inspection.
Long story short: there is ZERO difference between 0.999... and 1 😆
My take on this:
- Start with two definitions:
(a) 0.9999... = lim_(n->∞) [(0.9)_n]
(b) (0.9)n + 1/(10^n) = 1
- Now let's do some work:
lim_(n->∞)[(0.9)_n + 1/(10^n)] = lim_(n->∞)[1] (take the limit as n->∞ from both sides of (b))
lim_(n->∞)[(0.9)_n + 1/(10^n)] = 1 (the limit of 1 as n->∞ is 1)
lim_(n->∞)[(0.9)_n] + lim_(n->∞)[1/(10^n)] = 1 (separate the limit of a sum into the sum of the limits)
lim_(n->∞)[(0.9)_n] = 1 (the limit of 1/(10^n) as n->∞ is 0)
0.9999... = 1 (apply definition (a))
Once you have definition (a) in place the rest is straightforward. The key point that people don't understand (because so often it is not stated) is that the notation 0.999... means this. It's the limit of something, not 'equal'.
You could also say 0.999….=0.111….+0.888…
0.111…=1/9
0.888…=8/9
Then, 0.999…=1/9+8/9=1
@@matthewleitch1 This is a critical point.
@Bruh Bruh I just stated it, but I guess you would like something more authoritative! I found that Wikipedia spreads it over two pages. The idea of the recurring notation being an infinite series is here (en.wikipedia.org/wiki/Repeating_decimal#Repeating_decimals_as_infinite_series) then they explain that the infinite series notation really means a limit (when there is one) and they explain that here (en.wikipedia.org/wiki/Series_(mathematics)).
This "debate" was something settled long ago when developing the rudiments of real analysis, and was one of the introductory demonstrations the professor performed for us when I took real analysis.
I accept that 0.9 repeating is 1. But for the people that don’t, I don’t think the argument at 3:17 is convincing. It assumes that 0.000...1 = 0, which introduces a whole new problem
It's not 0.000...1, as that implies there's a 1 at some arbitrary finite point. It's 0.000... and you then keep writing zeros forever and never stop. And 0.000... is zero.
@@oenrn that’s true, I kinda abuse notation over there, but I still want to emphasize that assuming 0.000...1 = 0, although true, follows the same argument of .999... = 1. If there are still people who aren’t convinced that the latter is true, neither will they with the former. That’s why I am pointing it out. Since this video targets those people, we can’t argue like this, it’s circular reasoning. What do you think
1 - (9/10) - (9/10^2) - (9/10^3) ... = ?
@@GuiSilva1I guess it's philosophy then, and not math, because 0.000...1 is not 0. 0 means there is nothing at all, so to have even an infinitesimal would be greater than 0.
Math may say so, but .9999 repeating to me isn't 1, because philosophically speaking, .9999 repeating can be removed from 1 to get a number that is less than 1.
me at primary school:
10 / 3 = 3.3333...
3.333... × 3 = 9.999...
10 = 9.999...
1 = 0.999...
yes
We dont say 3.333333 is equal to 3.33333334 either
This is a cool proof that I haven't seen before! Thanks!
Glad you like it!
It was never a “debate”. It’s just true
Phenomenal channel really really like the way you explain things not just this but things in general especially regarding integrations and infinite sums. My only wishbis that you go deeper into more difficult math
unfortunately there will always be people who can't get past the fact that there are different shapes on either side of an equals sign
Everyone:- *debating*
Me:- *BRING SOME CHIPS!!*
Do you have some pop-corn though?
I GOT DA SOUR PATCH KIDS!
My math teacher in like 5th grade when I told them about this: "It's just inaccurate". That's an usual response if you don't wanna accept something.
a 5th grade math teacher is not gonna know this bruh😭😭
This was never a debate. It was settled ages ago. The debate is about why every high school grad doesn't know it.
After watching this 0.9999 is definitely not equal to 1. Impossible.
That's right. It's 0.999... = 1 and that is definitely a math fact. Summary:
0.999... := lim n->oo 0.999...9 (n 9s) = lim n->oo 1 - 1/10^n = 1
doesnt this also mean that any Number With a .999... at the end is equal to the whole Thing Like 3.999...=4?
Correct. Also 0.25 = 0.24999...
This is a super weird but clever approach that I have never even bothered to think or question. Super cool.
I'm a bit confused.
I agree that this does make sense in theory, but in reality, there is no reason why it should be true.
There is no infinity in our world.
No mater how close you get, you can never reach the 1.
Math as nothing to do with reality and is not constrained by it. Even the number 1 doesn't exist in [physical] reality - no mathematical object does.
"No mater how close you get, you can never reach the 1." What does that mean??? If yo mean that if you start with 0.9 the "append a 9 to get 0.99 then append another 9 to get 0.999, etc. Then you will never get to either 1 or 0.999...
@@Chris_5318 Don't take what I said a day ago too seriously.
I've been enlightened and changed to a better person.
Two numbers are inequal when you can stick another number between them. Therefore two numbers are equal when there is no number between them.
@@melvinwarmpf1264 Your second paragraph was an assertion. It is correct, but can you prove it, and can you prove there is no number between 0.999... and 1?
@@Chris-5318 Everthing is true until it's disproven. Can you disprove that there is a number between 1 and 0.999...?
You can't prove that something equals something, because you can't even actually prove that 1+1=2 or rather: we know that we can't prove it, this is what makes it the foundation of mathematics.
@@melvinwarmpf1264 "Everthing is true until it's disproven".
Utter rubbish.
"Can you disprove that there is a number between 1 and 0.999...?"
Of course *I* can. The question is, can you? Also, can you prove that if there is no number between a and b, that means that a = b? (Again, I can).
"you can't even actually prove that 1+1=2".
It is true by definition / the Peano axioms.
Plot twist: numbers aren’t real
My favorite way to think of this is with the fraction 1/3.
