I watched this and other videos in my three years of high school, so I decided to do a Maths major to understand and work with those fancy number sets (which I know now they are rings, fields, vector spaces...) and get farther in the study of those numbers. Three years later, I can watch this video while I understand the logic behind every argument and it's like a high 4.9999999... from HighSchool!me. Thank you for these videos, Vi Hart.
+TheDoubleAgent the first time I heard your Ravioli/Black and Yellow remix I did a spit take. yeah I was seeing how many people would fall for it and think it was 9:99 ;-)
More specifically on Reason 7, we are guaranteed for any two numbers (call them x and y) that are not equal (say, x < y) then the average of the two numbers lies between them (so x < [x+y]/2 < y). So if we believe that 1 =/= 0.999... then we ought to be able to average these two numbers to get a number in between them. We add them together to get: 1 + 0.999... = 1.999... But now if we try to divide 1.999... by 2 (to get the average), it becomes apparent that 1.999... / 2 = 0.999... This is equal to one of the two numbers we started with, which means we have failed to find a number in between 0.999... and 1. Therefore, 0.999... and 1 have to be the same number, because there is no number between them.
Here is another reason: There is an old trick you can do to get any series of infinite decimals, for example, 0.123123123 is completely viable You just simply take the number you want to repeat, and divide it by the same number of digits. For example, 324/999 = 12/37, which equals 0.324324324324... You might not be able to reduce it, but you will still always get that infinite decimal number. Lets say we want infinite 9s. Well thats easy! 9/9 = 1/1, and 1/1 = 1. i mean look at how many 9s there are there! so many 9s! more than graham's number 9s! of course i'm kidding. this is the most simple proof i can think of.
the note she added is exactly my problem 1/3 doesn't let itself be written down correctly (1/3)*3=1 0.3333..*3=0.9999.. but that only proves to me that the decimal representation of 1/3 was wrong to start with :P 1/3 'should be' infinitely tiny bit bigger than 0.333.. in a way that 3x that infinitely tiny bit will make the difference between 0.999.. and 1 but if you like your math to be practical, ignore me, just remember that math uses assumptions and rounding when dealing with infinity ^^
+rondowar It uses limits which you have not used. Consider e = | 1 - 0.999..9_n | as n -> infinity. e -> 0. Therefore in the limit 1 = 0.999... Let me be honest, if you think this is wrong you seriously need to revisit what you know about mathematics: read up on limits, series, and real numbers.
UteChewb I said I had a problem with decimal notation :), not that the math used is incorrect edit: problem as in, I dislike it, as it it's not good (in my opinion!) at accurately holding information using f(x) = 1 - x lim[x->1] f(x) = 0 this means that the difference approaches 0 note that "approaches" isn't the same as "equals" the limit equals 0; the difference approaches 0 0.999.. approaches 1; the limit equals 1 (using lim[x->1] f(x)=x) I'm not saying what you said is wrong
Except that the value of a recurring decimal is defined in the same way as an infinite sum, which is what limits are for. The limit of the sum 0.9 + 0.09 + 0.009 + 0.0009 ... is the same as 0.9 recurring (as pointed out by the video). We already know that if you wanted a real and useful answer to the sum, you take it's limit, and since the recurring decimal notation just means the same thing in an easier to see form, then we have to concede that it means the same thing. This leaves us with a very useful and accurate representation of real numbers.
I like reason 4 the most. It demonstrates why the infinity of the 9s is important. In decimal, multiplying by 10 shifts all the digits over the the left once. When you have finite digits, this leaves a zero on the right, but with infinite digits there is always a number to take its place. 0.333… * 10 = 3.333… and we don’t bat an eye. When you apply the same logic: 0.999… * 10 = 9.999… then the answer is clear. A little bit of algebra and it comes together in a very mathematical way. I like this proof the most because it feels the least hand-wavy to me. So long as you can grasp the concept of 0.333… * 10 = 3.333… and 3.333… - 0.333… = 3, you can accept the proof.
+Kyle, they're worse. The flat Earther's have some quite nice arguments. Unlike with 0.999... = 1, a human invention, whether or not the Earth is flat is not our choice to make. We can only make observations and measurements to determine whether or not the Earth is flat.
+EmperorZelos by using the paradox "this statement is false," you're telling the truth, however if the statement is true then it must be false but if it's false then it must be true... see it infinitely repeats.
Ian Kim ah it wasn't clear from what you said which statement you were refering to. Thought for a moment you were one of hte idiots who opposed the 0.999... = 1
Here's a demonstration I use with 4th graders: I write the following three equations on the board, 1/9 = .111 ... 2/9 = .222 ... ... 8/9 = .888 ... Then I show that each fraction equals its paired decimal expansion by replacing it with its long division form and showing my work. Finally, I ask them to come up with a fraction that can be paired with .999 ... They go wild! They love it! They start asking very insightful questions. NB: Most of the nearly 300 4th graders I have worked with over these past ten years only need me to write these three equations. A few have needed me to write the intervening equations.
@@ronanjm Wow replying to a 2-year-old comment... a young child in elementary doesn't need a college level mathematically rigorous proof to see why this is true as long as it gets their brains thinking for themselves. Just do as the person described and open up your calculator app and plug in all 9 equations from 1/9 to 9/9, It checks out. is it a rigorous mathematical proof stating all axioms and steps meant for adult mathematicians? Hell no. Does it work and is it simple enough for a child to understand without being factually incorrect? Yeah pretty much.
That’s really lazy of you, 1/9 does not perfectly and exactly equal 0.111... it’s a near number. A number that has a close estimated value. You “proof” is just saying 1/9=0.111... very very closely so that means *9 and 1=0.999... (closely).
its not an assumption if you divide 2 by 9 by hand you will continiou getting 2s aftr the 0 indefinetely. also if you divide in a calculator you get the same thin too
Every repeating decimal can be written as a fraction with an error infinitesimally close to zero but not necessarily zero. It doesn't mean that it is completely accurate, just that for finite calculations the difference is considered negligible and therefore zero. Essentially, her tenth "reason" which is... "it works"... therefor "it is."
9/9 belongs to the same equivalence class as all other fractions whose decimal value is 1. 9/9 is not special and certainly does not represent .9999.... any more accurately than, say, (3pi)/(3pi).
+jon AL Can you define a number which is infinitesimally close to zero but not necessarily zero? Is it a real number? What are the implications of such a number? Can you define 0.9999... ? Is your definition consistent with the existing mathematical definition? If you think really hard and do research on these questions, you will find the truth. What you say isn't completely unreasonable, but it requires a different kind of framework than the one we commonly use.
If it seems suspicious to you that all the definitions of things in the real numbers seem fine-tuned so that 0.999... = 1, it's because they kind of are. When mathematicians defined the real numbers, they wanted that number system to have a few specific properties. Among other things, it had to be a field, and it had to have the Archimedean Property. To put it simply, a field is a number system where every non-zero element has an inverse. This means that for any number x, there must exist y, such that x * y = 1. The Archimedean Property is that for any real number x, there must exist a natural number n such that x < n. So if the real numbers have those two properties, then 0.999... = 1. Because suppose not. Then you have 0 < ε = 1 - 0.999... = 0.0...01, as some people suggested. But what would be the inverse of ε? Positive infinity? What's the natural number that's greater than positive infinity?
caperUnderscore26"as someone stated here: ..." Yes. that person was me. I stated it. But I also stated the definition of a number, and it's not what you're saying it is. I stated the definition of a decimal number, and it's not what you're saying it is. 0.999... represents the equivalence class, which is the same thing as the limit. Natural numbers are equivalent to quantity (or rather, cardinalities of finite sets), but real numbers aren't. "Use in applications" is irrelevant because you're always going to use a rational number in an applications. This is pure mathematics, where "physical interpretations" and "applications" are mostly seen as accidental byproducts. "The mathematician does not study pure mathematics because it is useful; he studies it because he delights in it and he delights in it because it is beautiful." -Poincaré
caperUnderscore26 anything we can write iwth decimals are real numbers by definition :) finitely or infinitely many decimals. But before each specific value of a decimal place is always finitely many other decimals, which means notions like 0,(0)1 does not work because there aren't fintely many 0s before the 1 but 0,(9) works because for every nine we look at there are finite amount of 9s prior to it but infinitely many behind it.
Anybody else notice that Vi's voice sounds a bit distorted in this video, compared to both newer and older vids? If you listen very carefully, you can hear little audio artifacts that make it sound like she's been slowed down slightly. It's the same kind of distortion you hear when watching a TH-cam video at a slower speed. I think she must have slightly slowed down the recording to artificially stretch it out to be 9.999... minutes long. At least, that's the most logical explanation I can come up with. I mean, a suspiciously convenient runtime, a time-related audio distortion... yeah, I can put 1 and 0.999... together.
***** Sorry, I can't find the thread. He may have deleted it because I asked him to Google "limit of the sequence of their partial sums", or similar. He would have discovered that what I was saying was correct and that what he was saying was nonsense. He won't have liked that (because he's a fool).
Jason Zhao Where is your disproof? Later: I realise that's supposed to be it. That is not a disproof. All you've done is assert that 0.999... < 1. You have said that 1 != 1 because 1 < 1 is possible.
Well. I'm terrible at math, and I won't lie, while this is probably not enough to elevate me to a level of equal understanding (or even near that, really, lost cause), I do have to say it is a blast to listen to, because there's a clear joy in the workings of mathematics and numbers at play here. I do grasp some of it (felt somewhat silly at not knowing the fraction-proof previously) but enough to appreciate it nonetheless! Excellent video :)
I need to get SOME math teacher to watch Vi Hart. Then others will realize I watch math videos in my free time... oh well... Also, I give this a 9.999../10!
This video was mind blowing. I had no idea that 0.999 could even equal 1. Through ViHart's proof, especially the algebraic one that started with x=0.999 and ended with x=1, I understood that through simple algebra, a number could be equal to another number. I also liked the part when she explained that if 0.999 went on infinitely, then what number is more than 0.999 but less than 1? Therefore, 0.999 has to be 1. I didn't know split octonions, surreal, and hyperreal numbers existed, as my knowledge only go as far as imaginary numbers. When ViHart talked about split octonions, surreal, and hyperreal numbers, I was really confused to the whole concept of how and why those numbers existed. The ending of this video shows how math can really evolve, how things that 'didn't or can't exist' can exist if one can think of a way to prove that it can exist. This inspired me to maybe do the same and think outside the box of math. I really liked this video, and because of it, I have learned many new things.
"I understood that through simple algebra, a number could be equal to another number." I would say they are the same number (same value) that are written differently, like 0.5 and 1/2, 5 and 20/4, etc.
For any geometric progression where | r | < 1, the infinite sum exists and has the following value: Σ a_i , i=1 to ∞ is equal to a/(1-r) As .999... is the geometric progression of a=9/10, r = 1/10 and therefore, .999... = (9/10) / (1 - 1/10) = 1 This proof is found in MANY college text books and easily found online. Caper again promotes his own version of mathematics as he doesn't accept REAL MATH with REAL mathematical sources for validation. I used to think he just had HUGE conceptual errors, but I now realize he doesn't use established mathematics. If someone doesn't want to use real mathematics, that is fine...but see how far it gets you in college (if you even make it that far).
