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There is no proof that will ever satisfy a person dead-set against this. Ever since I brought this home from school as a child, my whole family ribbed me mercil

There is no proof that will ever satisfy a person dead-set against this. Ever since I brought this home from school as a child, my whole family ribbed me mercilessly for it. If you tell a person that 3/6 = 1/2, they'll believe you - because they have been taught from an early age that fractions can have multiple "representations" for the same underlying amount. People mistakenly believe that decimal numbers don't have multiple representations - which, in a way is correct. The bar or dot or ... are there to plug a gap, allowing more values to be represented accurately than plain-old decimal numbers allow for. It has the side effect of introducing multiple representations - and even with this limitation, it doesn't cover everything - Pi can't be represented with an accurate number, for example. But it also exposes a limitation in humans: We cannot imagine infinity. Some of us can abstract it away in useful ways, but for the rest of the world everything has an end. I wonder if there's anything I can do with my children to prevent them from being bound by this mental limitation?

traderjane 6y ago
You must go up to something like limits to make ... meaningful.
- deleted 6y ago
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tom-thistime 6y ago
(STATEMENT OF PERSONAL IGNORANCE [SOPI]: Anyone who actually understands this stuff please correct my mistakes below. Thanks.) In the real numbers, which are not always simple or intuitive, 0.99... = 1. That's true and I seem to understand the proof. But the real numbers aren't the only system that might be sitting behind "0.99..." and "1" when I write those symbols down and talk intuitively to people in my family. The reals are just the system we're taught first. I believe there are other systems (I think the surreals are an example) that work just as well for everyday purposes, but where ( my understanding is that) there are numbers that differ from 1 by a value that approaches zero, yet those numbers are not equal to 1. (I've played with the surreals but only as a hobbyist.) If you do calculus in these other numbers, I think physically meaningful problems will still yield the same answers. (For example Zeno's Paradoxes are still not an excuse for failure to attend school.) But it isn't a law of nature, I think, that all number systems that can hold 0.999... and 1 must make them equal.
- Scarblac 6y ago
  A number is just a number, it doesn't approach anything. A series can approach something, but a number can't. In any system where 0.99... is valid notation for a number, it doesn't approach anything.
  - tom-thistime 6y ago
    Sure. Instead of "0.99..." please substitute lim n->inf sum(1..n)(9 times 10^-n). The point I'm making is that the "obvious truth" 0.99... = 1 that we're all talking about depends on the assumption that we're working in the real numbers. I claim that the real numbers are not something intuitively obvious to every sufficiently intelligent person; instead they are kind of weird and technical. I go on to claim, though I'm more unsure of this, that the real numbers are not even the only way to make calculus work.
    - ramshorns 6y ago
      Well, a limit of a series is just a number and doesn't approach anything either. If the series approaches something, we say the limit exists and is equal to that thing. Anyway, in the surreal numbers you could probably make up a notation where 0.999... actually denotes 1 - ε or something. But I daresay it might not be very useful because then how do you denote 1 - ε/2 or anything else.
      - ramshorns 6y ago
        Oh, I figured it out, kind of. [I don't really know what I'm talking about either.] 1.000...0 = 1 1.000...05 = 1 + ε/2 1.000...1 = 1 + ε 0.999...8 = 1 - 2ε 0.999...9 = 1 - ε 0.999...98 = 1 - ε/5 . . .
        saalweachter 6y ago
        > 0.999...8 = 1 - 2ε > 0.999...98 = 1 - ε/5 cough
WilliamEdward 6y ago
This is essentially just a matter of limits, without which the world wouldn't make any sense. So you must explain that if you move your hand closer to an object, technically you are halving the distance infinitely many times, but if 0.999... != 1 then your hand would never touch anything.
lmkg 6y ago
It's more fundamental: People seem to have the intuition that the decimal representation of a number is a number. I don't know if it's because decimals resemble the ntural numbers or what, but decimals seem to have a primacy for people that fractions do not. The idea that there's a gap between the symbol for a thing and the thing itself is the stumbling block.
- Sharlin 6y ago
  Yes, but at the same time it is common for people to insist that 0.999… only "approaches" unity as if it were a series approaching a limit, rather than an unique number. Intuition is a funny thing.
  - tom-thistime 6y ago
    Saying "a number isn't a limit" is true, but it's only really relevant if you're talking to someone who genuinely has no idea what limits are. In actual math the number 0.99... can be defined as the limit of a series.
    - Sharlin 6y ago
      Every number is a limit, yes, but people think of 0.999… as a series. Not rigorously, of course, but that's a common argument even by people (perhaps especially by people) who have a highschool or even math-minor level understanding of series and limits.
- tom-thistime 6y ago
  Possibly people are looking at two different symbols and asking "can you show me logically why those are equal." If they're given a definition of "equal" and they still object, that's a different problem. I have this problem every time I play with group theory again. You get the axioms for a group, which say there is some identity but don't explicity require the identity to be unique. You can easily prove that the identity of a group is unique ... so long as you define "unique" to mean "if element e1 and element e2 are equal, then we say they are the same element." You could count things differently and say the identity is "not unique", it would just lead to a lot of stupid and un-illuminating consequences.
gspr 6y ago
> There is no proof that will ever satisfy a person dead-set against this. Indeed. I've torn my hair out trying to convince smart people with PhDs in hard sciences and had to give up in frustration. I usually find that the most success can be had by kicking the ball to them immediately and having them define what they actually mean when they say "0.999…". If we're going to debate whether that thing equals another thing, we better make sure we know what we're talking about. Inevitably, this either causes the dead-set person to give up, or give a myriad of definitions that are either meaningless, ill-defined, or causes them to realize that they don't actually know what "0.999…" means (or what they want it to mean). It is hard to have the patience to chase down the consequences of their ill-fated definitions, though.
- 2OEH8eoCRo0 6y ago
  Ask for a number between .9 repeated and 1
  - chris_wot 6y ago
    So does this mean that an infinitely small number is zero? As in 1/∞ ?
    - thaumasiotes 6y ago
      You can read about this in any work on nonstandard analysis. ("Nonstandard" is just the name, much like "imaginary" numbers.) An infinitely small number is zero when projected onto the real number line. If you introduce an infinitesimal quantity to the reals, then for every number there is a unique real number to which that first number is infinitely close (that is, the difference between them is infinitesimal). You can use that real number as a (good) approximation of all the nonstandard numbers in its halo. (As long as you're comparing it to other real numbers.)
    - selectodude 6y ago
      In calculus, yes.
    - GolDDranks 6y ago
Igelau 6y ago
What's interesting is that people pretty quickly become comfortable with the idea that 1/3 = 0.333… So using that as a foothold, we can express 1/3 + 1/3 + 1/3 as 0.333… + 0.333… + 0.333… and it should be pretty easy to digest. At once we can see that in this little zone we've defined, 1 and 0.999… mean the same thing. Not a rigorous proof, and one or two people will probably bring up whataboutisms like "that's just because the calculator can't do stuff!" but it should at least be proof of comfort for most people.
- tom-thistime 6y ago
  This is a really good point. Maybe the problem is how we define equality. What's the test for when two numbers are equal? People accept that 1/3 = 0.333333... The same people don't always seem to accept that 3*0.33333... = 1. Well, how are we defining "equals"? If we can give that definition in black and white, I think that may help.
  - scrimps 6y ago
    If 30.33333... = 1, then 30.33333... != 0.9999..., then 0.9999... != 1
  - esmi 6y ago
    > What's the test for when two numbers are equal? I put this in another comment but: For the reals: eliminate a < b and a > b then conclude a = b.
TomGullen 6y ago
I find the algebraic way convinces most people: x = 0.9999.. 10x = 9.9999... 10x - x = 9 9x = 9 x = 1
- solotronics 6y ago
  black magic! I wonder if there are any programming languages that are able to handle this properly?
  - 2OEH8eoCRo0 6y ago
    https://www.wolframalpha.com/input/?i=.9...+%3D+1 https://www.wolframalpha.com/input/?i=.9...+%3D+1
    - tygrak 6y ago
      https://www.wolframalpha.com/input/?i=10*0.9...-0.9.. https://www.wolframalpha.com/input/?i=10*0.9...-0.9...
  - brlewis 6y ago
    There are programming languages with rational number types, but none that I know of that represent numbers as repeating decimals.
- taylodl 6y ago
  That's how I was taught in my Algebra class
- thenewwazoo 6y ago
  Because I wondered, it’s not a trick: x = 0.444444... 10x = 4.444444... 10x - x = 4 9x = 4 x = 4/9 = 0.444444...
