9.999... reasons that .999... = 1 

my math teacher spent the whole class attempting to disprove vihart

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.

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.

If only this video was 2 seconds shorter

Screw this, I'll just calculate in base 12. 1/3=0,4 3/3=1

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.

I gave you .9999999999... likes!

My favorite reason it the simplest one. .333...=1/3 X3. X3. .999...=1

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.

I saw this when it released 9 years ago. Coincidence? I think NOT!

The people that say .999... =/= 1 are the flat earthers of mathematics.

(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

so 0.999... is... 1 derful

"Mathematics is about making up rules and seeing what happens"... I think I love you :D

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

This video is exactly my cup of tea. Simple, full of delight mathematics and puns! Great job!

Every repeating decimal can be written as a fraction. For .9999... that fraction is 9/9.

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.

forget "this is statement is false" GLaDOS should of used this instead.

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 RU-vid 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.

can you believe this is the same person who counts down from random microwave numbers... vi hart is truly an amazing and wonderful person.

If you could explain 8-dimensional electron theory that would be awesome!

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?

Those Lords are spot on.

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).

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 :)

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 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!

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?

02:13 , divide by 9 factorial? oh lord

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!

I'm gonna do this on a test! I'll then show her this video when she marks my test wrong.

This video makes me so happy. THIS is the beautiful part of math.

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.

Was I the only one that saw the video was EXACTLY 10:00 minutes?

funny lord of the files joke

I never understand a single word this girl is talking about, yet I keep watching the videos....

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.

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.

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.

People complaining at 'made up rules.' ALL rules in maths are made up, they are eventually accepted as the norm.

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!

Omg I just realized Formula Fourmula Three-point-nine-repeating-mula

The way you do your videos is just great, the ending in this one is treasured!

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 ?

the video is exactly this long (in minutes): 9.999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999...(repeating)

[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.

I love you. I honestly love you. I fall for your april fools before i watched this video... just... thanks so much for sharing

[holds back tears yelling] SOME INFINITIES ARE BIGGER THAN OTHER INFINITIES

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

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!

Vi Hart, you never cease to amaze me.

Too many people who refuse to believe that .9999999... = 1 even though it is largely accepted my the math community and math professionals.

Is there even a thing like -0? Isn't it just...well...0?

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.

.9 repeating is the limit of an infinite series, makes it pretty simple. Our number shorthand is syntactic sugar for more well defined constructs.

I came in to this video wanting to deny it but you literally just blew. My. Mind!

Sooo... 1/3 = 0.33333... 1/3 x 3 = 1 and 0.3333... x 3 = 1? *brain dies*

I HATE MATH BUT I LOVE WATCHING THESE VIDEOS

Really, the fact that almost all decimals define a unique number is much more surprising to me.

9:43 why can't x^y=0! After all, 0!=1

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...

I'm so glad I accidentally found this channel ♥

I don't know if she said that in the video, but if she did I must have overlooked it(sorry!): The formula for a geometrical progression of infinite terms is a1(first term) divided by 1 minus "q"(a number that, if multiplied by a certain term on that progression, will equal to the next one). Now, 0.999... can equal to a geometric progression like 9/10, 9/100, 9/1000 and so on. The "q" for that equation would be 1/10. Applying this information to the formula equals to, surprisingly... 1.

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.

This is why calculus exists

is there space between 12pm and 0am? (or 24:00 and 00:00)

I've been using the line "suspiciously practical" for years and just today realized i must have stolen it from this video

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 two numbers can you divide by to get .999...

My phone battery is 99% and I'm offended

wasn't expecting a math video where they suddenly bust out 10 lords-a-leaping

So... there infinity numbers between 1 and 2?...

You didn't use the convergent sequence proof!

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.)

Thanks, Vi. You blew my mind again. Plus, I really like your art.

im not smart enough for this

This video should be .99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999.... seconds shorter

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!"

A dotted dotted half note is not a hole note. Because a hole note gets 4beets, a half note gets 2beets, a quarter note get one beat, and an eighth note gets have a beat a beat. A dotted note gets that notes value+1/2 of that. A dotted half note gets 3 beets, and a dotted dotted half note gets 4.5 beats (or a hole note + an eighth note). Not trying to be rude. I am a big fan, but this peaked my challenging senses. Your videos are great and help me look almost as smart as my boyfriend. :P

saying that .999... = 1 is directly implying that precision=accuracy

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.

9:53 Wikipedia here I come.

Its very satisfying to me that this video is exactly 9.999... minutes.

Yup I used to believe 1 - 0.9̅9̅ is the infinitesimal and that 0.9̅9̅ ≠ 1 and that 1 ÷ 0 = ∞ but now I know!

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

Wouldn't it be better to say that .9999... has a limit of 1 as the number of 9s approaches infinity?

when you multiply .333(not repeating) you get .3.33. The digits after the decimal point are subtracted by one if the number has at leas tone decimal point. If you multiply .999 by ten the nines would be infinitely long minus one digit, because 9.9 repeating qualifies, at the end of infinity or 9.9 repeating -9(10^∞). This says the difference between 9.9 repeating and 1 is 9(10^∞).

I want that Venn graph-o-chart please

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

9/n = 0.nnnn... 9/9 = 0.9999... 9/9 = 1 Proof.

Video: "A Mathematician's Perspective on the Divide" Is deep and so.... well though out Vihart.

I think the resolution to that arrow paradox is that stuff is quantized.

999 dislikes . . .

Point 1 , I know its a joke lol Point 2 , you said its not a reason yourself Point 3 1:23 , If the number infinitely close to 1 is not 0.9 recurring then what is it? Point 4 I don't disagree with what the "proof" seems to show but instead I think the problem is that you can't perform ordinary maths with infinite numbers in the same way you can't draw a perfect 3D shape on a 2D piece of paper, the can't be combined without some error or assumptions Point 5 supports my reasons for point 4 that you cant mix infinite numbers and the reals together Point 6 could be argued that if the 9 repeats infinitely you would never get to 1.0 put you jumped the gap to prove that but not when 0.000..001 Point 7 the argument that 0.9 recurring is not equal to 1 is that 0.9 recurring is not a real number so that point is irrelevant Point 8 the argument that 0.9 recurring is not equal to 1 is the same as 0.3 recurring is not equal to 1 third (exactly) Point 9 If you were to plot the infinite series on a graph you would get a equation of the curve with no solution where the 1 should be Point 10 it is only consistent because we treat it like it is P.S. don't hate on me I'm not trying to prove you wring I'm just trying to understand why because none of the points have really convinced me and no I haven't done hyper-reals or stuff like that yet so I can't comment on that but thanks for the video anyway.

well the sum of an infinite series in a geometric progression is : a/1-r where a= first term of progression r= common ratio 0.999999....= 0.9+ 0.09+0.009...... where the common ratio is 1/10 substitute In the formula, 0.9/(1-1/10)= 1

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

Best and simplest reason, are you ready! Look: If "x" over 9 is 0.xxxxx... then 9 over 9 is 0.99999... but 9 over 9 is 1. So than 0.99999... is equivalent to 1. Now go save some time and don't watch the video!

Vi Hart 0.999... = 1 is an angreement and not a mathematical fact.... we kinda have to agree to it, that algebra works in that case... but what you really doing here is dealing with infinite sums... 0.999... is 9/10 + 9/100 + 9/1000 and so on... and those are NEVER easy to handle... So to me 1 / (1 - 0.999...) = +∞ is more likely than 0.999... = 1

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.