0.999...= 1
en.wikipedia.org
en.wikipedia.org
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?
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.
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.
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.
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
.
.
.> 0.999...98 = 1 - ε/5
cough
I don't know whether repeating decimals are useful for testing equality of surreal numbers.
All I'm trying to say is we're all talking about anything and everything except the definition "when are two real numbers equal?"
And then we're saying people who don't understand the consequences of that definition are kind of dummies ... while we continue to not actually say what the definition is.
That claim is certainly true. The proof is that calculus (Analysis) exists for the complex number system too. Although I don’t think that’s what you meant and I doubt the complex number system is “more intuitive.” Just out of curiosity, have you heard of real analysis? How do you define calculus?
I'm eager to be corrected if you can tell me something I said that's wrong. I'm not interested in gradually upping the ante with you until it's clear who really has more math background.
https://en.wikipedia.org/wiki/Nonstandard_calculus
It is based on the hyperreal numbers:
https://en.wikipedia.org/wiki/Hyperreal_number
Practically speaking, I don't think it buys you anything over traditional calculus/analysis. It's just pointing out that there are alternative approaches to formalizing calculus.
All I'm saying is [SOPI below] it's all a little more technical than the junior high school proof. For example if 23+6\epsilon = 23, then how do I define 23 + 6\epsilon - 23? I can choose different approaches here, but "zero" is going to be pretty inconvenient when I go to do an integral.
[SOPI] Statement of Personal Ignorance. I don't quite know what I'm talking about. If you do know, please step in and help correct me.
So if we go and define some other funky "infinitesimal" \epsilon != 0 in the surreals, we have to be careful. Apparently people have done that kind of thing successfully, but it took a long time after Cauchy for that to happen.
0.999... + 0.000...1 = 1
0.000...1 = 1/∞
0.999... = 1 - 1/∞
1/∞ = 0
Then you accept that
∞ * 0 = 1
But the definition of 0 is exactly that anything multiplied by it must be 0. So this cannot be true.
To take a more verbal route: you cannot take nothingness and repeat it. Repeating (or multiplying) nothingness (or 0) is fundamentally nonsense.
Programmer explanation: one cannot loop through `null` even once, let alone a large number.
Quote:
There are also representations like
{ 0, 1, 2, 3, … | } = ω
{ 0 | 1, 1/2, 1/4, 1/8, … } = ε
where ω is a transfinite number greater than all integers and ε is an infinitesimal greater than 0 but less than any positive real number. Moreover, the standard arithmetic operations (addition, subtraction, multiplication, and division) can be extended to these non-real numbers in a manner that turns the collection of surreal numbers into an ordered field, so that one can talk about 2ω or ω − 1 and so forth.
If it terminates it’s not an infinite series. In any case, if you add any finite number to .99999... it’ll be equal to 1 + that number.
1 / 3 = 0.33333....
2 / 3 = 0.66666....
Some people don't like that one. They might like this one better:
1 / 11 = 0.0909090909...
10 * 0.0909090909... = 0.90909090...
10 * 0.0909090909... + 0.0909090909... = 0.90909090... + 0.0909090909... = 0.9999999999...
This works for any repeating fraction. You can do it with 1/7 and 6/7. You add the decimal representations of the numbers up and the value will be 0.99999...
Technically, it works for any repeating fraction in any base. This is great because a lot of fractions are only repeating fractions in certain bases. So if 0.1 in base 10 is a repeating decimal in base 2 (it is) then you can show that (in decimal) 0.1 + 9 * 0.1 will represent (in binary) 0.11111...., which is equal to 1.
The issue is that 1 / 11 + 10 / 11 (in decimal) must still equal 1 in ALL bases. Well, guess what? In Base 11 the decimal looks like:
0.1 + 0.A = 1.0
In surreal numbers there's a number called ε a number infinitely close to 0 (but larger than it), so what you would think that 0.9999.. represent is actually written 1-ε maybe?