  - phaemon 6y ago
    All the single digit repeating decimals are x/9. 0.111... == 1/9 0.222... == 2/9 ... 0.888... == 8/9 0.999... == 9/9 :)
Okx 6y ago
I usually say "If and only if two numbers are different, then you can find a number between them". People often accept this axiom. Then, I offer them to find a number between 0.999... and 1.
- Falling3 6y ago
  That's a great approach
- tom-thistime 6y ago
  This works because you have defined what equality means. We all wanna talk infinity because it sounds more exciting, but I think everybody gets "infinitely close to 1" pretty well intuitively. What they don't get is whether "infinitely close to 1" means "equal to 1". That could happen because these people are stupid.[note] But it could also happen if nobody has defined equality. [note]for example even highly educated people maybe don't listen, which is functionally a lot like being stupid.
- solotronics 6y ago
  is there any difference between a black hole and nothing? (somewhat joking but I was thinking of a physical analogy of the limit approaching zero)
  - caspper69 6y ago
    A black hole has mass. Nothing has no mass. (also a somewhat joking answer)
  - raattgift 6y ago
    Yes, there's a difference. Firstly, though, there are multiple different types of black hole, from the theoretical to the astrophysical. We must narrow your question down to have any hope of a good answer. The simplest theoretical black hole, the Schwarzschild black hole, has one variable -- the central mass -- which must be positive. If we set the central mass in a Schwarzschild spacetime to zero, then we have Minkowski spacetime: no curvature, no horizon, no black hole. The Schwarzschild spacetime is completely empty except for the central mass, which is constant and located at an infinitesimally small point at all times. The Minkowski spacetime is completely empty everywhere and at all times. The symmetries of Schwarzschild and Minkowski spacetime are different, and if one were to probe the spacetimes in question with a Synge curvature detector [1], we would quickly discover which we were probing if our probes happened to be placed close to the central mass, and eventually if they were placed far from the central mass. If one placed the probes infinitely far from the central mass, it would take an infinitely long time to distinguish the presence of the central mass (which makes spacetime non-Minkowski); but these spacetimes are eternal anyway, so that's OK. So that's almost a "yes" to there being a theoretical black hole analogy between (1-) 0.999... and (1-) 1. I would not call this a physical analogy since neither Minkowski spacetime nor Schwarzschild spacetime is at all physical. Nature is full of stress-energy (gas, dust, ...) any of which breaks the vacuum condition of these spacetimes, there seem to be a lot of astrophysical black holes at the centres of galaxies and individual/binary stars that have become black holes, and even a two-black-hole universe is markedly different than a Schwarzschild spacetime. Additionally, these astrophysical black holes are not eternal, unlike Schwarzschild. In particular, the stellar mass ones were once stars, and the galaxy-centre ones at least had less mass in the past. These last conditions alone are substantial deviations from Schwarzschild that are even more obviously not Minkowski (e.g. if you put probe finitely but sufficiently far away, you could see an image of the radiant precursor star rather than the black hole!). Finally, in our physical galaxy the answer to your question is a big "yes!". The observed orbits of these stars [2] would be noticeably different if the central mass in the Milky Way's central parsec were anything but a black hole, and would be even more different if that central mass were not there at all. - -- [1] Synge, J.L., _Gravitation. The General Theory_, ch. XI §8, "A five-point curvature detector". [2] http://www.astro.ucla.edu/~ghezgroup/gc/animations.html http://www.astro.ucla.edu/~ghezgroup/gc/animations.html and http://www.astro.ucla.edu/~ghezgroup/gc/blackhole.html http://www.astro.ucla.edu/~ghezgroup/gc/blackhole.html
sorokod 6y ago
People mistakenly believe that decimal numbers don't have multiple representations - which, in a way is correct. It is correct if you take the limit, people usually do not.
0xff00ffee 6y ago
Yep. Agree 100%. It is like the blue dress. I think the problem is the repeating function. Infinite things are non-intuitive and should be presented differently. Even here on HN you still see people confused about "convergence" and "identity". 0.999... doesn't CONVERGE, it literally is 1. I suspect this persists even with students that have had second year college calculus that discusses convergent series and sums.
- tom-thistime 6y ago
  Fine. Define 0.999... as the limit of the series sum(n=1 ... N)(10^-n), as N-> infinity. This is standard high school calculus. "Number" and "series" and "limit" and "convergence" don't all mean the same thing. However this number is defined as the limit of a convergent series. So the question really is meaningful. (One clue that this question is meaningful is the amount of space introductory calculus textbooks use to address it.) Because I can still ask, in black and white, what law of "equality" do I use to establish that my limit equals 1? (It does, if I import the definition of "equality" from the real numbers. That's what they do in calculus class. )
  - 0xff00ffee 6y ago
    I'm not sure if you are asking for an answer or a rhetorical question? I'll assume the former. Your terms are bit jumbled, so let's keep it simple: you're asking how to prove if an infinite sum converges and what its value is. Convergence proofs require analytic thought: meaning there may not be an immediate look-up. You need to convert the problem into the known corpus of convergent sums or use one of many tests (bounds test, integral test, etc) to show it converges analytically. Which you only learn through experience and memorization (unless you want to re-prove hundreds of series... maybe you do!) Fortunately this one is easily re-written as a known convergent sum. First, you missed a term in your sum (9), re-written here: sum(n=1..inf) 9 * 10^-n Step 1: you pull out the 9 and it becomes 1/10+1/100+1/1000... Step 2: Then we shift to n=0 by subtracting 1/10^0 from the series so that it is in the form n=0..k-1 1/10^0 + 1/10 + 1/100 + 1/1000 + ... + 1/10^-n - 1/10^0 Step 3: Now we've got ourselves a geometric series of just 1/10^n .. wikipedia does a great job explaining the sum convergence for GS from n=0...inf: https://en.wikipedia.org/wiki/Geometric_series https://en.wikipedia.org/wiki/Geometric_series Step 4: compute geometric convergence (1-r^n)/(1-r) = (1-(1/10)^n)/(1-1/10) = 1/(1-1/10) = 10/9 So we have 10/9 as the solution to Sum[n=0...inf](1/10^n) Step 5: the remaining arithmetic Now subtract our 1/10^0 ... and then * 9 = 1
dubcanada 6y ago
I feel like most people are not answering this, they are giving the proof in a different way. Answering how to teach a child not to be bound to 100%'s is really hard. I personally would say just let them explore on their own, teaching a person that 0.99999 = 1 results in the same as if you taught them that 0.99999 != 1. You need to teach them that all sciences and maths are changing constantly, what might be a fact today could change tomorrow. You need to teach them that anything can become wrong or right as we progress and to be open about accepting new information, while being hesitant enough not to not succumb to false/fake information. That's a very hard lesson to learn and an even harder one to practice. But one that I think a lot of people need to learn.
lukifer 6y ago
I didn't grok infinity until I started thinking in terms of verbs rather than nouns. As a static number, the concept of infinity makes no sense; but once reimagined as a process (start counting up from 1, and never stop), all apparent paradoxes disappear. This is the inverse problem: it could just as easily be reframed as 0.000...0001 = 0. Defined as static nouns (does such a thing exist in nature?), it's seemingly paradoxical, and fascinatingly debatable in a "is a hot dog a sandwich" sort of way. But reframe it as a process (or as code), and all confusion disappears: for how many loops would you like to proceed? If you never stop, 0.99999... clearly approaches 1, without ever reaching it, and asking if they're the "same" is as academic as asking if the Ship of Theseus is the same ship, or if an electron is the same entity from one picosecond to the next.
- ta1234567890 6y ago
  > As a static number, the concept of infinity makes no sense; but once reimagined as a process Super insightful. That's the key right there. The same concept can also be applied to the physical world. Things are not static, they are in constant flux, everything is a process in motion.
  - thom 6y ago
    Yes, although for me this conception of infinity as a process also captures why there are probably no actual infinite things in the universe, only in silly games with numbers.
    - effie 6y ago
      Could you elaborate? Why infinite process cannot be actual thing in the universe? It's not like we know the start and the end date of the universe...
- TuringTest 6y ago