1/3 = 0.333..., so 1/3 * 3 = 0.999... right? but 1/3 * 3 = 3/3 = 1. So 0.999... = 1.
Far more understandable.
(maybe)
1/9 = 0.111...
2/9 = 0.222...
8/9 = 0.888...
9/9 = 0.999... = 1
This makes SO much more sense to me. lol
but 1/3 does not really = 0.33333.... 0.3333.... is just the closest representation you can make with a decimal.
@@Elrog3 1/3 equals .3 repeating numbnuts
The first 30 seconds were a roller coaster of not understanding nothing.
The fact that it isn't 1 but it's 0.999999 means it will eternally fall short of actually being 1. Whatever 1 is, 0.99999 is slightly less. You can do all the mental gymnastics you want, but they are not equal.
There are several million degreed mathematicians that disagree with you and your uninformed opinion. You are actually talking about the sequence 0.9, 0.99, 0.999, ... that doesn't get to 1. Here's the thing, it doesn't get to 0.999... either.
@@Chris-5318 1 is 1 and .9999 is .9999
@@firstnamelastname2552 and 2/5 + 3/5 is 1 and 0.999... is 1 and that is an actual fact.
@@Chris-5318 several million degreed mathematicians jerking their egos over semantics, 0. whatever is not 1, you can do twisty logic bs all you want but the number 1 does not start with 0
"oh but its notation" ok but thats not what we're talking about, maybe i wanna write 1 as 45 and call it another way to write 1, but that doesnt make it 1 lmfao
@@elleeVee LOL. You need to learn the difference between a numeral and a number. Numbers are pure abstract, and they do not begin with anything. You are an ignorant muppet and you do not have a clue about mathematics. Try again after you've finished eighth grade. 2/5 + 3/5 does not start with 1, yet it also equals 1.
If anyone doubts that they are equal just ask what is 1 - 0.9999…. with an infinite string of 9s.
What would you say to 1 - .9 = .01, 1 - .99 = .01, 1 - .999 = .001, so 1 - .99999... = 1.0000...1? I would accept that this expression doesn't disprove 0.99999... = 1.
@@chrisg3030 lim as n tends to infinity of 10^-n = 0.0000... with an infinite string of zeros.
1 minus an infinitesimal amount of 0.99999 will yield something infinitesimally smaller than 1. It would be infinitesimally close to 1, but isn't 1.
.9 repeating isn't 1. 1 is the destination you reach, a complete one of amount. Having .9 repeating for infinity would yield something infinitesimally close to 1, but, still not 1. 1 is a number with no .9 repeating.
Mathematically speaking sure, it can be "proven" to a degree. But I think it's leaving out the infinitesimal amount that would still exist between .9 repeating And complete 1.
@@superbuu122 how close to infinity is this infinitesimal amount?
@@padraiggluck2980 the infinitesimal I'm imagining wouldn't be located on the number plane at all. .9 repeating would never end, so there would be no infinitesimal to place at the end to turn it into 1.
In a sense, something else occurs to .9 repeating to " change it " or " evolve" into a complete 1. Something occurs to .9 repeating that isn't addition, but more of an evolution, that causes it to become 1.
It is easy and rational to demonstrate it with geometric series.
But I mean: it's provable in a thousand different ways. Which is the debate?
It depends on how you define numbers and sets.
In the practice and in logic the difference between 1 and 0.999...9 (and lets say we have 10 sextillion nines, so it has a finite number of 9) may as well be 0. But it is not 0, it only approaches 0.
I like it this way :
1/9 = .1 repeating
2/9 = .2 repeating
.......... with the symmetry
9/9 = .9 repeating
But wait 9/9 = 1
so 1 = .9 repeating
:))
We're still accepting that 1/10^n effectively reaches zero, which is the same kind of assertion required to assume 0.999... = 1.
For my part, 0.66666... = 6/9, 0.77777... = 7/9, etc. so 0.99999... = 9/9 = 1. That's all I needed.
1/10^n doesn't actually reach 0 as n -> oo. Your proof doesn't give insight into e.g. limits.
There's a big issue here in that someone who doesn't accept that 0.999...=1 is never going to accept the much less intuitive result that the real numbers are dense.
I honestly find it very intuitive that the real numbers are dense.
Since I can always use the average (x+y)/2 to show that there is a third number between x and y (element of) IR, with x =/= y.
Doesn't the density follow from a set of axions we defined about IR? (I'm not super knowledgable about proving such concepts.)
Are we still in the field of "real numbers" IR we use in our everyday lives, or not?
If so, then the real numbers are dense, no discussion.
If not, then please, dear critics, prove to me *why* we are no longer in the algebraically defined field of IR!
I feel like those people should make up their mind!
@@romano-britishmedli7407 The real numbers are dense, yes.
@@RaRa-eu9mw Yeah, I honestly don't get what the critics' problem is.
While 0.99... = 1 may sound unintiutive at first, I think that most people will accept that IR is dense, if we explain density to them.
And what convinces me and hopefully most others is that the difference between 0.99... and 1 is exactly 0, since I can't define an ε > 0 which describes the difference between 0.99... and 1; so the equation
0.99... + ε = 1
is false for all ε > 0.
(I don't get the concept of infinitesimals and don't want to dabble into this area of mathematics. I stay in my orderly real and complex numbers, thank you very much.)
@@romano-britishmedli7407 Do you really think the majority of people are still on board when you bring an epsilon into proceedings?
@@RaRa-eu9mw Ah, I went too much into university-mathematics, for the geeks here in the comment-section.
No, of course you are right.
I'll limit my "proof to common people" to the concept of density and "the difference between 0.99... and 1 has to be 0" (and explaining this in layman-terms).
This kinda applies to math but in wood working there is never the correct measurement. Just how much you want it to be off by.
The way i look at this particular concept is the same way we think about infinity. If you try to add anything to infinity, you just get infinity back. Effectively no change. Same applies here. You can say that there is no number between 0.999… and 1 because if you tried to add another nine at the end, you still end up with the same value. Your distance to 1 is infinitely small.
