9.999... really is equal to 10
2024 ж. 10 Мам.
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Is it possible to explain that 9.999... = 10 in a way that convinces 99.999...% of all the people in the audience? With the help of some clueless participants of the reality show Total Drama Island the Mathologer gives this math communication challenge his best shot.
Enjoy :)
Two friends, an engineer and a mathematician, hear that there is a damsel held in a castle's tower. They climb the steps to the tower where they are met by a guard who tells them that freeing her is simple. It takes but a kiss on her cheek. There is, however, one caveat: you can only approach her by continuously traversing half the distance, pausing, then continuing another half the distance, pausing, etc. After a few seconds the engineer begins to take his first step and the mathematcian exclaims "What are you doing!? You can never reach her by the limit set upon us!" To this the engineer responded, "no, but I can get close enough!"
Im an Engineer. I like that one.
close enough.
This blew my mind. Thank you!
Well, so the engineer understood that the trick was about the time... everyone walks or runs traversing half the distance between the start and the end, but nobody cares cause shorter is the segment and equally shorter is the amount of time needed to pass it. traduced in your case, the request doesn't bind the speed to the length the engineer is going to traverse cause there isn't a bond between the time lapse which has to be considered (example: length=2m; the engineer runs the 1st m in 1 sec, then, 1/2 m in 1/2 secs, then 1/4 m in 1/4 secs... so he moves always at the same speed which is 1m/sec), like "achilles and the tortoise".
@@antog9770 You have to pause between each stage, so his average speed gets slower and slower. I'll give you that the length of the pause wasn't defined.
Okay, new policy on this channel. Any comment with a disabled reply button will be removed :)
Mathologer I like it
+Mathologer Keep in mind that once there are a lot of replies (last I checked, it is 500), no more replies can be made.
Never knew :)
Mathologer I frequent some videos that are heavily commented on, and the highest number I've seen was 500 replies. and other comments made by this person (typically the uploader) were open to replies.
+Mathologer why are you hating on disabled buttons?
Lol I love your interpretation of reality shows "where they put a bunch of clueless people on an island and they do silly things"
kzhead.info/sun/bNuzYdyhgph8ia8/bejne.html Why this number doesn't exist?
Well... Mathologer got this one wrong. It's not 9.9 recurring to infinity. It's 9.9 recurring only to nine one hundred thousands of a millionth (or thereabouts) So yeah... he lost. It's not 10.
...because if it was really 9.9 recurring to infinity, the clock would have just rounded it to 10 anyway, just like every calculator rounds it to 10
It might be true that there were a few more extra 9s at the end there which the clock couldn't display because it ran out of digits on its display... however we can reasonably assume that the 9s don't recur to infinity, because 9.9 recurring to infinity is the SAME exact number as 10. So why did the clock bother displaying all those 9s in the first place? Whyyyyyyy didn't it just show 10? It would be same number and it would have therefore rounded it up to 10, if it was indeed that number Except it wasn't. I'd imagine the real number was more like 9.9999999999999999999999999123. Lol
Note that this all comes down to the accuracy of the measuring device. How small of a time interval the clock can measure. In practical terms, there's a limit to how accurate a measuring device can be. Let's say the clock can only measure to that many decimal places. Well then 9.9 recurring to that many decimal places would look EXACTLY THE SAME to the clock as 10 would. There would be NO distinction the clock could make between the two numbers, even if the number doesn't acually recur to infinity. So the clock would still display it as 10, because that's what it looks like to it. So I'm assuming the clock to be at least one more decimal place accurate than the number of digits it can display, because the event didn't look to it like it took exactly 10 seconds. The measurement device saw a small discrepancy there. That's why it displayed it like that. The only way mathologer could be correct is if the clock is completely unable to simply display whole numbers like "10" due to the way it interprets its measurement results OR if it just always displays whole numbers in this way by convention OR if it sometimes displays it this way and sometimes the simple way, and we just happened to see an occasion when it displayed it the recurring way. Lol ...but none of these possibilities are very likely, and no electronic device I've come across works like that. So we can safely assume mathologer to be incorrect about this 😁😂
Even though this is an old video, I'm leaving behind this comment. So, when I first learnt this I did not want to believe this. I saw the reasonings, still my gut feeling of these two being different didn't go away. Something felt odd... why does 9 do it but not other numbers? The thing that finally made everything clear was realizing this was just an artifact of us using decimal number representation. When you cjange your representation system, there's a good chance something that wasn't repeating, becomes repeating in another representation. For example, if you try to write 0.2 in binary, you would get a repeating number: 0.0011 0011 0011... And something that has infinite representation in decimal, might have finite representation in another format. For example, 0.3333.... in decimal is just 0.1 in trinary number system. So the "number" itself is different from its "representation". And it's entirely possible that may be a number has 2 different representation in a system. And 9.999... & 10 happens to be 2 differemt representation of the same number. We want math to be independent of representation. Otherwise it won't be universal. So this is basically just an artifact of decimal representation. This also helped me understand a huge difference between rational and irrational number. I used to think why lump repeating decimals with rational numbers? I mean irrational numbers doesn't end, neither does repeating numbers. Why not lump with irrational numbers? My approach to answer this would be that any repeating rational numbers end if you take a suitable numnber as base. So in that base representation it would definitely be a rational number. On the other hand, there is no base representation where an irrational number would abruptly stop or ever repeat same pattern. So if you lump repeating numbers with irrational numbers it would be very representation based and not universal. By the way, of course this was inevitable because this is synonymous to rational numbers being represented by a fraction of integers. But a slightly different way to look at the same thing.
I was kind of hoping Mathologer would bring this up in the video. Once a person understands how numeric bases work, I think it becomes much easier for them to accept the fact that every number with a terminating representation in a given base also has a non-terminating representation in that base. The "math in slow motion" segment hints at this, but I think it really helps to dig into the sort of idea you mentioned with rational numbers. I had a similar "don't want to believe" moment when I first started to work with continuous probability distributions: I was really put off by the idea that the probability of the random variable to take any single, specific value is 0. I thought, "How could it be 0? If that's true, then you could never get any result at all!" I wanted it to somehow be expressed as different from 0, but at the same time I had to admit that a different expression wouldn't actually have different properties. As someone who isn't deeply involved in math, it's unsettling to come across the infinite and infinitesimals when you aren't really expecting them. Another way to look at both situations, perhaps, is simply: In what contexts are the two representations meaningfully different? The answer is that the representation itself is the only such context, but that's a very important thing for most people, which I think is what makes it so difficult to let go. After all, if our calculators and computers started printing out infinite sequences instead of terminating ones whenever they could, I think we'd all be a bit frustrated with them.
It's not trivial to be intelligent. I've known a lot of fellow students in many advanced classes who simply don't "get" things which are trivially easy to understand by the brightest students. However, each one has his limits, and there are certain sticking points for everyone. We'd like to think that the brightest mathematician can balance his checkbook and win the lottery and solve the national debt, etc. But they usually have their limits. For 0.999... and other similar concepts, a bright student gets to run the shortcut inside his brain, and then he can explain it 2 or 3 ways to his peers, hoping it will click for them. The genius gets to see around a puzzle like this and shortcut around the sticky points. When I took certain standardized tests like the SAT, I noticed quite soon that the test writers were fishing for this brilliance in a handful of questions. They separated the men from the boys, as it were. The questions required 3D construction of geometric shapes, manipulating them under pressure, and needed a lot of hard thinking about symmetry and visualizing novel constructions. Your 110, 120 IQ folks just can't typically do this, and if they get the right answer on one of these, the test results will "know" that it was a guess based on the rest of their test answers. Such a person would need to also guess right on other questions for the test to score them higher. And if someone aced all the hard questions but flunked the easy ones, the score would be low or disallowed due to statistical uncertainty to stamp out perceived (or actual) fraud.
we use the decimal system because we have 10 fingers right?
@@AllAmericanGuyExpert I don't what relevance your comment has with my topic, but just classify as genius and non-genius is never good enough. If that were the case, only prodigies would rule the world. But in reality a lot of prodigies just phase out. But it's not like genetics is not a factor at all, it is. But it's not the only factor as well. If your logic was only geniuses will understand geniuses, and others will never understand them, then I'm not in your favor. People mostly don't understand topics because they didn't have a better teacher. If you want to teach someone, you have to try to break it down in his language, how he interprets things. The problem is most teachers can't see/articulate the same thing in different perspective. That's where students eager to learn struggle. I can garantee you, if we had better teachers, we will have better students. If genius was the only factor, you would never need a good teacher. So I say, yes, try to make it click for others and don't sigh because they don't click
@@midas01tw Not necessarily. It's more like historically algebra was invented in eastern part of the world where they used 10 digits. Back then, In different parts of the world different number of digits were used. Some used 12 digits, some 6, some 20 etc. And before that, roman numbers were used which is... you know... gets messy. It just happens to be that most of the modern algebraic mathematics comes from the eastern part of the world and they used 10 digits. But, then again, may be they used 10 digits because it was easier to keep track of using fingers, who knows? Tom Scott has a video on this topic. You could check it out
im supposed to be proving that 9.999 = 10, but this one mathematician keeps kicking my ass.