If 0.999... is not equal to 1, then what exactly is 1 - 0.999... (ie. their difference)? If the difference is 0, then they are equal, by definition. If the difference is not 0, then what is it?
TheRealUbehage If you can argue why the limit of a sum doesn't agree with the sum itself. In other words, why lim(n -> inf) sum(1,n) n does not give the same value as sum(1, inf) n
***** What are you two talking about? Nothing about math is arbitrary or made-up. If it were, it wouldn't have any real-world applications, just like the following is a grammatically correct question but is not actually meaningful: "Why are unicorns hollow?" You can ask this question, but no meaningful answer is possible, therefore the question itself is meaningless. If math were made-up and applied to reality rather than being derived from reality, we would have this same problem.
John Pryce Acctually it is about arbitrary rules. The thing is it is usually picked in such a manner that it has applications in the real world due to the reason you said. Some mathematical things starts out as purely a curiosa where someone asks "What happens if I do this?" and goes along and later they find there is a real world application of it. Definitions and axioms are arbitrarily picked and from it we derive everything. The axioms and definitions are usually picked in such a manner they are the most useful though but you can just aswell construct mathematics where usefullness = 0 and it would still be mathematics
John Pryce Numbers are made up, symbols are made up, the rules (meaning of +, =, etc.) are made up but once they are set, they never change. Unless you're trying to change the meaning of + or the meaning of 1 and 2, 1 + 1 = 2.
Araqius The symbols we use to represent these things are arbitrary, agreed. But the rules are not, just as the rules in physics are not arbitrary (they aren't well understood, and sometimes we think we've understood them and we haven't, but they aren't ARBITRARY). So no, you are mistaken.
the video is exactly this long (in minutes): 9.999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999...(repeating)
ohhh smart math people referencing popular rap music.... what will you do next?! I have to give this girl props, I've never sat through 10min of math without falling asleep, let alone be really interested or intrigued. Love the doodels girl, keep it up!
the infinent is infinent and what you mean is that is infinent of nubers betwen 0 and 1 and the SOME infinent is all numbers betwen -infinent to +infinent
You're correct, natural numbers and real numbers are two different infinities but the infinity behind the decimal point is always the same size, Aleph-0
***** ja jag menar ju att det finns olika oändligheter men inom fysik så fins det dem två oändligheter som jag pratade om. tex:en boll studsar ner motmarken och åker upp med hällften av sin längd för vart ända studs kommer den någon gång sluta studsa? och svaret är nej för att det finns en oändlighet mellan 0 och 1. och det andra är med hela tal som i matte. då går det att räkna 1 2 3 4 5... i oändlighet
Yew Wood ℚ = {m,n} | (m,n) = 1 and m,n ∈ ℤ where n ≠ 0} where the ordered pair (x,y) is equivalent to gcd(x,y) therefore any two integers m.n where the GCD(m,n) = 1 and n ≠ 0 would answer your question. (i.e. 1/1, 2/2, 3/3, 4/4 ect)
Yew Wood Think he means 3 * 1/3 =.999... = 3/3 = 1 as they are all equivalent relationships and classes just like my example such that 1/1=2/2=3/3=4/4 = .999... = 1 All completely mathematically correct equivalences. All different notations to express the same value of as 1.
LOLOLOLOL This video is AWESOME ViHart is obviously very highly intelligent but also even more highly and amazingly creative. Just like a true mathematician should be ! Yeay ViHart - now I'm going to have to watch all your vids.
Some people can't understand that 9.999... is the same as 9.9999... That is where the problem lies. The concept of a number reoccurring does not make sense to those people. I think they are unfamiliar with the notation. Perhaps they think that all numbers eventually reoccur with 0s. They must think our decimal system is perfect. I still believe that there are 1.111... types of people in this world, although Binary is not perfect either.
The most intuitive example that makes you understand: If 0.999... and 1 aren't the same number there has to be some number between them right? In other words 1 - 0.999... shouldn't equal 0 So what number is between them? Let's look at it more closely. 1 - 0.9 = 0.1 1 - 0.99 = 0.01 1 - 0.999 = 0.001 It seems as the number between them arrives when the 9s end, and that makes sense. So which number is between 0.999... and 1? Well it looks to be like this: 1 - 0.999... = 0.000... As you can see, the 9s never end, and because they never end the final 1 can never arrive. And if there is no final 1 the number between 1 and 0.999... is 0.000..., which is the same thing as 0. So there is no number between 0.999... and 1. This means that they must be the same!
Here's a proof I found when I was bored: Anything divided by 9 is point that number repeating. Thus: 1/9 = .111..., 2/9 = .222..., 3/9 (1/3) = .333..., 4/9 = .444..., 5/9 = .555..., 6/9 (2/3) = .666..., 7/9 = .777..., 8/9 = .888..., and here's the grand finale! 9/9 = .999... *Says in an overly sarcastic tone:* Whaaaaat?!?! But 9/9, by definition alone, is undoubtedly 1! How can .999... = 9/9 = 1?!?! DOES THAT MEAN THAT .999... = 1?!?! :O Now, you may be asking: "But Nin, what about 10/9? That doesn't equal .101010..." And in reply I say that it stops working after you go into double digits, but knowing you, you won't take my word for it, so look at this: Let's split up the fraction into two fractions that have single digit numerators: For example: 8/9 + 2/9 = .888... + .222... = 1.111... You can also write it as: 9/9 + 1/9 = 1 + .111... = 1.111... Thus, .999... easily equals 1.
Technically 1/3 doesn't equal .33333333... the proof was never complete because the mathematician who tried to prove it died writing 3s before he ever got to 9/9 = .999... His best friend, Professor Round, changed the last digit of his proof for 6/9 = .66666666666... to .66666666667 before he was buried and now we all "Round" numbers up (that are larger than .5) so we don't suffer the same fate.
jmdj530 I personally think that story's bullshit. If we rounded up, let's say, 2/3 to .6666666667, then 2 (6/3) would be 2.0000000001, which it isn't. Rounding and approximating makes it inaccurate, which is pretty useful if you're doing something that doesn't required exactly accurate math. And besides, you won't die if you write: "0.333...", the ... shows that it repeats until infinity. You can also write it as 0.3 (with a bar over the 3), or 1/3, neither of which are fatal. :P
jmdj530 "Technically 1/3 doesn't equal .33333333..." Please provide ONE single mathematical source that says it is not in any algebra or calculus or any actual mathematical text? They ARE equal technically or otherwise.
jmdj530 "1 = .999... is one too." Now that was very witty! (mathematically speaking it is funny 1 = .999 ... is "one" too...) But mathematically 1=.999... is no joke :P
I found something similar to what you were saying in your fourth argument on my own in 7th grade, and I brought it to my math teacher and she said that I did something wrong because two numbers of different values can't equal each other. So I just completely forgot about it until I watched this. I had no idea I was actually right. Props 7th grade me!
You have awoken to the great disservice of our overly regimented schooling, especially regarding math. Your teacher lost the sense of a kid and steered you away from yours. Hopefully you have re-embraced your innate genius and realized that sometimes, you are the only one who is right. Also, sometimes it is good to not expose your creation to naysayers. If you do, be prepared to not believe them, though you may need to "play along to get along". Good thinking.
Congratulations! You're one of those ultra rare people who AREN'T so stuck to their mind that they block out anything else. I applaud you. (this sounds sarcastic but isn't i swear)
[In response to all the responses to my comment below]: A number cannot be equal to another number in the same base unless you decide to use some weird set of number. Hence, 0.999... = 1 means that "0.999..." and "1" are two different expressions for the same number. "1" is the canonical representation of that number, the simplest one. "0.999..." is a shorthand notation for "the sum of n = 1 to n going to infinity of 9 / 10^n" which evaluates to the number whose natural symbol is "1". "0.999.." is of the same essence than "0.6 + 0.4" in the sense that it is just an expression that evaluates at one point to an irreductible symbol "1". You can of cource decide that "0.999... " cannot be replaced by "1" in all algebraic formulas. In that case you would have to rigorously justify that there exist x such that x = 1 - 0.999... and x is not 0. That number x would be the smallest value different than 0 in your set of number. That's fine, but you cannot use the set of real numbers for that. By the very construction of this set there are no smallest value because for any candidate x as the smallest value you can find a smallest one, say x / 2. Hence because you cannot provide the existence for that x you have to decide on two branches: allow an undefined value in your set and solve all the potential problem and inconsistencies that will come with this decision or construct a theory of convergence of infinite series. If you choose the second branch you have something which starts to look like the set of real numbers and which is probably more useful for practical applications.
There's nothing weird about decimals. There's something very weird about your ideas though. Like it or not, 0.999... = 1is a fact and there's nothing you can do about it (except cry).
Chris Seib An awful lot of so called "paradoxes" stems from simple error in judgement about what you reason about, for instance the dx in an integral, how can something infinitely thin be added to something that has a length ? That's because you are taking shortcuts, before using dx you have to define a proper mesure on your space, once you have a mesure you don't need to think in term of dx being an infinitely small quantity but as a notation for the mesure of your space and you stop giving a fuck about something which was not rigorous and paradoxal in the first place. It just become doing a stupid and simple thing.
+Martin Adams "Oh man, dx isn't an infinitesimal." Of course it is infinitesimally small. Didn't you know, infinitesimals now are known to exist? Except from zero, their absolute values are smaller than any rational positive number and still not zero. They are special numbers common to non-standard analysis (and 'infinitesimal' calculus), where you also have infinite numbers, bigger than any finite number, but still of different sizes (not equal to the ∞ concept of standard mathematics).
Martin Adams That's exactly what I am telling you, are you stupid ? Can't you read a fucking sentence properly ? The entire issue is actually your incapacity to understand a sentence correctly.
I feel that school and the pressure to do "well in life" limits me.I wish I could live in a cave and just think about numbers all day long but I can't let myself do that .I cant even.
Anil Radhakrishnan true maybe school is just presenting in the wrong that in a way that makes me really bored and makes me want to draw so maybe just if the schools and tech it in a funny and awesome way then i will remember and LIKE math
Dear god the more I keep coming back to this video and looking at the comments I lose at least a trillion brain cells. Like I can admit possibly debating for a bit of the student's intuition(although not with the number .999...) but this...
Chris, you need to know your audience. Answer the question in a way that the person will understand. Making yourself look smart will accomplish nothing other than confusing the person you are answering to.
PotaTox, I'm sooooo sorry that facts hurt your butt. Do you only want to see ignoramuses makes comments? Did you even watch the video? See reason 7 starting at around 7:40
Yes, I did watch the video. Yes, I agree that 0.9 repeating equals one. Let me ask you a question: Let's say that there is something you don't know about, and you ask someone to explain it to you. Do you want that person to explain it by stating a ton of facts using vocabulary that you don't understand, like some kind of dictionary, or do you want to have the person explain it in a way that states why something is the way it is and say it in a way that makes sense to you? Do you expect a kindergartner to understand what a derivative is just because you gave them the function of a derivative? I hope you never become a teacher.
