why 0.9999... is NOT equal to 1.
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- Опубликовано: 3 окт 2024
- im sick and tired of people saying that 0.9999... is equal to 1... so i decided to make a video on why its not 1, disproving a few kinda common proofs that exist out there. (well trying)
also sorry for not uploading in 2 months
join the discord server here lol. i do epic stuff on there: / discord
ALSO join this channel with memberships for some cool stuff
/ @tristantheoofer2
current sub count: 22781 
Игры
i just realised that around like 6 minutes i accidentally "prove" its 1 fuck 😭😭😭😭
anyways hopefully yall enjoyed this, so much shit has happened in the last 2 months, so im glad i was able to make something for yall lmao. and also this video was more my take on the 0.999... = 1 thing than anything else lmao
also ofc sources and music are below
sources
en.wikipedia.org/wiki/0.999 - 0.999.. = 1 "proofs"
www.askamathematician.com/2011/01/q-%CF%80-4/ - pi = 4
music
atlantis - audionautix
part 1 - douglas holmquist (from smash hit)
cliffside hinson - c418
total drag - c418
beyond space - chill carrier
a slow dream - emily a. sprague
CORRECTIONS:
~8:10 i accidentally said "as n goes to infinity" instead of "as k goes to infinity" sorry-
edit: this video has an 89.3 like to dislike ratio now- why am i not surprised lmao, especially with a topic thats so controversial
thanks for clarifying this is not the first
help i’m being araved
@@ItzSneakyMinecraft???
@@tristantheoofer2 why does IT SAY IT WAS FROM 1 DAY AGO THIS WAS UPLOADED AN HOUR AGO??!!
pov what you hear when your math teacher wants to say something
As a tristangent fan, I can confidently say that I understood about 0.999% of this video.
why is this so accurate bro 😭😭😭😭😭😭😭
same
@@tristantheoofer2 probably because I am, in fact, a fan of yours lol
"I'm breaking up with y-" BABE SHUT UP TRISTANGENT UPLOADED
lmaooooooo
@@tristantheoofer2 did you know uranium glass is safe as long as you dont grind it up and snort it?
@@jakfjfrgnei worlds most wild acid trip
@@jakfjfrgnei but that is why uranium glass is fun
Relatable
if it equalled 1 then it would be called 1
i mean ok then-
Fr 0.999999… is not 1 bc it's
0.000000…001 smaller than 1 therefore it's not 1
@@HFIAPY Except that the 1 at the end of those infinite zeroes doesn't exist because it's infinitely small. What else is infinitely small? 0.
2-1 isn't called 1 tho
@@POverwatch0.000...0001 /still/ is infinitely small, however, it is still bigger then 0.
0.999 repeating does equal 1, at least when working with real numbers. Easiest way is the 10x argument that you just "debunked", let x=0.999999999 repeating, then consider 10x. 10x=9.999999999 repeating. Now consider 10x-x. it is 9.99999999999 - 0.99999999999, which is precisely 9. as 9x=9, x=1. Contrary to what you're saying, it does not assume 0.9999999=1 to begin with, we simply let it be x and prove that x is 1.
Again, we are working with real numbers, so the argument that 1-0.99999999 is an infinitesimal number does not work. Infinitesimal does not exist in the real numbers, therefore 1-0.9999999999 is regarded as 0 in the reals. 0.9999999999=1.
Also, you are fundamentally misrepresenting the concept of limits. Look up the epsilon delta definition of limits. Using the definition, we can prove that limit of 1/10ⁿ as n goes to infinity does, in fact, equal to precisely 0, not some really really small number (again, there are no infinitesimals in the reals)
TLDR 0.999999=1 in the real numbers, by the 10x argument and limit argument. the 10x argument doesnt assume 0.999999=1, and when in doubt, limits shouldn't be done intuitively, but rigorously using definitions.
So what now? I'm right and you're wrong? Not exactly. I have just enough knowledge in the real numbers to confidently say that I'm correct and 0.9999999=1 in the reals, but you bring up an interesting concept: infinitesimals. Introduce that to the reals and you get new number systems, including surreal numbers and hyperreal numbers. And I don't know anything about them, and you may be proven correct in those number systems. You may be right afterall, just not in the real numbers.
honestly fair enough with that, especially the whole limits thing i suppose. this also would mean we are both correct but in seperate ways. in that case what would 0.999... be? a stupid representation of 1??
While I fundamentally disagree with you on this video, please keep in mind that this isn't personal, and I think it really echoes back to the community post you made about one or two months ago: that 1+2+3+... equals -1/12.
It's a genuinely interesting to think about it and considering different worlds or definitions where the equality will be true. Same thing here. 0.999999=1 in the reals, but what if it isn't? We get infinitesimals and new number systems. You may have made some interesting points that while sadly doesn't work in the real numbers, work in some other systems.
@@tristantheoofer2 after some research, you will be right in the hyperreals, and I think indeed, there are infinitely many numbers between the two.
@@tristantheoofer2 exactly that, it's a stupid way to represent 1 (although it can have uses - a common way to give each decimal a unique representation is to choose a non-terminating sequence)
ok so this is a technically correct kinda deal
...This isn't a debate.
This is a mathematically proven fact.
Literally the only thing this video can possibly be about is how you don't understand it, so I guess I watch that. lmfao
"I don't feel this is true." This is literally where you start. Your conclusions are flawed because they are based on your biases.
Ok, and the first "argument" that you "debunk" is just an illustrative example and not an actual proof. And it's looking like the second one is too.
I agree that there are flaws with these examples, but disproving them doesn't affect the larger argument in any way.
Yep, third argument too.
These are not proofs.
You need to start with learning what a mathematical proof is, and how to understand them, because you clearly don't even have the basics of the background to be approaching tackling this problem.
These are not proofs, they're illustrations, and yes, they're poorly constructed ones.
If these are the only arguments you've seen, and you've never seen the actual proof... no wonder you don't understand or believe this.
ok no i am not going to start a argument again i am NOT
@@andynilsennot4329 ok but it is literally not an opinion?
@@andynilsennot4329 It's not an opinion it's a wrong answer.