9.999 = 10 - 0.001 < 9.999... = 10 though
Tell him he doesn't know what he's talking about. Show him this video of a 100% CERTIFIED PROFESSOR OF COMPLEX DIFFERENTIALS AND MATHEMATICS (check him out on wikipedia to make sure its not some random old youtuber) telling you that 10 = 9.999... and 1 = 0.9999.... And then show them that math is this wonderful magic they never saw. Watch them weep in your power to understanding this that they so lack. Those fools. In all seriousness though this is totally real. There's paper's about it.
is it the Mathematician27?
I heard that TheMathematician27 once divided a number by 0
I heard that TheMathematician27 once solved a calculus equation in a single line.
plot twist, the number stops right where the clock does
Not necessarily.
I remember in the original TV show airing the clock actually did go off the screen but it was taken down and changed when they later found out that 9.99 repeating was equal to 10.
plot twist the number was 9.9999999999999999999999995278
He is trying to make it interesting, and takes a tv show as an example.
Eleanor Drapeaux 😂
9.99999999 DOES NOT = 10 9.999... Does.
almost got me :P
@@EmperorZelos me too :)
Agree, the decimals nines musti infinity of number
9.(9) as parenthesis mean periodic decimal, "infinite repetition" of that digit
OmnissiahZelos i have argued about this in the yt comments before
Wouldn’t nine and nine one hundred thousands of a millionth mean 9+9*(100000/0.000001) which equals 900000000009?
No. The word "of" implies multiplication. For example, 2/3 of 5 is (2/3)*(5) = 10/3
@@williamwilliam4944 Yes. So, you multiply by a millionth; which is the same, as dividing by million. So, William Li is correct.
Yes, it would. In fact, Mathologer pointed this out, toward the beginning.
@@PC_Simo no, he isn't correct. He did not divide by a million. He divided by a _millionth_ .
@@williamwilliam4944 Oh, right! I must have misread his comment. Yay, for sleep deprivation! 😅
Missed an opportunity to call your mystery number X instead of M. Already the ancient Romans knew that X=10!
:)
X=10! X=1*2*3*4*5*6*7*8*9*10 😂😅
@@Mathologer 3 years later and still answering comments? now that's some dedication.
Ancients Romans new M = 1000.
@Gary H, because nothing can be more equal than infinitely close values.
I'm amazed that mathologer went through a whole video on recurring numbers and didn't use the word "recurring" once!
Tycho Photiou also non-terminating
He's German he prefers saying "nein nein nein nein"
Cinema Rat lol
@@csicee wow ! You played with my brain for a while
I though about this as a child in school. I must have been like 6 or 7. My thought though was if 1/9 is .1111... then 9/9 should be .9999... and i asked my teacher if .999... was equal to 1 and they said it was a good question and i have never had an answer. Thank you for answering my question 22 years later.
i hope you turned out allright. '-)
you are comparing two different worlds, thats not working :P but if you thought about such stuff in so young ages - respect fo' dat! yo :D
@@jackcarpenters3759 underrated
wait, you didnt learn that in school
I always get it like this: What about ⅓? It can be written as 0.3333(3) multiply by 3, and get 0.9999(9). Wait, you multiplied a third by three... Oh, that's a one...
Honestly this is one of the coolest things in math I’ve seen in a while and you present it so well thanks
Please guys, don't suppress the reply button when you post comments. How am I or anybody else supposed to get back to you. Have to admit that I always feel tempted to just delete comments like that but I have resisted the temptation so far. Anyway, here is one of these comments that was posted a couple of hours ago: "I agree that 9.9999999999... is 10. You can also say that 3.333... is equal to 3 1/3. 2 x 3 1/3 is 6 2/3. And finally, 3 1/3 x 3 is 10. But one thing I'm skeptical about, it has nothing to do with 9.9999... equaling 10, and it's this; how do you know that the 9.999999999999... on Total Drama Island is infinite? For all we know, it could be just 9.99999999999999999999999999999999999999999999999999 and that's it." Here is my reply, which obviously has a chance close to 1-0.999... to reach the person who posted this comment: You are absolutely right, following the last 9 on the screen could really be anything. I just use fun clips like this and conjured up unusual scenarios like the infinitely many 9s as hooks to get people interested in what I really want to talk about. No deeper meaning intended as far as I am concerned. I am just having a bit of fun to get things rolling :)
+jqbtube What you mean is that there are no infinitessimal real numbers. There are of course number systems, such as the surreal numbers, which have infinitessimal elements, so that if e is infinitessimal, then e0. We can then say that 9.999...=10-e < 10. Of course, we lose a lot of the applicability of the real numbers when we extend them to the surreals, but it's still a valid set, so you shouldn't be so cut and dry about there being no infinitessimals.
+Alexander Jones Yes, there is the possibility to allow an infinitesimal number ε which is smaller than every number. But 9.999… would still be equal to 10 and not 10-ε.
conepictures Suppose there exists a number e such that for any positive real number x, 0 < e < x. Now consider the sequence a_0=9, a_1=9.9, a_2=9.99, etc. Any sequence which converges does so to a limit L such that L
+Alexander Jones "Suppose there exists a number e [...]" Well this ε isn't really a number. It is more like a concept which tries to incorporate the concept of infinitesimal calculus into the field (ℝ,+,*) of real numbers, which are then called "surreal numbers" - which is kind of high mathematics. Even most mathematicians don't bother to do that, since calculating with real numbers and than looking at the limit yields the same practical results and is less error-prone. But have a look at the wikipedia article, it is a fascinating concept. But lets look at your proof anyway: Your sequence has now the nice form a_n = Σ_k=0,...,n 9*10^-k With each a_n < 10. This is of course correct. Unfortunately, while in ℝ there always is a least upper bound (often that is how they are defined) this is not the case in the surreal numbers. That is kind of the point of those and how they are constructed. For example we have for each a_n: 10 > 10-ε > 10-2ε > 10-3ε > 10-ε² > 10-2ε² > 10-3ε²-2ε > ... > a_n So you won't find an upper pound anymore. I'm not even sure you can take a limit in the surreal numbers at all.
conepictures Just take S to be any upper bound, and we still have that a_n S then there's a contridiction in the definition of a limit since there is a real number between L and S and so there is a neighborhood of L which contains no points in a_n, so it's not a limit point.
Reading some of the comments I thought that maybe it is worth spelling this out in BOLD LETTERS: There is really no discussion as to whether what I say in this video is correct or not. In my day job I am a math prof at university and I assure you that everything I am saying here is what EVERY professional mathematician knows to be true: 10=9.999.... Also please watch the WHOLE video before you post any comments :)
AcoAegis No, 10 = 9.999... EVERY mathematician, EVERY physicist, EVERYBODY who knows anything about numbers at a deeper level agrees on this, there is simply no discussion. There is a precise mathematical definition of what exactly 9.999... means (the sum of an infinite geometric series) and any meaningful discussion of 9.9999... needs to be conducted in terms of this definition. Just because 9.9999... superficially looks like 9.9999 suggests that both may behave similarly in some respects but it does not prove that they have to behave the same in every respect. You simply cannot argue that something has to be the case for 9.9999... just because it is true for 9.9999. If you don't want to take my word for it, please look it up in ANY real analysis book ever written. Anyway, here is another argument that sometimes works for people who have trouble accepting that 10 = 9.999...: 10-9= 1 10-9.9=.1 10-9.99=.01 etc. This implies that whatever number 10-9.999... is, it has to be smaller than any positive number, that is, it has to be 0. In turn since this difference is not negative this means that 10=9.999... In other words, if 10 was not 9.999... there would be a number that is smaller than every positive number but larger than 0. At least in the standard model of the real numbers such a number does not exist. Also, if 10 was not equal to 9.999..., all mathematicians would be idiots.
There are no infinitesimal numbers in the Real number system, this is known as the Archimedes Principle. Since there are no infinitesimal numbers between 9.99999... and 10, the difference between them is zero. Hence they really are equal.
bpdav1 That is correct. I am trying avoid mathematical terminology that only experts will be familiar to keep things as accessible as possible both in the video and in my comments. As I said in the video some members of my audience are only little kids :)
Yeah fair enough, maybe a better way to put it is if there was a number between 9.9999... and 10, it would be infinitely small. Since no infinitely small numbers exist, there are no numbers between 9.9999.... And 10, hence they are equal.