Same goes for 0.000…1
In fact, thats what you would need to add to 0.999… in order to get one. You have to add together 2 infinitely precise values to reach the whole.
You started off so well. You gave the reason why 9 * 0.999... = 9.999... - 0.999... = 9 and so 0.999... = 1. Then you blew it.
0.000...1 is a terminating decimal whose value is greater than 0. If you add it to 0.999... you will get 1.000...0999... where the last 0 is in the place where your 1 is. Your claim is not a fact, it is something that you made up.
Note that 0.999...9 + 0.000...1 = 1 where the ... indicates the same finite number of repetitions in both cases.
Decimals cannot represent non-zero infinitesimals.
The fact is that 0.999... = lim n->oo 0.999...9 (n 9s) = lim n->oo 1 - 1/10^n = 1 and not 1 - infinitesimal.
If 0.000...1 is supposed to represent an infinite number of 0's followed by a 1, then that is not a (real) number. What can follow an infinite amount of something? That doesn't make any sense*. 0.000...1 does not exist*.
You should think of numbers and the notation for numbers as separate. Numbers are abstract concepts that are represented by symbols. If I write "1", then I did not write a number. I wrote a symbol representing a number. And a number can be represented by multiple symbols.
The difference between 0.999... and 1 is not just infinitesimally small*. There is none. They are two symbols representing the same number. 0.999... is actually equal to 1. You can write one instead of the other in any context you like without changing the value*. It's just like -0 is equal to 0. They are two different symbols, but they represent exactly the same number and are completely interchangeable.
* in the real numbers
@@sciencetube4574 Just to boost your armoury:
If 0.000..1 is supposed to represent infinite number of 0s followed by a 1, then it is not a valid decimal. You cannot have a 1 at the end of an endless string of 0s. Also every decimal place is indexed by a natural number, That 1 cannot be indexed by a natural number.
Lightstone extended decimals can represent infinitesimals e.g. 0.999... ; ...999... is a valid hyperreal number. I gather that 0.000... ; 1 is not a valid hyperreal but 0.000... ; ...1... is valid.
@@sciencetube4574 .000001 points for explaining this without taking extra shots at someone with a misunderstanding. So many guys here making good points and clearing up confusion but being a tool while doing it. Anyway, the push to think of numbers and their notation as different is probably simple for you guys but kind of a mind blower for me.
@@BStack That is not a lot of points, but it is a strictly positive amount, so I'll take it :)
Yeah, maths can be confusing. 99.9% of the time, an intuitive understanding is completely sufficient. It's only the weird cases, like 0.999..., where this understanding breaks and you have to look at the formal definitions. And most people don't bother to do that, since it will never again help them in their life. But yes, formally, there is no mathematical preference for any way of writing a number. They are technically just entries in a set that fulfills certain properties. Writing them in the conventional way might help people understand what you mean, but there is nothing stopping you from calling them anything else.
I like this proof:
1 - 0.999999 = 0.0000000…. 1
However the number on the right hand side can NEVER exist. Think about it like this. There are a lot of zeroes and the 1 is waiting for its turn, but it never gets its turn so it is constantly 0.00000, and thus just 0.
Anything - 0 = anything, therefore,
1-0(0.0000…1) = 1 and rearrange to make:
0.9999 = 1
I'm fascinated by this aspect of the topic too, and like the way you put it, that the 1 never gets its turn.
But what if you generate the right hand side with the series 1 - 9/10 - 9/10^2 - 9/10^3 ... Here the 1 gets its turn straight away when you do the first step, 1 - 9/10 = 0.1 and just starts getting pushed along by the zeroes with 1- 9/10 - 9/10^2 = 0.01 and so on. In this case shouldn't we say 1 - 0.999... = 0....001? With the recursion dots not between the zeroes and the 1, but between the decimal point and the zeroes to indicate that's where new zeroes are forever being formed? (I like to think of them kind of bubbling up there).
I'm not sure what consequences this has for your proof, or for the proposition that 0.9999... = 1. I dare say it leaves them both intact, which is fine by me.
Will you do a video on the infinite series of all natural numbers? Despite its divergence, people say sometimes it's -1/12
There is a nice Numberphile video about this.
@@user-xx6pr1te7q I would suggest watching the Mathologer video though, as the sum of all natural number isn't -1/12. That video clears it up, but also goes really in depth about ways to associate divergent series to a number.
I will suggest you to see mathologer video
@@Jotakumon okay thanks.
@@sangeetaraisane2141 sure thanks
In your "formal proof" you say just at the beginning that "I doubt that you think that 0.999... is greater than 1". This is more intuitive than formal.
0.999... isn't greater than 1.
Even if it's not the case we can still say the deference between them is some value x where 0≤x≤1, which is all we need to start this proof.
For me even as a kid it always boiled down to the following - If I can just add nines ad infinitum, is there even a point in talking about a number n between 0.(99) and 1? The short answer is no, obviously, but it seems to rub people the wrong way lol
And if you add a positive number of any size to .9999.. it will blow past 1, because there is always 9s.
@@suddenllybah precisely
Well it should, because this kind of confused thinking has often lead to misconceptions that lasted for millennia and probably cost millennia in time that could have been spent on productive thinking.
@@divisionzero715
Why?
How the fuck would you know what should be the result of "adding" anything to 0.99... if you don't have a clear idea of what 0.99... even means? It's pure nonsense.
If you follow the common definition it's trivially equal to 1.
Forget 'adding' something to .999 repeating.... simply find the smallest non zero positive number and subtract it from 1. If you believe there must be such a number... then you know the answer is .999 repeating. If you believe such a number doesn't exist, that the smallest possible number greater than 0 is just a concept. Then you have to accept that the greatest number less than one is also just a concept. But they are not equal. So they are two different ideas. Not equal.
It approaches 1, but never actually reaches it.