  I find that "infinite as an endless process" concept intuitively very heplful as well. However, reading Gödel, Escher, Bach[1] showed me that there's another, more static logical interpretation of infinity which also comes handy. In an infinite process, you can always take "one more step" to create the item after that. Let's assume there exists a "final" mathematical object that goes after every finite item in the generation process (i.e. it is higher than any item in the list, or smaller, or has happened after all of them)... This object doesn't really belong to the infinite generative sequence, it's an item outside all of them, and can't be reached by completing the sequence; it merely exist outside the process and happens to have the property of "dominating" all the items in it. You can assume the existence in the same way you assume the existence of a number which is the square root of -1, or how you define triangles whose angles add up to more or less than 180 degrees. If you do that, this object "at the infinite" can be formally defined and treated axiomatically to find out what mathematical properties it possesses. [1] https://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach https://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach
zests 6y ago
The real issue is that we don't define the real number system before we use it. The fact that 0.999... = 1 is a consequence of a formal definition of decimal numbers. We can create a new definition of decimal numbers that does not satisfy this equation and use it in place of our current one. Let's imagine a new decimal number system with some vague notion of infinitesimal numbers. We lose some properties we enjoy in our current system but all of those properties still hold for numbers with no infinitesimal part. We can still use our every day numbers like nothing has changed yet we also have a notion to describe infinitesimal values. We can make statements like 1/3 is infinitesimally less than 0.333... and carry on like nothing else has changed. Now let's sit someone down, start with the rational numbers, introduce Dedekind cuts to define the real numbers and prove that in the real number system that 0.999... is exactly equal to one. Let's also convince them that the real numbers are the unique complete ordered field and that each of these properties are indispensable. Then they will believe that 0.999... should be equal to 1.
adunk 6y ago
One way to introduce the idea that a number represented as decimal digits can have multiple representations is to talk about the numbers 1 and 1.0 being exactly the same. And that 1.00 is the same as 1. Just like 0, 0.0, and 0.00 are the same number. Most people would agree at this point. Then keep stretching the number of zeroes to 0.000... - which, again, is exactly the same as 0. From there, it is not a huge stretch to be able to go from that 0.000... is another way to write 0, then 0.999... is another way to write 1.
- andrewprock 6y ago
  The real question is what do you get if you add: 0.999 ... infinite number of 9s ... 9 and 0.000 ... infinite number of 0s ... 1
  - aw1621107 6y ago
    > 0.999 ... infinite number of 9s ... 9 > 0.000 ... infinite number of 0s ... 1 Wouldn't these be ill-defined? You can't say "infinite number of 9s" then have the numerical representation terminate with a 9. If the decimal representation terminates, then by definition it isn't infinite.
    - andrewprock 6y ago
      Whether or not a specific construction is well defined depends on the system you are using. This representation is certainly well definable. I think your objection here is in the same vein as those who object to the notion that 0.999... = 1.0 For most people, the concept of an infinite representation is not well defined.
      - aw1621107 6y ago
        > This representation is certainly well definable. Do you mind defining it more formally for me then or pointing me to explanations/systems that would permit such a definition? I'm not really sure what such a formulation might look like, but then again I'm not nearly well-acquainted enough with mathematical topics past what is commonly taught. Does it involve the hyperreals, surreals, or one of the other systems beyond the "standard" reals as mentioned by other comments in the thread? > I think your objection here is in the same vein as those who object to the notion that 0.999... = 1.0 > For most people, the concept of an infinite representation is not well defined. I think (or at least hope) it's a bit more nuanced than that. I understand that some reals with infinite decimal representations can be well defined as an infinite series, and that defining 0.999... using such a series allows other manipulations to be done to complete the proof. However, adding the concept of an "end" to said infinite series kind of breaks that understanding. The translation to an infinite sum no longer seems to hold, so I'm at a bit of a loss. It's also somewhat counterintuitive to have an "end" to infinity, but as the rest of the thread shows intuition isn't always reliable for this kind of thing, especially for those who aren't particularly familiar with more detailed bits of math.
mrighele 6y ago
What set in stone the equality for me was learning about limits and series, because 0.999... is essentially a funny way to represent a serie. Before that, despite accepting the proofs that were given to me, there was always something in the back of the brain telling me "mmmm there is something wrong in that". The only thing close to that was a reasoning like the following: 1 divided by 3 = 1/3 = 0.333..., but then 3 * 0.333... = 0.999... so 1 = 0.999... This comment in the wikipedia page nails it down: "The lower primate in us still resists, saying: .999~ doesn't really represent a number, then, but a process. To find a number we have to halt the process, at which point the .999~ = 1 thing falls apart. Nonsense."
jdoliner 6y ago
When I was a child I was convinced by a pretty simple conversation with my father: Me: 0.9999... is not the same as 1 Him: Well if it's not the same is it more than 1 or less than 1? Me: Less Him: Okay then how much less is it? At this point I started trying to do 1 - 0.999..., using the methods I'd been taught, and after a few iterations of "borrowing" the 1 I realized the answer was 0.000... which I was pretty convinced was equal to 0.
- hota_mazi 6y ago
  Hehe... smart man. Another one is that 1 / 3 * 3 = 1 <==> 0.333... * 3 = 1 <==> 0.999... = 1
  - umvi 6y ago
    "yes but 1/3 does not equal .333... it's just an approximation since there's no perfect way to represent 1/3"
    - jdoliner 6y ago
      If 1/3 doesn't equal .333... then how much do they differ by?
      - brutt 6y ago
        (1/3)/∞
  - brutt 6y ago
    In base 3, 1/10*10=1, 0.1*10=1 What the problem?
analog31 6y ago
I've had the same experience, even debating this topic with engineers. I think there are actually two hang-ups. 1. People have had it drilled into their heads that humans can't comprehend infinity. It was taken for granted by philosophers, that an "infinite regression" is a logical fallacy (e.g., used in a proof by Thomas Aquinas), and that tricks such as infinity and the infinitesimal were not rigorous. Mathematical infinity has been a settled matter for all practical purposes since the early 20th century AFAIK. 2. Related to the above, most people also believe that there is always a gap in any knowledge, and something hiding in that gap. Thus it's perfectly natural to believe that there's something hiding between 0.999... and 1, that we just haven't found yet. Knowing for certain that there is nothing between 0.999... and 1 is regarded as a kind of arrogance. I think the way to approach this with children is to teach math as an abstract topic, that's not necessarily rooted in the objects of everyday life. For instance there's no physics experiment that can test the necessity of any math being carried beyond roughly the 15th decimal place. Yet we enjoy exploring it anyway.
- ff317 6y ago
  Couldn't you formulate a problem for extreme decimal-place accuracy based on the multiplication of errors in a physical process that's repeated in ways that multiply small errors into bigger ones?
  - analog31 6y ago
    That would certainly be an interesting study, I just have never been able to come up with any concrete idea on my own.
- happythomist 6y ago
  > It was taken for granted by philosophers, that an "infinite regression" is a logical fallacy (e.g., used in a proof by Thomas Aquinas)... Aquinas specifically objected to the notion of an essentially ordered infinite causal series. He had no objection to an accidentally ordered infinite causal series or other kinds of infinite series. This distinction is extremely important for the purposes of understanding his proofs of God's existence, and people often unfairly reject his arguments because they conflate the two. More reading here: http://edwardfeser.blogspot.com/2010/08/edwards-on-infinite-causal-series.html http://edwardfeser.blogspot.com/2010/08/edwards-on-infinite-...
hota_mazi 6y ago
Not only can numbers have more than one representation, but they can also have zero! Looking at you, irrational numbers.
- effie 6y ago
  Irrationals have a unique decimal representation in the mathematical sense: given a definition such as $x^2 = 2 $, any digit of the decimal expansion of $x$ can be determined.
  - hota_mazi 6y ago
    > Irrationals have a unique decimal representation in the mathematical sense Not all of them do. Actually, so many don't that mathematically, the number of them that do is zero. Sure, there are exceptions like sqrt(2) and sqrt(3), but there is an uncountable infinity of irrationals between these two numbers that just don't have a representation.
    - effie 6y ago
      For irrationals, the problem with infinite sequence of digit 9 does not occur. So for any given irrational (given its definition), any digit is determined by this definition, because any irrational number can be approximated with arbitrary precision by a rational number (which has unique decimal expansion). If you think otherwise, where is the problem? This is not affected by the fact that irrationals cannot be counted. Given the irrational, a rational close enough exists which has the same decimal expansion for the first n digits, for any n.
coherentpony 6y ago