+Rebecca Trudgeon +Vi Hart We know that 1/3 = 0,33333... We also know that 1/3 x 3 = 1 and 0.33333... x 3 = 0,9999... From that, we can conclude: 0,9999... = 1 It only depends on the way you calculate it, if you use the fractions (a way to represent racional numbers with 2 integer numbers) you get 1. Else, if you use the decimal numbers (a racional value represented by a "broken integer" number) you get 0,9999... We must always remember that a number is just a way to represent values. Our world is full of values that we do not choose, we can only choose the numbers that represent them. I guess we can say that 1 is equal to 0,9999... but depending on the practical situation you need to use these numbers, we might consider them different. If you need to measure something very tiny, for example, 0,9999... might be significantly different from 1. Did you get it? Sorry if i've made some grammar (or math) mistakes, english is not my native language. (and i'm a computer engineering student, not a mathematician).
No not the infinitesimal - the split octonians. Also, I figured that 0.0̅0̅1 just isn't small enough there is still 0.0̅0̅05 and it must just be beyond our numeric system like its inverse, ∞. (is infinity the inverse of infinitesimal or is it called something else?)
I was referring to the 8-dimensions. They are abstract dimensions, not physical dimensions. Pure math makes no claims about the physical world and Vi wasn't suggesting it did either. 0.000...1 is meaningless. The 0s don't stop repeating, there is no end at which to place the 1.
Duh! 0.999... doesn't approach anything. It is a series that has a sum. That sum is the limit of the sequence 0.9, 0.99, 0.999, ... and that turns out to be equal to 1.
My one and only problem with the .999 repeating = 10 is the .0000...1 that you never get to. I don't want to drop that 1 Just because I never get there doesn't mean it doesn't exsist.
The .000...1, with the 1 on the infinite digit decimal place is called an infinitesimal, and it is smaller than all finite positive real numbers and bigger than zero, however it doesn't work in the reals. When using the standard real number system, you only use one infinitesimal: zero, (and no infinitely big values are allowed as well). So all infinitesimals are equated to zero in the real numbers, and all decimal numbers (a finite sequence of digits, then a dot, and finally a sequence of digits to the right) are all defined to be real numbers. You can't have infinitely many digits to the left (due to the real number definition) or other than zero as an infinitesimal in real numbers, also because of the definition. If you want to calculate with other infinitesimals than zero or infinitely big numbers, you have to use a non-standard number system, like hyperreals or surreal numbers, but they use different kinds of notations, not the simple decimal notation the reals use).
So when Hand Sanitizer says "removes 99.999% of germs" it is technically 100% of germs are gone which is completely and utterly false so I'm confused, then again I am horrible at math so there ya go. Also | - (-1) | = 1 isn't that amazing?!?! (no) oh yeah and this whole video is completely and utterly confusing... oh yeah and right now in math class I am learning sequences so everyone try and figure out this sequence: 2, 1, 2/3, 1/2, 2/5... -figured it out in class and got me a box of sour patch kids like a boss -my math teacher is a boss because he gave me a box of sour patch kids because i did something amazingly epically awesomely amazing, epic, and awesome and mathematical in class that no one else knew how to do, even though (like i said) i am not good at math -finally, to wrap this up neat and tidy into a bow or whatever, i really like to ramble... (OH REALLY???) -I talk so much that I am starting to use sarcasm against myself in my ramblings -What am I talking about right now -Bye -Oh yeah and first one to figure out the sequence I gave to you so kindly earlier wins my box of sour patch kids (not really because I already ate it, good day, sir.)
That's different. 99.999 isn't 99.999... They write 99.999% because, as far as they know it kills 100% but they can't say that because in the odd chance that one lone bacteria escapes people could sue them for false advertising. So that's the gist of it.
Arthur Cote If there was a single type of bacteria who could resist the hand sanitizer, they would have changed the recipe, it is as far as they know effective against 100%. 99.99% is just a way to cover their own backs legally.
atomheartother A quick search found me this www.ncbi.nlm.nih.gov/pubmed/20429659 showing that alcohol based cleaners like purell are not effective against C difficile.
I really wonder: are the limits even taught in US schools? I was not familiar with this 0.999... notation until I moved to the US a few years ago, and I really am puzzled about the amount of "proof that .999...=1" videos I can see on youtube right now. Because either you know nothing about maths, in which case 0.999 (not even repeating) is one, or you DO know about maths, and you should know limits. So if you write H(n) = 0.999....9, which counts n "9"s, you have H(n) = 1 - (1/10)^n, and then 0.999.... = lim(n->∞,H(n)) = 1 - lim(n->∞,(1/10)^n) = 1 - 0 = 1 Am I missing something ?
Limits are taught in American schools, albeit not particularly well most of the time. Your proof is perfectly valid, but I have to say that I think it's slightly more complex than the algebraic one which goes x = 0.999... 10x = 9.999... 9x = 9.000... x = 1.000... = 1 The thing with the way limits are taught in American schools is that it seems that they are not taught conceptually. Students are told that lim(x->infinity) is where the function goes towards as x goes to infinity, without explaining the actual concept of infinity from the perspective of number theory or set theory. Without a firm grasp on the concept of infinity, it's easy to make mistakes by trying to utilize the rules which one learned for finite numbers or numbers with finite decimal representations or whatever.
As so called moderator Caper has decided to delete our comments even if it is only about math and without any abuse then I will write it here: He thought that mathematical induction can be applied to prove 0.999... != 1 by writing: 0.9 != 1 0.99 != 1 ... 0.999... != 1 - from his logic. Then if you apply same logic you should also get: 0.9 != 0.999... 0.99 != 0.999... ... 0.999... != 0.999... - FAIL. He is so weak in arguing about math with his 5 year old brain that he prefers to delete the comments instead of writing that he was wrong.
This video should be .99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999.... seconds shorter
1123aka Saying .999... = 1 means someone understands basic fundamental facts of mathematics. There are absolutely equal as per mathematical definitions as well as dozens of other aspects of real mathematics. It is absolutely no different than saying 1/2 equals .5 are are they some how different too because they use different types of numbers? 1/2 is fractional notation, .5 is decimal notation, and .999... is infinite repeating decimal notation... 1/2 * 1 = 1/2 , .5 * 1 = .5 and .999... * 1 = 1
can you define infinity? no. then you cannot use the "=" sign. It's like saying relative is absolute and absolute is relative. No matter how "infinite" 0.99 goes,it will still be smaller than 1.
1123aka Infinity is a concept in ℝ and isn't defined as a real number. .999... isn't infinite, it is quite a finite number as it exists between other real finite numbers such as 0 < .999... < 2. It has an infinite decimal expansion but it's value is finite. What is the mathematical definition of a repeating decimal in the form of an infinite convergent series? I'll give you a hint: It's .999... ≜ Σ 9/10^n, (n=1 to ∞) and Σ 9/10^n, (n=1 to ∞) is an infinite convergent series and by the definition of an infinite convergent series Σ 9/10^n, (n=1 to ∞) =1 therefore .999... = 1 You don't get to make up your own definitions in math...you either use real mathematics or you don't... you don't.
"infinite" is not defined. concept of convergence is perfectly suited to real world applications as well as our own need to understand the universe in terms of "knowns" and "finites" . You cannot define infinity. NO matter how many expressions you throw at me,mathematics is still a child of philosophy and scientific method. If you wanna feel like you know "infinite-ness" ..in any manner or way,go ahead and close your mind. Convergence has nothing to do with absoluteness. You are good at what you were taught, but you pathetically lack the "how" of thinking. This is why we need to teach people how to think,while we feed them with mathematics of which they can never grasp the true nature
1123aka So basically you reject modern mathematics and want to teach people things which are not true based upon your limited and incorrect understanding of infinity and real mathematics? Is that basically your position?
Wait, shouldn't it be like this? 0.99999......9999=x 9.99999......9990=10x 9x=9.0000....001 x=9.0000....0001/9 Even if it is infinite numbers, we know that the "end" of 0.999... is 9 and when you multiply by 10, there should be a zero at where there would have been a 9? Or does the whole infinite = infinite + 1 deal ruin it all.
+German Eagle Infinity implies endless. There is no end, and so there is no last digit to be a 9. Your ....s necessarily imply a finite number of 9s. Food for thought - what is the last digit of the decimal representation of Pi? Each 9 corresponds to a counting/natural number. There is no last natural number. For clarity, there are infinitely many natural numbers, yet none of them is infinity.
Chris Seib But we know the last digit is nine, no matter how much times we do it. Pi has no logical patern that we know of, so it cannot not be compared to this. In between those nines there are infinite 9, no it does not go against anything.
German Eagle The last digit of 0.999... does not exist, so it cannot be 9. Your use of the phrase "no matter how much times we do it" gives the game away - it implicitly assumes a finite number. Just for a laugh, what is the last digit of 0.1212121... where 12 keeps on repeating? Compare your answer with the last digit of 0.1212121... where 21 keeps on repeating. What do you mean by "in between those nines"? I hope that you aren't referring to the non-existent last 9. Do you think that 0.999... really is 0.999...9? Infinity means endless. Saying that there are infinitely many nines between two nines is meaningless. Either there is a last 9 or there isn't. Infinity is not a very large number as you seem to be treating it as - it is not a number. Try this. The set {1} has size 1 and 1 is in the set. The set {1,2} has size 2 and 2 is in the set. What is the size of the set {1,2,3,...} consisting of all the natural numbers. Is that number in the set? (If not, then why not?) Before responding, look up aleph-0.
Chris Seib To answer your question, in a sense, I do see it as so. But before I explain, I see why this does not work. I could say that 9.999...9 is the same as 9.999... .Because there is still infinite nines in there, but for cases like 121212 or any variation, it wouldn't really work. As I said, I understand what you are saying, and see why I am wrong.
German Eagle A problem with the notation 0.999...9 is that that last 9 is at the oo + 1 decimal place. So that is treating oo as a natural number. But oo cannot be a natural number, because it would have to be the last one, so there cannot be a number after oo. The axioms state that if n is a number, that n + 1 > n yet we must have that oo + 1 = oo and so it breaks the rules. Incidentally, that's why aleph-0, the size of the set of the natural numbers, cannot be in the set of the natural numbers. Crudely speaking, aleph-0 is the smallest kind of infinity there is. The number of real numbers is larger than aleph-0 and is written aleph-1. It is common to see aleph-1 = 2^aleph-0. But this does not mean what it seems to mean. It is a statement about cardinalities.
If you don't happen to be an accomplished mathematician, stop pretending to know what you're talking about when it comes to abstract, counter-intuitive mathematical ideas, such as the equivalence between .999... and 1. The people who study this stuff for a living pretty much unanimously agree on the equivalence, and it's just frustrating whenever uninformed people act like their idea of how math should work is better than the valid, PROVEN ideas developed and discovered by actual mathematicians.
I love list ning to you to fall asleep, it takes me back to my high school math classes when I could still let myself drift off to math. College doesn't let me do that anymore :/ . I love you Vi, you make my life so much better.
+Acre00 Nope .0.999... already has infinitely many 9s and already is the limit of the sequence 0.9, 0.99, 0.999, ... In fact that's what 0.999... is defined to be. It is obvious that 1 also is the limit of that sequence. As the limit must be unique (by definition of limit) we have that 0.999... = 1
No? Why would there be? At least if we are taking about the infinitesimal moment between each day, there is no distinction for the same reasons described in the video. Real numbers which are infinitely similar are the same in our system.