Parents no love 😂
prove it
Ill explain why youre wrong here on the numberline
The argument that "there must be a 1 at the end" of 0.999... misunderstands infinity. 0.999... means the 9’s repeat forever, so there is no end where a 1 could be placed. Infinity doesn’t work like that- you can’t finish an infinite sequence and then add something afterward. The idea of a "1" at the end (like 0.000...1) is nonsensical, as no such number exists in the real number system. Algebraically, 0.999... = 1, and there’s no gap between them. The supposed 1 "at the end" is simply not possible. The idea of a "1 at the end" of 0.999... is impossible because there is no end to an infinite sequence. By definition, the 9's go on forever, so there’s never a point where you can add a 1. The argument assumes infinity is something you can eventually reach, but infinity doesn't work like that-it keeps going without stopping. The concept of 0.000...1 (infinite zeros followed by a 1) is mathematically invalid because you'd never actually reach the 1 after infinite zeros. Plus, in real number math, 0.999... equals 1 exactly, with no gap. The same goes for you trying to disprove the algebraic proof, you cant jus stick a 1 at the end of an infinite series.
and the problem with the calculus argument is already at the start..
the idea that 0.999... is only "approaching" 1 but never "reaches" 1 misunderstands how limits and infinite series work in calculus. Yes, 0.999... is an infinite decimal that gets closer and closer to 1, but the key point is that in the limit, it equals 1. In calculus, when we say a number "approaches" a limit, it means the value gets arbitrarily close to the limit and eventually equals it. There's no difference between 0.999... and 1 because the infinite sum converges to 1, meaning they are mathematically identical, not just "close."
fuck i thought approaching didnt mean legit eventually equaling something- but in that case... how would is equal EXACTLY 1?? and wouldnt that definition of approaching essentially mean "this is close enough to where we can say its this"? like thats a genuine question. because arbitrarily close doesnt seem like it *could* ever equal anything specifically
@@tristantheoofer2 "approaching" a number means that as you get closer and closer, the difference between the numbers becomes so small that it's effectively zero. When u say 0.999... "approaches" 1, it doesn’t just get close to 1-it becomes 1 exactly because there's no real number between 0.999... and 1.
It’s not a case of “close enough”-in the case of an infinite sequence like 0.999..., the sum converges exactly to 1. When we say "arbitrarily close," we mean that for any tiny gap you imagine, 0.999... will eventually fill that gap entirely.
The difference between 0.999... and 1 is not just small-it’s zero. So, in mathematical terms, they are equal, not just approximately the same. It might seem weird, but that’s how infinite sequences work: they reach their limit, and in this case, the limit is exactly 1
Holy essay
@@tristantheoofer2 it is not true in all cases, like lim(x->0)1/x=1/0 is not true, but in the case of lim(x->infinity)1/(10^x)=0 is true because there is no paradox.
> tristangent uploads
> watch video
> understand nothing
> still happy and joyful
lmao that is so real
real asf
@@tristantheoofer2 Also you can not consider infinite as a number, 0,999... does not have a number as a gap, there are bigger infinites but they are still infinites
A lot of your arguments here rely on the claim that 0.0repeating1 is greater than 0. So lets assume this is true. What would happen if we add this number to itself, which is the same thing as doubling it. We would get 0.0repeating2. Now lets repeat this process again. We would get 0.0repeating4. Now lets do it again, and again, and again. If this number really is greater than 0 like you say it is, it should eventually reach a number above 1 performing this doubling process a finite number of times. But it doesn't. No matter how many times you complete this process, it will have infinitely many zeroes before its other digits, meaning that it is less than 0.
"less than 0" how would this be less than 0? anyways.. i see how you have some point here actually... im starting to see how i may be wrong actually
what is 0.000...1 times infinity then? would that just be 0.000...99999...??? how would having two repeating sequences in a decimal even make sense whatsoever???
@@cater_piler You don't multiply things by infinity in mathematics. Infinity is not a number, but an idea. But if you were to multiply it by infinity, any number above 0 would be a valid solution, which is a problem, because you could then "prove" that any positive number is equal to any other positive number.
NEEEEEEEEEEERRRRRRRDDDD
@@Bill_W_Cipheralon Amit quora post Debunked you
Maybe the real 0.r9 is the friends we made along the way
Pixel. I didn't expect you to comment here.
@NeutronGD_OFFICIAL this guys been watching me for a while
i love how confident you talk in this video about such controversial topic, prob my favorite from all of those vids you have
fair enough lmao. and somehow im wrong to about 60% of the comment section-
these comments are just diabolical😭
FR BRO 😭😭😭😭😭
Howdy, what you are describing appears to be the hyperreal numbers. While this is a valid number system it is a completely different number system to the one that most people usually use (standard real analysis). So the real answer to this question, like many in math is yes and also no. Yes, you can technically use Infinitesimals to get this result but saying that it doesn't equal 1 is probably a bit of a weird take in my opinion to call the more common math system "wrong", but it could be fair to conclude that in some ways it is kinda both (math can be weird like that). I'm not the most knowledgeable in this area so I would recommend looking into both systems to see the differences and how everything works for yourself. What I do know however is limits and a LOT of math relies on similar usages of limits which are considered by the entire math academic community to be well proven. Most of your arguments aren't necessarily the most sound and come from a misunderstanding of limits. To disprove limits you have to look at the reasons why limits exist and why they work and then disprove something there, which I would recommend to be an extremely tall task as something nearly unanimously agreed upon by mathematicians.
Math often has situations like this, people assume its one field or that there is one true way to do math when this really isn't the case.
*disclaimer* I am nearing the end of my second year as a math major in university and consider myself fairly decent at math, however, this is not a field I have studied. I looked into it a bit after watching this video but I could be incorrect, don't take my word as law, I would recommend looking into hyperreal and standard analysis yourself and seeing the differences there
before i dropped this i didnt even know the hyperreals were a thing-
and also, why cant hyperreals and reals be in the same system? would it just... break shit? or what?
and the fact that the actual answer is both is somehow not surprising to me actually... cus of course it is
I'm curious, what fields of math do you study? I need some advice in becoming a math major, so I'm wondering what courses I should take in University.