> There is no proof that will ever satisfy a person dead-set against this. Yes there is. There is a proof that uses only fundamentals of first year university analysis. When you see 0.99999.... this can be written as an infinite sum \sum_{i=0}^\infty 0.9 x 10^{-i}. Truncate the sum and set S_n = \sum_{i=0}^n 0.9 x 10^{-i} and now simply use the rules of arithmetic progressions to get the limit out: 0.1 S_n = \sum_{i=0}^n 0.9 x 10^{-i-1} S_n - 0.1 S_n = 0.9 - 0.9 x 10^{-n-1} 0.9 S_n = 0.9 ( 1 - 0.1^{n+1} ) S_n = 1 - 0.1^{n+1} Now let n tend to infinity to find the limit, which is 1. You don't need to imagine infinity to do any of this.
naringas 6y ago
just tell them to write out the complete infinite sequence of 9's when they are done they will have undesrtood. chuckle
WORLD_ENDS_SOON 6y ago
I wonder if this is related to "intuitionist" math. This is an alternative formulation of math which doesn't have the law of excluded middle, recently discussed on Hacker News relating to this physics research: https://www.quantamagazine.org/does-time-really-flow-new-clues-come-from-a-century-old-approach-to-math-20200407/ https://www.quantamagazine.org/does-time-really-flow-new-clu...
- black_knight 6y ago
  If you want to work with the real numbers intuitionistically (or constructively), you quickly find out that infinite decimal expansions are not what you want. In classical mathematics all the usual definitions of real numbers (decimal, Cauchy sequences and Dedekin cuts) are equivalent. If you overthrow the Law of Excluded middle, these are all different. Infinite decimal expansions are bad intuitionistilcally for several reasons. The first one which come to mind is that you cannot add numbers together. Imagine your numbers started 0.33333 and 0.66666. OK, so far it would seem that the sum would start 0.99999, but somewhere down the line one the firs number could contain two 4s, making a 1 carry all the way up and leaving one behind, so that it should in reality be 1.00000000001… On the other hand there could also show up a 2 later, making it 0.999999998. Thus, you cannot decide weather the first decimals should be 1.00 or 0.99 without looking at infinitely many decimals. And the fact that 0.999… = 1.000 will not help you out, since 1.00000000001 ≠ 0.999999998. Being able to define addition on decimal expansions is equivalent for constructivists to solving the halting problem. It cannot be done. It turns out Cauchy sequences are better behaved, and (with a bit computational improvement) you can make a lot of things work out. See Bihshop's book, Foundations of Constructive Analysis, for details.
  - WORLD_ENDS_SOON 6y ago
    Thank you for the reference! Your explanation makes sense to me. I'm curious if students that struggle with learning decimal expansions and limits are just subconsciously rejecting the non-constructive foundations of the math they're learning. Probably not, but it's fascinating regardless.
jonfw 6y ago
Is this a mental limitation, or is it a simple defense mechanism against diving into rabbit-holes of thought with no end and no real productivity? It seems much easier to come to the conclusion that .99999... and 1 are different numbers, is it really worth the effort to consider otherwise? We create these abstractions to simplify our thought- and analyzing or over-analyzing these simplifications can have the opposite effect.
laichzeit0 6y ago
> We cannot imagine infinity Now try imagining that some infinities are bigger than others: https://en.wikipedia.org/wiki/Aleph_number https://en.wikipedia.org/wiki/Aleph_number
- LunaSea 6y ago
  This is one of my pet peeves in maths. Although I do understand the concepts presented, the notion of "greater" makes no sense when applied to something without boundaries. Yet it's used all the time.
  - petters 6y ago
    Personally, I think it makes perfect sense. Take two sets A and B. If we can assign every element in A to a different one in B, we say that |A|≤|B|. Makes perfect sense for normal, finite sets, right? As it happens, this definition extends to infinite sets as well.
    - erodommoc 6y ago
      Right, but many things make sense for finite sets that don't make sense for infinite sets. Just because you can extend that definition doesn't mean that it's "true" for infinite sets.
  - laichzeit0 6y ago
    Things quickly fall apart if you rely on your intuition. Let's imagine all the odd numbers: 1, 3, 5, etc. Now imagine all the even numbers: 2, 4, 6, etc. Can we agree that there is an "infinite" amount of numbers in each of those groups? Now imagine all the odd numbers and even numbers together. That's also infinite right? Would you say there are more "all numbers" than just "all odd numbers", or would you say that there are an equal number of them? (hint: the answer is equal).
    - erodommoc 6y ago
      Well, the answer is equal because of how you define equality of infinite sets (one-to-one and onto). It's a very useful definition, but it's hardly the only possible definition.
addicted 6y ago
There isn’t a proof because it’s actually kind of arbitrary that 0.999... = 1. Fundamentally, this is true because we chose it to be true. Now, there are good reasons we chose it to be true, and that’s what people usually use as proofs. If it’s not true then a bunch of mathematical expressions become more inconvenient. But there is no reason as such why 0.999... could not have been defined as something that was always < 1. Fundamentally, 0.999... has no intrinsic meaning, and it’s value depends on the meaning we decide to give this representation.
- twiceaday 6y ago
  .999... and 1 exist on a continuous line. If they are different numbers, name a number between them.
  - chottocharaii 6y ago
    ? Just because there is nothing between two numbers does not mean the two numbers are equivalent. What nonsense is this
    - MockObject 6y ago
      Either two numbers are equal, or they're not equal. (Unless we're calling into question the law of excluded middle, but must we?)
    - momentbruh 6y ago
      That is true for the reals
    - Dylan16807 6y ago
      Without using 9999..., please name any two real numbers that don't have a number in between.
  - ben509 6y ago
    { 0.999... | 1 } using surreal number notation.
  - 6y ago
mcguire 6y ago
"The intelligibility of the continuum has been found–many times over–to require that the domain of real numbers be enlarged to include infinitesimals. This enlarged domain may be styled the domain of continuum numbers. It will now be evident that .9999... does not equal 1 but falls infinitesimally short of it. I think that .9999... should indeed be admitted as a number ... though not as a real number."
omgwtfbyobbq 6y ago
I wonder if there's anything I can do with my children to prevent them from being bound by this mental limitation? I would try to explain to them that numbers are a framework for us to understand both the observable universe and abstract ideas, depending on what we're using them for. Like you said, it's hard for people to understand that numbers have multiple representations and to grasp the implications of those representations. I think that if you can communicate that different representations can have the same meaning, accepting those representations when they come across them may be easier. Or, if they're experienced enough with math, I think going through Euler's identity in addition to the link could help. https://en.wikipedia.org/wiki/Euler%27s_identity https://en.wikipedia.org/wiki/Euler%27s_identity
grawprog 6y ago
I'm willing to believe, but every proof on that page I read came down to basically, it might as well be 1, so it is 1. The way i see it, it comes down to accuracy like any of our measurements and falls under rounding error. There's no way we can ever actually measure the infinite amount of space between .999... and 1 so effectively they're the same. As far as math and anything practical and even theoretical is concerned, they're the same, but...conceptually in my brain it just feels that little bit smaller. I know I'm wrong for all intents and purposes, but I dunno, that.
nialv7 6y ago
Infinity will forever be an abstraction, as there is no infinite physical actions you can carry out. Which means what infinite actions actually constitue is purely by definition, as you cannot experimentally verify it. And that's why under some definition it makes sense to say 1+2+3+4+5+... = -1/12
mninm 6y ago
I too was unable to convince my family but based on your comment I just thought of a new (to me) example I might have tried. It leverages the grasp of fractions that you mentioned people already have. Everyone knows that 1/3 = .333... and it can be pretty easily shown that 1/3 + 1/3 = 2/3 = .666... So I would ask them that since .333... + .333... = .666... does it make sense that .333... + .333... + .333... = .999...? And since .333... = 1/3 isn't .333... + .333... + .333... = .999... the same as saying 1/3 + 1/3 + 1/3 = 3/3? And since 3/3 = 1 and 3/3 = .999... it makes sense that 1 = 3/3 = .999... This might work on your kids but in my experience recalcitrant people will either act bored as if they don't care or will try to claim that somehow they understood it all along.
webninja 6y ago
Maybe try expressing it in the form of money? Let's say a gallon of gas is 99 cents with infinitely repeating 9's. 0.99999999999 cents. You're still going to end up paying a dollar a gallon for it because eventually it's going to get rounded off. No gas store operator is going to try to cut a penny for you and give you a fraction of a penny. They could argue that the fraction of a penny becomes infinitely small and that giving you a shaving off the side of a copper penny would be infinitely too large. Now don't mind me while I open up a store where every price tag ends in 0.99...repeating and have a poor college student at the checkout lane with a penny shaver to calm down any rowdy customers he or she can't explain away.