+Logan Gantner. Even in the surreals and Abraham Robinson's hyperreals (both of which include infinitesimals) 0.999... = 1. I see no reason to speculate that there is even an infinitesimal gap between the days. The idea is bizarre, to boot.
about a month ago, he couldn't multiply 2 numbers. today, he can't add 2 number. i wonder what will happen, if we wait another month :-) somebody should help this poor creature, he shows clear symptoms of cerebral atrophy!
He tries to "Cite" a source, and what does he do?`...youtube video of another crank who think there is a conspiracy to not see the "flaws" the crank sees.
***** He, just already, lose the ability to remember what he said. upic.me/show/50142054 (He say n can not equal infinity.) upic.me/show/50142056 (He say "for "x = ∞"") And he also just admit himself to be an idiot. upic.me/show/50142134 "You are too much of an idiot to see what I did there, dufus. lol" (If anyone who see what he did there is an idiot, then he himself who see what he himself did there is also an idiot too.)
There are 1.9999...... kinds of people in this world: 1) Those that understand algebra and aren't too stubborn to understand what all mathematicians understand. 2) Those trying to figure out how much of their body they should cut off to have 0.9999...... left.
This seems really sketchy to me, you are using your conclusion to prove that its right, which is making up stuff, like in proof 3.999... you subract .9999... from 10 and get to 9, but that isn't a proof, as that would assume that you've estabalished that .9999... = 1, which you never do because you use that assumption in all your proofs
No she isn't she is not once using the conclusion to prove it is right. and she isn't subtracting 0,(9) from 10 she is subtracting it from 9,(9) 9,(9)-0,(9)=9=10x-1x=9x No assumption of the conclusion anywhere
Ghork1 Caper is a mathematical moron while Emperor teaches Mathematics ..I would listen to Emperor. Caper- "9.(9) - 0.(9) = 9 assumes that 9 = Σ 81/10ⁿ " Bullshit...no assumption is drawn here. 9.5 - .5 = 9 9.6 - .6 = 9 9.8888 - .8888 = 9 9.9999... 0.9999.... = 9 See the pattern? It's basic math. Also we don't have to "prove" that .9999... = 1 as by definitions they are equal. That is just mathematical fact. There are however proofs which do show they are equal, but we wouldn't even ever need to assume they are equal as by definition they ARE EQUAL. You can't make an assumption about something when the definition is already provided. It is is like saying x = 1, now prove x + 1= 2 by assuming x = 1. Well we already know x=1 so why we we need to "assume" x =1 ? Same thing here...in real mathematics 1=.999... So Caper's diatribe about having to make assumptions is just plain idiotic.
***** Caper seems to be right thought, the thing is that its a agreed upon thing to make it work for certain equations, however 1 and .999 repeating are different values, one is 1, and another is approximating 1, almost the same thing but still different
Ghork1 He isn't right though. "however 1 and .999 repeating are different values" Nope, from definition of real numbers they are the same " one is 1, and another is approximating 1, almost the same thing but still different" Again, nope, they are the same, if you want I can cite sources where the definitions are and then show you how they are equal
Ghork1 its pretty much the same as the proofs used to make 1+2+3+4+5...... = -1/12 which is based around shifting positions of infinite series, like this is with infinite numbers, the thing is by shifting its making it a different finite number, just as this is with moving the decimal place once. But have a place in math for use of relative infinities calculations etc. But that doesn't make it fact, what it does make is make is make it a tool used for calculation, but like with series, and with a number infinitely close to 1 even a child could tell you that those aren't facts, however they are fine tools
my math teacher spent the whole class attempting to disprove vihart
whuuut? How did he try that?
he showed it in class
Alex Dillon but how did he try to disprove it?
***** joke?
nope not kidding
I watched this and other videos in my three years of high school, so I decided to do a Maths major to understand and work with those fancy number sets (which I know now they are rings, fields, vector spaces...) and get farther in the study of those numbers.
Three years later, I can watch this video while I understand the logic behind every argument and it's like a high 4.9999999... from HighSchool!me.
Thank you for these videos, Vi Hart.
Sometimes I feel as if Im smart. Then I watch a Vi Hart video. ._.
+Geometry Dash Jake if you major in mathematics, then one day you too can have an idea what vi is talking about!
+TheGrammargestapo1 c;
+Konhat Lee Sakurai same thing here numberphile and these videos inspired me to do maths and now i am already in the first year of my university
Which universities are you guys going to?
If only this video was 2 seconds shorter
+TheDoubleAgent the first time I heard your Ravioli/Black and Yellow remix I did a spit take.
yeah I was seeing how many people would fall for it and think it was 9:99 ;-)
you mean 1.9999... shorter
2?
59.4 seconds is 99% of a minute so...?
+Andy Rosene no its not thats a finite series of ones youtube doesnt have enough data space to type it out hahaha
Screw this, I'll just calculate in base 12.
1/3=0,4
3/3=1
then you get 0;24972497... = 1/5 and a similar thing for 0;EEEEEE... (E being eleven).
@@meta04 r/woooosh
@@maxsch.6555 hey this isn't reddit
3/3 = B not 1 in base 12
gamecoolguy619 im sorry i know it’s an old video and comment but what???
3/3 =1 in any base (that has 3s)
and B = 11 (in bases 12 and up)
I gave you .9999999999... likes!
I gave you 0.9999999 likes
ANd, in return, others have given you 71.999... likes!
Others have given you 99.99999...... likes!
@@Dekeullan TH-cam doesn't let you do a fractional like.
Chris Seib.
leave her alone, it was a simple mistake. no need to be rude, dude. stop.
More specifically on Reason 7, we are guaranteed for any two numbers (call them x and y) that are not equal (say, x < y) then the average of the two numbers lies between them (so x < [x+y]/2 < y). So if we believe that 1 =/= 0.999... then we ought to be able to average these two numbers to get a number in between them. We add them together to get: 1 + 0.999... = 1.999... But now if we try to divide 1.999... by 2 (to get the average), it becomes apparent that 1.999... / 2 = 0.999... This is equal to one of the two numbers we started with, which means we have failed to find a number in between 0.999... and 1. Therefore, 0.999... and 1 have to be the same number, because there is no number between them.
your maths is a bit dodgy. try 0.9...5. that os in-between 1 and 9
*0.99..
@@pepthebabslasonge2551 no. in between 0.999 and 1. 0.99 isn’t in between 1 and 0.999 it comes before 0.999 therefore you’re the one being dodgy.
@@PeyPeySupreme yeah, I wasn't the smartest cookie when I wrote that
@@pepthebabslasonge2551 me just realizing this comment was 3 years ago 💀
Here is another reason:
There is an old trick you can do to get any series of infinite decimals, for example, 0.123123123 is completely viable
You just simply take the number you want to repeat, and divide it by the same number of digits.
For example, 324/999 = 12/37, which equals 0.324324324324...
You might not be able to reduce it, but you will still always get that infinite decimal number.
Lets say we want infinite 9s.
Well thats easy!
9/9 = 1/1, and 1/1 = 1.
i mean look at how many 9s there are there!
so many 9s!
more than graham's number 9s!
of course i'm kidding.
this is the most simple proof i can think of.
When Niko stays in the library for too long be like
hi niko
"9 divided by 9 = .9 repeating" said no one ever
My favorite reason it the simplest one.
.333...=1/3
X3. X3.
.999...=1
the note she added is exactly my problem
1/3 doesn't let itself be written down correctly
(1/3)*3=1
0.3333..*3=0.9999..
but that only proves to me that the decimal representation of 1/3 was wrong to start with :P
1/3 'should be' infinitely tiny bit bigger than 0.333.. in a way that 3x that infinitely tiny bit will make the difference between 0.999.. and 1
but if you like your math to be practical, ignore me, just remember that math uses assumptions and rounding when dealing with infinity ^^
+rondowar It uses limits which you have not used. Consider e = | 1 - 0.999..9_n | as n -> infinity. e -> 0. Therefore in the limit 1 = 0.999... Let me be honest, if you think this is wrong you seriously need to revisit what you know about mathematics: read up on limits, series, and real numbers.
UteChewb
I said I had a problem with decimal notation :), not that the math used is incorrect
edit: problem as in, I dislike it, as it it's not good (in my opinion!) at accurately holding information
using f(x) = 1 - x
lim[x->1] f(x) = 0
this means that the difference approaches 0
note that "approaches" isn't the same as "equals"
the limit equals 0; the difference approaches 0
0.999.. approaches 1; the limit equals 1 (using lim[x->1] f(x)=x)
I'm not saying what you said is wrong
Except that the value of a recurring decimal is defined in the same way as an infinite sum, which is what limits are for. The limit of the sum 0.9 + 0.09 + 0.009 + 0.0009 ... is the same as 0.9 recurring (as pointed out by the video). We already know that if you wanted a real and useful answer to the sum, you take it's limit, and since the recurring decimal notation just means the same thing in an easier to see form, then we have to concede that it means the same thing. This leaves us with a very useful and accurate representation of real numbers.
Yep.
I like reason 4 the most. It demonstrates why the infinity of the 9s is important. In decimal, multiplying by 10 shifts all the digits over the the left once. When you have finite digits, this leaves a zero on the right, but with infinite digits there is always a number to take its place. 0.333… * 10 = 3.333… and we don’t bat an eye.
When you apply the same logic: 0.999… * 10 = 9.999… then the answer is clear. A little bit of algebra and it comes together in a very mathematical way. I like this proof the most because it feels the least hand-wavy to me. So long as you can grasp the concept of 0.333… * 10 = 3.333… and 3.333… - 0.333… = 3, you can accept the proof.
The people that say
.999... =/= 1 are the flat earthers of mathematics.
+Kyle, they're worse. The flat Earther's have some quite nice arguments. Unlike with 0.999... = 1, a human invention, whether or not the Earth is flat is not our choice to make. We can only make observations and measurements to determine whether or not the Earth is flat.
Simply Curious see there are a lot of people that didn't learn this so they maybe very skeptical.
OK wise guy. redo viharts proof with X but use 0.11... instead. you will prove that 0.11... is equal to 0.0123... .
triangle earthers*
@Yoyo Dong I am sorry but that was a mistake with my maths. I have retracted all my statements I made. I am sorry.
"Mathematics is about making up rules and seeing what happens"...
I think I love you :D
!Lol! yep, all of us do, she's 0.999...derfue
Hello
I saw this when it released 9 years ago. Coincidence? I think NOT!
I saw it when it came out 10 years ago, but i guess that doesn't make a difference
Hey I saw your comment from 9 months ago
I saw it when it released 9.9999… years ago
@@peepock7796 hey, me 1.999...!
Don't you mean 8.9999999999999999999.... years ago?
This video is exactly my cup of tea. Simple, full of delight mathematics and puns! Great job!
so 0.999... is... 1 derful
Carly Santos I just got that! Thanks for helping me with the joke
Carly Santos LOL XD
Olivia Grant moving swiftly 3.999...ward (4ward)
"According to this 3.999...mula..." lol
wau
forget "this is statement is false" GLaDOS should of used this instead.
+Ian Kim What statement is false?
+EmperorZelos by using the paradox "this statement is false," you're telling the truth, however if the statement is true then it must be false but if it's false then it must be true...
see it infinitely repeats.
Ian Kim ah it wasn't clear from what you said which statement you were refering to. Thought for a moment you were one of hte idiots who opposed the 0.999... = 1
+EmperorZelos actually it was a reference to a game.