N E R D
Breaking News: Random af roblox youtuber solves mathematical arguement that has been going on for years.
looking at the like/dislike ratio (its 74%) it seems not- lol
Watching veritasium's video about infinity will explain this question
Bro proved litterally nothing 😭😭
3:27
You're completely right in this part of your argument. After a finite number of iterations, no matter how incredibly large that number would be, you would always arrive at a number that is above zero. However, this process is not supposed to be finite. If you were to do this process infinitely many times, you would arrive at zero. However, you might object to this logic saying that you can' complete an infinite process. And that's a perfectly valid statement. So, let's try doing this process a finite number of times, like 3 times. You'd get 0.001. With 4 iterations you would get 0.0001. With 5 iterations you would get 0.00001. As you can see, we're continually subtracting numbers from 1, so we're either converging on a number or drifting off to negative infinity. We can prove that we are not approaching negative infinity with a pretty simple proof. Lets represent this process with the equation 1-x=h. x represents the number we are subtracting by and h represents the result. x is always going to be smaller than one, since all the digits to the left of the decimal place are always zero. And when you subtract any positive number by another smaller positive number, the result will always be positive. Therefore, this process can not drift off to negative infinity. The only other option is that it is converging on a number, and the question is, what is that number. Since the number in this process is getting continually smaller, once we drop below a given number, we will never again reach it. This implies that 0.1 is not the answer, since we get below this number on the 2nd iteration, with the result being 0.01. But that isn't the answer either, since we get below that on the 3rd iteration with 0.001. And neither is that the answer since we drop below that on the 4th iteration. So this means that if we can prove that
1. Zero is the highest number it will never drop below
and
2. Once you drop below a number, you will never reach it or a higher number again.
we have proven that 0.9 repeating is equal to 1. (This next part gets a bit difficult to follow) Lets take another look at that equation from earlier (1-x=h). What we need to show here is that h can never drop below zero given that x is a number between 1 and 0. This given statement can be written a bit more algebraically with 1>x>0. Well, since x is always less than 1, if we plug 1 into the equation, we should get a result that is less than or equal to zero. And if you do that, you get 1-1=0, which is obviously true. This implies that plugging in a number greater than 1 will give you a negative number, which is not allowed. Therefore, zero is the highest number it will never drop below. That just leaves us with the second statement to prove. Well, by definition of the problem, each number we plug in for x is larger than the number in the previous iteration. And if you take a constant and subtract it by a number that is getting larger, the difference will be getting smaller. Therefore, we have proven the second statement true. And just like that, we have proven that 0.9 repeating is precisely equal to 1. It's not an approximation to one. Its exactly one.
but how do you get EXACTLY zero??? like, genuinely exactly 0. i could see how its a very small amount ABOVE that, but not EXACTLY that
@@tristantheoofer2 You never do. No matter how many iterations you complete, it will always be barely above zero. And you can't finish an infinite process. But you can get arbitrarily close to 0, and remain at least that close to 0.
.. but thats not equal to zero in that case. how specifically is 0.999... 1 if zero doesnt *necessarily* mean zero in that case?
@@tristantheoofer2 I'm not exactly sure how to best explain this. I'm a calculus student, not a calculus teacher. All I can say is that limits have different axiom systems than normal arithmetic and algebra. I understand if you don't feel that answer is fully satisfying and coherent. I just don't know how to better explain it.
very intresting
3:31 do not let bro know about calculus
There are infinitely many numbers between 0.99... and 1
Like what...? Genuine question
0.99... < x -> ∞ < 1
i mean if you think about it, theres ALWAYS slots to put in more numbers even if the decimal goes on infinitely. always. you can always add 1 more number to the end.
@@tristantheoofer2 You should watch veritasium's video about infinity
infinity goes on an infinite amount of time
think of it like this:
there are two types of infinity: quantitative infinity and un-quantitative infinity. quantitative infinity is infinity you can count, like 1, 2, 3, etc. un-quantitative infinity is infinity you can't count, like the number of unique decimals between 0 and 1. if you try, what should the first number be? should it be 0 is 0.r0...1? but you can add infinitely many zeroes before adding a 1. it's kind of the same logic with 0.r9. you cannot stop between that number and 1, otherwise it isn't infinite. and there are infinitely many numbers in between 0.r9 and 1 because you can always add more.
I'd also like to point out that infinity is not really a number, it's more of a concept. conceptually, there are infinitely many numbers between 0.r9 and 1, but no one can truly prove for or against that because these abstract concepts do not have real value due to the very nature of infinity.
Genuinely it is the difference between theoretical and practical. Like theoretically 0.r9 does not equal 1. Practically it can, at least in a statistical sense . A probability of 0.r9 for example would be represented as “approximately 1” or “approaching 1”, and generally a probability cannot be 1 in any practical sense. The theory is sound that they are not equal, but of course practically approaching 1 is practically equivalent to 1.
ofc ofc, and i agree. 0.r9 PRACTICALLY is 1. but 0.r9 is not TRULY 1
@@tristantheoofer2 agreed
@@tristantheoofer2 practically and truly 1
Every time you post a video like this, I understand little to nothing upon watching it, then it all suddenly clicks two days later when I’m trying to fall asleep at 11:00 at night - great work as usual, I always learn something new whenever you share these sorts of things! :)
ty lmao- ive had a shit ton of people roast me in the comments though cus apparently im kinda wrong or something
@@tristantheoofer2 even though you might be wrong in their eyes, I found I still learned a lot anyways :)
There is NO WAY that 0.999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 repeating = 1
the wish for perfect precision takes another life...
real
the way he glitches and the skybox turns black while clicking the y pos×2 😂 1:07
someones brain didnt understand infinity again and it shows
disagree, i'd not let that slide so,
0.999 .. . / me * me =No
No = maybe
maybe= icecream
icecream = 3
3 = 1
which means 0.999 . .. = 1
its that simple
proof by words?
@@tristantheoofer2 its a fact
fuck, that's a good proof
real????