But there's another number ε/2 that is between 0 and ε; 1-ε/2 is even closer to 1 than 1-ε is. Indeed, there are infinite numbers infinitely close to 1! (and none is really represented by 0.999...)
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.
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.
e.g. 5 = 5 is true under judgemental and propositional equality, whereas x + 2 = 2 + x is only true under propositional
On the other hand, hypothetically, if it were the case that we were missing a key definition that's needed for some proof, and someone didn't believe that proof, then maybe we would't quite know enough yet to decide that the person is in mental kindergarten. A better first step for us might be to supply the missing definition.
Lack of fingers was another big spur to the development of camel intellect. Human mathematical development had always been held back by everyone’s instinctive tendency, when faced with something really complex in the way of triform polynomials or parametric differentials, to count fingers. Camels started from the word go by counting numbers.
Computer programmers (and historians!) have a similar problem with dates, and in particular with issues like daylight saving time and time zones. I think a lot of the problem is that again there's no way to talk about a particular instant of time without adopting some necessarily arbitrary and relative nomenclature like “January 17, 1706 at 09:37 local time in Boston”. But when was this _really_? Unfortunately there is no “really”. (“Oh, you mean Ramadan 1117 AH, now I understand.”)
...which is true for any base-n representation where n is a natural (even rational) number. And that's kind of implied most of the time, so it seems like a useful definition. Where would this lead to problems?
Or a rational number whose decimal representation doesn't repeat?
It's not like those people haven't worked with an irrational base before, either! Radians have an irrational base. When we talk about 2π radians, or 1/4π radians, that's exactly what we're doing.
Usually we define it like this: an irrational number is one that isn't a quotient of two integers. Starting from that definition, we then prove the _theorem_ that the decimal representation a number repeats if and only if the number is rational.
It's much easier to start from the intrinsic properties, and use those to prove things about the representation, than the other way around. But if you don't distinguish the representation from the thing itself, you can't tell which way you are going.
The proof that the usual definition is equivalent to the representation is fairly straightforward and easy, no matter which side you picked as the definition. And once the equivalence is established, all other proofs proceed naturally. It therefore matters a lot that we pick one as a definition and know which one we picked, but not so much which one we picked.
Now in fact the quotient definition is by far more interesting mathematically. There is also a clear foundational reason to prefer it, namely that you can easily construct and prove things about the rational numbers long before you construct the real numbers. However it is unlikely that anyone who is confused about the definition of a rational number has a clear understanding of how the reals are constructed, so that is not a particularly important consideration for them.
Furthermore the fact that foundational considerations argue for one construction over another has little bearing on what is pedagogically preferable. As a famous example, the easiest way to rigorously define logarithms is through the integral of 1/x. However explaining logarithms that way to someone who doesn't know them is a pedagogical disaster.
Seldom do we prove that a number is irrational by inspecting its decimal expansion. This would be in most cases a very unnatural proof. Since irrationality is a negative property (meaning, one arising out of a negation: the number is not a ratio), most of the time you prove it by contradiction. But people who just know the "digits don't repeat" definition expect us to somehow be able to list all of the digits of an irrational number and show that this infinite list doesn't repeat, which is, of course, an impossible task.
The equivalent property, that a number is irrational if it's not equal to m÷n for any integers m and n, is much simpler. So we use that as the definition, and from that simple and intrinsic definition, we prove the _theorem_ that the decimal representation of an irrational number never repeats.
I think the numerals and numbers issue is more complex because numerals are fundamentally hard to reason about. Even the question "what is a number?" is deceivingly deep.
When people receive a time, they may (usually) want it in their own time zone, but they might instead want it in the time zone of the entity they're getting the time from, if they're subsequently going to talk to that entity about the time. When they talk about meeting someone else, when they convey the time of the meeting, they usually mean whatever that time means in the place where they meet, which might be different from the current location of either. It might even be different due to political changes around time zones and daylight saving if the meeting is far enough in the future.