traderjane 6y ago

You must go up to something like limits to make ... meaningful.

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tom-thistime 6y ago

(STATEMENT OF PERSONAL IGNORANCE [SOPI]: Anyone who actually understands this stuff please correct my mistakes below. Thanks.) In the real numbers, which are not always simple or intuitive, 0.99... = 1. That's true and I seem to understand the proof. But the real numbers aren't the only system that might be sitting behind "0.99..." and "1" when I write those symbols down and talk intuitively to people in my family. The reals are just the system we're taught first. I believe there are other systems (I think the surreals are an example) that work just as well for everyday purposes, but where ( my understanding is that) there are numbers that differ from 1 by a value that approaches zero, yet those numbers are not equal to 1. (I've played with the surreals but only as a hobbyist.) If you do calculus in these other numbers, I think physically meaningful problems will still yield the same answers. (For example Zeno's Paradoxes are still not an excuse for failure to attend school.) But it isn't a law of nature, I think, that all number systems that can hold 0.999... and 1 must make them equal.

Scarblac 6y ago
A number is just a number, it doesn't approach anything. A series can approach something, but a number can't. In any system where 0.99... is valid notation for a number, it doesn't approach anything.
- tom-thistime 6y ago
  Sure. Instead of "0.99..." please substitute lim n->inf sum(1..n)(9 times 10^-n). The point I'm making is that the "obvious truth" 0.99... = 1 that we're all talking about depends on the assumption that we're working in the real numbers. I claim that the real numbers are not something intuitively obvious to every sufficiently intelligent person; instead they are kind of weird and technical. I go on to claim, though I'm more unsure of this, that the real numbers are not even the only way to make calculus work.
  - ramshorns 6y ago
    Well, a limit of a series is just a number and doesn't approach anything either. If the series approaches something, we say the limit exists and is equal to that thing. Anyway, in the surreal numbers you could probably make up a notation where 0.999... actually denotes 1 - ε or something. But I daresay it might not be very useful because then how do you denote 1 - ε/2 or anything else.
    - ramshorns 6y ago
      Oh, I figured it out, kind of. [I don't really know what I'm talking about either.] 1.000...0 = 1 1.000...05 = 1 + ε/2 1.000...1 = 1 + ε 0.999...8 = 1 - 2ε 0.999...9 = 1 - ε 0.999...98 = 1 - ε/5 . . .
      - saalweachter 6y ago
        > 0.999...8 = 1 - 2ε > 0.999...98 = 1 - ε/5 cough

WilliamEdward 6y ago

This is essentially just a matter of limits, without which the world wouldn't make any sense. So you must explain that if you move your hand closer to an object, technically you are halving the distance infinitely many times, but if 0.999... != 1 then your hand would never touch anything.

lmkg 6y ago

It's more fundamental: People seem to have the intuition that the decimal representation of a number is a number. I don't know if it's because decimals resemble the ntural numbers or what, but decimals seem to have a primacy for people that fractions do not. The idea that there's a gap between the symbol for a thing and the thing itself is the stumbling block.

Sharlin 6y ago
Yes, but at the same time it is common for people to insist that 0.999… only "approaches" unity as if it were a series approaching a limit, rather than an unique number. Intuition is a funny thing.
- tom-thistime 6y ago
  Saying "a number isn't a limit" is true, but it's only really relevant if you're talking to someone who genuinely has no idea what limits are. In actual math the number 0.99... can be defined as the limit of a series.
  - Sharlin 6y ago
    Every number is a limit, yes, but people think of 0.999… as a series. Not rigorously, of course, but that's a common argument even by people (perhaps especially by people) who have a highschool or even math-minor level understanding of series and limits.
tom-thistime 6y ago
Possibly people are looking at two different symbols and asking "can you show me logically why those are equal." If they're given a definition of "equal" and they still object, that's a different problem. I have this problem every time I play with group theory again. You get the axioms for a group, which say there is some identity but don't explicity require the identity to be unique. You can easily prove that the identity of a group is unique ... so long as you define "unique" to mean "if element e1 and element e2 are equal, then we say they are the same element." You could count things differently and say the identity is "not unique", it would just lead to a lot of stupid and un-illuminating consequences.

gspr 6y ago

> There is no proof that will ever satisfy a person dead-set against this. Indeed. I've torn my hair out trying to convince smart people with PhDs in hard sciences and had to give up in frustration. I usually find that the most success can be had by kicking the ball to them immediately and having them define what they actually mean when they say "0.999…". If we're going to debate whether that thing equals another thing, we better make sure we know what we're talking about. Inevitably, this either causes the dead-set person to give up, or give a myriad of definitions that are either meaningless, ill-defined, or causes them to realize that they don't actually know what "0.999…" means (or what they want it to mean). It is hard to have the patience to chase down the consequences of their ill-fated definitions, though.

2OEH8eoCRo0 6y ago
Ask for a number between .9 repeated and 1
- chris_wot 6y ago
  So does this mean that an infinitely small number is zero? As in 1/∞ ?
  - thaumasiotes 6y ago
    You can read about this in any work on nonstandard analysis. ("Nonstandard" is just the name, much like "imaginary" numbers.) An infinitely small number is zero when projected onto the real number line. If you introduce an infinitesimal quantity to the reals, then for every number there is a unique real number to which that first number is infinitely close (that is, the difference between them is infinitesimal). You can use that real number as a (good) approximation of all the nonstandard numbers in its halo. (As long as you're comparing it to other real numbers.)
  - selectodude 6y ago
    In calculus, yes.
  - GolDDranks 6y ago

Igelau 6y ago

What's interesting is that people pretty quickly become comfortable with the idea that 1/3 = 0.333… So using that as a foothold, we can express 1/3 + 1/3 + 1/3 as 0.333… + 0.333… + 0.333… and it should be pretty easy to digest. At once we can see that in this little zone we've defined, 1 and 0.999… mean the same thing. Not a rigorous proof, and one or two people will probably bring up whataboutisms like "that's just because the calculator can't do stuff!" but it should at least be proof of comfort for most people.

tom-thistime 6y ago
This is a really good point. Maybe the problem is how we define equality. What's the test for when two numbers are equal? People accept that 1/3 = 0.333333... The same people don't always seem to accept that 3*0.33333... = 1. Well, how are we defining "equals"? If we can give that definition in black and white, I think that may help.
- scrimps 6y ago
  If 30.33333... = 1, then 30.33333... != 0.9999..., then 0.9999... != 1
- esmi 6y ago
  > What's the test for when two numbers are equal? I put this in another comment but: For the reals: eliminate a < b and a > b then conclude a = b.

TomGullen 6y ago

I find the algebraic way convinces most people: x = 0.9999.. 10x = 9.9999... 10x - x = 9 9x = 9 x = 1

solotronics 6y ago
black magic! I wonder if there are any programming languages that are able to handle this properly?
- 2OEH8eoCRo0 6y ago
  https://www.wolframalpha.com/input/?i=.9...+%3D+1 https://www.wolframalpha.com/input/?i=.9...+%3D+1
  - tygrak 6y ago
    https://www.wolframalpha.com/input/?i=10*0.9...-0.9.. https://www.wolframalpha.com/input/?i=10*0.9...-0.9...
- brlewis 6y ago
  There are programming languages with rational number types, but none that I know of that represent numbers as repeating decimals.
taylodl 6y ago
That's how I was taught in my Algebra class
thenewwazoo 6y ago
Because I wondered, it’s not a trick: x = 0.444444... 10x = 4.444444... 10x - x = 4 9x = 4 x = 4/9 = 0.444444...
- phaemon 6y ago
  All the single digit repeating decimals are x/9. 0.111... == 1/9 0.222... == 2/9 ... 0.888... == 8/9 0.999... == 9/9 :)

Okx 6y ago

I usually say "If and only if two numbers are different, then you can find a number between them". People often accept this axiom. Then, I offer them to find a number between 0.999... and 1.