Mathologer It's easier to write 0.999.... as a fraction - 0.999...=9/9=1
The people saying it isn’t true are basically math’s flat earthers
I promise you that some are far worse than that.
flatearth sounds easy and "normal" against all this stuff :D
The claim is true, the proof is not. mCoding has a good video on it.
@@unknown3158 as lord_ne said on that video "The proofs are all correct, the justifications are simply left as an excercise to the reader"
That's not true. It's called "finitism" and it's a legitimate form of mathematics that excludes any form of infinity considered to be incoherent, non mathematical objects. Any mathematician is a finitist if you ask them if "infinity" is a number. Flat erthers are just idiots.
Mathematicians: "Does 09.99999999999999999... = 10?" Scientists "9.99 = 10 +/- 0.01. That's already within 0.1% error. Close enough!"
There is no close enough about it. 9.999 . . . = 10 precisely. Scientists know that for a fact. Unlike you they learn math at university.
@@Chris_5318 Dude, it was a joke.
@@mstech-gamingandmore1827 Dude, It was a bad bad joke.
@@Chris_5318 doesn't mean you have to act like a smartass about it, let the dude enjoy have his fun youtube comment sections
@@gdtrilogy9060 You speak from ignorance.
Imagine watching sponge bob with this guy
lmfao
lmfao
lmfao
what
Lmfao
The hardest part about this is the realization that there is no irrational number between 9.99repeat and 10. Just chew on that.
Yeah well they are equal so this isnt such a tough realisation
I’m totally disagree: 10 is not equal to 9.9999...and it’s so because 10 is rational number (even integer) meanwhile 9.999999... is irrational number! Really, you can not express 9.9999... by fraction of two integers.
@@stanbondarev9256 except 9.99999... IS a rational number. all irrational numbers when expressed as decimals neither terminate nor repeat. and 9.999999... clearly has a repeating "9". As for what it is as a ratio of two integers: the general rule of expressing a repeating decimal as a ratio involves taking the repeating digits, and dividing them by the same number of "9"s. so for example, 0.555555... would be 5/9. 0.282828.... would be 28/99. 0.147147147147... would be 147/999. and so on. if you were to take that same logic with 0.999999... you'd get 9/9 which is 1 (and thus 9.99999... would be 9 + 9/9 which is 10). if you can't accept that, you either need to show a different ratio for all these repeating decimals, or you need to prove ALL repeating decimals aren't rational. good luck with either
@@stanbondarev9256 9.999...=x 99.999...=10x 9x=90 x=10
morfowt not all infinite decimals are irrational but with repeating 9. Look: there isn’t “infinite decimals” in definition of rational numbers, but only a «fraction of two integers». And you can’t express 9.999(9) as a such fraction, you need to put irrational number in numerator or denominator. And I try to describe that number in the next comments.
This guy is a trip love that lil evil laugh lol. Keep the videos coming
I’m digging your math videos, which break things down nicely!
I timed the clip you showed and it actually come out to 11.1045 seconds, so even if Chris would still be cheating whether he believes that 9.99999... = 10 or not
I am still trying to get to the bottom of why so many people feel so strongly that 9.999 should not be equal to 10, contrary to what I and the rest of mathematics says. While replying to somebody just a minute ago it occurred to me maybe the explanation is that there seems to be a fairly widespread misunderstanding of what 9.999... really stands for. Maybe the following helps to set things right. At some point in time, a long time ago, mathematicians defined and made sense of what exactly 9.999... means, we "own" the definition of 9.999.... The expression 9.999... is nothing god given, we defined it. Whatever anybody thinks 9.999... should stands for is irrelevant, what counts is how its creators, the mathematicians, defined it. Many people seem to be under the impression that 9.999... stands for an infinite neverending process of adding up more and more terms (sort of the expression growing with more 9s being added at the end all the time). Such a process can obviously never reach 10, I completely agree with that. However, this is NOT what 9.999... stands for, this is not how we defined it. 9.999... stands for a number that is the COMPLETE sum of all the infinitely many terms. This number is greater than any of the partial sums of the infinite process (9, 9.9, 9.99, etc.) but cannot be greater than 10. Therefore it has to be equal to 10. There is a lot more to all this. In fact, it takes a fair amount of work to lay the proper foundations for the real numbers, but that's the gist of it. The minute part of the whole theory of real numbers that I am trying to capture in the video is to show how we prove the following: IF (the complete infinite sum) 9.999... is a number, THEN this number is equal to 10. Anybody who wants to really understand how infinite sums, real numbers, limits, etc. work has to bite the bullet and hit the maths books.
Mathologer I buy this stuff, but I just wonder, who taught a calculator how to use infinity?
***** I did :)
Mathologer I think you might enjoy checking out Norman J. Wildberger's channel here on KZhead. From what I understand, he's a mathematician who argues that real analysis fails to properly lay the foundations of the real numbers. I don't necessarily agree with everything he claims (based on the little bit of real analysis I know), but he raises some interesting points. Also, I'm not sure what he would say about the 9.999... issue. Perhaps even the most "heretical" mathematicians out there are in agreement on that one. Oh, and p.s. I love your videos :)
Mathologer Perhaps there is a reason why Mathematicians are not believed (not all the time anyway). Mathematics isn't really a science even though the discipline is inseparable from all scientific fields. Mathematics is an abstract construction of the human mind. Even with something as rudimentary as Numbers, there are at least 3 philosophical viewpoints with respect to the existence of numbers (Nominalism, Fictionalism and Platonism). Mathematicians are the Philosophical Priests selectively exploited by Scientists and Engineers....
globe0698J Mathematics would still be in the dark ages without all those (sane) heretics who dared to do the "imaginary", the "complex", the "infinite"... I've hear of Norman Wildberger but don't know any details. So, you really think his stuff is worth a look?
I actually figured out thus myself one time. I was converting decimals to fractions, and you can figure out pretty easily that you get weird stuff when you try to convert 9/9.
Exactly, i wish someone could explain this to all the people in the comment section who don't understand the recurring digit rule
This guy makes me hate every math teacher I ever had. Great content, thank you for all of your work!
I guess I should repost this message every once in a while to make sure that the comments by some of the two or three confused people hogging this comment section don't take over: Let's spell this out again in BOLD LETTERS: There is really no discussion as to whether what I say in this video is correct or not. In my day job I am a math prof at university and I assure you that everything I am saying here is what EVERY professional mathematician knows to be true: 10 and 9.999.... are simply two different ways of expressing the same real number in decimal notation. In other words 10 = 9.999... If you've got any doubts that's perfectly understandable, anything to do with infinity is tricky. In any case, if you've got any doubts please relax and watch the video again to the end, keeping an open mind all the way through.
I'll just put this here. If you divide two numbers you will get the ratio between those numbers, like 3/2=1.5 states that 3 is 1.5x the size of 2 or 50% bigger. 1.0... / 1.0... = 1.0... Same 0.9... / 0.9... = 1.0... Same 1.0... / 0.9... = 0.9... Opps Zero defines the center of the number line and is not positive or negative. Positive(+) numbers are larger then zero. Negative(-) numbers are smaller then zero. Lets Subtract 1.0... And 0.9... To see if they stay at zero. 1.0...-0.9...= 0.0... Maybe 0.9...-1.0...= -0.0... Opps Using the long form, divide 1 into 6 parts, this is the same as dividing 1 into 2 parts and then into 3 parts. You should get 0.16666... Now take that number and start multiplying by 6 and you will see that wherever you start, the first number produced is 6 followed by 9s up to the decimal point. This shows the information that is lost when converting back because we are taught that the 6 is not there because the 9s stretch to infinity. But a computer will keep up with a 7 in 1/6=0.166...7 to keep track of this lose. 1/6 = (1/2/3) = 0.1666... 0.1666... * 6 = 0.999...6 And finally, I can't give you a number between 1.0... And 0.9... but the equation is very simple ((1.0... + 0.9...) / 2.0...) And when someone says there is no number between them, well first off I would hope our infinite number system wasn't so finite.