(yes i know its sarcasm)
@@tristantheoofer2 sarcasm? idk what you taling abt
why did i understand everything...
cus youre very smart fr
i understood all of it up until the huge equation
cause i havent done it in school yet.
man, that one bit was confusing.
let me clarify first and foremost that i am all for critically developing mathematical intuition as it is one of my very own favorite occupations, however as it stands this video is next to dangerously misleading. going into any problem with the predetermined rejection of the result is a cardinal logical fallacy and may lead to viewers stumping their curiosity on a topic, stubbornly clinging to opinionated denial versus open-minded interest in learning.
far were it from me to say i understood algebra, so maybe as a layman i can suggest the following gateway: 0.99... = 1 "for all intents and purposes". it is not exactly a fundamental principle of math, but more so a conclusion of set proofs. hence even disagreeing with their practices, if you wanna get philosophical about it, what they really proof is that in the respective mathematical fields there is simply no known reason whatsoever to detest the conclusive assumption for the sake of progressing research. furthermore having such baselines enables accessibility and an overarching agreement through which scientific findings can be compared and linked. the argument about how infinities work is also to be made, as others here have pointed out. there just is no end to the 0s where we could eventually put the 1. that's precisely why there is no gap to be found.
i see where you're coming from with the argument regarding approaching terms - as far as numbers in-between go, it stands, but in presence of unfathomably large or the abstract infinitely sized, we circle back to the safety net of necessity.
i really hope to not come across as condescending or so here, i truly enjoyed the vibe of this video! so hey maybe if you can find an instance wherein there is an important distinction to be made between 0.99... and 1, that could be quite revolutionary! it is still an ongoing field of research after all :)
if you like, i can search and link some videos that i found helpful before as well
actually if you could link some videos that would be pretty awesome
also i did NOT mean to dangerously mislead people holy fucking shit-
@Spectral_RotD bruh im literally a nerd myself tf are you on about-
you also literally walked into what is essentially an argument over a fucking number, of course the whole comments section will be nerds
Well my brain just left…
lmao mine almost did making this thing
Never expected this from a 16 yr old, a very strong and valid argument right here, great work
Lmao this video come out 3am in my country
lmao ty.. though i apparently know less about calculus than i thought
@@tristantheoofer2 Welp calculus is a huge topics, there's Calc 1, calc2 and calc 3. Calc 1 would involve limits like you stated, differentiation and integration, they are very basic and general, which is the level im studying right now, The next semester i will be studying Calculus 2, which involves in differential equations, hopefully it's not that bad for me
@@tristantheoofer2 Conclusion is 0.999 repeating converges to 1. Which is just equal to 1. (1 - [limit of n approaching infinity 1/n]) means you substituted a number that get closer and closer to infinity until you substituted the ACTUAL infinity itself, which is defined to be zero. Since infinity is not a number, 0.999.. doesn't actually exist. It just become 1 due to there is no more real between 1 and 0.999... repeating. You did a very good job on explaining, as a grade 12 math major i am satified
@@tristantheoofer2 forget about the like ratio, no one can judge your point of view
Now do one about how [1+2+3...∞] doesn't equal -1/12.
that one honestly is way more annoying because that cant really be *defined.8 with the real numbers i think? i just know it has smth to do when you set the reimann zeta function to -1
Part 1: The 4 Arguments
Part 2: The 1/3 Argument: this one is basically an argument about Fractions (e.g. ⅓, ⅔, ⅙ etc.) Being miscalculated/estimated and not the real answer.
Part 3: The Numberline Argument: this one where you have a numberline with 0.r9 and 1. When you subtract these 2 (1-0.r9) = 0.r0...1
the 10x argument
the calculus argument
the video was enjoyable, but i think the 10x argument has a slightly different reasoning:
let x = 0.r9 then 10x = 10 * 0.r9 = 9.r9
if we subtract x from 10x we get 10x - x = 9x = 9.r9 - 0.r9 = 9 implying x = 1
but this also implies that youre kinda just tacking another digit on the end of the end of the string of digits... though i guess infinity doesnt care does it
i think that argument is stupid regardless
imo 0.r9 is a real number, except it physically cannot be represented and therefore cannot be imagined due to the 9's being infinite, kinda like a 4D space, while it (or more the concept of it) exists, it's physically impossible to represent and imagine it because we're in a 3D space, hopefully that makes sense
ok this is probably the best take on it so far honestly. same with 0.r3 and actually really *any* infinitely recurring decimal that close to 1 in any base. theyre real, but cant be represented as anything. so stuff like 0.r1 in binary, 0.r2 in trinary, 0.r3 in quaternary, 0.r4 in quinary, 0.r5 in seximal, etc etc
Bro is just challenging the global math 🙏
These are good arguments if you assume infinitesimals, however the real numbers do not have infinitesimals. If you have a system of arithmetic with infinitesimals, you can arrive at 0.999…≠1. To properly work with this, we have to define what we mean by decimal expansions. If we define the decimal expansion of 0.a_1 a_2 a_3…=Σ_{i=1}^{infinity}(a_i/10^i), and we also assume that the real number the sequence of partial sums converges to is the value of the infinite sum, we end up with 0.999…=1. If we change from the real numbers to a system of infinitesimals, then we could have the sum not converge to any value and therefore not exist, or the sum might converge to 1-ε where ε is an infinitesimal number. The proofs that you said were not correct are false in the axioms you were using, however they are true in the standard axioms of the real numbers.
ok i see
1
also the fact that you didnt align at the tightropes on f2 ToIE just HURTS DKGLRNXMZLRMDJ4SK
lmao i have no sympathy for you for seeing that
Will Achilles ever pass the tortoise
Awesome video as always. Have been here for a long time.
There are plenty of other sources that might help with this problem. Overall, the main line of thought is that there exists no number between 0.9…9 and 1. I think least upper bound is an interesting concept. Thus, mathematically, I would say it’s one. Philosophically, however, you would be right to call it “different”. So I wouldn’t exactly say your thoughts are wrong per se as they reflect a more philosophical and metaphysical definition rather than a purely mathematical one.
“Man I got 9+ notifications to check out”
First thing I see is this video and I immediately watch it, thank you for consistently making absolute bangers
> tristangent uploads
> happi
> watch video
> understand nothing
> happi
lmao
The thing is to me, whenever I was bored at school, I actually thought of something, if you placed a brick very close to a wall, and you can't fit any bricks, you make the brick smaller, and place a smaller brick, but there's still a finite space, so you keep adding smaller and smaller bricks, seemingly so close, but never touching, literally 0.999 ≓ 1. You always have a finite space, tyring to fill it in, but you never seem to actually reach the wall.
i mean if 1/infinity worked amazingly like that then yeah theoretically that would totally work
Nice video, Keep it up!