Portal 2 to be specific.
Ian Kim My bad
your .9 repeating jokes around 9:00 were hilarious, first math video that genuinely made me laugh, subbed and liked, 10/10
edit: 9.9999../10 haha
jay-z 99 problem reference
can you believe this is the same person who counts down from random microwave numbers... vi hart is truly an amazing and wonderful person.
Here's a demonstration I use with 4th graders:
I write the following three equations on the board,
1/9 = .111 ...
2/9 = .222 ...
...
8/9 = .888 ...
Then I show that each fraction equals its paired decimal expansion by replacing it with its long division form and showing my work.
Finally, I ask them to come up with a fraction that can be paired with .999 ...
They go wild!
They love it!
They start asking very insightful questions.
NB: Most of the nearly 300 4th graders I have worked with over these past ten years only need me to write these three equations. A few have needed me to write the intervening equations.
Éammon Síoċáin wow what 4th graders are you teaching. When I was in 4th grade I was probably eating rocks or something
Éamonn Síoċáin this isn’t a valid proof.
@@ronanjm Wow replying to a 2-year-old comment... a young child in elementary doesn't need a college level mathematically rigorous proof to see why this is true as long as it gets their brains thinking for themselves. Just do as the person described and open up your calculator app and plug in all 9 equations from 1/9 to 9/9, It checks out. is it a rigorous mathematical proof stating all axioms and steps meant for adult mathematicians? Hell no. Does it work and is it simple enough for a child to understand without being factually incorrect? Yeah pretty much.
I hate fractions XD
That’s really lazy of you, 1/9 does not perfectly and exactly equal 0.111... it’s a near number. A number that has a close estimated value. You “proof” is just saying 1/9=0.111... very very closely so that means *9 and 1=0.999... (closely).
(I didn’t finish the video) but i always thought if
2/9=0.22222...
3/9=0.33333...
Then 9/9=0.99999... but also 1
That's a good one, too, but it does have the assumption that N/9 = 0.NNNN... for all integer values of N, which you'd first have to prove to use this.
exactly i thats my favorite proof of the 0.999..=1
its not an assumption if you divide 2 by 9 by hand you will continiou getting 2s aftr the 0 indefinetely.
also if you divide in a calculator you get the same thin too
Emre, your calculator has infinitely many digits!!! I don't think so.
All, you cannot assume that 1/9 = 0.111... It just happens that it is true.
That's how I learned how .999999... was equal to one.
Every repeating decimal can be written as a fraction. For .9999... that fraction is 9/9.
Every repeating decimal can be written as a fraction with an error infinitesimally close to zero but not necessarily zero. It doesn't mean that it is completely accurate, just that for finite calculations the difference is considered negligible and therefore zero. Essentially, her tenth "reason" which is... "it works"... therefor "it is."
Ava Nicole yup!
+jon AL Nope. 0.999... = 1 is exact.
9/9 belongs to the same equivalence class as all other fractions whose decimal value is 1. 9/9 is not special and certainly does not represent .9999.... any more accurately than, say, (3pi)/(3pi).
+jon AL
Can you define a number which is infinitesimally close to zero but not necessarily zero? Is it a real number? What are the implications of such a number? Can you define 0.9999... ? Is your definition consistent with the existing mathematical definition? If you think really hard and do research on these questions, you will find the truth. What you say isn't completely unreasonable, but it requires a different kind of framework than the one we commonly use.
If it seems suspicious to you that all the definitions of things in the real numbers seem fine-tuned so that 0.999... = 1, it's because they kind of are.
When mathematicians defined the real numbers, they wanted that number system to have a few specific properties. Among other things, it had to be a field, and it had to have the Archimedean Property.
To put it simply, a field is a number system where every non-zero element has an inverse. This means that for any number x, there must exist y, such that x * y = 1.
The Archimedean Property is that for any real number x, there must exist a natural number n such that x < n.
So if the real numbers have those two properties, then 0.999... = 1. Because suppose not. Then you have 0 < ε = 1 - 0.999... = 0.0...01, as some people suggested.
But what would be the inverse of ε? Positive infinity? What's the natural number that's greater than positive infinity?
caperUnderscore26"as someone stated here: ..." Yes. that person was me. I stated it. But I also stated the definition of a number, and it's not what you're saying it is. I stated the definition of a decimal number, and it's not what you're saying it is. 0.999... represents the equivalence class, which is the same thing as the limit.
Natural numbers are equivalent to quantity (or rather, cardinalities of finite sets), but real numbers aren't. "Use in applications" is irrelevant because you're always going to use a rational number in an applications. This is pure mathematics, where "physical interpretations" and "applications" are mostly seen as accidental byproducts.
"The mathematician does not study pure mathematics because it is useful; he studies it because he delights in it and he delights in it because it is beautiful."
-Poincaré
Fthagen
You will soon discover that the swine do not appreciate your pearls.
caperUnderscore26 anything we can write iwth decimals are real numbers by definition :) finitely or infinitely many decimals. But before each specific value of a decimal place is always finitely many other decimals, which means notions like 0,(0)1 does not work because there aren't fintely many 0s before the 1
but 0,(9) works because for every nine we look at there are finite amount of 9s prior to it but infinitely many behind it.
If you could explain 8-dimensional electron theory that would be awesome!
Those Lords are spot on.
Lord Voldemort, Lord Jesus Christ, Lord Vader, Lord of the Flies, genius
+Demassify Lord of the rings? Maybe?
And lord c'thulhu
King Wa Siu Yep, Sauron's on there
@@katherineshideouslaughter Yup, that one's an important Lord one should not forget.
Anybody else notice that Vi's voice sounds a bit distorted in this video, compared to both newer and older vids? If you listen very carefully, you can hear little audio artifacts that make it sound like she's been slowed down slightly. It's the same kind of distortion you hear when watching a TH-cam video at a slower speed.
I think she must have slightly slowed down the recording to artificially stretch it out to be 9.999... minutes long. At least, that's the most logical explanation I can come up with. I mean, a suspiciously convenient runtime, a time-related audio distortion... yeah, I can put 1 and 0.999... together.
I can put 2 and 1.999999990 together
You watched to many vihart
Or she was slow because before recording she drank 99.9999... bottles of beer
This video makes me so happy. THIS is the beautiful part of math.
Somesh, you are a trolling buffoon.
Somesh, the math is 100% spot on, you cannot construct them being distinct in mathematics.
***** He can't even say why he says it's wrong. He just claims it's a mathematical conjuring trick / fraud. He's an idiot.
Chris Seib Where has he done that?
***** Sorry, I can't find the thread. He may have deleted it because I asked him to Google "limit of the sequence of their partial sums", or similar. He would have discovered that what I was saying was correct and that what he was saying was nonsense. He won't have liked that (because he's a fool).
I remember watching this in elementary school. I didn't believe 0.999... = 1, and I didn't understand the math. Now, knowing calculus, I believe it.
I'm gonna do this on a test! I'll then show her this video when she marks my test wrong.
Shockingly, many math teachers say it's wrong.
+Chris Seib My disproof:
Given x
Jason Zhao Where is your disproof?
Later: I realise that's supposed to be it. That is not a disproof. All you've done is assert that 0.999... < 1. You have said that 1 != 1 because 1 < 1 is possible.
Did you really think asserting something counts as a proof in mathematics?
***** I dont understand your issue with my proof. I established a parameter x
The way you do your videos is just great, the ending in this one is treasured!
Well.
I'm terrible at math, and I won't lie, while this is probably not enough to elevate me to a level of equal understanding (or even near that, really, lost cause), I do have to say it is a blast to listen to, because there's a clear joy in the workings of mathematics and numbers at play here. I do grasp some of it (felt somewhat silly at not knowing the fraction-proof previously) but enough to appreciate it nonetheless!
Excellent video :)
I need to get SOME math teacher to watch Vi Hart. Then others will realize I watch math videos in my free time... oh well...
Also, I give this a 9.999../10!
I never understand a single word this girl is talking about, yet I keep watching the videos....
Hi
Its very satisfying to me that this video is exactly 9.999... minutes.
funny lord of the files joke
This video was mind blowing. I had no idea that 0.999 could even equal 1. Through ViHart's proof, especially the algebraic one that started with x=0.999 and ended with x=1, I understood that through simple algebra, a number could be equal to another number. I also liked the part when she explained that if 0.999 went on infinitely, then what number is more than 0.999 but less than 1? Therefore, 0.999 has to be 1. I didn't know split octonions, surreal, and hyperreal numbers existed, as my knowledge only go as far as imaginary numbers. When ViHart talked about split octonions, surreal, and hyperreal numbers, I was really confused to the whole concept of how and why those numbers existed. The ending of this video shows how math can really evolve, how things that 'didn't or can't exist' can exist if one can think of a way to prove that it can exist. This inspired me to maybe do the same and think outside the box of math. I really liked this video, and because of it, I have learned many new things.
"I understood that through simple algebra, a number could be equal to another number."
I would say they are the same number (same value) that are written differently, like 0.5 and 1/2, 5 and 20/4, etc.
they are the same number, just written in two different ways. 1 = 0.99~ = 1/1 = one = 99999^0 = uno = I
For any geometric progression where | r | < 1, the infinite sum exists and has the following value: Σ a_i , i=1 to ∞ is equal to a/(1-r)
As .999... is the geometric progression of a=9/10, r = 1/10 and therefore, .999... = (9/10) / (1 - 1/10) = 1
This proof is found in MANY college text books and easily found online. Caper again promotes his own version of mathematics as he doesn't accept REAL MATH with REAL mathematical sources for validation. I used to think he just had HUGE conceptual errors, but I now realize he doesn't use established mathematics. If someone doesn't want to use real mathematics, that is fine...but see how far it gets you in college (if you even make it that far).
Really, the fact that almost all decimals define a unique number is much more surprising to me.
02:13 , divide by 9 factorial? oh lord
that means x=8!
+Víctor Ordz. I hope you aren't serious, but if you are, it's not 9 factorial, it's 9 (exclamation point).
If 0.999... is not equal to 1, then what exactly is 1 - 0.999... (ie. their difference)?
If the difference is 0, then they are equal, by definition. If the difference is not 0, then what is it?
well, it would be
0.000...0001
which is 0
but it is harder to convince people that that is true
First Last
It can't be that because it implies there's a finite amount of zeros there.
TheRealUbehage Naughty naughty. That -1/12 is a Ramanujan sum and shouldn't be written that way. But it's quite interesting nevertheless.
TheRealUbehage
If you can argue why the limit of a sum doesn't agree with the sum itself.
In other words, why
lim(n -> inf) sum(1,n) n
does not give the same value as
sum(1, inf) n
lim(n -> inf) n/n = 1
but inf/inf = undefined.
Omg I just realized
Formula
Fourmula
Three-point-nine-repeating-mula
I've been using the line "suspiciously practical" for years and just today realized i must have stolen it from this video
People complaining at 'made up rules.' ALL rules in maths are made up, they are eventually accepted as the norm.
Indeed, math is all about making up rules and follow them to their logical conclusion
*****
What are you two talking about? Nothing about math is arbitrary or made-up. If it were, it wouldn't have any real-world applications, just like the following is a grammatically correct question but is not actually meaningful: "Why are unicorns hollow?"