And even if there was not, that does not make them the same number, if you take just the whole numbers 6 and 7, there is no whole number between them but that doesn't make them the same. Im sure I could come up with more. These are just a few (I will say) questionable situations that show 1.000... And 0.999... As separate identities that I have come up with while debating this with others. I don't call them proofs because thats a bold word and new information can always come available to modify my understanding of how (math, science, physics, ext...) works. I do have a problem with mathematicians using the word proof when they shove this in your face... M = 0.999... M10 = 1.999... = M*10 M9 = M10-M = 9 M9/9 = M = 1 Nice trick M=0.999... M3=M+3=3.999... MX=M3-M=3 What happened to all my nines? Oh yeah smh. You say your a professor at a university, ok then I would like you to attempt to convince me that 0.999... = 1.000... exactly. Not because EVERY mathematician in the world says so. Thats only going to work on someone how can't think for themselves. Just because we are not "mathematicians" doesn't mean we are just going to blindly follow everyone without testing it for ourselves. if you are really a professor than this should be an easy task. I asked you for "proofs" and besides the one in the video, all I have read is "every mathematician believes it" last I checked, that doesn't prove anything. Thanks for your time and I hope to get an intelligent reply from you soon.
Jason Walker Jason, I am afraid you are very confused, you clearly do not know what you are talking about and you are unable to learn from anything I or others have said to you. Very sad really because you are so passionate about all this very important mathematics which is such a rare thing. I think it would be a waste of my time to engage with you any further, especially since I can make a real difference for so many other people in this comment section.
Mathologer What you and the others have said? Other then defend your position that they are equal, no one has put forth any actual proof. I have looked and besides taking advantage of subtracting to erase the recurrences, I can find any real evidence that they are the same. You want to define them as the same number then fine, but don't argue when someone calls you out because you insist they are exactly equal.
Jason Walker This is a video aimed at a general audience including little kids. There is really no point whatsoever in discussing the subtleties of the theory of real numbers in this comments section. Also, I don't feel any need to prove anything to you much beyond the scope of this video, as anything more detailed has already been said very eloquently in zillions of maths books. If you are looking for proofs look there. They convince me and all my colleagues. Maybe they'd convince you, too, if you just approached the whole thing with an open mind. Anyway, over and out from me, second semester is about to start here in Australia and as usual a lot of students need to be convinced that 1 really is equal to 0.999...
This is why most people might be skeptical : 1 - 0.9 = 0.1 1 - 0.99 = 0.01 1 - 0.999 = 0.001 1 - 0.9999999... = 0.00000000...1 The thing is the last line doesn't follow since if the 9s go on FOREVER then the zeros go on forever and therefore that "1" never appears!
OTOH 0.999... - 0.9 = 0.0999... 0.999... - 0.99 = 0.00999... 0.999... - 0.999 = 0.000999... So maybe they should also conclude that 0.999... - 0.999... = 0.000...999... or something equally ridiculous. Some people actually do "think" that's OK (Anon Wibble knows who I mean).
@@Chris_5318 Hey you scumbag still 0.999... is not 1. And your IQ is still in the two digit zone. Cheers!
@@Deibler666 You mean my IQ is two digits more than yours.
@@Chris_5318 That has no sense. You've deteriorated.
@Tycho that "1" that never appears, that's exactly why 0.999... cannot be equal to 1.
My argument to that, when I needed to convince a friend was (I' m sure someone beat me to it, but, here it is): If 10 is different from 9.9... then x=(10+9.9...)/2 must be between them, i.e. less than 10 and more than 9.9... If we must represent this number in decimal, then it cannot have an integer part of 10 (x is less than 10 and anything with integer part of 10 is not less than 10). It cannot have an integer part less than 9 because it must be greater than 9.9... and anything with integer part of 8 or less is less than 9.9... So its integer part is 9. Its 1st decimal point digit cannot be less than 9 (otherwise it would not be greater than 9.9...) so it must be 9 (there is no greater digit than 9). And so for every other point. So x cannot differ, in its decimal representation, from 9.9... in any digit. so x=9.9... but if 10 and 9.9... are different then x must be greater than 9.9... That did it, at least for my friend.
It si a very cute and accurate argument.
@@EmperorZelos Thank you
Nice use of proof by contradiction.
@@hanztimbreza6217 Thanx
nice one
1 is a very good approximation for 0.999..., this may have something to do with the fact that 1 equals 0.999...
This is 0.9999..derful
RobarthVideo I don't know why you don't have 999.999... thumbs up.
NoriMori Thanks of saying so. I am sure you'll be able to appreciate this: www.qedcat.com/total_drama_number.jpg
RobarthVideo 34.9999th like
Wait a second. Just extrapolating from this, 3/3rds would logically be both 0.99999999... and 1? HAHA, TAKE THAT MR ROBINSON IN 8TH GRADE! I TOLD YOU I WASN'T CRAZY
@xxGodx thanks for being such a buzzkill lol
To be more accurate, 1 and 0,999... represent the same number
well a comma is not very accurate
you better check your work again 1/3 is not 0.3
Yes, that's the arithmetic proof for 0.999... = 1. You can prove it with every branch of math.
Most helpful Home Depot employee ever!
The other way of seeing, like my high-school math teacher told me, is to see that there are no numbers in-between the two. However, it is my contention that this is a malformed equation (1=0.999...) because of a domain error. You're crossing two domains of number (integer and reals) and using an equal sign. This is philosophically unsound. Consider this equation 1.0=0.9999.... Are you as happy to accept this? Also, just for fun: What's the next number after pi? But if you were to ask: What's the next number after 0.999..., it would need to be 1.0? So it cannot after and equal, yes?
There is a number in between which is 0.0000000......1
@@user-vu4hr4fk3t you can not put something after Infinity
@@katokianimation I honestly don't understand why this idea of infinity, nobody has ever added to infinity, you always add up to a certain number of terms. A sum is always a finity numbers of terms. And in this case the difference between X and 10 will be 1/10 ^ n n being how far you went with your decimal terms. for me "..." is a lie, cause no one has ever done it, in fact it is more serious, no one can ever do that.
@@katokianimation a sum is by definition something that must have an output, therefore a finite process. Therefore there's no infinity sum of terms. Seems like I'm joking but I honestly think this way at this point and can't see otherwise. Maybe there's an explanation why to think of these sums as inifinity and not only going till a very big number and stop somewhere.
@@luiz_46487 have you heard of infinite sums?
Here's an old math joke: can four nines equal one hundred? Yes it surely can: 99 + 9/9!
I thought this had a factorial for a few seconds
9 factorial, what? XD
@@EpicMathTime same but also... wait what are you doing here? Such a specific video, I'm really only here for the comments and I would guess you are too?
or 99,99...
@@benjaminojeda8094 9 factorial = 9! 9! = 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 362 880
This guy's laugh is amazing
WHY for the fuck's sake every time this topic is touched there are armies of pseudo-intellectuals who go out of their way to disprove something that every mathematican on Earth agrees on?
I'm sure that psychologists will be able to explain it. I suspect that it might have something to do with half the population having less than the average IQ.
@@Chris_5318 obtuse people who suffer from Cognitive Dissonance and the Dunning Kruger effect
I never gave that much thought. I just accepted that 9.9999... was infinitely close to 10 and so the dif was negligible. I am now convinced. Thank you.
Funnily enough, logically the very expression "infinitely close" also gives it away... after all, the only you be infinitely close to something, is to be the same thing.😂
Actually you were much more right than Mathologer was in his video. I made my own contribution about this now. Please have a look at it. Maybe I can "convince you back"?
Can we talk about his hilarious giggling pls?😂
Hepi geff
Klaus Dieter finally someone who wrote it xD
No, let’s not. I used to have a good friend from Germany, totally butch, totally straight, and he used to giggle like this too.
If you say pls because it is shorter than please, I will say no because it is shorter than yes.
I was giggling like that after I read and understood the thumbnail
Ah, I finally get it. Explaining it as an infinite sum was exactly what I needed! Much thanks. (four years after the video was posted lol)
I love this channel so much like correcting everything
I had a good friend who was poet and did not know much of math. This came up or actually the 1 = 0.9 9 recurring, and I said to him that it's true or math is broken. He said that that's not enough and I asked does he want the kindergarden proof or the graduate student proof and he wanted both. We had nice evening with lots of beer and mathematics. Next day he phoned me and thanked.
Ну на самом деле 0.(9) и 1.3 это различные объекты. Под 0.(9) автор видео понимает бесконечный ряд, а 1.3 это число записанное в десятичной системе счисления. И если все строго выводить, делать соответствие число - десятичная запись, то у 10 будет только одна запись 10. Для ряда 9.(9) не будет в соответствие числа.
Man, I never understood why people have such a problem with this. Look, if 2 numbers are different then you should be able to find a 3rd number inbetween them. If you can't, they are the same number. So, if you think 10 and 9.99999999.........are different numbers, please name the number that is inbetween them.
thats not named the strong mathematical proof for most of us its easy to understand why pi is irrational and any other number when divided-multiplied or whatever-does something with pi is irrational too, but we need a strong proof that will, perhaps, show some universe secrets to us
+Алексей Перкутов What? Universe secrets? You clearly know what you are talking about.