I don't think there are many areas of math where ∞ is a number. In my experience people have a whole other problem with ∞, thinking that it is some sort of huge concept defined globally in math, where it is just a notation shared by various non-mystical definitions across subjects (e.g. bijection-based definition of infinite set, epsilon-based definition of convergence, etc)
How is this a mistaken belief?
Every rational number winds up in a repeating decimal representation and every number with a repeating decimal representation is a rational number. We learn algorithms to go back and forth between the two in elementary school.
Therefore irrational numbers cannot have repeating decimal representations. Conversely numbers with decimal representations that don't wind up repeating cannot be rational and so must be irrational.
I wonder if there's a way of teaching this kind of distinction and issue well in a way that would make sense for most students.
I think Feynman said somewhere that the New Math explicitly taught base representation and base conversions, probably as a way of trying to underscore the idea that "123" is a representation of a number rather than a number. Feynman found this to be of questionable value and thought that most students didn't manage to get the point.
Edit: there's a similar issue in linguistics because you have words, phonemes, phones, graphemes, and glyphs. You could say that "dog" isn't a word, but is rather the standard way of writing a particular word in the standard writing system for English (which would sometimes be indicated by <dog> in linguistic contexts). This idea lets you refer to <alright> and <all right> as ways of writing the same word, or <color> and <colour>, or in the case of languages with multiple writing systems <हिन्दुस्तानी> and <ہندوستانی>, or <אַ שפּראַך איז אַ דיאַלעקט מיט אַן אַרמיי און פֿלאָט> and <a shprakh iz a dialekt mit an armey un flot>.
The responses are comical, even here on HN, such as not being able to wake up or not knowing when morning is or when meals are. Let’s not forget that time zones are a human invention less than 200 years old.
What's the point of that statement? Before the introduction of formal time zones, we had thousands of informal ones, one for each settlement, calibrating noon to the zenith of the sun.
We still do not. China and India are examples of large geographies spanning across a vast amount of longitude and yet each are a single time zone. Time zones are a political entity only, an unnecessary complexity. The absence of time zones will not halt business or communication over long distances.
> For most people talking about time in their day to day lives it's far more useful to communicate a relative time of day
People have done this for thousands of years without modern chronometers. Examples: dusk, dawn, morning, midday, afternoon, evening, twilight.
I've heard there's some pushback against the Chinese policy in that some people in the west keep an unofficial local time which is widely understood and quoted (though presumably not for things that are sufficiently official or relevant to other regions). Apparently there's currently an ethnic conflict over the time zone status in Xinjiang:
https://en.wikipedia.org/wiki/Xinjiang_Time
Maybe this conflict has now been pushed underground by force?
> In 2018, according to Human Rights Watch, a Uyghur man was arrested and sent to a detention center because he set his watch to Xinjiang Time.
The Circus Animal's Desertion, W. B. Yeats
I've often wondered if there is some alternate base or mathematical system entirely that would be "better" in these respects. The thought usually comes up thinking about why pi is such an "ugly" number in base-10 decimal.
Come to think of it, this is one way of thinking about the relationship between symbols and geometry.
It's the one issue I have with the metric system... But that ship has sailed :) look up the dozenal society if you're curious how fervent some supporters might be.
The numerals are not distinctly varied like our Arabic numerals. Quite the opposite, they are repetitive and completely systematic and require 80% less effort to remember.
https://commons.wikimedia.org/wiki/File:Babylonian_numerals....
Now sure, make a square using those measures and measure the diagonal, like an awkward mathematician - "see, see, we need irrationals!" - but then we can just cut another measure that's exactly that length ... stupid mathematicians!
Yeah representation is not reality.
Now, says the ratio of those two measuring sticks ...
Continued fractions give very approachable representations for common irrational numbers like e and sqrt(2). While π doesn't have a good continued fraction, it has some very well-behaved generalized continued fractions.