You can ask this question, but no meaningful answer is possible, therefore the question itself is meaningless. If math were made-up and applied to reality rather than being derived from reality, we would have this same problem.
John Pryce Acctually it is about arbitrary rules.
The thing is it is usually picked in such a manner that it has applications in the real world due to the reason you said. Some mathematical things starts out as purely a curiosa where someone asks "What happens if I do this?" and goes along and later they find there is a real world application of it.
Definitions and axioms are arbitrarily picked and from it we derive everything. The axioms and definitions are usually picked in such a manner they are the most useful though but you can just aswell construct mathematics where usefullness = 0 and it would still be mathematics
John Pryce Numbers are made up, symbols are made up, the rules (meaning of +, =, etc.) are made up but once they are set, they never change.
Unless you're trying to change the meaning of + or the meaning of 1 and 2, 1 + 1 = 2.
Araqius
The symbols we use to represent these things are arbitrary, agreed. But the rules are not, just as the rules in physics are not arbitrary (they aren't well understood, and sometimes we think we've understood them and we haven't, but they aren't ARBITRARY). So no, you are mistaken.
the video is exactly this long (in minutes): 9.999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999...(repeating)
no. I DISagree!
Vladimír Vozár then you, good sir, have been proven wrong.
ThePCguy17
You're wrong. It's 9.9 (repeating) minutes long and .9 (repeating) seconds long. (TH-cam says 10:01 on the bar thing.)
it said just 10:00 on mine...so...youtube is lying
***** You're using Firefox.
ohhh smart math people referencing popular rap music.... what will you do next?! I have to give this girl props, I've never sat through 10min of math without falling asleep, let alone be really interested or intrigued. Love the doodels girl, keep it up!
She is every nerds dream girl :)
wasn't expecting a math video where they suddenly bust out 10 lords-a-leaping
[holds back tears yelling] SOME INFINITIES ARE BIGGER THAN OTHER INFINITIES
the infinent is infinent and what you mean is that is infinent of nubers betwen 0 and 1 and the SOME infinent is all numbers betwen -infinent to +infinent
*****
And I men infinities I'm swedish so I dont speak so good english hehe sorry
You're correct, natural numbers and real numbers are two different infinities
but the infinity behind the decimal point is always the same size, Aleph-0
***** Jag är också svensk, det stavas "infinite" och annie har rätt, det finns olika stora oändligheter men troligen inte på det sättet hon menar.
*****
ja jag menar ju att det finns olika oändligheter men inom fysik så fins det dem två oändligheter som jag pratade om.
tex:en boll studsar ner motmarken och åker upp med hällften av sin längd för vart ända studs kommer den någon gång sluta studsa? och svaret är nej för att det finns en oändlighet mellan 0 och 1. och det andra är med hela tal som i matte. då går det att räkna 1 2 3 4 5... i oändlighet
What two numbers can you divide by to get .999...
Yew Wood ℚ = {m,n} | (m,n) = 1 and m,n ∈ ℤ where n ≠ 0} where the ordered pair (x,y) is equivalent to gcd(x,y) therefore any two integers m.n where the GCD(m,n) = 1 and n ≠ 0 would answer your question.
(i.e. 1/1, 2/2, 3/3, 4/4 ect)
3.(1/3) gives you 0.999..., but it can also give you 3/3, that's 1.
celo7531
That is a good one. You are a genius. :)
Yew Wood Think he means 3 * 1/3 =.999... = 3/3 = 1 as they are all equivalent relationships and classes just like my example such that 1/1=2/2=3/3=4/4 = .999... = 1
All completely mathematically correct equivalences. All different notations to express the same value of as 1.
caperUnderscore26
Nice work. That is very true.
In addition to the 1/3 proof, you can do the same with 1/9. 1/9 =.1111...
1/9x9=1 and.1111...x9=.9999... therefore.9999...=1
Dylan Gibson. mmmmmk
LOLOLOLOL
This video is AWESOME
ViHart is obviously very highly intelligent but also even more highly and amazingly creative.
Just like a true mathematician should be !
Yeay ViHart - now I'm going to have to watch all your vids.
Was I the only one that saw the video was EXACTLY 10:00 minutes?
+Lupa Nadia youtube adds a second
+Evan Knowles TH-cam added 0.999... sec
+poppe191 lol the truth
+poppe191 My trailer in my channel when I export it in Movie Maker is 1 minute 0 second and 0 jiffy. And TH-cam wrote it as 1:01!
EXACTLY 9.999999999 REAPEATING MINUTES
Some people can't understand that 9.999... is the same as 9.9999... That is where the problem lies. The concept of a number reoccurring does not make sense to those people. I think they are unfamiliar with the notation. Perhaps they think that all numbers eventually reoccur with 0s. They must think our decimal system is perfect. I still believe that there are 1.111... types of people in this world, although Binary is not perfect either.
The most intuitive example that makes you understand:
If 0.999... and 1 aren't the same number there has to be some number between them right?
In other words 1 - 0.999... shouldn't equal 0
So what number is between them? Let's look at it more closely.
1 - 0.9 = 0.1
1 - 0.99 = 0.01
1 - 0.999 = 0.001
It seems as the number between them arrives when the 9s end, and that makes sense.
So which number is between 0.999... and 1?
Well it looks to be like this:
1 - 0.999... = 0.000...
As you can see, the 9s never end, and because they never end the final 1 can never arrive. And if there is no final 1 the number between 1 and 0.999... is 0.000..., which is the same thing as 0.
So there is no number between 0.999... and 1. This means that they must be the same!
ywecur_ CHEERS! Thank God there is hope for humanity here! (Besides EmperorZelos, Chris Seib, and twee Weekes!
Spot on my friend...spot on! :)
Thanks, Vi.
You blew my mind again.
Plus, I really like your art.
Here's a proof I found when I was bored:
Anything divided by 9 is point that number repeating. Thus:
1/9 = .111..., 2/9 = .222..., 3/9 (1/3) = .333..., 4/9 = .444..., 5/9 = .555..., 6/9 (2/3) = .666..., 7/9 = .777..., 8/9 = .888..., and here's the grand finale!
9/9 = .999...
*Says in an overly sarcastic tone:* Whaaaaat?!?! But 9/9, by definition alone, is undoubtedly 1! How can .999... = 9/9 = 1?!?! DOES THAT MEAN THAT .999... = 1?!?! :O
Now, you may be asking: "But Nin, what about 10/9? That doesn't equal .101010..."
And in reply I say that it stops working after you go into double digits, but knowing you, you won't take my word for it, so look at this:
Let's split up the fraction into two fractions that have single digit numerators:
For example: 8/9 + 2/9 = .888... + .222... = 1.111...
You can also write it as: 9/9 + 1/9 = 1 + .111... = 1.111...
Thus, .999... easily equals 1.
Technically 1/3 doesn't equal .33333333... the proof was never complete because the mathematician who tried to prove it died writing 3s before he ever got to 9/9 = .999...
His best friend, Professor Round, changed the last digit of his proof for 6/9 = .66666666666... to .66666666667 before he was buried and now we all "Round" numbers up (that are larger than .5) so we don't suffer the same fate.
jmdj530 I personally think that story's bullshit. If we rounded up, let's say, 2/3 to .6666666667, then 2 (6/3) would be 2.0000000001, which it isn't. Rounding and approximating makes it inaccurate, which is pretty useful if you're doing something that doesn't required exactly accurate math. And besides, you won't die if you write: "0.333...", the ... shows that it repeats until infinity. You can also write it as 0.3 (with a bar over the 3), or 1/3, neither of which are fatal. :P
jmdj530 "Technically 1/3 doesn't equal .33333333..." Please provide ONE single mathematical source that says it is not in any algebra or calculus or any actual mathematical text? They ARE equal technically or otherwise.
Steve McRae It was a joke. A very clever joke.
1 = .999... is one too.
jmdj530 "1 = .999... is one too." Now that was very witty! (mathematically speaking it is funny 1 = .999 ... is "one" too...)
But mathematically 1=.999... is no joke :P
I found something similar to what you were saying in your fourth argument on my own in 7th grade, and I brought it to my math teacher and she said that I did something wrong because two numbers of different values can't equal each other. So I just completely forgot about it until I watched this. I had no idea I was actually right. Props 7th grade me!
You have awoken to the great disservice of our overly regimented schooling, especially regarding math. Your teacher lost the sense of a kid and steered you away from yours. Hopefully you have re-embraced your innate genius and realized that sometimes, you are the only one who is right. Also, sometimes it is good to not expose your creation to naysayers. If you do, be prepared to not believe them, though you may need to "play along to get along". Good thinking.
I came in to this video wanting to deny it but you literally just blew. My. Mind!
Congratulations! You're one of those ultra rare people who AREN'T so stuck to their mind that they block out anything else. I applaud you.
(this sounds sarcastic but isn't i swear)
Vi Hart, you never cease to amaze me.
[In response to all the responses to my comment below]: A number cannot be equal to another number in the same base unless you decide to use some weird set of number. Hence, 0.999... = 1 means that "0.999..." and "1" are two different expressions for the same number. "1" is the canonical representation of that number, the simplest one. "0.999..." is a shorthand notation for "the sum of n = 1 to n going to infinity of 9 / 10^n" which evaluates to the number whose natural symbol is "1". "0.999.." is of the same essence than "0.6 + 0.4" in the sense that it is just an expression that evaluates at one point to an irreductible symbol "1".
You can of cource decide that "0.999... " cannot be replaced by "1" in all algebraic formulas. In that case you would have to rigorously justify that there exist x such that x = 1 - 0.999... and x is not 0. That number x would be the smallest value different than 0 in your set of number. That's fine, but you cannot use the set of real numbers for that. By the very construction of this set there are no smallest value because for any candidate x as the smallest value you can find a smallest one, say x / 2. Hence because you cannot provide the existence for that x you have to decide on two branches: allow an undefined value in your set and solve all the potential problem and inconsistencies that will come with this decision or construct a theory of convergence of infinite series. If you choose the second branch you have something which starts to look like the set of real numbers and which is probably more useful for practical applications.
There's nothing weird about decimals. There's something very weird about your ideas though. Like it or not, 0.999... = 1is a fact and there's nothing you can do about it (except cry).
Chris Seib An awful lot of so called "paradoxes" stems from simple error in judgement about what you reason about, for instance the dx in an integral, how can something infinitely thin be added to something that has a length ? That's because you are taking shortcuts, before using dx you have to define a proper mesure on your space, once you have a mesure you don't need to think in term of dx being an infinitely small quantity but as a notation for the mesure of your space and you stop giving a fuck about something which was not rigorous and paradoxal in the first place. It just become doing a stupid and simple thing.
+Martin Adams "Oh man, dx isn't an infinitesimal." Of course it is infinitesimally small. Didn't you know, infinitesimals now are known to exist? Except from zero, their absolute values are smaller than any rational positive number and still not zero. They are special numbers common to non-standard analysis (and 'infinitesimal' calculus), where you also have infinite numbers, bigger than any finite number, but still of different sizes (not equal to the ∞ concept of standard mathematics).
Martin Adams That's exactly what I am telling you, are you stupid ? Can't you read a fucking sentence properly ? The entire issue is actually your incapacity to understand a sentence correctly.