Алексей Перкутов Either you didn't understand my post, or I am not understanding your post, but what you said has nothing to do with what I said.
With all due respect, but that is BS as a proof. You are assuming the answer. I agree there are no numbers between 10 an 9.999..., but that is not a proof that they are the same. You made the job a lot more difficult by now having to prove that numbers cannot be adjacent. I know this is the case, but me knowing this is not a proof..
isn't 0.000...1 in between it. It may extremely small but it's not 0
Loving these videos - keep up the good work!
10TurtleStack Glad you do and thank you very much for saying so.
I just thought about how can you cut a physical object into three equal pieces, by percentages how many would each part be? It made no sense to me so I googled it and someone on reddit thought the same thing as me two years ago and in the comment section there someone linked this video. It helped a lot that’s fascinating! Thank you(:
Hi @Mathologger, I've always wondered if there is a good way to convince myself if the two infinities (before and after multiplying by 10) are actually the same 'size'. One reason I struggle is because there are infinite sets which can be greater than other infinite sets (e.g.: all points on a line infinity > infinity of natural numbers)
The immediate answer to your question it the bit where he does it with the numbers written as fractions, as the term of the sums perfectly correspond to each other. However, there are one more answer: it's through the definition of the infinite sum through limits. It's that part where he promises that it works. I'm not sure I can expand it further on a KZhead comment. Basically, he (legally) approaches the infinite process by a finite one where all steps are legal. It's the whole point (and the whole beauty) of Analysis.
To be honest, the reason people aren't convinced of this is that the crux of the issue is a mathematical subtlety most people will never encounter in their lives - the notion that notation is not inherently meaningful (in a rigorous sense) until we *define what it means.* Most people go about using repeating decimal notation without ever having given it a formal treatment. This can obviously "work" for a lot of tasks, but it leaves one totally unequipped to even begin to tackle the problem in question, since one cannot in that situation say clearly what a repeating decimal means! Simple algebraic arguments like the first one presented in this video can be helpful in explaining why the notion that .999...=1 is useful and consistent with the rest of our mathematics, but it's very hard to shake the nagging feeling that they fall short of fully justifying that it well and truly *is* that way; it leaves people with some nagging ontological doubt that is very hard to dispel. And this is for good reason - they *don't* justify that it well and truly "is" that way, because there's no "true definition" floating around in abstract platonic space that makes it so; it's "true" because we define our notation to make it true, and we define our notation to make it true because, per the simple algebraic arguments, that is the most useful/consistent way to define our notation. To a non-mathematician, this is an at-best unsatisfying and at-worst incomprehensible idea. It *feels* like ".999..." is an inherently-meaningful and well-defined expression; the notion that this is *at base* just a meaningless collection of symbols and that we get to *choose* what it means requires a paradigmatic shift in the way one views mathematics. Of course, once one makes this leap, the problem becomes a triviality; it remains clear that the obvious and natural definition to use is the one satisfying ".999...=1", and there is no more nagging ontological worry because the ontological question of "what is it *really*" dissolves completely.
This is an excellent analysis. Thanks for sharing.
A lot of people know roman numerals and are therefore 'aware' that a number can be written in different ways.
That's a long way away from realizing that asking "what does .999... equal?" without having first decided on a definition for infinite decimals is an ill-posed question. It's not a matter of "a number can be written in different ways," it's a matter of "this collection of symbols doesn't mean anything by itself; we have to agree on how to interpret it before we can ask questions about it."
That is indeed a very long way but I still think that it can open the mind a bit.
EXACTLY! The problem is not the students questioning 0.9... being equal to 1.0..., the problem is rather not questioning what "..." is supposed to mean at all. Every shorthand explanation you find in the comments is not only wrong, but utterly meaningless. Once one establishes what infinite decimals are actually supposed to mean, no explanation is necessary anymore and 0.9... = 1.0... becomes as natural as 1/2 = 2/4. The problem is that "convergence" or "equivalence classes" don't enter the discussion. I'm with the students rejecting the lazy off-hand explanations. Those are less than wrong.
The root of this whole problem is convincing people that infinity -1 = infinity.
Amir Abudubai But...isn't that obvious?
Amir Abudubai which is actually untrue imo ik, mathematical limits work by just haveing that statement true, but to me those 2 infinities are different, just like infinite is smaller than 3 to the power of infinite (that's mathematically accepted) sorry for my bad english, it is not my first language
Amir Abudubai the problem with that statement is that infinity really isn't a number, it's a term to explain something that goes forever. so really you can't take away from it at all
You are mixing two different objects. 3^inf>inf is a statement you would find in the theory of cardinal numbers, where there are many infinities (inf-1=inf there too). In the extended real line, or in the equivalent statements in the reals, inf-1=inf, and also 3^inf=inf, in that case there are only two infinities +inf and - inf. Context (definitions) are central in mathematics. Discussing whether such context is adequate is another thing. In the real numbers 0.999...=1, period, and most people would not take the time to understand why the real numbers are defined in that way. In other systems the equality does not hold. For instance, in the 10-adic numbers, 0.9999....=-1
Sir, If N=infinite, N!=N.
I remember I had this question when I was a kid. Glad that someone gave me comprehensive answer finally.
I REMEMBER ASKING MY GRADE 8 MATH TEACHER THAT
This is one of the greatest channels on youtube. You're like the cool German uncle I never had.
Mathologer: *states a fact* *several people are typing*
i would accept an F from him if would be my math teacher, because he is so friendly and charismatic
I'd love to get foundation in the theories to understand this more intimatly, but would have no chance of getting a second degree in a university.. can you recommend an online course?.. I already have a degree in computing science so it's not all foreign to me at all if that helps make a recommendation 😉
read up on real analysis or www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=&ved=2ahUKEwiqtuLV-q3rAhVUEncKHdUSAOsQFjAAegQIBRAB&url=https%3A%2F%2Fwww.math.wustl.edu%2F~kumar%2Fcourses%2F310-2009%2Fpeano.pdf&usg=AOvVaw1z9yMXEIPhFxo7M8RZMIl3
When you cut a cake in the 3 peices, the last 0.001 parts of the cake, is on the knife; And Thats a fact
That isn't a fact and it's irrelevant. ---- EDIT: People, I am and was aware that the original post might have been a joke. Except for this first comment, all of my subsequent responses have been dealing with trolling buffoons like "The Hermit", not the original post. So STFU instead of telling me it was a joke. OTOH, I know for a fact that many people use arguments similar to the OP as a serious attempt at "proving" that 0.333 . . . is not 1/3.
Kay
@@Chris_5318 It might not qualify as 'fact' but it is highly relevant.
@@TheSimonScowl It is totally irrelevant. The topic is pure math, not domestic science or home economics. In what way do you think that there is any relevance?
@@Chris_5318 Because math is math. A bit of cake left behind on a knife is a fraction (however small) of the whole cake. Do you not understand that?
To any doubters out there: If you ask people, is it true that 10 / 3 equals 3.3333... everyone would say YES. If you then ask people, is it true that 9.9999... / 3 equals 3.3333... they would also say YES. NOW how can you get the same result by dividing both 10 and 9.9999... by three, unless they are the same number? Don't be fooled by the fact that THEY LOOK SO DIFFERENT. They are the same number, just written in a different way.
Tiikuri Another pretty neat explanation
Tiikuri this is one of the few area where the imperial system rules supreme, ten divided by three, is three and one third
***** Fractions are in the metric system too.
Tiikuri well i feel like an idiot now
***** It seems i may have been wrong. Fractions should not be used with metric measurements. Fractions themselves don't really have anything to do with imperial or metric systems anyway, now that i think about it.
Outstanding video - thank you!
I never liked to work with math because I'm really stupid and I missed a lot of how math works in school. I was quite often absent in school because of my disability. But I love things like this, its not hard to understand the concept even if you have no previous knowledge about certain math's related things and its really fun :) The Futurama theorem video was fun too
We can also show it with the sum of geometric series. 9/10+9/100+9/1000+...=9(1/10+1/100+1/1000+...)=9*((1/10)/(1-1/10))=9*(1/10)/(9/10)=9/9=1 Right?
Wrong
@@osdever care to reason?
@@apratimtewari4288 Yea
Sander, please don't disable the reply button in your comments. Anyway 1.111... =10/9, 2.222...= 20/9, 3.333... = 30/9 etc. 8.888...=80/9 9.999...=90/9=10 To show that any of these identities hold you can argue exactly like I did for 9.999...=10.