I love the use of your unidentified "it".
The sequence 0.9, 0.99, 0.999, ... approaches 0.999... and 1, that's why 0.999... = 1.
0.999... is constant and doesn't approach anything. It is 1, it always has been 1 and it always will be 1.
@@Chris-5318 "9 is 1" read your own comment and think what you just said lmfao
@@elleeVee What are you on about you muppet? You should learn to read before you write.
I like best:
(0.999...)*10-9=0.999...
1*10-9=1
Meaning both are solutions to the first degree equation 10x-9=x. Since such equations have only one solution, 0.999...=1.
If 0.999... has infinite decimal places, then 10 times that has infinite decimal places plus one more decimal place. Your first equation is wrong. Unless you can somehow define equality such that a number with one more nonzero significant digit is equal to a number without that additional digit. Which I think is part of what real mathematicians do using set theory. Making their own goofy elaborate definition of a fundamental concept to just make their problem go away.
@@thereaction18 Infinity is weird like that. Infinity + 1 is still infinity. The problem is that infinity isn't a number, so it doesn't behave intuitively.
@@personperson6827 If infinity isn't a number, how can an infinite series equal one? One is a number. Infinity isn't. They can't be equal. Unless you create another definition of "equal" tailor made to fit. Which seems to be what mathematicians have done.
@thereaction18 infinity ≠ infinite series
The problem in this proof lies in 0:57
It assumes 0.999... is a number. Specifically a real number.
Then it shows there is no difference between it and 1, but the conclusion is not the only possible conclusion.
Another possible conclusion is that 0.999... is not a real number. That is, it has no precise value.
And indeed that is exactly what basic arithmetic tells us.
For example, the exact value 1/3 cannot be expressed using decimals. Dividing 1 by 3 always leaves a reminder, even if we calculate endlessly.
Therefore 0.3333... cannot have a definite/finite value, and therefore is not a number.
The same with 0.999...
In fact, most representations of decimal real numbers are not real numbers. They never achieve the exact value they suppose to represent.
We lose sight of this fact because we cheated.
We decided to define the limit of a converging infinite series as its sum, but this definition of a sum is not consistent with the definition of the sum of a finite series - that has no limit.
That is the number 1.0000000.... cannot be obtained with the new definition of a sum of an endless series, as 1 + 0.0 + 0.00 + 0.000 has no limit.
You assert many false claims.
The standard definition is that 0.999... represents a real number. Further, every infinite series of reals terms represents a real number, iff (if and only if), it is convergent. Further, most mathematicians that work with e.g. Robinson's hyperreals, retain that fact. It is a small minority that unnecessarily muck things up by, e.g., redefining 0.999... to be 1 - 1/10^H where H is a particular hypernatural number.
You said, "And indeed that is exactly what basic arithmetic tells us."
- You are wrong. Basic arithmetic tells us that 0.999... = 1 and 0.333... = 1/3. So does Calculus 2 (real analysis).
In fact ALL decimals represent real numbers and ALL real numbers have one (or two) decimal representations.
Series do not have a limit, the have a sum if they are convergent. That sum is the limit of the sequence of partial sums (iff the limit exists).
The definition of the sum of an infinite series is a generalisation of the sum of finitely many terms. e.g. 1 + 2 + 3 = 1 + 2 + 3 + 0 + 0 + 0 + ... and has the sequence of partial sums: 1, 3, 6, 6, 6, ... and that has the limit is 6. Hence the sum of 1 + 2 + 3 is 6.
The sequence of partial sums of 1.000... is 1, 1, 1, ... and that has the limit 1. Hence 1.000... = 1.
If you are going to contradict a mathematician (i.e. the mathematical community in this case), you should at least try to check the facts beforehand. First you need to learn the difference between series and sequences, because you are muddling them up.
Hey Chris,
Thank you for your thorough response.
1) Yes, I know the standard definition is that decimals with infinite digits after 0 represent a real number, but definitions in math cannot be arbitrary. They must be logically consistent with previous assumptions and conclusions. For example, if in a decimal system, you cannot express 1/3 exactly with any N digits, and you want to define that with infinite digits it is 1/3, then your definition is not consistent with your previous assumptions unless you explain logically how one leads to the other. Alternatively, you can redefine any concept you want arbitrarily, but then you cannot use both the old definition and the new one. You will get contradictions.
2) You wrote: You are wrong. Basic arithmetic tells us that 0.999... = 1 and 0.333... = 1/3.
Yea... please divide 1 by three in a decimal system and come tell me when the reminder has disappeared.
2.1) In calculus 0.333... = 1/3 because of our decision to define the sum of infinite series as its limit.
Have you ever asked yourself why all real numbers have one or two representations?
Shouldn't a consistent system have only one?
Well, as I understand it, it is because this is a system that defines numbers in two different ways.
The old definition can only define rational numbers.
The newer one defines all real numbers.
This is why rational numbers have two representations, and irrational numbers have only one representation.
3) Yes, I know The definition of the sum of an infinite series is a generalization of the sum of finitely many terms, but I claim this generalization is not logically consistent.
If you want it to be consistent you need to ask a simple question:
Will an infinite series with a limit reach the limit?
For example, will the series 0.9 + 0.09 + 0.009 ... reach 1 if we sum an infinite amount of its members?
Well, consider the group of all partial sums of the series.
This group represents the summation of infinite members of the series.
Now let's ask ourselves: is 1 a member of this group?
@@Chris-5318
@@Lucidthinking The defintion is consistent with all of mathematics.
You can't conclude that 1/3 = 0.333... by using the division algorithm alone. You would need to use limits too.
1/3 = 0.3 + 1/30 = 0.33 + 1/300 = 0.333 + 1/3000 = ... = 0.333...3 (n 3s) +(1/3)/10^n = ...
Taking the limit we get 1/3 = 0.333...+ 0 = 0.333...