Falling3 6y ago
That's a great approach
tom-thistime 6y ago
This works because you have defined what equality means. We all wanna talk infinity because it sounds more exciting, but I think everybody gets "infinitely close to 1" pretty well intuitively. What they don't get is whether "infinitely close to 1" means "equal to 1". That could happen because these people are stupid.[note] But it could also happen if nobody has defined equality. [note]for example even highly educated people maybe don't listen, which is functionally a lot like being stupid.
solotronics 6y ago
is there any difference between a black hole and nothing? (somewhat joking but I was thinking of a physical analogy of the limit approaching zero)
- caspper69 6y ago
  A black hole has mass. Nothing has no mass. (also a somewhat joking answer)
- raattgift 6y ago
  Yes, there's a difference. Firstly, though, there are multiple different types of black hole, from the theoretical to the astrophysical. We must narrow your question down to have any hope of a good answer. The simplest theoretical black hole, the Schwarzschild black hole, has one variable -- the central mass -- which must be positive. If we set the central mass in a Schwarzschild spacetime to zero, then we have Minkowski spacetime: no curvature, no horizon, no black hole. The Schwarzschild spacetime is completely empty except for the central mass, which is constant and located at an infinitesimally small point at all times. The Minkowski spacetime is completely empty everywhere and at all times. The symmetries of Schwarzschild and Minkowski spacetime are different, and if one were to probe the spacetimes in question with a Synge curvature detector [1], we would quickly discover which we were probing if our probes happened to be placed close to the central mass, and eventually if they were placed far from the central mass. If one placed the probes infinitely far from the central mass, it would take an infinitely long time to distinguish the presence of the central mass (which makes spacetime non-Minkowski); but these spacetimes are eternal anyway, so that's OK. So that's almost a "yes" to there being a theoretical black hole analogy between (1-) 0.999... and (1-) 1. I would not call this a physical analogy since neither Minkowski spacetime nor Schwarzschild spacetime is at all physical. Nature is full of stress-energy (gas, dust, ...) any of which breaks the vacuum condition of these spacetimes, there seem to be a lot of astrophysical black holes at the centres of galaxies and individual/binary stars that have become black holes, and even a two-black-hole universe is markedly different than a Schwarzschild spacetime. Additionally, these astrophysical black holes are not eternal, unlike Schwarzschild. In particular, the stellar mass ones were once stars, and the galaxy-centre ones at least had less mass in the past. These last conditions alone are substantial deviations from Schwarzschild that are even more obviously not Minkowski (e.g. if you put probe finitely but sufficiently far away, you could see an image of the radiant precursor star rather than the black hole!). Finally, in our physical galaxy the answer to your question is a big "yes!". The observed orbits of these stars [2] would be noticeably different if the central mass in the Milky Way's central parsec were anything but a black hole, and would be even more different if that central mass were not there at all. - -- [1] Synge, J.L., _Gravitation. The General Theory_, ch. XI §8, "A five-point curvature detector". [2] http://www.astro.ucla.edu/~ghezgroup/gc/animations.html http://www.astro.ucla.edu/~ghezgroup/gc/animations.html and http://www.astro.ucla.edu/~ghezgroup/gc/blackhole.html http://www.astro.ucla.edu/~ghezgroup/gc/blackhole.html

sorokod 6y ago

People mistakenly believe that decimal numbers don't have multiple representations - which, in a way is correct. It is correct if you take the limit, people usually do not.

0xff00ffee 6y ago

Yep. Agree 100%. It is like the blue dress. I think the problem is the repeating function. Infinite things are non-intuitive and should be presented differently. Even here on HN you still see people confused about "convergence" and "identity". 0.999... doesn't CONVERGE, it literally is 1. I suspect this persists even with students that have had second year college calculus that discusses convergent series and sums.

tom-thistime 6y ago
Fine. Define 0.999... as the limit of the series sum(n=1 ... N)(10^-n), as N-> infinity. This is standard high school calculus. "Number" and "series" and "limit" and "convergence" don't all mean the same thing. However this number is defined as the limit of a convergent series. So the question really is meaningful. (One clue that this question is meaningful is the amount of space introductory calculus textbooks use to address it.) Because I can still ask, in black and white, what law of "equality" do I use to establish that my limit equals 1? (It does, if I import the definition of "equality" from the real numbers. That's what they do in calculus class. )
- 0xff00ffee 6y ago
  I'm not sure if you are asking for an answer or a rhetorical question? I'll assume the former. Your terms are bit jumbled, so let's keep it simple: you're asking how to prove if an infinite sum converges and what its value is. Convergence proofs require analytic thought: meaning there may not be an immediate look-up. You need to convert the problem into the known corpus of convergent sums or use one of many tests (bounds test, integral test, etc) to show it converges analytically. Which you only learn through experience and memorization (unless you want to re-prove hundreds of series... maybe you do!) Fortunately this one is easily re-written as a known convergent sum. First, you missed a term in your sum (9), re-written here: sum(n=1..inf) 9 * 10^-n Step 1: you pull out the 9 and it becomes 1/10+1/100+1/1000... Step 2: Then we shift to n=0 by subtracting 1/10^0 from the series so that it is in the form n=0..k-1 1/10^0 + 1/10 + 1/100 + 1/1000 + ... + 1/10^-n - 1/10^0 Step 3: Now we've got ourselves a geometric series of just 1/10^n .. wikipedia does a great job explaining the sum convergence for GS from n=0...inf: https://en.wikipedia.org/wiki/Geometric_series https://en.wikipedia.org/wiki/Geometric_series Step 4: compute geometric convergence (1-r^n)/(1-r) = (1-(1/10)^n)/(1-1/10) = 1/(1-1/10) = 10/9 So we have 10/9 as the solution to Sum[n=0...inf](1/10^n) Step 5: the remaining arithmetic Now subtract our 1/10^0 ... and then * 9 = 1

dubcanada 6y ago

I feel like most people are not answering this, they are giving the proof in a different way. Answering how to teach a child not to be bound to 100%'s is really hard. I personally would say just let them explore on their own, teaching a person that 0.99999 = 1 results in the same as if you taught them that 0.99999 != 1. You need to teach them that all sciences and maths are changing constantly, what might be a fact today could change tomorrow. You need to teach them that anything can become wrong or right as we progress and to be open about accepting new information, while being hesitant enough not to not succumb to false/fake information. That's a very hard lesson to learn and an even harder one to practice. But one that I think a lot of people need to learn.

lukifer 6y ago

I didn't grok infinity until I started thinking in terms of verbs rather than nouns. As a static number, the concept of infinity makes no sense; but once reimagined as a process (start counting up from 1, and never stop), all apparent paradoxes disappear. This is the inverse problem: it could just as easily be reframed as 0.000...0001 = 0. Defined as static nouns (does such a thing exist in nature?), it's seemingly paradoxical, and fascinatingly debatable in a "is a hot dog a sandwich" sort of way. But reframe it as a process (or as code), and all confusion disappears: for how many loops would you like to proceed? If you never stop, 0.99999... clearly approaches 1, without ever reaching it, and asking if they're the "same" is as academic as asking if the Ship of Theseus is the same ship, or if an electron is the same entity from one picosecond to the next.