Mathologer I totally agree with everything in the video, but above, you said that 90/9 = 9.999... and 10, but that is not true. 90/9 only equals 10.
Josiah Swanson But if you agree with everything in the video then you agree with the fact that 10 = 9.999... . But then 9.999...=10 = 90/9. What are you worried about here? Basically you can take any of those one-digit repeating decimals and use the method in the video to figure out what fraction it corresponds to: Set M=d.ddd... where d is your digit of choice. Multiply by 10 to get 10 M = dd.ddd.... Subtract to get 9M= d0. And therefore M=d0/9. Summarize d.ddd...=d0/9 and so 0.000...=0/9=0 (just for fun) 1.111... =10/9, 2.222...= 20/9, 3.333... = 30/9 etc. 8.888...=80/9 9.999...=90/9
Mathologer You got me! Good job
Mathologer Sir, you ask (somewhat incredulously) that you cannot understand why people can still agree with your mathematical processes and still append all that with "But I still can't see how you can overlook that 10 is not the same as 9.99999999..., I mean "10" is 2 digits, the 1 is in the TENS place and 0 is in the ONES place, that other thing is 9 in the ONES place and followed by all the Nines that could ever be written in the universe" The thought then follows (almost invariably) that one should "listen to Sesame Street! ♫One of these things is not like the other♫ See, the digits NEVER make the "crest" up to the tenS level." [whew] The disparity comes from most folks being so grounded in their analog world that the digital just doesn't "jive" with them. Actually I had the same conundrum when I first took Calculus. It was a matter of (desperately) hanging on to the thread of what I truly comprehended - Geometry, and the sacrosanct definition of "Asymptotic." Here's the short version: Professor: OK, we all know that the expression (choose a letter)=n² will give us a graph which is asymptotic to the (choose a letter) axis. Which means what? Me: (envision Arnold Horshack's "Ooooh! Ooooh!) That it could never reach it. Professor: But what if it could? Me: "But it can't." Professor: Yes, but what if it could? Me: "But it CAN'T" Professor: (again) Yes, but what if it could? Me: "But it CAN'T!""That simply is the Definition of Asymptotic." You can see my intellect clawing at the face & eyes of CALCULUS as I drown in crippled logic. And, as it was just about 4:51, the looks of my classmates as I hijacked all of Their time in class. I _truly _ considered withdrawing from the course. This same kind of thing happens when the typical (OK, maybe not SO typical) American who has (absorbed pretty much by osmosis) an 'inkling' of an understanding of Newtonian Physics is exposed to Quantum Theory. They just kinda go "Nahhhh." It's just too much for their experiential basis, living in the macro world, to adjust to. So what does the physicist do when the guys down at the bowling alley scoff at Vacuum Energy and Quantum Tunneling? He get's out his Quantum Leatherman (otherwise known as Math) and proceeds to melt their cerebellum.Making their lizard brain run all the way back to Sesame Street and say, "But just Look - they're DIFFERENT!" I'm not trying to insult anybody, but THAT is why they say "OK, I followed that, but can't you see they're different?" Ok, that and just like the folks in Los Angeles "don't trust any air they can't see" a whoooole lotta folks don't trust any math higher than the stuff they use to balance their checkbooks... [Sorry I got loquacious]
Thanks a lot for that. That was a fun read :) Actually, I can sympathise with everybody who finds this stuff weird. Not to understand something the first (or 2nd, or 3rd...) time you see is perfectly okay with me and I am happy to discuss things further. What gets me is the people who come barging in claiming that this is all nonsense, that I am lying or worse. Then confronted with the fact that this is actually what all mathematicians subscribe to they declare that all mathematicians are wrong, etc. Check out some of the threads in this comment sections (we must be approaching some KZhead record with some of them).
He wouldnt be a good teacher if his equipment didnt malfunction during the lesson leading to an awkward laugh. I would've loved to have this guy as a math teacher
Always love these videos. It was correct to do all the algebra with M but it was done with the assumption the series converged but no proof of the convergence was done.
The convergence stuff was omitted because the video was for kids. Besides, it is pretty "intuitive" that even infinite decimals represent a number.
9.999... can't equal 10 because that's a bunch of nines and nines don't look like tens ha I'm right you are wrong
and 1/2 and 3/6 cannot be equal, one has 1 and 2, the other has 3 and 6, ergo different.
My comment was purely sarcasm. I've looked through the comments for this video and I do believe that 9.999... equals 10
Aleph Null Belief is required for religion, not math.
Chris Seib If you believe a proposition that just means that you think it is true. This can apply to any proposition, including mathematical ones. You probably say this because you think belief implies faith, which it doesn’t.
Well, if you ever want to be a smart ass, you could say that A does not equal A because the first A is on the left and the second A is on the right.
Great work, love your stuff! Another way to think about this, if 10 and 9.999.. are not the same number, then you could find a number between the two!
Answer Evaded Your comment presents a nice argument that I wish I had thought about before. To write it out in a (slightly) more rigorous fashion: Let T(N) = Sum(9/10^k, k = 0 to N) (that is, T(5) = 9.99999, with T(N) converging to T = 9.999... as N->infinity), and let e(N) = |10-T(N)|. Since e(N) is decreasing and bounded below by 0, it must converge to some limit k. Therefore, we have a real number x = T + k/2 such that T
***** That's exactly right.
Mathologer So if my address is 100 and my neighbor's address is 101, since there are no houses in between, my address is 101? That argument holds no water. I call 9.999... and 10 "next door neighbors" or "the discrete interfering with the continuous due to the introduction of infinity." I assert that there exists an infinitesimally small number that is between 9.999... and 10. I can't define it or write it using our numerical system, but that doesn't mean it doesn't exist. I can bound it with the lower value T(N)=sum(9/10^k, k=0 to N) as given by MyOverflow, and the upper value of S(N)=10-1/10^N. At N=infinity, that difference becomes zero and they both converge to 10. But you can't go the place I called "N=infinity." It doesn't exist. It's a concept. As N becomes infinitely large or N approaches infinity, the difference becomes infinitesimally small. 9.999... is just as mathematically malleable as infinity itself (as opposed to the set of integers, for example), and so, for the purposes of any approximation, yeah, sure, call 9.999... basically 10. But it isn't. From what I can tell through the comments section, though, 9.999... == 10 because it was defined that way and for no reason more. Seems pretty petty to me, but I can accept that reasoning. If you're going to pull that argument, then you can't treat the 9's as numbers, because you've made 9.999... into a symbol and not a number, by defining its equivalency to another number. You've seen the proofs that you can use infinity to completely mess up sums without actually breaking any math laws (sum of all whole numbers -1/12, etc) so you can't actually give a mathematical definition of 9.999... because it isn't a number.
Marshall Curry The addresses are integers. We're talking about real numbers.
It still doesn't hold water. I defined an upper and lower bound for the number whose value exists between 9.999... and 10. Those two bounds become infinitesimally close together as N gets large, but they're never QUITE equal. The same can be said for (2/3)^N. It gets really close to zero as N gets large, but it never quite gets there. (2/3)^N > 0, not >= 0.
I'm KAZAKH 🇰🇿 KZ:Сабақтарыңыз өте қызық🔥 Ары қарай жылжып, дами беріңіз. EN:Your lessons are very interesting.move on and develop.
I think many people have a bad prospective on this. You shouldn't think of a real number as "being" a string of digits. Real numbers are defined regardless of their decimal expansion, which is just a way to write a number down, it's like a name. It turns out that you can't name most number using only a finite amount of digits, so we came out with a way to represent numbers using infinite digits that allows us to represent every real number (sometimes in different 2 ways). When you see 0.9999... you should think "those are symbols that are ment to represent some numbers". With the agreed upon rules, it turns out that it actually represents the number one, that is also represented by 1.0000... This can be shown by first understanding what are those agreed upon rules, and then by using only very elementary and fundamental properties of real numbers (supremum, completeness, etc...). For those thinking that there is a number smaller than 1, that is bigger than every other number smaller than 1 (that should supposedly be written as 0.999...), there is no such number! This is very, very much like the fact that there's no biggest number, you can always go higher. For those thinking that 1-0.999... is not 0, but a positive number smaller than all other positive numbers, again there is no such number. Also it's decimal representation cannot be something like 0.000...1, since this is not a decimal representation in the first place. A decimal representation is a way to write down a number, but 0.000...1 is not a decimal representation, it's meaningless. Also you wouldn't say that the decimal representation of pi, 3.14159265... does not represent pi but something a bit smaller just because it always comes just a bit short if you cut it at some finite amount of digits. For extra clarifications and a full understanding of this, don't skip your topology class! :)
Not only people think that numbers "are" their representation, they think that the representation (i.e. symbols such as "0.999..." or "9/9") is somehow god given and it's up to us to ponder about their "real" meaning.