Martin, ignore MisterLi, he's a born again crank. Sadly about a year ago, he was quite sane.
Too many people who refuse to believe that .9999999... = 1 even though it is largely accepted my the math community and math professionals.
I'm so glad I accidentally found this channel ♥
I feel that school and the pressure to do "well in life" limits me.I wish I could live in a cave and just think about numbers all day long but I can't let myself do that .I cant even.
0:10 . 9 . . . d e r f u l
I HATE MATH BUT I LOVE WATCHING THESE VIDEOS
YO THIS IS MATH AND YOU LIKED IT QED
U just don't like the way it is presented to u
Anil Radhakrishnan true maybe school is just presenting in the wrong that in a way that makes me really bored and makes me want to draw so maybe just if the schools and tech it in a funny and awesome way then i will remember and LIKE math
+Panda_Emperor917 try some MOOCS out they may serve you better
Or get into programming math will make a lot more sense then
thxs man
Is there even a thing like -0? Isn't it just...well...0?
Every real number has an additive inverse: x + (-x) = 0. 0 is no exception. it is only unusual in that 0 = -0
***** -0 is just the additive inverse of 0. -0 = 0
There is a thing like 0.999... It represents the number 1. But that does require a proof.
By the decree of the Institute of Electronics Engineers, there is! And it *is* equal to 0! But only sometimes.
***** Please explain. Seriously.
Chris Seib Look up IEEE floating point
I love you. I honestly love you. I fall for your april fools before i watched this video... just... thanks so much for sharing
Dear god the more I keep coming back to this video and looking at the comments I lose at least a trillion brain cells. Like I can admit possibly debating for a bit of the student's intuition(although not with the number .999...) but this...
You have seen nothing yet. Look into the comments on virtually any video about 6/2(1+2).
So... there infinity numbers between 1 and 2?...
There are aleph-0 rationals and aleph-1 reals between every pair of (non-equal) numbers.
Chris, you need to know your audience. Answer the question in a way that the person will understand. Making yourself look smart will accomplish nothing other than confusing the person you are answering to.
PotaTox, I'm sooooo sorry that facts hurt your butt. Do you only want to see ignoramuses makes comments?
Did you even watch the video? See reason 7 starting at around 7:40
Yes, I did watch the video. Yes, I agree that 0.9 repeating equals one.
Let me ask you a question: Let's say that there is something you don't know about, and you ask someone to explain it to you. Do you want that person to explain it by stating a ton of facts using vocabulary that you don't understand, like some kind of dictionary, or do you want to have the person explain it in a way that states why something is the way it is and say it in a way that makes sense to you? Do you expect a kindergartner to understand what a derivative is just because you gave them the function of a derivative?
I hope you never become a teacher.
iiVia _ also original commenter, yes, there is
Sooo... 1/3 = 0.33333... 1/3 x 3 = 1 and 0.3333... x 3 = 1? *brain dies*
+Rebecca Trudgeon +Vi Hart
We know that 1/3 = 0,33333...
We also know that 1/3 x 3 = 1 and 0.33333... x 3 = 0,9999...
From that, we can conclude: 0,9999... = 1
It only depends on the way you calculate it, if you use the fractions (a way to represent racional numbers with 2 integer numbers) you get 1. Else, if you use the decimal numbers (a racional value represented by a "broken integer" number) you get 0,9999...
We must always remember that a number is just a way to represent values. Our world is full of values that we do not choose, we can only choose the numbers that represent them.
I guess we can say that 1 is equal to 0,9999... but depending on the practical situation you need to use these numbers, we might consider them different. If you need to measure something very tiny, for example, 0,9999... might be significantly different from 1. Did you get it?
Sorry if i've made some grammar (or math) mistakes, english is not my native language. (and i'm a computer engineering student, not a mathematician).
+Rebecca Trudgeon Think of it like this:
1/9 = 0.11111...
2/9 = 0.22222...
3/9 = 1/3 = 0.33333...
4/9 = 0.44444...
5/9 = 0.55555...
6/9 = 2/3 (2 x 1/3) = 0.66666...
7/9 = 0.77777...
8/9 = 0.88888...
9/9 = 3/3 (3 x 1/3) = 1 = 0.99999...
Hope this helps a little bit.
+Gabriela de Carvalho what would you measure that is .99... long? just wondering, couldn't think of a scenario where you would find that
Sam Brownlow Please do a research on the different roles of infinte divisibility for economics, quantum mechanics, and order theory :)
.9 repeating is the limit of an infinite series, makes it pretty simple. Our number shorthand is syntactic sugar for more well defined constructs.
My phone battery is 99% and I'm offended
9:43 why can't x^y=0!
After all, 0!=1
Aeroscience I think the ! was meant to be an exclamation mark, not a factorial sign.
Whoops! You're probably right, I was kinda slow on that :P
Aeroscience LOL
KAuser 2094
you cant divide by 0 silly
выдео тупе
This is why calculus exists
This video has now been around for 9 years
Yup I used to believe 1 - 0.9̅9̅ is the infinitesimal and that 0.9̅9̅ ≠ 1 and that 1 ÷ 0 = ∞ but now I know!
yup of course 8-dimensions I predicted it would have to do with physics.
It has nothing to do with physics. I'm not sure how you came to realise that 1 - 0.999... is not an infinitesimal, but you are right about that.
No not the infinitesimal - the split octonians.
Also, I figured that 0.0̅0̅1 just isn't small enough there is still 0.0̅0̅05 and it must just be beyond our numeric system like its inverse, ∞. (is infinity the inverse of infinitesimal or is it called something else?)
I was referring to the 8-dimensions. They are abstract dimensions, not physical dimensions. Pure math makes no claims about the physical world and Vi wasn't suggesting it did either.
0.000...1 is meaningless. The 0s don't stop repeating, there is no end at which to place the 1.
Yes, and infinitesimal is 0.0̅0̅1 as infinity is 10̅0̅.0 am I right?
You didn't use the convergent sequence proof!
What is it? If it's an iterative process, that doesn't really count. It never reaches the answer, just converges TO it.
@@adamh2077 It qualifies as a convergent geometric series so you can actually evaluate it by definition as (9/10)/(1-(1/10)) = 1
I want that Venn graph-o-chart please
It got more and more complex as the video reached the end... my knowledge of existing numbers exploded... :O
I think it comes down to the fact that .9 repeating is the definition of a limit as it approaches 1 from the left.
Duh! 0.999... doesn't approach anything. It is a series that has a sum. That sum is the limit of the sequence 0.9, 0.99, 0.999, ... and that turns out to be equal to 1.
im not smart enough for this
My one and only problem with the .999 repeating = 10 is the .0000...1 that you never get to. I don't want to drop that 1 Just because I never get there doesn't mean it doesn't exsist.
The .000...1, with the 1 on the infinite digit decimal place is called an infinitesimal, and it is smaller than all finite positive real numbers and bigger than zero, however it doesn't work in the reals. When using the standard real number system, you only use one infinitesimal: zero, (and no infinitely big values are allowed as well). So all infinitesimals are equated to zero in the real numbers, and all decimal numbers (a finite sequence of digits, then a dot, and finally a sequence of digits to the right) are all defined to be real numbers. You can't have infinitely many digits to the left (due to the real number definition) or other than zero as an infinitesimal in real numbers, also because of the definition. If you want to calculate with other infinitesimals than zero or infinitely big numbers, you have to use a non-standard number system, like hyperreals or surreal numbers, but they use different kinds of notations, not the simple decimal notation the reals use).
But look, 1-.999... isn't .000...1, .999... is a real number and thus is contained in it.
But look, 1-.999... isn't .000...1, .999... is a real number and thus is contained in it.
En realidad ese ".000...1" no existe.
So when Hand Sanitizer says "removes 99.999% of germs" it is technically 100% of germs are gone which is completely and utterly false so I'm confused, then again I am horrible at math so there ya go. Also | - (-1) | = 1 isn't that amazing?!?! (no)
oh yeah and this whole video is completely and utterly confusing...
oh yeah and right now in math class I am learning sequences so everyone try and figure out this sequence:
2, 1, 2/3, 1/2, 2/5...
-figured it out in class and got me a box of sour patch kids like a boss
-my math teacher is a boss because he gave me a box of sour patch kids because i did something amazingly epically awesomely amazing, epic, and awesome and mathematical in class that no one else knew how to do, even though (like i said) i am not good at math
-finally, to wrap this up neat and tidy into a bow or whatever, i really like to ramble...
(OH REALLY???)
-I talk so much that I am starting to use sarcasm against myself in my ramblings
-What am I talking about right now
-Bye
-Oh yeah and first one to figure out the sequence I gave to you so kindly earlier wins my box of sour patch kids (not really because I already ate it, good day, sir.)
.9 repeating does not equal 1.
That's different. 99.999 isn't 99.999...
They write 99.999% because, as far as they know it kills 100% but they can't say that because in the odd chance that one lone bacteria escapes people could sue them for false advertising. So that's the gist of it.
atomheartother It's also because that brand of hand sanitizer is not effective against all species, but is effective against 99.999% of species IIRC.
Arthur Cote If there was a single type of bacteria who could resist the hand sanitizer, they would have changed the recipe, it is as far as they know effective against 100%. 99.99% is just a way to cover their own backs legally.
atomheartother A quick search found me this www.ncbi.nlm.nih.gov/pubmed/20429659 showing that alcohol based cleaners like purell are not effective against C difficile.
made 9 years ago. beautiful.
I really wonder: are the limits even taught in US schools? I was not familiar with this 0.999... notation until I moved to the US a few years ago, and I really am puzzled about the amount of "proof that .999...=1" videos I can see on youtube right now. Because either you know nothing about maths, in which case 0.999 (not even repeating) is one, or you DO know about maths, and you should know limits. So if you write H(n) = 0.999....9, which counts n "9"s, you have H(n) = 1 - (1/10)^n, and then 0.999.... = lim(n->∞,H(n)) = 1 - lim(n->∞,(1/10)^n) = 1 - 0 = 1
Am I missing something ?
Limits are taught in American schools, albeit not particularly well most of the time.
Your proof is perfectly valid, but I have to say that I think it's slightly more complex than the algebraic one which goes
x = 0.999...
10x = 9.999...
9x = 9.000...
x = 1.000... = 1
The thing with the way limits are taught in American schools is that it seems that they are not taught conceptually. Students are told that lim(x->infinity) is where the function goes towards as x goes to infinity, without explaining the actual concept of infinity from the perspective of number theory or set theory. Without a firm grasp on the concept of infinity, it's easy to make mistakes by trying to utilize the rules which one learned for finite numbers or numbers with finite decimal representations or whatever.
As so called moderator Caper has decided to delete our comments even if it is only about math and without any abuse then I will write it here:
He thought that mathematical induction can be applied to prove 0.999... != 1 by writing:
0.9 != 1
0.99 != 1
...
0.999... != 1 - from his logic.
Then if you apply same logic you should also get:
0.9 != 0.999...
0.99 != 0.999...
...
0.999... != 0.999... - FAIL.
He is so weak in arguing about math with his 5 year old brain that he prefers to delete the comments instead of writing that he was wrong.
imgur.com/zeAcExq
for all to see aswell!