ta1234567890 6y ago
> As a static number, the concept of infinity makes no sense; but once reimagined as a process Super insightful. That's the key right there. The same concept can also be applied to the physical world. Things are not static, they are in constant flux, everything is a process in motion.
- thom 6y ago
  Yes, although for me this conception of infinity as a process also captures why there are probably no actual infinite things in the universe, only in silly games with numbers.
  - effie 6y ago
    Could you elaborate? Why infinite process cannot be actual thing in the universe? It's not like we know the start and the end date of the universe...
TuringTest 6y ago
I find that "infinite as an endless process" concept intuitively very heplful as well. However, reading Gödel, Escher, Bach[1] showed me that there's another, more static logical interpretation of infinity which also comes handy. In an infinite process, you can always take "one more step" to create the item after that. Let's assume there exists a "final" mathematical object that goes after every finite item in the generation process (i.e. it is higher than any item in the list, or smaller, or has happened after all of them)... This object doesn't really belong to the infinite generative sequence, it's an item outside all of them, and can't be reached by completing the sequence; it merely exist outside the process and happens to have the property of "dominating" all the items in it. You can assume the existence in the same way you assume the existence of a number which is the square root of -1, or how you define triangles whose angles add up to more or less than 180 degrees. If you do that, this object "at the infinite" can be formally defined and treated axiomatically to find out what mathematical properties it possesses. [1] https://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach https://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach

zests 6y ago

The real issue is that we don't define the real number system before we use it. The fact that 0.999... = 1 is a consequence of a formal definition of decimal numbers. We can create a new definition of decimal numbers that does not satisfy this equation and use it in place of our current one. Let's imagine a new decimal number system with some vague notion of infinitesimal numbers. We lose some properties we enjoy in our current system but all of those properties still hold for numbers with no infinitesimal part. We can still use our every day numbers like nothing has changed yet we also have a notion to describe infinitesimal values. We can make statements like 1/3 is infinitesimally less than 0.333... and carry on like nothing else has changed. Now let's sit someone down, start with the rational numbers, introduce Dedekind cuts to define the real numbers and prove that in the real number system that 0.999... is exactly equal to one. Let's also convince them that the real numbers are the unique complete ordered field and that each of these properties are indispensable. Then they will believe that 0.999... should be equal to 1.

adunk 6y ago

One way to introduce the idea that a number represented as decimal digits can have multiple representations is to talk about the numbers 1 and 1.0 being exactly the same. And that 1.00 is the same as 1. Just like 0, 0.0, and 0.00 are the same number. Most people would agree at this point. Then keep stretching the number of zeroes to 0.000... - which, again, is exactly the same as 0. From there, it is not a huge stretch to be able to go from that 0.000... is another way to write 0, then 0.999... is another way to write 1.

andrewprock 6y ago
The real question is what do you get if you add: 0.999 ... infinite number of 9s ... 9 and 0.000 ... infinite number of 0s ... 1
- aw1621107 6y ago
  > 0.999 ... infinite number of 9s ... 9 > 0.000 ... infinite number of 0s ... 1 Wouldn't these be ill-defined? You can't say "infinite number of 9s" then have the numerical representation terminate with a 9. If the decimal representation terminates, then by definition it isn't infinite.
  - andrewprock 6y ago
    Whether or not a specific construction is well defined depends on the system you are using. This representation is certainly well definable. I think your objection here is in the same vein as those who object to the notion that 0.999... = 1.0 For most people, the concept of an infinite representation is not well defined.
    - aw1621107 6y ago
      > This representation is certainly well definable. Do you mind defining it more formally for me then or pointing me to explanations/systems that would permit such a definition? I'm not really sure what such a formulation might look like, but then again I'm not nearly well-acquainted enough with mathematical topics past what is commonly taught. Does it involve the hyperreals, surreals, or one of the other systems beyond the "standard" reals as mentioned by other comments in the thread? > I think your objection here is in the same vein as those who object to the notion that 0.999... = 1.0 > For most people, the concept of an infinite representation is not well defined. I think (or at least hope) it's a bit more nuanced than that. I understand that some reals with infinite decimal representations can be well defined as an infinite series, and that defining 0.999... using such a series allows other manipulations to be done to complete the proof. However, adding the concept of an "end" to said infinite series kind of breaks that understanding. The translation to an infinite sum no longer seems to hold, so I'm at a bit of a loss. It's also somewhat counterintuitive to have an "end" to infinity, but as the rest of the thread shows intuition isn't always reliable for this kind of thing, especially for those who aren't particularly familiar with more detailed bits of math.

mrighele 6y ago

What set in stone the equality for me was learning about limits and series, because 0.999... is essentially a funny way to represent a serie. Before that, despite accepting the proofs that were given to me, there was always something in the back of the brain telling me "mmmm there is something wrong in that". The only thing close to that was a reasoning like the following: 1 divided by 3 = 1/3 = 0.333..., but then 3 * 0.333... = 0.999... so 1 = 0.999... This comment in the wikipedia page nails it down: "The lower primate in us still resists, saying: .999~ doesn't really represent a number, then, but a process. To find a number we have to halt the process, at which point the .999~ = 1 thing falls apart. Nonsense."

jdoliner 6y ago

When I was a child I was convinced by a pretty simple conversation with my father: Me: 0.9999... is not the same as 1 Him: Well if it's not the same is it more than 1 or less than 1? Me: Less Him: Okay then how much less is it? At this point I started trying to do 1 - 0.999..., using the methods I'd been taught, and after a few iterations of "borrowing" the 1 I realized the answer was 0.000... which I was pretty convinced was equal to 0.

hota_mazi 6y ago
Hehe... smart man. Another one is that 1 / 3 * 3 = 1 <==> 0.333... * 3 = 1 <==> 0.999... = 1
- umvi 6y ago
  "yes but 1/3 does not equal .333... it's just an approximation since there's no perfect way to represent 1/3"
  - jdoliner 6y ago
    If 1/3 doesn't equal .333... then how much do they differ by?
    - brutt 6y ago
      (1/3)/∞
- brutt 6y ago
  In base 3, 1/10*10=1, 0.1*10=1 What the problem?

analog31 6y ago

I've had the same experience, even debating this topic with engineers. I think there are actually two hang-ups. 1. People have had it drilled into their heads that humans can't comprehend infinity. It was taken for granted by philosophers, that an "infinite regression" is a logical fallacy (e.g., used in a proof by Thomas Aquinas), and that tricks such as infinity and the infinitesimal were not rigorous. Mathematical infinity has been a settled matter for all practical purposes since the early 20th century AFAIK. 2. Related to the above, most people also believe that there is always a gap in any knowledge, and something hiding in that gap. Thus it's perfectly natural to believe that there's something hiding between 0.999... and 1, that we just haven't found yet. Knowing for certain that there is nothing between 0.999... and 1 is regarded as a kind of arrogance. I think the way to approach this with children is to teach math as an abstract topic, that's not necessarily rooted in the objects of everyday life. For instance there's no physics experiment that can test the necessity of any math being carried beyond roughly the 15th decimal place. Yet we enjoy exploring it anyway.

ff317 6y ago
Couldn't you formulate a problem for extreme decimal-place accuracy based on the multiplication of errors in a physical process that's repeated in ways that multiply small errors into bigger ones?
- analog31 6y ago
  That would certainly be an interesting study, I just have never been able to come up with any concrete idea on my own.
happythomist 6y ago
> It was taken for granted by philosophers, that an "infinite regression" is a logical fallacy (e.g., used in a proof by Thomas Aquinas)... Aquinas specifically objected to the notion of an essentially ordered infinite causal series. He had no objection to an accidentally ordered infinite causal series or other kinds of infinite series. This distinction is extremely important for the purposes of understanding his proofs of God's existence, and people often unfairly reject his arguments because they conflate the two. More reading here: http://edwardfeser.blogspot.com/2010/08/edwards-on-infinite-causal-series.html http://edwardfeser.blogspot.com/2010/08/edwards-on-infinite-...

hota_mazi 6y ago

Not only can numbers have more than one representation, but they can also have zero! Looking at you, irrational numbers.

effie 6y ago
Irrationals have a unique decimal representation in the mathematical sense: given a definition such as $x^2 = 2 $, any digit of the decimal expansion of $x$ can be determined.
- hota_mazi 6y ago
  > Irrationals have a unique decimal representation in the mathematical sense Not all of them do. Actually, so many don't that mathematically, the number of them that do is zero. Sure, there are exceptions like sqrt(2) and sqrt(3), but there is an uncountable infinity of irrationals between these two numbers that just don't have a representation.
  - effie 6y ago
    For irrationals, the problem with infinite sequence of digit 9 does not occur. So for any given irrational (given its definition), any digit is determined by this definition, because any irrational number can be approximated with arbitrary precision by a rational number (which has unique decimal expansion). If you think otherwise, where is the problem? This is not affected by the fact that irrationals cannot be counted. Given the irrational, a rational close enough exists which has the same decimal expansion for the first n digits, for any n.

coherentpony 6y ago

> There is no proof that will ever satisfy a person dead-set against this. Yes there is. There is a proof that uses only fundamentals of first year university analysis. When you see 0.99999.... this can be written as an infinite sum \sum_{i=0}^\infty 0.9 x 10^{-i}. Truncate the sum and set S_n = \sum_{i=0}^n 0.9 x 10^{-i} and now simply use the rules of arithmetic progressions to get the limit out: 0.1 S_n = \sum_{i=0}^n 0.9 x 10^{-i-1} S_n - 0.1 S_n = 0.9 - 0.9 x 10^{-n-1} 0.9 S_n = 0.9 ( 1 - 0.1^{n+1} ) S_n = 1 - 0.1^{n+1} Now let n tend to infinity to find the limit, which is 1. You don't need to imagine infinity to do any of this.