The simplest definition is: a finite decimal ak ... a1.b1 ... bh is defined to be a fraction and an infinite decimal is defined to be a limit. You'd still have to define what a limit is, but that is somewhat more intuitive.
There are TWO standard definitions of the real numbers. Namely Dedekind cuts and Cauchy sequences. (They are completely equivalent.) The usual decimal representation of a number turns out to be a Cauchy sequence.
The "simplest definition" that you provide turns out to be rather non-simple in practice. Try proving that multiplication is commutative to see the difficulty.
There are plenty of other number systems out there. Try https://en.wikipedia.org/wiki/Surreal_number or https://en.wikipedia.org/wiki/P-adic_number or the complex numbers.
Since the fundamental thing most people want to do with numbers is see which one is bigger, they favour decimal expansions. And since decimals worked so well for fractions, why not use them for everything else?
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.
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.)
That means that the "infinitely small" doesn't exist; "smallest apart from zero" doesn't exist either.
Zero. Just zero. The difference is zero. 0. Because 0.999… = 1.
I think people intuitively see that infinitely zero equals zero.
If people accept the former, and that the RHS of the former is in fact 0, they've already also accepted that 0.999…=1. I don't see what the discussion is at that point
I don't have a direct computation for making the latter obvious, just indirect ones like 1 - 0.999... and 3 x 0.333...
How can they compute 1-0.999… when they clearly have no idea what 0.999… is?
Put 1.0 on top, 0.9 on the bottom. Start subtracting from left to right, and keep writing nines on the bottom as you go to the right. In no time you'll see that the answer is infinite zeros.
How do they know that that's a real number?
What does "infinitely small" mean?
> As in 1/∞ ?
What notion of division are we talking about here? The division most people expect is that of real numbers. ∞ is not a real number, so you'll have to specify what you mean.
If you've taken Calculus, you've already worked with math that requires the infinitesimal to exist.
It's not a value you can meaningfully write out, but you can't write out pi, e, phi, root 2, 1 / 3 in base 10, root -1, etc. "I can't write it down" isn't a particularly unique property for numbers.
Had you actually meaningfully studied this subject, or did you just link to a Wikipedia article you half-heartedly skimmed one day?
There is in the surreal and hyperreal number systems. I got that from skimming wikipedia though....
Not at all. Standard calculus uses standard real numbers, for which there is no infinitesimal. One may well speak of infinitesimals as a mental tool when building a mental model for calculus, but those infinitesimals are not actual real numbers (or a well-defined mathematical object at all - in standard calculus).
The problem is that if we include such a number in our formal system of math, we quickly find contradictions and the whole system falls apart. So such a number is incompatible with any formal system of math (though I guess you could start building one which does include such a number and see what properties it has).
Herein lies the problem, the people you are talking with do not use a form system. There system of math has something similar to the same flaw of their system of grouping of things, which would include the whole grouping that contains every grouping that doesn't contain itself. People rarely deal in formal systems and thus they can handle completely illogical statements fine as long they are protected from seeing the consequence of it.
Well only to the extent that you don't want to throw away any of the other axioms. Sometimes you do and there are some fun systems of math, but few have any practicality and those that do are often so advanced that even someone with an undergraduate focus in math can't appreciate those systems.
It is much the same with computer science. I personally enjoyed playing around with formal concepts of computation and adding some extras to see what happens. For example, what happens to a Turing machine if part of the machine can time travel or has access to an oracle. Does this make concepts like time travel inherently contradictory to our notion of computation?
But the practicality of these exercises does not exceed their entertainment value.
Assume x is the smallest real number greater than 0. Then x/2 is also a real number and is greater than 0 but less than x. Therefore, x can't be the smallest real number greater than 0.
However, basic arithmetic taught to children requires that adding trailing zeros does not change the value of a number. You'll have a hard time doing arithmetic once you change that assumption.
0.999... + 0.000...1 = 1
0.000...1 = 1/∞
0.999... = 1 - 1/∞It's 0.999... and not 0.999...0
In the same way, it's 0.000... and not 0.000...1.