Much simpler is to use the 8th grade "proof":
10 * 0.333... = 3.333...
expand each side
9 * 0.333... + 0.333... = 3 + 0.333...
subtract the isolated 0.333... from each side
9 * 0.333... = 3
divide through by 9
0.333... = 3/9 = 1/3
A convergent infinite series has a sum, not a limit. The limit of the sum is the sum. That's because lim x->a c = c where c is a constant and x and a are arbitrary.
You asked "Will an infinite series with a limit reach the limit?"
The question is nonsensical. You need to learn the difference between series and sequences, as I have already suggested. It is the main reason that you are getting everything wrong. You keep on talking about the unchanging series 0.999... as if it was the sequence 0.9, 0.99, 0.999, ... that is in turn a representation of the discrete function 0.999...9 (n 9s) that may be written as 1 - 1/10^n. The sequence of partial sums of 0.999... is 0.9, 0.99, 0.999, ... and that does not reach [the sum of] 0.999... or 1. The limit of that *>> sequence
@@Lucidthinking You said, "Have you ever asked yourself why all real numbers have one or two representations?"
I am perfectly well aware that rationals of the form N/10^n have two representations. It is only terminating decimals that have two decimal representations. It is obvious why that is so.
You asked, "Shouldn't a consistent system have only one?"
Obviously not. It would be kinda nice if it was so, but at most it's minor flaw.
You said, "Well, as I understand it, it is because this is a system that defines numbers in two different ways."
Your understanding is wrong. There are infinitely many representations for every number. That has nothing to do with how numbers are defined. Numbers are not their representations. Numbers are pure abstract and only exist as concepts, as is the case for all mathematical objects.
See the Wiki "Construction of the Real Numbers" to see that each number is defined to be the equivalence class of the sequences that actually have that number as a limit. (At least, that's the Cauchy construction). There are uncountably infinitely many sequences in that class, not merely 2.
Thank you Chris,
1. The reason I confuse between series and sequence is that English is not my native language. I know the difference between them.
2. As I see it, the main issue lies in this sentence:
"You asked "Will an infinite series with a limit reach the limit?" The question is nonsensical."
If you think it is nonsensical you have not understood my claim.
My main problem is with the definition that the sum of an infinite series is the limit of its partial sums.
I claim that this definition must either be deducted logically from the behavior of the sum of a finite series, or not.
If it is not, it is an arbitrary definition, and then this whole video is not necessary. You don't need to "prove" 1 = 0.999...
Just say they are equal by definition.
3. I claim that you cannot sum an infinite series. Not if you keep the same meaning of a sum you have on finite series.
For mathematicians, the inconsistency I refer to is, as you say, a minor flaw.
This is because when you deal with the abstract, you don't "pay", for such minor errors. But in computer science or physics, this becomes significant.
For example, you probably know Zeno's paradox of Achilles and the turtle. Many say that calculus solves the paradox, but it is wrong.
Calculus doesn't *know* how to sum a process with infinite steps. It just redefines the limit of the process as the sum.
But in the physical world, there is no real continuum, no infinities, and no singularities. Many physical theories may reach such results but one should not confuse the math with the reality it suppose to describe.
4. By the way, I am familiar with modern ways of constructing the reals, like Cauchy sequences, and Dedekind cuts.
I don't think seeking answers in more abstract math would give me an answer. Maybe I'm wrong, but as much as I know, the definition of a sum of an infinite series was created long before Cauchy sequences, Dedekind cuts, or Topology.
I never heard that back then this definition was controversial, so the justification for this definition should be found in calculus and not in Set theory or Topology.
@@Chris-5318
Amazing video!!
Thank you!!
(0.9)n
So what!? The only real number x that satisfies (0.9)n
Moral of the story: uncountable infinity is weird
This guy gets it.
mhm
I get the joke: "We are all set"
A set is basically some collection of numbers, and the real numbers are all numbers if we ignore the complex numbers, but they use two axis, and the real numbers are the set of all numbers on one axis
To be fair, 1/10n cannot "be zero" at 3:25 - it can only approach 0 as n tends to infinity (i.e. a limit is not necessary an equality). So perhaps you've only demonstrated that (0.9)n as n tends to infinity has the limit of 1?
but "0.999..." is n being infinity
He never claims 1/10^n is equal to 0, he claims that there is only one non-negative real number which is less than 1/10^n for all natural numbers n, and that number is 0
@@anshumanagrawal346 You can't exhaustively show that. You have to use a proof by induction, which works for practical applications but it does take a logical leap.
@@Elrog3 😂
@@anshumanagrawal346 Guess I've wasted time on this one.
Guys he played the trick at 3:19
Using limits or roughly calculus is forbidden because we first denied proof using theorems of calc but still I thought that kind of proof is non necessary.😶
you're so underrated, these videos are just awesome
Since when was this ever considered a "debate".
Besides, the common base-10 decimal notation is just a symbolic system for writing down abstract concepts. A certain way to write something could mean anything that we designate.
0.333 repeating forever is not thing that actually exists. The concept of "one third" is a thing that exists however. We have just decided to devise a symbolic system where we designate the concept of 0.3 repeating to actually mean the concept of "one third" (equivalent to 1/3). In this context 0.999 repeating isn't anything special.
I think this debate comes from a difference of semantics. When we see something like 'lim n -> infinity of 1/10^n' we say that's equal to zero. And sure, that's right, since that's as it's been defined. And this proof in this video relies on that. 1-x
I agree with this. We are much more inclined to say "is equal to" than "is"
If 0.99... =1, so does 0.99...98=0.99... meaning 0.99...98=1, of you repeat this indefinetly, you get 0=1, so 0.99... ≠1
0.999...98 is a terminating decimal and it is less than 0.999...
In fact 0.999... - 0.999...98 = 0.000...01999... = 0.000...02
3:22 people who assert 0.999... =/= 1 also assert that you can't put a 0 on the right side of the inequality; you put 0.0000...0001.