I’d personally say you are correct, and that the majority of these proofs are merely the result of infinity not being a number.
The way I see it, if a number involves infinity in any form, it is not a real number (including repeating digits). 1 is a real number, while .9r is not, so therefore they are not the same. I especially like your point about 0.3r being an approximation rather than a literal representation of 1/3. Infinitely repeating digits like that are the result of decimal representation rather than a genuine infinite real number (in base 3, “1/3” would be 0.1; no repeating numbers required.)
At the end of the day, math is in many ways an abstract construct. Focusing on semantic concepts like 1=0.9r is much less useful than pragmatically finding the answer. 1 does not equal 0.9r because there should only be a single real representation of each number, and having redundant symbols for numbers will only cause confusion.
agreed honestly, thats why i suggested at the end that 0.r9 doesnt exist. just another thing to think about. personally i say it still isnt 1, especially cus of that possibility that it doesnt exist.
also also with the fraction thing, 0.r9 simultaneously is and isnt rational lmao
(I'm not exactly proving your arguments wrong as 0.99999 being 1 is a somewhat controversial "fact" in mathematics. I do believe it is not, but i will take a more neutral approach and not let my biases play here)
1. I dont exactly know what you mean by "cant be expressed in decimal forms" for 0.r666 and 0.r333, Both of these are rational numbers (they match the prerequisites for a rational number, it can be expressed in form p/q and it is either a whole number, a non-infinite decimal number or an infinite repeating decimal number, like 0.r3333 and 0.r9999), but you only need 1 of these to define that a number IS a rational, so a repeating decimal can be expressed in form p/q, (where p and q rational numbers, this works because the group of rational numbers are closed in the case of division). Also 0.r999 does exist, atleast in the set of rationals, reals and complexes. This does open another door in the fact that if adding some numbers repeatedly until n number of times is not the same as multiplying that number by n, so your argument might make more problems.
2. Now i'm going to be kinda philosophical for this one, because atleast in this case, there is a barrier of "should make sense" in mathematics. If we define a number that is endless and say at some point in it's end that it has a different value, we are basically contradicting ourselves. Philosophically, infinity is an amount of items that is endless. So if we say that something endless has an end, we are contradicting ourselves. Therefore, the infinitesimal is basically just 0. It should have no end as 1, therefore it is basically just 0. Mathematicians still consider it more than 0, for the case of calculating the instantaneous rate of change of specific physical things like velocity and acceleration, you might also know about the derivative, used to calculate the rate of change at the
infinitely small change of delta x for a function.
3. I'm not going to check this because i don't exactly like this argument. (The 0.r999 = 1 argument)
4. The same argument from the numberline proof extends here but another thing is that the values of a rational function (like 1/10^n) approaches 0 when n approaches infinity IF the function has a denominator greater than its numerator. it's still only infinitesimally close to 0 though.
Now, even if the infinitesimal is greater than 1, The philosophical barrier combined with the logistics of calculus makes it basically 0. Another thing is that the infinitesimal does not exist in the real numbers set OR the complex numbers set (sets with irrational and complex numbers respectively) because to definite the infinitesimal, you have to define the first infinite ordinal, or omega onto the real number/complex number system. Because the infinitesimal will be 1/omega. We normally make functions with both the domain and co-domain sets all containing real numbers, so having this system won't exactly make sense for most functions, so we just approximate it to 0 because THAT is what it is (for confusion, refer to my numberline argument). But all of this could be wrong, im no mathematician just a dude who does math and talks about math for a hobby.
POV: guy debunks 250 year proof from the greatest mathematician of all time while playing Roblox.
I like to call those "math bugs". Like 1/0 which sometimes can be infinite and its confusinf or √1 can technically be -1 and 1 and trust me as someone who uses graphing calculators a lot it can get annoying that it doesnt equal to -1. Im just a math nerd and a computer science nerd ig.
Learning what are "math bugs" can be useful since you can actually accept both values.
Trust me, as a math graphing calculator nerd, you sometimes gotta accept both values. 1/0 is a great example. I work on a lot on math. Sometimes 1/0 can be infinite and sometimes its not. Really complex.
Now, about your point, well I get it. However your videos does has some mistakes. While the argument of 4 is pi is false, it's really hard to explain what is acceptable and what is not. No hate at all. Some stuff are just "math bugs". Both are right. I totally get you. Accept math bugs sometimes. They are weird and beutiful.
I'm not gonna go to the very details of your mistakes but I have a lot of experience on math. So yeah you're both right and wrong since it's a "math bug".
By the way, this is all my theory. It's not proven that math bugs are a thing but theres lot of things that I know from my experience that aren't proven. I just use a lot of math.
fair enough and i completely agree. 1/0 according to computers is infinity but according to people its undefined. its like 0/0 which is somehow 1, 0, and infinity at the same time which counts as undefined cus theres multiple values it can represent.. just like 0.99...
other people have said its just a different representation before which i dont really buy, cus that just feels like a cheat code to say theyre the same tbh. yk?
honestly any recurring decimal i think could be said to be some kinda glitch in decimal representation, we just use other ways of number representation (eg fractions) to deal with that lol
@@tristantheoofer2 1/0 is undefined "according" to people because it goes to both infinity and negative infinity. You can see this visually if you graph 1/x
@@kahafb how the fuck does it go to negative infinity??
@@tristantheoofer2 he's got a point. Graph it yourself. It can also be -inf. I don't say it's undefined I say technically it can also be -infinite
if theres an asymptote coming infinitely closer to 1 and it reaches 0.99999999... it still isnt 1
You always choose the best music for your videos. Love the Smash Hit soundtrack cameo in there!!
lol ty
0.999999999... doesn't equal 1 in my opinion because look! theres literally a 0 at the beginning of the number! how could they mess that up!?
I worked for Mathis RV, he's a frauf
1÷3? 0.333333333... so 0.33... + 0.33... + 0.333... is 1? It's 0.99...