0.999... = 0.999...9
0.999...9 + 0.000...1 = 1
0.999...0 + 0.000..1 = 0.999..1
0.000...1 = 1/∞
0.999...9 = 1 - 1/∞
0.999...0 = 1 - 1/∞ - 9/∞ = 1 - 10/∞
If x/∞ = 0, then 0.999...x = 1.
If x/∞ ≠ 0, then 0.999...x ≠ 1.If Universe is finite, then finite number of elements can make only finite number of combinations, thus this discussion is repeated infinite number of times again. Why I should waste my time again?
0.000...1 = 1/10^∞ = 1/∞Here John Conway explains them: https://www.youtube.com/watch?v=1eAmxgINXrE
0.999...1 = 1 - 1/∞ - 8/∞ = 1 - 9/∞And what does this mean? I will remind you that for an integer d between 0 and 9, 0.ddd… means the limit of \sum_{i=1}^N d/10^i as N tends to infinity.
0.000...1 = 1/∞Fine by me. Define whatever notion you're using. You can't just throw out non-standard things and expect people to know what you mean.
Which is the point in using the word obvious, obviously. Namely, using it to feel superior or to not provide a better argument.
Anyone who uses the word differently is doing it wrong.
I will never accept that they are the same. The difference between 0.9 repeating infinitely and 1 is infinitely small, but it isn't zero.
Is 9999..... the same as infinity?
What is 1.0 - 0.99999.... = ?
What does it mean to say X is a number, if you can't subtract it from another number and get a number as an answer?
What is an infinitely large number?
> What does it mean to say X is a number, if you can't subtract it from another number and get a number as an answer?
By that logic, 0.99 repeating isn't a number at all, and therefore can't be equivalent to 1, because you can't subtract it from 1. So my understanding that they are different is correct.
> What is an infinitely large number?
Neither is a well-defined concept within the standard reals, and completely unnecessary for understanding that 0.999…=1.
> > What does it mean to say X is a number, if you can't subtract it from another number and get a number as an answer?
> By that logic, 0.99 repeating isn't a number at all, and therefore can't be equivalent to 1, because you can't subtract it from 1. So my understanding that they are different is correct.
0.99… is a real number. The sequence (a_n)_{n positive integer} with a_n = 9/10^1 + 9/10^2 + … + 9/10^n has a limit (do you want me to prove that?). 0.99… is defined as that limit. That limit is 1. Therefore 0.99… = 1.
I think you're struggling to grasp the definition here. The defintion of 0.ddd…, where d is an integer between 0 and 9, is the limit of the above sequence with 9 replaced by d. That limit always exists, and the definition is therefore OK. In the case of d=9, the limit is 1.
0.9 is not equal to 1,
0.99 is not equal to 1,
0.999 is not equal to 1,
0.9999 is not equal to 1,
0.99999 is not equal to 1,
0.999999 is not equal to 1,
Saying that if you add enough "9"s it suddenly equals 1.0 makes absolutely no sense to me, and I seriously doubt that anyone will be able to convince me that it does make sense. I've read every single post in this thread and none of you have gotten me any closer at all to believing or understanding that 0.9 repeating equals 1.
Maybe I'm too old to understand this "new math" where all numbers are equal to each other.
No finite representation of repeating 0.9s can equal 1.0
The ask that people accept infinite representations as valid is a big one.
> 0.99 is not equal to 1,
> 0.999 is not equal to 1,
> 0.9999 is not equal to 1,
> 0.99999 is not equal to 1,
> 0.999999 is not equal to 1,
> and so on, ad infinitum.
You are correct about all of these, and all finite strings of the above form.
> Saying that if you add enough "9"s it suddenly equals 1.0 makes absolutely no sense to me, and I seriously doubt that anyone will be able to convince me that it does make sense. I've read every single post in this thread and none of you have gotten me any closer at all to believing or understanding that 0.9 repeating equals 1.