naringas 6y ago

just tell them to write out the complete infinite sequence of 9's when they are done they will have undesrtood. chuckle

WORLD_ENDS_SOON 6y ago

I wonder if this is related to "intuitionist" math. This is an alternative formulation of math which doesn't have the law of excluded middle, recently discussed on Hacker News relating to this physics research: https://www.quantamagazine.org/does-time-really-flow-new-clues-come-from-a-century-old-approach-to-math-20200407/ https://www.quantamagazine.org/does-time-really-flow-new-clu...

black_knight 6y ago
If you want to work with the real numbers intuitionistically (or constructively), you quickly find out that infinite decimal expansions are not what you want. In classical mathematics all the usual definitions of real numbers (decimal, Cauchy sequences and Dedekin cuts) are equivalent. If you overthrow the Law of Excluded middle, these are all different. Infinite decimal expansions are bad intuitionistilcally for several reasons. The first one which come to mind is that you cannot add numbers together. Imagine your numbers started 0.33333 and 0.66666. OK, so far it would seem that the sum would start 0.99999, but somewhere down the line one the firs number could contain two 4s, making a 1 carry all the way up and leaving one behind, so that it should in reality be 1.00000000001… On the other hand there could also show up a 2 later, making it 0.999999998. Thus, you cannot decide weather the first decimals should be 1.00 or 0.99 without looking at infinitely many decimals. And the fact that 0.999… = 1.000 will not help you out, since 1.00000000001 ≠ 0.999999998. Being able to define addition on decimal expansions is equivalent for constructivists to solving the halting problem. It cannot be done. It turns out Cauchy sequences are better behaved, and (with a bit computational improvement) you can make a lot of things work out. See Bihshop's book, Foundations of Constructive Analysis, for details.
- WORLD_ENDS_SOON 6y ago
  Thank you for the reference! Your explanation makes sense to me. I'm curious if students that struggle with learning decimal expansions and limits are just subconsciously rejecting the non-constructive foundations of the math they're learning. Probably not, but it's fascinating regardless.

jonfw 6y ago

Is this a mental limitation, or is it a simple defense mechanism against diving into rabbit-holes of thought with no end and no real productivity? It seems much easier to come to the conclusion that .99999... and 1 are different numbers, is it really worth the effort to consider otherwise? We create these abstractions to simplify our thought- and analyzing or over-analyzing these simplifications can have the opposite effect.

laichzeit0 6y ago

> We cannot imagine infinity Now try imagining that some infinities are bigger than others: https://en.wikipedia.org/wiki/Aleph_number https://en.wikipedia.org/wiki/Aleph_number

LunaSea 6y ago
This is one of my pet peeves in maths. Although I do understand the concepts presented, the notion of "greater" makes no sense when applied to something without boundaries. Yet it's used all the time.
- petters 6y ago
  Personally, I think it makes perfect sense. Take two sets A and B. If we can assign every element in A to a different one in B, we say that |A|≤|B|. Makes perfect sense for normal, finite sets, right? As it happens, this definition extends to infinite sets as well.
  - erodommoc 6y ago
    Right, but many things make sense for finite sets that don't make sense for infinite sets. Just because you can extend that definition doesn't mean that it's "true" for infinite sets.
- laichzeit0 6y ago
  Things quickly fall apart if you rely on your intuition. Let's imagine all the odd numbers: 1, 3, 5, etc. Now imagine all the even numbers: 2, 4, 6, etc. Can we agree that there is an "infinite" amount of numbers in each of those groups? Now imagine all the odd numbers and even numbers together. That's also infinite right? Would you say there are more "all numbers" than just "all odd numbers", or would you say that there are an equal number of them? (hint: the answer is equal).
  - erodommoc 6y ago
    Well, the answer is equal because of how you define equality of infinite sets (one-to-one and onto). It's a very useful definition, but it's hardly the only possible definition.

addicted 6y ago

There isn’t a proof because it’s actually kind of arbitrary that 0.999... = 1. Fundamentally, this is true because we chose it to be true. Now, there are good reasons we chose it to be true, and that’s what people usually use as proofs. If it’s not true then a bunch of mathematical expressions become more inconvenient. But there is no reason as such why 0.999... could not have been defined as something that was always < 1. Fundamentally, 0.999... has no intrinsic meaning, and it’s value depends on the meaning we decide to give this representation.

twiceaday 6y ago
.999... and 1 exist on a continuous line. If they are different numbers, name a number between them.
- chottocharaii 6y ago
  ? Just because there is nothing between two numbers does not mean the two numbers are equivalent. What nonsense is this
  - MockObject 6y ago
    Either two numbers are equal, or they're not equal. (Unless we're calling into question the law of excluded middle, but must we?)
  - momentbruh 6y ago
    That is true for the reals
  - Dylan16807 6y ago
    Without using 9999..., please name any two real numbers that don't have a number in between.
- ben509 6y ago
  { 0.999... | 1 } using surreal number notation.
- 6y ago

mcguire 6y ago

"The intelligibility of the continuum has been found–many times over–to require that the domain of real numbers be enlarged to include infinitesimals. This enlarged domain may be styled the domain of continuum numbers. It will now be evident that .9999... does not equal 1 but falls infinitesimally short of it. I think that .9999... should indeed be admitted as a number ... though not as a real number."

omgwtfbyobbq 6y ago

I wonder if there's anything I can do with my children to prevent them from being bound by this mental limitation? I would try to explain to them that numbers are a framework for us to understand both the observable universe and abstract ideas, depending on what we're using them for. Like you said, it's hard for people to understand that numbers have multiple representations and to grasp the implications of those representations. I think that if you can communicate that different representations can have the same meaning, accepting those representations when they come across them may be easier. Or, if they're experienced enough with math, I think going through Euler's identity in addition to the link could help. https://en.wikipedia.org/wiki/Euler%27s_identity https://en.wikipedia.org/wiki/Euler%27s_identity

grawprog 6y ago

I'm willing to believe, but every proof on that page I read came down to basically, it might as well be 1, so it is 1. The way i see it, it comes down to accuracy like any of our measurements and falls under rounding error. There's no way we can ever actually measure the infinite amount of space between .999... and 1 so effectively they're the same. As far as math and anything practical and even theoretical is concerned, they're the same, but...conceptually in my brain it just feels that little bit smaller. I know I'm wrong for all intents and purposes, but I dunno, that.

nialv7 6y ago

Infinity will forever be an abstraction, as there is no infinite physical actions you can carry out. Which means what infinite actions actually constitue is purely by definition, as you cannot experimentally verify it. And that's why under some definition it makes sense to say 1+2+3+4+5+... = -1/12

mninm 6y ago

I too was unable to convince my family but based on your comment I just thought of a new (to me) example I might have tried. It leverages the grasp of fractions that you mentioned people already have. Everyone knows that 1/3 = .333... and it can be pretty easily shown that 1/3 + 1/3 = 2/3 = .666... So I would ask them that since .333... + .333... = .666... does it make sense that .333... + .333... + .333... = .999...? And since .333... = 1/3 isn't .333... + .333... + .333... = .999... the same as saying 1/3 + 1/3 + 1/3 = 3/3? And since 3/3 = 1 and 3/3 = .999... it makes sense that 1 = 3/3 = .999... This might work on your kids but in my experience recalcitrant people will either act bored as if they don't care or will try to claim that somehow they understood it all along.

webninja 6y ago

Maybe try expressing it in the form of money? Let's say a gallon of gas is 99 cents with infinitely repeating 9's. 0.99999999999 cents. You're still going to end up paying a dollar a gallon for it because eventually it's going to get rounded off. No gas store operator is going to try to cut a penny for you and give you a fraction of a penny. They could argue that the fraction of a penny becomes infinitely small and that giving you a shaving off the side of a copper penny would be infinitely too large. Now don't mind me while I open up a store where every price tag ends in 0.99...repeating and have a poor college student at the checkout lane with a penny shaver to calm down any rowdy customers he or she can't explain away.