0.r9 can be represented as an infinite geometric sum S = 0.9 + 0.09 + 0.009 + ... on into infinity. A bit of high school math will tell you that a geometric sum with a common ratio r < 1 can be found using the formula a / (1 - r), where a is the first term. In this case, the common ratio is 0.1 (because we take the first term, 0.9, and multiply it by increasing powers of 0.1, such that the sequence becomes 0.9 x 0.1^0 + 0.9 x 0.1^1 + 0.9 x 0.1^2 etc.), and the first term is 0.9, obviously. By using the formula, a / (1 - r) , we find that the sum S = 0.9 / (1 - 0.1) = 0.9 / 0.9 = 1
Therefore, by proof of the sum of an infinite geometric series, 0.r9 is equal to 1
Also, with the "10x" argument, it actually states that:
x = 0.r9
10x = 9.r9
10x - x = 9.r9 - 0.r9
9x = 9
x = 9 / 9 = 1
@Caerwyn-z5o ... but that shouldnt even work due to the number having literal infinite digits, thats just tacking on 1 more digit to the end of it for the funnies then subtracting the original number. its like making up the very slight 0.r0...9 difference by saying it basically doesnt even exist in the first place
@@tristantheoofer2 By saying the number has "literal infinite digits", while yes 0.r9 has infinitely many 9s after the decimal, this doesn't mean there is a final digit to get to. The infinite sequence of 9s means that as you go further and further to the right, the difference between 0.r9 and 1 becomes infinitesimally small, effectively 0.
Also, you saying "tacking 1 more digit to the end" suggests that the belief that adding another digit to an infinite sequence changes its value, but it doesn't. In a repeating decimal like 0.r9, there isn't a final digit to "tack on" because the 9s go on forever. If we did tack on a 1 onto the end for example, there would be a long string of 9s followed by a 1, but not an infinite string. In calculus, an infinite sequence can converge to a specific value, which in this case is 1. So, there is no "extra" or "missing" digit, 0.r9 is another way of representing 1.
Finally, by saying there is a "slight 0.r0...9 difference", you are implying that there is an infinitely small difference between 0.r9 and 1, but that's not true. 0.r0...9 is an attempt to express an infinitesimally small number, but mathematically, this is exactly 0 because there is no measurable difference.
There is no extra digit or tiny difference being ignored, it just simply doesn't exist because it can't. And when there is no difference between two numbers, they are equal.
Hence why 0.r9 is exactly 1, and not "infinitely close but not quite"
this... this was a debate..? tbh i just called it 1 cause i was taught to round the thing for ease of seeing it in problems tbh
0.99999999 is not equal to 1 because they look like seperate numbers. i still have a lot to learn in math though, as i am not even halfway through high school.
anyways this video was fun and interesting to watch even though i only understood about half of it.
i mean in math something looking like diff numbers doesnt really mean anything
ofc ofc. once you get into calculus itll likely be easier to understand. my teachers have said that in calculus, limits are essentially just a number getting so close that you can essentially consider it as what you are looking for, which isnt really equaling anything now is it.
@@tristantheoofer2 you're so confused 😭😭😭😭
If you use a different base like base 3, one third multiplied by 3 = 1. It's really just a side effect of using base 10.
shit i actually agree with this
in base 3, one third is represented by 1/10(base 3), and multiplying it with 3, 10(base 3) will be 1 so 1/3 multiplied by 3 is still 1. in any base 1/a * a is always 1 that is fundamental property and definition of division and multiplication. base doesnt matter with this problem
"0.999... = 1" has ruined a point i was trying to make about something completely different in math, thank you sir (also i forgot the point NOOO-)
lol rip to that- honestly the issue with this debate is you can really go either way. i just took the "it doesnt = 1" side cus it makes more sense to me
@@tristantheoofer2 yea same
Here’s an easier way to explain your point
0=0 ✅
1=1 ✅
0.9999=1 ❌
conclusion: 0.9999 isn’t 1, just dont think too much 👍
real???
@@tristantheoofer2real
Add infinitly more 9s then its 1
@strumblers3701 its still not
@@tristantheoofer2 Actually,
0.999... (repeating infinitely) is mathematically equal to 1. Here's a simple way to understand it: Let 𝑥 = 0.999...
x=0.999..., then multiplying both sides by 10 gives 10𝑥=9.999...
10x=9.999.... Subtracting 𝑥=0.999... from 10x leaves you with 9𝑥=9, and dividing both sides by 9 gives x=1. So, 0.999... isn't just close to 1; it is 1 when you consider it as an infinite series.
(Didn't watch the video yet, sorry if you covered these already)
My argument: 0.999... doesn't even exist. What does it even mean for a number to repeat forever? It's just a list of better and better approximations. Sure, it may converge to something, but doing so doesn't mean it is equal to that thing.
For example: Let's try to approximate Pi. Start with a circle of diameter 1, and a square inscribing it. The circumference of the circle is Pi, and the perimeter of the square is 4. Now, move parts of the perimeter orthogonally, such that it approximates the circle ever more. What's the perimeter of the shape? 4. And it keeps being 4. The perimeter converges to 4, and so the circumference converges to 4. But, does that mean Pi is equal to 4? No! Of course not. Convergence does not mean equality.
Let's look back at 0.999... I think I can quite reasonably say that 0.999... converges to 1. However, as I have shown, convergence does not guarantee equality. 0.999... is not guaranteed to equal 1.
On a further note, is 0.999... even a number? All numbers can be expressed either in mathematical notation or verbally, without using infinities. For example, Phi is defined by "(1+sqrt(5))/2", and Pi with "The circumference of a circle over the diameter". How would you define 0.999...? "Infinitesimally smaller than 1"? "0.999..., with nines extending to infinity"? How would you define it without using infinities? I can't think of a way. Perhaps, in the end, 0.999... isn't even a number, who join the ranks of infinity and omega, as mere ideas. And since a non-number can't be equal to a number, I rest my case.
edit: did i just accidentally sum up the entire video
wtf yes you almost summed up the video in one comment lmao how
> be a mathmetition
> also be a tristagent fan
> see this video
> “save till later”
> realise that they might take offence to it
> oh shit
whar
@@tristantheoofer2 whar 🦭👍💧🧊0️⃣⚫️9️⃣9️⃣9️⃣🔁🟰🅾️®️🚫🟰1️⃣❓😰🏔️⛰️⚠️🧍🎼🤣✏️2️⃣🤫🧏
Also 100 divided by 3 = 33.33 repeating but 33.33 repeating x 3 = 99.99 repeating, LITERALLY WHAT?