I think it's because you, and a lot of other people in this thread, are turning the question on its head. The difficulty does not so much lie in figuring out whether 0.999… is equal to 1 or not, but rather in what we mean when we write 0.999….
I know I'm repeating myself from elsewhere in the thread, but I'll try again. Try to go through these step by step, and feel free to let me know where you lose the thread.
DEFINITION: A finite decimal representation of a real number is a finite string of the form `a_m a_{m-1} … a_0 . b_1 b_2 … b_n` where each `a_i` and each `b_i` is a natural number between 0 and 9 inclusive (a digit). We say that this finite decimal representation represents the real number
a_m*10^m + a_{m-1}*10^{m-1} + … + a_0 + b_1*10^{-1} + b_2*10^{-2} + … + b_n*10^{-n}.
EXAMPLE: The string `12.98` has `m=1`, `n=2` with `a_1=1`, `a_0=2`, `b_1=9` and `b_2=8`. It therefore represents the real number
1*10^1 + 2*10^0 + 9*10^{-1} + 8*10^{-2}
Within this standard framework, there is no way to ask "what is 0.999…?. It is not yet defined, because we have only defined what finite strings mean. The standard definition for what one means by 0.999… follows. (One can obviously also define these things 0.888…, 1.999…, etc., but let's stick to one case here).
DEFINITION: Let `(c_n)_{n natural}` be a sequence of real numbers (let me know if you need a definition of sequences!). We say that the sequence has the limit x as n tends to infinity (these are words, you don't have to ascribe meaning to "infinity" in that sentence – it's just a word, like "gnarf"!) if, given any real eps>0, there exists an M such that for all m > M, |c_m - x| < eps.
Definition (this is the definition you have to wrap your head around before continuing): Consider the sequence `(c_n)_{n natural}` where `c_n` is the finite sum
9*10^{-1} + 9*10^{-2} + … + 9*10^{-n}
"THEOREM": The limit defining `0.999…` does exist. It is `1`.
PROOF: You can fill this in. If you can't, I'm happy to do it.
As you can see, at no point in the above did feelings or beliefs matter :-)
Of course it's hard because in day to day life, even for the vast majority of STEM practitioners, the nuance of the proof that 0.9999... is 1 is not of much utility.
Whenever one sees a 0.999[... to however many digits] one can safely assume it's less than one or perhaps more realistically "almost 1". To say 0.999... with the very specific detail that the 9's go on forever is actually a strange thing to say and outside of most people's experience.
There are simple enough proofs of this that normal folks who paid attention in high school can follow, but I think it has to be framed more as a clever brain-teaser than as a proof.
Oh absolutely. I'm not expecting STE(no M this time!) practitioners to necessarily be aware of why 0.999…=1 in their daily lives, but I do expect them to have encountered enough situations in their field of expertise where scraping the surface using shallow intuition and gut feeling lead them wildly astray. I'm therefore surprised that they're willing to deny this basic fact to the face of mathematicians. The ones I've interacted with also don't happen to be the types that'll start arguing Anatomy 101 facts with a heart surgeon at a bar, but somehow arguing over basic calculus with mathematicians is fine.
Well, there's this:
wikipedia.org/wiki/Vaccine_hesitancy
wikipedia.org/wiki/Homeopathy
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.
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.
I put this in another comment but: For the reals: eliminate a < b and a > b then conclude a = b.
x = 0.9999..
10x = 9.9999...
10x - x = 9
9x = 9
x = 1
x = 0.444444...
10x = 4.444444...
10x - x = 4
9x = 4
x = 4/9 = 0.444444...
0.111... == 1/9
0.222... == 2/9
...
0.888... == 8/9
0.999... == 9/9 :)
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.
(also a somewhat joking answer)
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 and http://www.astro.ucla.edu/~ghezgroup/gc/blackhole.html
EDIT: The number above seems well defined. It's lim n->inf (10^-n). That's zero.