The answer is no. It's pretty much certain that there will be some disbeliever, so you won't be able to persuade 100% of the audience (or 99.999...% of the audience - same thing).
There are disbelievers even in 1+1=2.
Uh I love you joke xD
no one asked
@@XenophonSoulis Well, in fact, 1+1 could be different from 2: it only depends where they are defined. For exemple, in Z/2:{0,1} 1+1=0. :)
One person out of 10 trillion is not the same as 99.9999...% because the number stops somewhere. 99.9999...% goes on indefinitely, and that's what makes it equal to 100%
I always thought of a very similar truth in that 1/9 is .11111... repeating infinitely (try it using long division and you will see the pattern repeating endlessly). This happens with 2/9 as well being .222222... then 3/9 or 1/3 (simplified) is .3333333... repeating all the way to 8/9 being .888888... repeating indefinitely. Now, if you do 9/9, what do you get? Well, with the exception of 0, any number divided by itself is 1, thus continuing the pattern one might conclude that .99999... indefinitely is actually 1. I thought of this probably 10-15 years ago now.
Also, another way to rewrite the number .99999 repeating is 1-(1/x) as x approaches ∞
Then Why isn't 10/9 1.11111111111 oh wait...
As Mathologer said in the video, we'd first need to prove that 1/9 IS 0.111...
@@FFVison exactly
@@yanndebacker7163 You meet infinity way more often than you think :) Look up the dichotomy paradox, or even the fletcher's paradox (these assume you can move infinitesimally small distances and be in a difference place, which in physics you literally can't. It's called a planck distance, but the concept is still valid). You'll find that you're paradoxically both not moving and moving at every infinitesimally small unit of time :D
When I was in grade school I was able to convince myself with the following reasoning: Assume we let (x) = 10 - 9.999999999999999... and we then let (y) = 1 - (x) Now what is (x) ? but an infinite string of zeros (or noughts.) (x) = 0.00000000000000000000... Now perhaps we might initially find ourselves wanton for some long trailing 1 somewhere down the line, but of course there is none; we never arrive at such an infinitesimal degree. What is more when we evaluate (y) subtracting (x) from 1, we are subtracting nothing but an endless series of zeros.
Excelente. Your style of sound laughing remembers a friend who is my old friend of teen club of amateur astronomy , an askenazin from polland (but both we are Brazilians at the cradle). Sorry for my poor English but... I need to ask... If I can always exchange the last digit by itself minus one by "9..." because the operations will prove equivalence, could it be also true to "8..."? Or whatever? It is only an idea, I didn't test it.
Yes. e.g. 1/4 = 0.25 = 0.24999 . . . Only 999 . . . is special like that. Every terminating decimal has a dual that ends with 999 . . . 0.888 . . . = 8/9 0.24888 . . . (and the like) don't have another decimal form. It is equivalent to the rational 224/900 = 56/225. If 0.(N) represents the n digit string N being repeated, then 10^n * 0.(N) = N.(N) = N + 0.(N) (10^n - 1) * 0.(N) = N 0.(N) = N/(10^n - 1) e.g. 0.(428571) = 428571/999999 = 3/7
My favorite proof of this is to ask what the average between the two numbers is. If the average isn’t just one of the two numbers, then it must exist between them. There is no in between, so they can’t have an average that isn’t just one of the them.
Another important point is that there is a difference between a "value" and a "symbol". The "value" Ten is the value Ten, no matter how you represent it (and "ten" only is "ten" in English, there are hundreds of other words for it in other languages). The most common way of writing this in the western world is 10. However the Romans used X and the Chinese sometimes still use 十. Regardless, those are all just different symbols that represent the same value. We are also accustomed to other methods of indicating the value "ten", such as 20/2, sqrt(100)... etc. 9.99999... is simply yet another method of representing this same value.
Sqrt(100) is not 10 it's +/- 10 Also your whole thing is irrelevant to the result
@@guythat779 sqrt(100) is a covenant, not a simple operation, and this covenant says that with the notation "sqrt" you have to refere to a positive number...
@@guythat779 No, sqrt(100) is 10. sqrt(x), by definition, means the positive number that when multiplied by itself results in x. The same goes for x^(1/2). What your thinking of is x^2 = 100, where solving for x results in x = +/-10
Someone may have pointed this out already -- but speaking more generally: if you try to represent numbers in _any_ decimal-like system with place values before and after the decimal point that are all integer powers of some base (such as, say, 10), and then you try to represent some rational number a/b in this system but b has some prime factors that are _not_ prime factors of your base, then you will have no choice but to write it as a repeating decimal. But then if you allow this you might as well just allow _all_ repeating decimals, including the numbers that don't have to be written this way. And at that point you might as well just consider _all_ decimals to have an infinite number of digits (some just happen to end in repeating zeros). So then every decimal will have a non-repeating part (maybe infinite, maybe finite) which if it's finite will be followed by a repeating part. So you would have things like 8.3251251251... = 8.3 + 0.0251251251... = 8 + 3/10 + 2.51/99.9, but also things like 6.1 = 6.1000... = 6 + 1/10 + 0/90, which could also be written 6.0999... = 6 + 0/10 + 9/90. So then all 6.1000... = 6.0999... really means is that 6 + 1/10 + 0/90 = 6 + 0/10 + 9/90. That fact that a number can sometimes be written as a sum of fractions in two different ways should come as a surprise to no one.
@@vibaj16 I think you mean: ”by default”.
Achilles and the Tortoise paradox... :)
I've had friends who said that they thought 0.99999.... was just *slightly* less than 1. Here is how I "proved" that they are in fact equal: For any two unequal real numbers x < y, there always exists another real number that is between them -- for example, x < (x+y)/2 < y. Can you think of a decimal number that is greater than 0.99999... (with literally an infinite string of 9's) but less than 1? The answer is no, you can't.
A nice way of convincing someone 9.9999999.... = 10 is to ask them to find a number between 9.999999 and 10, since there are no numbers in between these two, they must be the same
KREATOR AT WORK someone gets it
Easy. Another .9
There are something between 9 and 10 in the natural nambers ?
1-(1 - 0.99999)÷ 2
@@richardsilva9488 Not in the naturals. 9.999... is not natural. Between 6.5 and 6.7 there might be 6.6, but between 9.999... and 10 there's not
This ... is how we subtly get kids to be curious about math :)
I came here for an wrong math and the entire video was total drama island
I like that you could have go by fractions but you decided to explained in another way.
Okay maths teacher... I am coming
This is all base-dependent. You can write 0.(3)=0.333... in base 12 as 0.4. And there are no repeating digits. It's the property of the number 10 that makes "one third" look how it looks. Because you can't divide it to have no remainder. So everything said in this video is right.
You're just dodging the problem. In base b+1 0.bbb... = 1.In base 12, 1/3 = 0.4 = 0.3BBB... where B stands for the digit that is 11 in decimal.
I'm just saying that if there is base b, in which you can write a number with repeating decimals (which you can always do), there is always another base in which the repeating digits go away because how the number is represented, but of course, the same problem comes up with that base too.
***** You mean that you can write numbers like 0.1 and 0.25 in decimal and they aren't repeating. Wow, I didn't know that.
Wow, I just made another proof to what is the question in the video. You clearly didn't read the whole thing.
+Chris Seib What problem is he dodging?
but to be fair, they didn't state in the episode that it's going 9999... forever, so even if it was extremely close and even if they had a more accurate clock, it still wouldn't've equaled 10
So what!?
I was interested in this in the primary school!
Get off my 9.99999999 squared ft lawn!!
+hukes Would be interesting to know whether anybody ever used 9.999... on a legal document :)
Yes, what would law judge? If it were my call, I'd say it is 9.9999..., not 10 squared ft. I know, for all practical ends, it equals 10, but in my poor (and old minded) head, it still is 9.9999. That little 0.000...1 must be respected. :)
There is no such thing as 0.000....1, which you have to respect, but I am also pretty sure that if you left the decision to a jury, even to a jury consisting of randomly chosen school teachers the verdict would be 9.999... not 10. Which, of course, is really a very sad state of affairs given that 9.999...=10 is really a fairly trivial result in the grand scheme of things.