Hello! Great question!! And I state that it isn't. "0.99999...9(9)" DOES NOT equal "1.0000...0(0)". It is one of fundamentals of HyperMath, which shows "our reals numbers" are presented no more accurate than "real" numbers within CPU.
If you think that a rational (1/3) number is "0.333..3(3)" (decimal notation), you might be wrong.
Just as "0.1111..1(1)" (decimal notation) is not rational (1/9).
A calculator is right, when it says that "0.333333"x3 = "0.999999", not "1.000000".
No matter how many "3"s are in "0.333333" and how many "9"s are in "0.999999", these items
are not (1/3) and 1 ("1.00000").
If you think that you can approach to a real number (specific real number) as close as you want with a fraction, then you will be disappointed. You can't: your decimal fraction notation is limited. There's the same reason as why computers can't precisely handle decimal fractions. They can handle only 2-based fractions to a specific degree of precision. The same principle is for decimal fractions (even unlimited): they can't handle all rational items.
I'm not sure what you're proving here. They are equal. If you don't think so, give a formal proof.
yes but you aren’t taking the limit. You are trying to use finite limits to prove your points.
Well ... actually depends on your number system, even your model of the reals. It's true in the standard model of the reals, which is the completion of the rationals using distance defined by the absolute difference.
Here's a counterexample. Define a [real] "number" as a monotonous sequence of rational numbers, where two sequences sharing an infinite common subsequence* are considered the same "number". Algebraic operations are performed element-wise. Then, 0.999... converges to the sequence 0.9, 0.99, 0.999,... and the difference to 1 is 0.1, 0.01, 0.001, ... (or any monotonous sequences sharing infinitely many elements with these, respectively).
*) It's a bit more subtle than that, for instance (the existence of infinite sub-)sequences that keep overtaking each other infinitely many times would also have to be considered the same number.
The universe has the same problem. You can’t continue to separate something into smaller parts forever. Eventually you have to reach a base unit for which everything else is built or calculated from. Calling any undefined decimal a real number because it can be imagined as repeating to infinity is wrong in my head. So the problem is, I can’t accept any repeating decimal as a real number just because of our wonderful ability to imagine.
You: "The universe has the same problem. You can’t continue to separate something into smaller parts forever."
The universe is physics. The topic is math. Your second claim is too vague to be meaningful (in physics).
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You: "Eventually you have to reach a base unit for which everything else is built or calculated from."
Not in math. The sequence 1, 1/2, 1/3, 1/4, ... never ends, there is no last "base" term.
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You: "Calling any undefined decimal a real number because it can be imagined as repeating to infinity is wrong in my head. So the problem is, I can’t accept any repeating decimal as a real number just because of our wonderful ability to imagine."
0.999... is defined. Admittedly Brian didn't define it. The value of 0.999... is actually defined to be the limit of the sequence 0.9, 0.99, 0.999, ... (which is "obvious") and that turns out to be 1. The n th term of that seqeunce is 0.999...9 (n 9s) = 1 - 1/10^n, so:
[the value of] 0.999... := lim n->oo 0.999...9 (n 9s) = lim n->o oo 1 - 1/10^n = 1
So, ".99..." is equivalent to 1 in most contexts.
@@johnlabonte-ch5ul Karen. Here is a well known proven math fact: 0.999... = 1, precisely.
You do not know what the word "equivalent" means in mathematics.
0.999... = 1 in all context where the usual meaning of the symbols is involved, e.g., 0.999... is a decimal numeral, not a hexadecimal numeral.
I WAS LITERALLY ASKING MY FRIENDS ABOUT THAT YESTERDAY
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There's a logical solution to this as well. If we think about how to get .99 repeating to equal 1, we just need to add .000...1 to it, just like adding .01 to .99. In this case, however, we have infinite zeros with a 1 at the end, but since there are infinite zeros, that number is infinitely small, or equal to 0. Thus, 0.9 repeating is 0 away from 1.
that's literally the entire thing the video showed lol, .000...1 is 1/10^n as n approaches infinity
Great explanation 👌
Thank you 🙂
Engineers after watching this: Pfff, it's easy! You have to use the "ceiling" code to change from .999999999.... to 1! Easy, right? 😎
lol
of course 0.99... is the same as 1, because we have decided that it is. x also = 1 = 0.99... because 1 + infinity is still infinity.
Our whole modern math framework works in a context where we tell ourselves that we can imagine infinity easily by looking at 3 dots. 0.99... is basically a very weird and cryptic way of writing and defining 1. I think the problem is these 3 dots and what they represent. It seems like we're telling ourselves something that isn't actually true, that we can somehow imagine a completed infinity. Seems similar to making a symbol that means "square circle" and then telling ourselves that now we have a square circle.
I think the more correct way of dealing with numbers like 1/3 (numbers with unending digits after the decimal place), is not to use the 3 dots in the way we use them, but instead accept that we cannot write it in decimal form. writing for example (0.9)n makes sense to me, meaning something that could potentially get arbitrarily close to 1.
I think 0.9... should mean almost the same thing, a number that tends towards 1, the more 9s you imagine/write down, the closer you get, and you can get so close that you go beyond the planck length or whatever, but it is technically in a mathematical sense still not 1 and will never be 1. (I realize this is not what it currently means, but this is what I think it should mean).
The sad truth of math should be, imo, that we cannot translate certain numbers precisely from fractions to decimal numbers.
1/3 ≈ (0.3)n
2/3 ≈ (0.6)n
3/3 = 1
3/3 ≈ (0.9)n
"Imagining" completed infinities is also fine, in principal I have nothing against it. I just think it should be a special niche kind of math, a kind of experimental, creative kind of math, instead of the mainstream way, the accepted right way to understand math.
Complete infinity is pretty much mainstream math. I can't imagine why it troubles you at all.
In essence this is taking Dedekind cuts.
Boy do i want to have fun with this comment and something he said in the video about the other sets of numbers. Are you familiar with the surreal numbers?