The decimal representation of a Real number has to be indexed by Natural numbers, i.e. every decimal digit[n] has a well-defined index n which is a Natural number. Infinity is not a Natural number, so 0.00...1 is not a Real number either.
The limit of lim n->inf 10^-n is exactly 0, it is not 0.00...1.
I will say though that if your explanation of 0.999... = 1.0 requires that you explain the distinction between countable and uncountable infinities, that's a big ask for most lay people.
I'm no expert, but countability doesn't depend on how the set is ordered. It depends on whether the elements can be placed in 1:1 correspondence with the integers. 1,0,0,... has a countable number of elements, and so does 0,0,0,...,1. They can be put in 1:1 correspondence with each other. This definition of countability is described in your link.
I meant to ask, does "countable representation" some kind of detailed definition that I can look at?
0.9 -> 0.99 -> 0.999 -> ... -> ?
0.1 -> 0.01 -> 0.001 -> ... -> ?
9/10 + 9/100 + 9/1000 + ... + 9/10^n + ...
1/10 + -9/100 + -9/1000 + -9/10000 + ...
But I just meant them as series, not necessarily as sums.
1 - 9/10 - 9/100 - ...
1 - ( 9/10 + 9/100 + ... )
0.00...1 = 1 - 0.999...
The decimal representation of a Real number has to be indexed by Natural numbers, i.e. every decimal digit[n] has a well-defined index n which is a Natural number. Infinity is not a Natural number, so 0.00...1 is not a Real number either.
It is correct if you take the limit, people usually do not.
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.
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. )
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
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
sum(n=1 ... N)(9*10^-n).
I can't, uh ... fully endorse that comment, which is not entirely accurate and doesn't answer my question. But I sure did miss the '9'.
For ordinary math, though, using some criterion for equality (for example x>=y and y>=x) is basic and not controversial. So it seems unconvincing (to me) when you seem to imply the opposite.
You are putting words in my mouth.
And you clearly do not understand the answer.
I guess I'm not very good at ELI5 because I very clearly answered your question with your own proposal.
Maybe when you get to college a professor can do a better job explaining it to you (if you actually make it to college, because you're going to struggle very hard if that's how you think when an answer is spoon-fed to you).
I use applied math. I haven't taken a class in real analysis. But it's fun how often grinding out the solution to a "real world," practical PDE turns out not to actually be the nicest (simplest and/or clearest and/or sufficiently insight-producing) way to understand the (hopefully) corresponding physical problem in the lab.
Stripping off the "calculus" and replacing it by limits sometimes seems to help highlight alternate perspectives that the magic "integrals" and "derivatives" kind of conceal.
Even when it's not more effective, it's definitely more fun.
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.
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.
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.
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.
But it can't be, because there's nothing after "0.000..."; that ... goes on infinitely. It's literally "0s forever, never stopping". It's not a process of "keep adding 0s", it's the end result of never adding 0s. It's not a process, it is a noun.
But if one eschews that abstraction and looks at it purely as a process (I want to render 1/3 in decimal notation, then multiply that decimal notation by 3), there is always that niggling 0.000...1 remainder at every snapshot. The "never stopping" bit is what smuggles verbiness into the "0.999..." noun, while simultaneously pretending it's a static value.
Personally I find the following algebraic proof to be the most approachable:
x = 0.999…
x = 0.9 + 0.0999…
x = 0.9 + (0.999… ÷ 10)
x = 0.9 + (x ÷ 10)
x - 0.9 = x ÷ 10
10x - 9 = x
9x - 9 = 0
9x = 9
x = 1
0.999… = 1 x = 0.999...
x = 9/10 + 9/10^2 + 9/10^3 ... 9/10^inf
x = 9/10 + (9/10 + 9/10^2 + 9/10^3 ... 9/10^(inf-1))/10
x = 9/10 + (x - 9/10^inf)/10
x - 9/10 = (x - 9/10^inf)/10
10x - 9 = x - 9/10^inf
9x - 9 + 9/10^inf = 0
9x = 9 - 9/10^inf
x = 1 - 1/10^inf
x = 1 - 0.000...1 x = 9/10 + 9/10^2 + 9/10^3 ... 9/10^inf
x = 9/10 + (9/10 + 9/10^2 + 9/10^3 ... 9/10^(inf-1))/10
x = 9÷10 + 9÷10² + 9÷10³ + …
x = 9÷10 + (9÷10 + 9÷10² + …) ÷ 10
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.