*****
You're doing good job :)
Zuriah Heep Always make sure to screencap such things due to his dishonest ways
This video should be .99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999.... seconds shorter
This answered less questions than it made
10/10 would question my existence again
Will do
+Somesh Thakur for real
saying that .999... = 1 is directly implying that precision=accuracy
1123aka Saying .999... = 1 means someone understands basic fundamental facts of mathematics. There are absolutely equal as per mathematical definitions as well as dozens of other aspects of real mathematics. It is absolutely no different than saying 1/2 equals .5 are are they some how different too because they use different types of numbers? 1/2 is fractional notation, .5 is decimal notation, and .999... is infinite repeating decimal notation...
1/2 * 1 = 1/2 , .5 * 1 = .5 and .999... * 1 = 1
can you define infinity? no. then you cannot use the "=" sign. It's like saying relative is absolute and absolute is relative. No matter how "infinite" 0.99 goes,it will still be smaller than 1.
1123aka Infinity is a concept in ℝ and isn't defined as a real number.
.999... isn't infinite, it is quite a finite number as it exists between other real finite numbers such as 0 < .999... < 2. It has an infinite decimal expansion but it's value is finite.
What is the mathematical definition of a repeating decimal in the form of an infinite convergent series?
I'll give you a hint: It's .999... ≜ Σ 9/10^n, (n=1 to ∞)
and Σ 9/10^n, (n=1 to ∞) is an infinite convergent series and by the definition of an infinite convergent series Σ 9/10^n, (n=1 to ∞) =1 therefore .999... = 1
You don't get to make up your own definitions in math...you either use real mathematics or you don't... you don't.
"infinite" is not defined. concept of convergence is perfectly suited to real world applications as well as our own need to understand the universe in terms of "knowns" and "finites" . You cannot define infinity. NO matter how many expressions you throw at me,mathematics is still a child of philosophy and scientific method. If you wanna feel like you know "infinite-ness" ..in any manner or way,go ahead and close your mind. Convergence has nothing to do with absoluteness. You are good at what you were taught, but you pathetically lack the "how" of thinking. This is why we need to teach people how to think,while we feed them with mathematics of which they can never grasp the true nature
1123aka So basically you reject modern mathematics and want to teach people things which are not true based upon your limited and incorrect understanding of infinity and real mathematics? Is that basically your position?
Wait, shouldn't it be like this?
0.99999......9999=x
9.99999......9990=10x
9x=9.0000....001
x=9.0000....0001/9
Even if it is infinite numbers, we know that the "end" of 0.999... is 9 and when you multiply by 10, there should be a zero at where there would have been a 9? Or does the whole infinite = infinite + 1 deal ruin it all.
+German Eagle Infinity implies endless. There is no end, and so there is no last digit to be a 9. Your ....s necessarily imply a finite number of 9s. Food for thought - what is the last digit of the decimal representation of Pi?
Each 9 corresponds to a counting/natural number. There is no last natural number.
For clarity, there are infinitely many natural numbers, yet none of them is infinity.
Chris Seib But we know the last digit is nine, no matter how much times we do it. Pi has no logical patern that we know of, so it cannot not be compared to this.
In between those nines there are infinite 9, no it does not go against anything.
German Eagle The last digit of 0.999... does not exist, so it cannot be 9.
Your use of the phrase "no matter how much times we do it" gives the game away - it implicitly assumes a finite number.
Just for a laugh, what is the last digit of 0.1212121... where 12 keeps on repeating? Compare your answer with the last digit of 0.1212121... where 21 keeps on repeating.
What do you mean by "in between those nines"? I hope that you aren't referring to the non-existent last 9. Do you think that 0.999... really is 0.999...9?
Infinity means endless. Saying that there are infinitely many nines between two nines is meaningless. Either there is a last 9 or there isn't. Infinity is not a very large number as you seem to be treating it as - it is not a number.
Try this. The set {1} has size 1 and 1 is in the set. The set {1,2} has size 2 and 2 is in the set. What is the size of the set {1,2,3,...} consisting of all the natural numbers. Is that number in the set? (If not, then why not?) Before responding, look up aleph-0.
Chris Seib To answer your question, in a sense, I do see it as so. But before I explain, I see why this does not work. I could say that 9.999...9 is the same as 9.999... .Because there is still infinite nines in there, but for cases like 121212 or any variation, it wouldn't really work. As I said, I understand what you are saying, and see why I am wrong.
German Eagle A problem with the notation 0.999...9 is that that last 9 is at the oo + 1 decimal place. So that is treating oo as a natural number. But oo cannot be a natural number, because it would have to be the last one, so there cannot be a number after oo. The axioms state that if n is a number, that n + 1 > n yet we must have that oo + 1 = oo and so it breaks the rules. Incidentally, that's why aleph-0, the size of the set of the natural numbers, cannot be in the set of the natural numbers. Crudely speaking, aleph-0 is the smallest kind of infinity there is. The number of real numbers is larger than aleph-0 and is written aleph-1. It is common to see aleph-1 = 2^aleph-0. But this does not mean what it seems to mean. It is a statement about cardinalities.
If you don't happen to be an accomplished mathematician, stop pretending to know what you're talking about when it comes to abstract, counter-intuitive mathematical ideas, such as the equivalence between .999... and 1. The people who study this stuff for a living pretty much unanimously agree on the equivalence, and it's just frustrating whenever uninformed people act like their idea of how math should work is better than the valid, PROVEN ideas developed and discovered by actual mathematicians.
What an extreme hater
i feel like every time i encounter this question i am being asked how many lights i see
Lolol I feel so stupid looking at all the comments but we'll turn it into a positive by saying, "look how much is left to learn!"
9:53 Wikipedia here I come.
Kenneth Cheng yeah I thought the same
I think the resolution to that arrow paradox is that stuff is quantized.
No it's not. It's a math problem. The arrow etc., is just mind candy.
I love list ning to you to fall asleep, it takes me back to my high school math classes when I could still let myself drift off to math. College doesn't let me do that anymore :/ . I love you Vi, you make my life so much better.
Wouldn't it be better to say that .9999... has a limit of 1 as the number of 9s approaches infinity?
+Acre00 Nope .0.999... already has infinitely many 9s and already is the limit of the sequence 0.9, 0.99, 0.999, ... In fact that's what 0.999... is defined to be. It is obvious that 1 also is the limit of that sequence. As the limit must be unique (by definition of limit) we have that 0.999... = 1
No, that would be 0.9[n] has the limit of 1 as n goes to infinity, however 0.999.... has all those 9s so it is 1.
More precisely: 0.999... *is* the limit, and the limit is equal to 1.
is there space between 12pm and 0am? (or 24:00 and 00:00)
Well spotted. I like your thinking..
No? Why would there be? At least if we are taking about the infinitesimal moment between each day, there is no distinction for the same reasons described in the video. Real numbers which are infinitely similar are the same in our system.
+Logan Gantner. Even in the surreals and Abraham Robinson's hyperreals (both of which include infinitesimals) 0.999... = 1. I see no reason to speculate that there is even an infinitesimal gap between the days. The idea is bizarre, to boot.
No, because they mean the same thing or time.
imgur.com/CoOq2Yc
Caper fails even basic arithmetic, 1,1 isn't 1+0,1 in his world.
This is why 0,(9) must equal 1, otherwise you get contradictions
about a month ago, he couldn't multiply 2 numbers. today, he can't add 2 number. i wonder what will happen, if we wait another month :-)
somebody should help this poor creature, he shows clear symptoms of cerebral atrophy!
***** I think all that is left is the ability to compare two numbers.
He tries to "Cite" a source, and what does he do?`...youtube video of another crank who think there is a conspiracy to not see the "flaws" the crank sees.
*****
He, just already, lose the ability to remember what he said.
upic.me/show/50142054 (He say n can not equal infinity.)
upic.me/show/50142056 (He say "for "x = ∞"")
And he also just admit himself to be an idiot.
upic.me/show/50142134
"You are too much of an idiot to see what I did there, dufus. lol" (If anyone who see what he did there is an idiot, then he himself who see what he himself did there is also an idiot too.)
Araqius yepp, he is an idiot. his stuff is inconsistent so he can't keep track of it because of it.
There are 1.9999...... kinds of people in this world:
1) Those that understand algebra and aren't too stubborn to understand what all mathematicians understand.
2) Those trying to figure out how much of their body they should cut off to have 0.9999...... left.
9/n = 0.nnnn...
9/9 = 0.9999...
9/9 = 1
Proof.
you mean n/9
yep! sorry for my misleading comment.
Now prove that n/9=0.nnnnnnnnn for all n.
u can't because9 divides by 9 equal 1, fractions are just divisions
Ps: i'm french so sorry for the faults
What יותם ענבר means is that 1/9 is 0.11111, and 0.111111... x 9 = .999999. But also, 1/9 x 9 is 1. So .99999 is 1!
I'm not saying that "all numbers are equal" I'm saying that all zeroes could be unequal.
What do you have to say to that
I know someone who unironically believes that all repeating numbers equal infinity.
This seems really sketchy to me, you are using your conclusion to prove that its right, which is making up stuff, like in proof 3.999... you subract .9999... from 10 and get to 9, but that isn't a proof, as that would assume that you've estabalished that .9999... = 1, which you never do because you use that assumption in all your proofs
No she isn't she is not once using the conclusion to prove it is right.
and she isn't subtracting 0,(9) from 10
she is subtracting it from 9,(9)
9,(9)-0,(9)=9=10x-1x=9x
No assumption of the conclusion anywhere
Ghork1 Caper is a mathematical moron while Emperor teaches Mathematics ..I would listen to Emperor.
Caper- "9.(9) - 0.(9) = 9 assumes that 9 = Σ 81/10ⁿ "
Bullshit...no assumption is drawn here.
9.5 - .5 = 9
9.6 - .6 = 9
9.8888 - .8888 = 9
9.9999... 0.9999.... = 9
See the pattern? It's basic math.
Also we don't have to "prove" that .9999... = 1 as by definitions they are equal. That is just mathematical fact. There are however proofs which do show they are equal, but we wouldn't even ever need to assume they are equal as by definition they ARE EQUAL. You can't make an assumption about something when the definition is already provided. It is is like saying x = 1, now prove x + 1= 2 by assuming x = 1. Well we already know x=1 so why we we need to "assume" x =1 ?
Same thing here...in real mathematics 1=.999...
So Caper's diatribe about having to make assumptions is just plain idiotic.
***** Caper seems to be right thought, the thing is that its a agreed upon thing to make it work for certain equations, however 1 and .999 repeating are different values, one is 1, and another is approximating 1, almost the same thing but still different
Ghork1 He isn't right though.
"however 1 and .999 repeating are different values"
Nope, from definition of real numbers they are the same
" one is 1, and another is approximating 1, almost the same thing but still different"
Again, nope, they are the same, if you want I can cite sources where the definitions are and then show you how they are equal
Ghork1 its pretty much the same as the proofs used to make 1+2+3+4+5...... = -1/12 which is based around shifting positions of infinite series, like this is with infinite numbers, the thing is by shifting its making it a different finite number, just as this is with moving the decimal place once. But have a place in math for use of relative infinities calculations etc. But that doesn't make it fact, what it does make is make is make it a tool used for calculation, but like with series, and with a number infinitely close to 1 even a child could tell you that those aren't facts, however they are fine tools
fun fact:
this video is 9.999... minutes long