If 0.9 repeating = 1, then does 1/infinity = 0?
The limit of 1/x as x goes to infinity is indeed 0
In a sense. There is no number x such that 1/x=0, but as |x| grows larger, 1/x comes closer and closer to 0. In fact, it can be as close to 0 as you want, without actually equaling 0.
1/x converges to zero as x approaches infinity, but infinity is not a real number that x can equal. But the difference is 0.999... is a real number that x can equal and it converges to 1, which means this convergence implies 1=0.999... convergence implies equality for all parts of functions which are continuous between negative infinity and infinity. It doesn't imply such at either infinities or a discontinuity of a function.
The reals are dense in what?
Technically they aren't dense in the hyperreals, as for any none zero infinitesimal epsilon, there are no real numbers between epsilon and 2*epsilon.
Well, they're dense in themselves :^) just like any other topological space
That's extremely stupid Hyperreals (By definition) are not just the real numbers
@@anshumanagrawal346 I wouldn't call the Hyperreals stupid, let's just put that out there.
I studied them quite a lot and the construction depends on the Ultrafilter theorem which might sound controversial at first but then you realize that it's weaker than the axiom of choice which we use all of the time.
@@factsheet4930 The Hyperreals are not stupid, it's stupid to bring them when it's exclusively stated that we're talking about the real numbers
Guys quite saying half of mathematics is arguing over definitions, it certainly is not. Now, it is true that most of mathematical proofs are unpacking definitions, but that IS NOT ARGUING over the definitions. Get these ideas straight. Almost all mathematicians agree on almost all of the definitions pertaining to anything mathematical (and before you start arguing with me saying I don’t know anything, I am currently doing mathematical research and getting a mathematical degree, so I at least know a little bit, ok?)
This topic has a physic meaning: when you can't demonstrate a result with finite aritmetic, you may study the convergence. For example: the velocity of an object moving in a fluid is regularly slowed by viscosity but never really stopped. Despite this, the space reached is finite anyway.
P.S. love this channel you are great :)
@Rick Does Math Ho scoperto solo ora che sei italiano ahah. È bello incrociare i destini con le passioni. In che facoltà studi?
@Rick Does Math impressionante, io non ero così appassionato al liceo anzi, non lo sono mai stato prima di cominciare l'università. Comunque frequento il primo anno del politecnico di Torino, se vuoi ci scambiamo i social così parliamo senza intasare i commenti a sto povero cristo ahah. Su ig mi chiamo come qui su youtube, passa da me magari ti giro qualche appunto di analisi matematica :)
Two numbers are equal if you can't find any other number between them.
You can't find any real number on the real number line between 0.999999..... and 1.
So, they are actually equal.
Why can't you find a number between point nine repeating and one? Just take the average!
@@thereaction18 The average infact is 1.
@@kumaranmol3515 How many decimal points did you calculate that to?
You take the limit at infinity of the convergent series. You don't calculate to a number of decimal points.
@@marcdavies7046 See, right there you said "convergent series", not "number". Point nine repeating is a convergent series, not a number. We pretend it represents a number through the convention you refer to. But you literally cannot find it on the number line. You pretend it has a place there, then you assert that it makes the most sense to think of it as a limit, and you define a limit as a number which something approaches and may possibly equal but not necessarily.
4:00 It had this debate yesterday on bprp video, and now here? Oh my 😂 Nevertheless, great stuff I like it.
It was me who started that debate🤣
@@SidharthjainSingla this debate is older than you
"And that's a good place to stop"
@@Nothing-pg9qc We all have some culture there.
And that’s a good place to stop
A) (1-1)•X
B) (1-0.999...)•X
Something tells me that the limit of function A when X tends to infinite is 0, but function B is an indeterminate
0.999... := lim n->oo 0.999...9 (n 9s) = lim n->oo 1 - 1/10^n = 1. " := " means is defined to be equal to.
Thats actually really cool, but really weird
My life is a lie
🤯
-1/12
Actually this is a definition that is used to define real numbers when we define Cauchy sequences. So when you say .9999 repeating is a real number you already equated it to 1. You can read Terence Tao's analysis book for more details.
I know my high school math teacher doesn't care, but I'm still bringing this up if I ever see him again because I argued with him about this.
One still needs to show that that the two Cauchy sequences are equivalent. Not quite sure what you mean by "you already equated it to 1".
@@billh17 I don't know why, but this argument has been popping up a bit recently. "There's nothing to show because they're the same number." But how do you actually know they're the same number?! Do intro set theory students need to prove |N| = |Q|, or is there nothing to prove because they're "the same number"? Was Euler's famous solution to the Basel problem pointless because the sum of the reciprocals of the squares and pi^2/6 are the same number, so there's nothing to prove?
@@billh17 We define a relation R between two Cauchy sequences over the rational numbers {a_n} and {b_n} if the sequence {a_n - b_n} tends to 0 as n approaches infinity; in other words, they have the same end behavior. Turns out R fulfills the requirements to be an equivalence relation:
(1) it is reflexive, i.e. {a_n} R {a_n} since {a_n - a_n} tends to 0,
(2) it is symmetric, i.e. {a_n} R {b_n}, then {b_n} R {a_n}. a_n - b_n tends to 0, so since |a_n - b_n| = |b_n - a_n|, then b_n - a_n also tends to 0.
(3) it is transitive, i.e. If {a_n} R {b_n} and {b_n} R {c_n}, then {a_n} R {c_n}.
The distance between any rational number Cauchy sequence that represents 0.999..., for example (0.9, 0.99, 0.999, ...), and the distance between any rational Cauchy sequence that represents 1 (1, 1, 1, 1, ...) will converge towards 0. By the Cauchy sequence definition, the sequences have an equivalence relation.
0.99....=0.99....