i like math vids theyre very interesting. and btw video idea can you do a video about why cpus simply cant do decimal number math (for example cpus think 0.2+0.1 doesnt exactly equal 0.3) AKA the same reason the same jump in a gd level becomes harder the longer you are in the level
the reason computers cant do decimals is because they work in a different base (base 2) so there are a lot more recurring decimals that it trips over.
a different base means that instead of having the hundreds then the tens then the ones column in a number (e.g. 541 is 5*100+4*10+1) there is a 6*6s column then a 6s column then a ones column (this would be base 6) or a 16*16s column then a 16s column then a ones column (base 16 or hexadecimal). all the maths works the same but the numbers are written differently
Computers use base 2 so there is a ones column a 2s column a 2*2 (4)s column a 2*2*2 (8)s column and so on. For example 5 would be written as 101 as it is 4+1 and 28 would be 11100 as it is 16+8+4. decimals are the same so there is a half's column then a quarters column instead of a tenths then a hundredths
the issue they have in adding comes because 0.2 + 0.1 or 2/10 + 1/10 would be 10/1010 + 1/1010 in binary which would lead to recurring decimals. this would be fine but a computer can't store a recurring decimal (as it is infinitely long) so it rounds it (like how you might use 0.33 instead of 2/3) this leads to a nice sensible rounded answer in base 2 but when it converts back to base 10 (where the sum did not need rounding) the sum is still slightly off.
@@jameshulse1642 THANK YOU SO MUCH DUDE
stay out of my territory
Nah
i agree
nah, id trespass.
omg tristangent vid after watching a stream about my favorite game
this is amazing
I am very much confused, but thanks for he antvenom style video - cheezit
ngl i was for a bit too until i got the concept. i recommend watching this video by combo class about it cus i think he has the most unbiased take of all of them ruclips.net/video/PGRhYQN0QA0/видео.html
@@tristantheoofer2 nuh uh, math is for nerds
This kind of reminds me of how the area of circle is determined. As the slices increases, the curved side also becomes more straight (if you get what i mean).
yep which is why that pi = 4 makes no sense lol
This is genuenly interesting!
Damn, I now realize the flawed nature of periodic numbers
Also, this 0.r9=1 makes me think about 1/0 equaling an infinte number that is simultaneously positive and negative because, as you said in the first 2 arguments, there will always be a gap between 2 numbers no matter how small the difference would be.
yep. and 0.r9 is simultaneously rational and not, meaning the number cant be expressed as anything but a decimal, which ALSO doesnt work
@@tristantheoofer2 for a while, I've started to rethink math as a whole... How did we get here? Is 1/3 even 0.r3? And most importantly, what is 0.r9?
1/3 IS 0.r3... approxiamately. you can only represent recurring decimals 100% accurately as a fraction. same with 2/3, its 0.r6 APPROXIAMATELY. which also means 3/3 is 0.r9... approxiamately.
@@tristantheoofer2 that makes sense
chat what's the point of studying infinity it makes me feel small
one formal definition of the real numbers in math is equivalence classes of cauchy sequences, or if you haven't taken 2 years of mathematical analysis, basically the set of all ways to approximate a number using rational numbers. so cauchy sequences that could "belong" to the real number 0 could be (1, 1/2, 1/3, 1/4, ...), (0, 0, 0, ...), or (1, 2, 3, 0, 0, 0, 0, 0, ...). the fact that any finite decimal can't exactly equal most real numbers is practically built into this defintion, because mathematicians don't want to use a number system where there technically isn't 1/3, only approximations.
so when people say 0.999... = 1, they're using the formal definition of equality for real numbers: do the cauchy sequences approximate the same number? and yes, 0.999... and 1 both approximate 1, so they are the same real number
(in detail, the notation 0.999... is defined to mean the sequence (0.9, 0.99, 0.999, ...), which is cauchy because it's made of rational numbers and approaches 1)
ok
All parts:
Part 1: the 4 arguments
Part 2: the 1/3 argument
Part 1 -> 2 -> 3: the numberline argument
Part 4x: the "10x" argument
Part 5: the calculus argument
Part -: 0.r9... doesnt exist?
Well, in base 10, it does, which is the fact, but no, it doesn’t. It’s its own thing.
But 0.99999... + 0.1 should equal 1, right?
+ 0.99... + actually 1/10 terminatimg? no. that would be 1.0r9
@@tristantheoofer2 ah
i used the calculator and it said 0.9 repeating is 1, and the calculator cannot be wrong at all so like why are you trying
u cant pot another number after a repeating 0.r0=0 bc theres nowhere to put a 1 bc its occupied by a 0 same for 0.r9 so u cant put a number bettween 0.r9 and 1 or 4 and 3+1
my little brain hurts after watching this video
mine does after making it-
holy shit they uploaded, i can finally watch something while i run a bruteforce!
I will have to disagree with you on this video. This is not an argument. I encourage you to dive deeper into calculus, and especially something called Real Analysis. Starting from a few, universally accepted axioms, the calculus proof is valid and consistent with the framework of mathematics underlying it.
If you ever studied Real Analysis, heavily recommend it by the way, you will realize that limits, convergence, etc... are extremely rigorously defined concepts.
If something approaches 0 as, let's say, x→∞, then the limit just equals infinity as it cannot technically be anything else!
This is what the whole saga of ε-δ proofs are all about. They say that no matter how small of a value you throw, I can compute a formula that will always give you a smaller value.
Therefore there is proven to exist a formula that will, if applied iteratively, will always give you 0, if the limit is 0.