I know, it is trivial, but I obsess in such tiny details. ;)
But in this case the detail you are worrying about is so tiny that it even does not exist :)
0 divided by 0 is actually 27 because, if asphalt weighs 36.17 pounds per 1 square meter that means that cow is 71 years old. If she is 71 years old that positively means that my journal is green. Since my journal is green I can eat potatoes. I eat 1 potatoe in 47 seconds. So basically this formula lets me to get 27 from diving 0 by 0.
:)
0 divided by 0 can be whatever you want because 0 divided by 0 is undefined
That's what they said. 27.
There's also some infinity in it, because it depends on which potato you are eating - there are hundreds of types of potatoes. And some potatoes aren't even suitable for consumption.
thanks now ill get revenge on my teacher
Chris is the kind of guy who would say that anyone who corrects him is wrong or would eliminate them
If you are referring to Chris Seib, he sounds like a professor that got fed up with explaining the same thing over and over again to people that simply refuse any explanation with no counterargument given. His explanations are actually really good and those who bothered with them understood them.
I'd say that might be the case if he were arguing about science or philosophy or something else where people can have different interpretations, but in maths, if you have a set of axioms, what is true is true and what is false is false, regardless of what you want it to be.
In portugal we learn about this as a little fun fact when dealing with rational numbers and fractions in 8th grade maths
No numbers were harmed in making of this video..😂😂😂
There were several times in this video where I thought he was going to talk about binary floating point imprecision. I guess that would've been a cop out though lol. I love that Mathologer is always happy to exhaustively explain things that seem so counter intuitive at first! Also, I need to get me the timer from that cartoon! Precise to 19 places after the decimal point... very impressive!
Floating point is about clever computer engineering, but it has nothing to do with pure math.
@@Chris_5318 It’s basically a binary equivalent to the scientific notation (a*10^(+/-b)), which is also used in (spoiler alert!) science, much more often, than in pure maths; so, I wouldn’t be surprised that the same should be the case for the floating point numbers and computer science 🤔.
@@PC_Simo I know what scientific notation and floating point are about. (How wasn't that obvious?). Neither has anything to do with the video.
@@Chris_5318 When did I suggest you didn’t know that? Or, when did I suggest it had to do with the video? 🤔
@@PC_Simo Your entire previous comment suggested that I didn't know it. It actually came across as patronising. The fact that you posted on this page suggested that you thought your comment was relevant. Your last response was disingenuous.
Me following this video: "okay... Yep, I see... Ah! Okay... Mhm..." 4:04: Me: 😐
Fun fact, die Anzeige in der Serie ist auf die zehntausdendstel Femtosekunde genau. Eine Femtosekunde ist die Zeit in der sichtbares Licht schwingt. Zum Vergleich mit einer bekannteren Größe, eine Femtosekunde sind 1.000.000 Nanosekunden. D.h. damit die angezeigte Präzision sicher anzugeben ist, muss mindestens auf das 1000stel der Zeit die sichtbares Licht braucht um einmal zu schwingen gemessen werden.
(+)10=(+)9.9999... (+)10 *_"-"_* 9.9999...=? (+)10 *_"-"_* 9.9999...=0.00000000... because the digit one can never be placed in that calculation.
never ≠ after forever.
Mario Joia so is (+)10+(+)9.9999...=19.9999... or 20?
If the Romans had invented algebra, what would have been their go-to variable? I'm guessing it wouldn't be x.
.00001?
@@ExploringCabinsandMines I hope that was a joke. -Randy, there is no after forever. 9.(9) can't have anything at the end because there is no end.- Edit: oops, my bad, I misunderstood what you meant. Sorry. ArcticXII 20 = 19.(9)
Next video: Why Every Proof that .999... = 1 is Wrong lol
... and make sure you check out the day on which it was published :)
That video is an April fool's joke
You're an April fool's joke. =D
If you accept that 0.999999... is different than 10, you must also accept Zeno's paradox which is clearly wrong.
I actually wait to see such a video in November 32nd.
I was literally just thinking about how 0.999... = 1 while doing my math hw just now and this is the first video on my recommended
This is one of the best videos of mathologer
Not even close. He has many awesome videos.
@@Chris_5318 but i really liked it 😂
@@joeljose182 I'll stick with my first response.
You’re brilliant!
The way I justify "9.9999…=10" to myself is "9.9999… +(1/∞)=10". Since I know that any real number over infinity solves to zero (at least that's what I remember from calc), then the two numbers must be equal, because their difference is exactly zero. It's still an unusual concept for me, though.
1/oo means lim n->oo 1/n and that is 0, as you said.
_yes corect is from you notation with need period and 0,(9)9+0,(0)1=1 __the this callcul is correct and exactley and the this callcul is part of a new class of nmbr with name transsinphinite number theory__archaicxn lord
This is actually the one line of logic that convinced me. The video I still had my doubts.
-hey you are really good at math, right? -yeah, sure -can you explain me one thing? -of course! What is this? -when I cut 3 same peaces of cake one peace is 3.33% right? -yes it is. -so when I add all that peaces I get only 9.99% of cake. Where is my 0.01% of cake!! -it stays on the knife.
It’s 33.333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333%
@@masonhunter2748 No it's not. It's 33.(3)%
Nephew sorry my supercomputer ran out of space
All this proved to me is matth is imperfect and truly can't explain the universe.
Your comment doesn't make any sense. How did it prove that math is imperfect? Does 1 + 9 = 10 also mean that math is imperfect? What do you think 9.999 . . . would have to be to make math perfect to you? It is not the domain of math to make any claim about physical reality, that's the job of physics. If you look up the axiomatic field definitions of the real numbers, you won't see any mention of physical reality, such as "adding" apples together. Numbers are pure abstract.
@@Chris_5318 It's not that the comment doesn't make sense, it's that you don't understand it. To assume it makes no sense because you don't understand it as arrogance. Sometimes the problem lies within, not externally.
@@mattiefee I had accidentally posted my previous reply in the wrong thread. I understand your comment far better than you do. The imperfection is your understanding the nature math . Once again, math isn't physics and it doesn't claim to explain the universe at all. You do not know what you are talking about. You are arrogant for thinking that you are right and that anyone that disagrees is wrong. BTW thank you sooooo much for explaining what you meant. That was very helpful /s. PS Your original comment doesn't make any sense.
@@mattiefee You are correct, the problem lies within. Within you that is. Mathematics is perfect for what it is to do, itself. It has no interest in the physical universe.
@@EmperorZelos Ah yes, you the "all knowing" obviously knows it is perfect and has no interest in the physical Universe (are you sur e the Universe in its entirety is physical?? You must be with your level of conviction!) I yield to your absolute certainty and knowledge... hahaha. Have a good day.
Amazing video very well explained :D
and I think that some people don't belive 9.9999... is equal to 10 is because they just don't look the same numbers some people don't like to use their heads
yet they'll happily say 2/3 and 4/6 are the same but they look just as different.
10:26 Good luck with taking "every single one of those guys"
Luck isn't required.
All right, proceed to infinity then
@@tomaszkantoch4426 It was done in the video. You use deduction, something that requires a brain. There isn't a process being done one step at a time. The whole multiplication is done at once/in one go. Calculators aren't required either. This is math, not computer science. Try again, but this time try not to confuse the result with the explanation of result.
@꧁༒ⱤɨCʞƔ༒꧂ Says someone with that "name". The irony is palpable.
@꧁༒ⱤɨCʞƔ༒꧂ LOLOLOLOL. You should change your name to ꧁༒ɨⱤoNƔ༒꧂
So the issue I always have with this is the difference between 0.999... as an infinitely extending sequence of digits, and as a sequence of digits that extends *to* infinity. If it only extends towards it then there's always a rational residual. Some component that we haven't reached yet as we cancel terms, so these things aren't equal but only coincide at a limit. If the digits extend *to* infinity, then we have an end that we can unroll from, so we're all good, and 0.999... is indeed just an alternative way to notate 1. I've never seen a convincing case for which of these two models is the correct one, although I recognise that the latter is the more convenient one.
But the residual cannot be rational because for any rational number you give me, I can take more 9s and show that the residual is smaller. In fact, I can show that the residual is smaller than any _real_ number, so it must be 0 (unless you admit the existence of infinitesimals which are closer to 0 than any real number).
@@APaleDot exactly. The answer depends upon the number system.
@@mrpocock I'm not sure. Any number system which contains the real numbers as a subset would say they are equal. It's hard to imagine a number system which is capable of representing such small quantities, yet is not equivalent to the real numbers.
@@APaleDot well, any number system that augments real numbers with infinitesimals?
@@mrpocock I've been trying to find an answer online since I'm no expert. I found an answer on Quora about the hyperreal numbers which seems convincing, let's see if KZhead will let me post the link.
no 0.999... got this video as a KZhead recommendation.