The best way to prove that 0.99999 = 1 is to, ask what is smaller than 1 infinite times
There is a potential answer: 0
And there is a definitely known answer: 0.000000000001 (zeros go forever)
And since that, the gap between 1 and 0.999999999 is 0.0000000001 it means 0.000000001 Acts like 0
which means that there are no Gap between 1 and 0.99999999
I like to think of it this way. If I am trying to justify 0.9.... < 1 then basically I am saying that I have found a number that is less than one but there is no number between it and 1. Which is of course impossible.
But what about infinitesimals :"((
Irrelevant here.
Digital representations of numbers are based on sums of powers of a number. Limits of sums of powers of rational numbers will always necessarily be equal to real numbers. So 0.(9), given how the notation works, is necessarily equal to a real number. Infinitesimal quantities do exist in the hyperreal numbers and in the surreal numbers, but as hyperreal quantities cannot expressed by using the ordinary digital representaion, and as 0.(9) is a digital representation, 0.(9) is a real hyperreal number, not a nonstandard hyperreal number. This is, st[0.(9)] = 0.(9). So yes, infinitesimal quantities are ultimately irrelevant, and unless you actually reform the mathematical consensus on how the digital notation works, 0.(9) = 1 remains true even in the surreal number system.
The set of real numbers is defined (up to an isomorphism) to be the unique Dedekind-complete ordered field. Dedekind completeness (the property that every subset of *R* has a least upper bound) implies the Archimedian Property, which states that there are no infinitesimals in the real numbers.
It's not a debate, it's a fact
If you don't think 0.99999... = 1, explain this
9/10 + 9/100 + 9/1000 + ...= x
10(9/10 + 9/100 + 9/1000 + ...) = 10x
9 + 9/10 + 9/100 + 9/1000 = 10x
9 + x = 10x
9 = 9x
1 = x
What if we approach this from continuity and differentiability. So we can say that 0.999...... tends to 1
No wait it's true 0.999.... will eventually reach one provided the the 9's are infinite
It's a constant number and doesn't tend to anything. It just is exactly 1.
That goes, but there's a difference in meaning between tending to one, and equaling one. And we usually like it more when things equal each other, than tend to one another :p
@blackpinredpin
Legendary explanation
Blackpink in ur area!!
black *pin* red *pin* lmfao
I'll see you in THAT ONE
Oh my god XD
Didn't get the set joke though . . .
We are talking about the SET of Real Numbers 😎
@@BriTheMathGuy Jesus man XD
One was better :)
You underestimate the stubbornness and stupidity of the internet
ive allways seen fractions as a rounded up decimal so .9 repeating can round up to 1 every time
you've always seen it cuz the moment they invented the calculator, they used 0.(9) = 1 or 0.(1) = 1/9 or 0.(3) = 1/3. the lim system is there for a reason
Me when I'm less 0.25 cents of what I'm buying
0.25 cents? Did you mean 1/4 of a penny?
Just sharing my thoughts, (I am open to any discussions just don't try to find out my address and try to kill me)
Saying that 0.9999 recurring is 1 would not be true, since something that is not 1 cannot be 1. Imagine if I said 256 is equal to 1, everyone would look at me and think that I am crazy. But It is ultimately using the same logic because I am saying something that is not 1- in this case 256 for example- is 1.
I believe that the problem comes to maths. I think that in some ways, fractions is flawed. This is because how could 1/3 be 0.33333 recurring? A fraction that would make sense in math would not be recurring as it doesn't make sense. For example, 1/3 should be a third of 1, however a third of one is not 0.33333 although it is extremely close to it, it is not a third of 1. However it is impossible to get a third of 1 in decimal form, therefore it shouldn't exist. So if we can't find 1/3 in decimal form because it is impossible, then there is basically a flaw in maths.
The idea of 0.3333 recurring not being equal to 1/3 is intrigueing, no clue about the details but nice idea!
Your comment basically is saying, it's not 1 because it is not written like the symbol '1', I don't think you understand the notion of a repeating decimal. Take 0.33333... it is not 3 going on for a very long time, it is 3 going on *forever*, that is, there is no real number in between 0.33333.... and 1/3, they are *the same thing*. Similarly, 0.9999... and 1 have no number between them, they are 2 representations of the same thing, one is an integer, the other, an infinite sum, both of which correspond to the same, unambiguous and non-approximated value
@@brandonklein1 But my point is that 0.33333... Is infinite, therefore there is always going to be a number between them as it has no * limit * . This means that 0.33333... will never be 1/3 as a fixed 1/3 does not exist. Similarly, 0.99999... will always have a number between them as infinite is not limited, therefore 0.99999... will never be 1 as there is no bridge between them. It is unlimited. 0.99999 will never be * exactly * 1.
@@thepingusaver292 it has no finite length in our base 10 number system. What does .33... Mean? Well we can write it as lim x->\infty 3*sum_1^x (1/10)^n. This is not an approximated value, nor is it "effectively" the same thing as 1/3 it is completely, with no ambiguity 1/3.
Just because the way we write something has no end it doesn't mean that it isn't defined.
For example, is π defined? *Yes*. I can say π=4(1-1/3+1/5-1/7...) And this is *exactly* true, it is not approximately true or "effectively" true or "infinitely close" to being true it is the exact same thing. The only difference is that in the case of π we have a symbol we use to "stuff away" all of the messiness.
You are almost right, the problem is not math, it's the education system, we are led to believe the idea of "infinitely many digits" makes sense on its own when it doesn't, the real definition is slightly different and that makes it trivially obvious why .999... = 1
The argument that convinced me is that if they're different, you can find numbers between them!
Mathmetician: 1 = 0.99........
Me: no actualy 1 has to be a little bit bigger
Mathetician: ok proof it!
Me: umm...F@#k!!
If people agree 1/3 = .3 repeating then 1/3+1/3+1/3=.9 repeating and 1/3+1/3+1/3 equals 1 but it’s an interesting way to represent 1
Saying that 0.000 … 1 is equal to zero is same that saying we don’t exist if universe is equal to 1
Physics bros don't even have this issue, we just round the shit out of it