If you want to disprove these, you will need to disprove a few hundred years' worth of accepted theorems.
his knowledge about calculus is basic calculus 1 knowledge, i wont really recommend real analysis too much to a person at that level..
a
I mean, infinity is per definition not a defineable number, so 0.r0...1, while technically equalling zero bwcause there's literally an invalidly large amount of zeroes infront of the 1, is not actually a number as you can't define where the 1 is. So if you subtract an infinitely (so non-defineably) small number from one, you're *realistically* still left with one, but because the number can't be defined, the outupt value can't be defined either and is therefore invalid/non existant. that's my opinion at least.
also sorry if it doesn't make sense because firstly english isn't my first language and secondly i'm still only in 10th grade without really ever having learned about this sort of stuff
and yk what? youre def entitled to that opinion and while i dont agree, youre def able to believe that
This is exactly why 10\3 can’t exist
like as a decimal? i mean.. it can, just as an approxiamation
i agree and prove it outside of math with very simple logic all i need is a piece of paper or 1 if i rip the smallest atom of carbon off the paper is it still a whole sheet no, because its missing an atom and also if it was 1 than its use is not necessary however it might still be like in aerospace engineering for radar cross section were ft. 0.999r is the cross section this is important because of how small light or radio waves are that difference can be the difference between being blown out the sky or not taking a missile to the face (the reason light is important is radar)
sorry for the run on
also if your reading this Tris awesome vid i will def sub
but paper has a finite number of atoms. this number doesnt have a finite number of decimals
@@tristantheoofer2 paper dose have a limted number its just insaneky karge amount
@@zanesteichen a paper has an infinitely tiny amount of atoms when compared to infinity
@@jchevertonwynne fair
Why are you using infinitesimals? They aren’t real numbers they are hyper real numbers (which means that 0.R9/Infinitesimal = 0.R9/0 = Undefined) while 0.999999999999… is. Also 5:30 is finite because Just because a number has infinite digits doesn’t mean it’s infinite, I can add 1 to 999999999… and it will become 100000….. or multiply 1000…. By 2 to make 20000…
i just realized
this is pointless
have you ever heard about limit
ok, what's next? 0.5 ≠ ½? 💀🙏
Amid the comments pointing out mistakes, I want to say that asking questions and starting debates (with an open mind at least) is a good thing to do. Its an opportunity to think more critically and learn something new. I just hope people aren't to mean about it, and that you aren't discouraged from sharing what you think in the future. Its certainly not something I would be brave enough to do, and that's honorable in it's own right.
Bro 0.9999999… is not equal to 1 as long as you dont round
thats what im sayin ffs. apparently from some shit with limits and getting arbitrarily close to infinity though it is the same- which i dont get personally
roblox youtuber versus decades of renowned mathematicians.
eh not really. its just my take on the whole thing and even then the entire reason the debate exists is because of how you can interperet the proofs different than someone else. either way the algebraic proofs are bullshit
@@tristantheoofer2 at this point it's not really a debate among academics, more so the general population and mathematicians. the algebraic proofs are logically sound and easy to understand, but disagreeing with the logic of a proof obviously renders it false, anyone can do this. the algebraic proofs have their place, and I would certainly not discount them altogether. they serve as an easy way to conceptualize the issue based on widely accepted prior knowledge.
I like how all the proofs already require 0.999... to be equal to 1. I'm not gonna say I think that it is equal to 1 or not but I't fun to think about it.
I really like how you made this video so keep up the good work !
haha yeah i noticed that too lol. also ty ^^
I used to say it did equal 1 but there is just too much ‘evidence’ against it lol
ive never viewed it as equaling 1 simply because all the arguments really dont point to it being 1, even the thing where you cant "find" a number between 0.999.. and 1. in that case your options are 1. yes there is, or 2. theyre right next to eachother lmao.
So for the 10x example, I'm going to give what is in my opinion a better demonstration.
x = 0.999...
Multiply both sides by 10
10x = 9.999...
Subract x from both sides
9x = 9.999... - x
Substitute in 0.999... for x on the right
9x = 9.999... - 0.999...
9x = 9
x = 1
Also, in the calculus approach, they define 0.999... as
lim 0.{9} k times
k-> infty
This is the definition, and the definition says "this is a limit. The expression will never actually equal this value, but it will get infinitely close." The expression never equals 1, but its limit is 1 because it can get arbitrarily close to 1. Because repeating decimals are defined as this limit, this proves that 0.999...=1 given this definition of repeating decimals.
Also, I would like to add
This is arbitrary. All of math is arbitrary. Mathematicians pick the definitions that are most useful, and the most useful definition of recurring decimals defines 0.999... as equal to 1
i was gonna say lmao. i guess mathemeticians be mathin
ooooooh ok so limits 1 because it can get whatever "arbitrarily close" is??? tf
@@tristantheoofer2 Arbitrarily close means that whatever distance d you chose from 0, there will always be a value e where with e or more 9s, it is within that distance of 1 for any arbitrarily small value of d. That is how a limit is defined in calculus. In this case, it is guaranteed that with at least _ceil(log_10(1/d)) + 1_ 9s, the value is guaranteed to be within d of 1. That is how a limit in calculus is defined. 0.999... is equal to 1 because of the definitions that are standard in calculus and in mathematics.
Is it just me or this is the 2nd video on this
trust me theres more than 2
@@tristantheoofer2 also theres some random mod for minecraft called "big int" (i have no fucking idea) that just allows people to go like insanely far away (yes like further than the 2^63 limit). (i have no idea on how to install it tho bruh)
@ErrorAnimator687 yeah i have it
I heard somewhere that it's both depending on your perspective. In real numbers, they're the same because you can't put more decimals after the infinite decimals, but in hyperreal numbers, they're seperate numbers, because you can, so infinitely close doesn't mean the same as
In summary the whole debate is pointless (unless you count the decimal point :). )
ngl yeah the whole debate is pointless- i just wanted to have my take on it because i was interested lmao
Very hard math
not really, just some dumb proofs
@@tristantheoofer2 very dumb
@@tristantheoofer2 college graduates and math majors vs 11th grade roblox kid who makes bait video
Dude you are a really good teacher like i learn alot more from you than my teachers
lol ty
do not listen to this man please
@@tristantheoofer2 np
PLEASE don't let your mathematical knowledge be comprised entirely of a roblox youtuber's half-assed conjecture
while watching this video why did you speaking remind me of listening to a cat meow .v.
idk why but pretty funny :3
0.9r: who are you?
1: I'm you, but 0.0r1 more stronger than you.
real
I heard an argument that on the number line infinitesimals don’t exist so 0.99999999 = 1 (because it has infinite digits) I heard