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.
0.999 ... infinite number of 9s ... 9
and
0.000 ... infinite number of 0s ... 1
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."
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.
Another one is that
1 / 3 * 3 = 1
<==> 0.333... * 3 = 1
<==> 0.999... = 1
(1/3)/∞ 1/10*10=1, 0.1*10=1
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.
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-...
Looking at you, irrational numbers.
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.
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.
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.
when they are done they will have undesrtood.
chuckle
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.
We create these abstractions to simplify our thought- and analyzing or over-analyzing these simplifications can have the opposite effect.
Now try imagining that some infinities are bigger than others: https://en.wikipedia.org/wiki/Aleph_number
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.
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.
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).
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.
So let there be an ω with 0.999... = 1 - 1/ω. Then a number between 0.999... and 1 would be 1 - 0.5/ω.
(The Wikipedia article even reproduces that argument)
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.
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
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.
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.
The point is that in elementary school arithmetic, you define addition, multiplication, subtraction, division, decimals, and equality, but you never define "...". Until you've defined "...", it's just a meaningless sequence of marks on paper. You can't prove anything about it using arithmetic, or otherwise.
What the "arithmetic proofs" are really showing that if we want "..." to have certain extremely reasonable properties, then we must choose to define it in such a way that 0.999... = 1. Other definitions would be possible (for example, a stupid definition would be 0.999... = 42), just not useful.
What probably causes the flame wars over "..." is that most people never see how "..." is defined (which properly would require constructing the reals). They only see these indirect arguments about how "..." should be defined, which look unsatisfying. Or they grow so accustomed to writing down "..." in school that they think they already know how it's defined, when it never has been!
Sure, but the point of "elementary school" arithmetic is not "elementary" arithmetic, as a mathematician would define it :-)
The goal is to teach people to reason by matching patterns. Deductive/Inductive reasoning can slowly proceed from that, as they try to frame their intuition for patterns into increasingly more general abstractions.
Technically it’s an infinite series (sum of an infinite sequence), which is a finite number in certain cases like this one.
what are the properties that we would lose?
0.999… = 1 is a property of the way we write some rational numbers, not of the number system itself.
0.999... with infinite nines is equal to 1.
That is, for any number system I've seen, 1 = 1 + dx, and infinity = infinity + 100.
I've never considered them right at all. By saying something like
0.9... x 10 = 9.9...
and then saying that
9.9... - 0.9... = 9
you're basically just a priori defining 0.9... to be 1. In other words you're basically just defining 0.9... as a symbol to be some number x which has the property that 10x - x = 9. So you're basically just defining it to be 1.
I've never seen a proof of 0.9... = 1 using Peano arithmetic which made any sense to me. I doubt one actually exists in any true logical meaning. Unless you're making use of limits, completeness, or something equivalent I don't see how a proof could possibly make any sense.
Peano arithmetic only covers nonnegative whole numbers, so one will never exist.
Thank you for the pedantism. How about I replace "Peano arithmetic" with the "operations of multiplication/addition/division/etc. expressible upon the rational numbers"?
Real numbers are defined as an equivalence class such that if the differences of two infinite sequences of rationals tend toward zero, then they are equal. The difference between 0.999... and 1.000... clearly tends towards zero as it heads of to infinity, and so they are equal.
If you want to argue that it doesn't then you have to come up with some other definition for numbers which have an infinite decimal expansion.
(Technically, of course, 1 is a rational number, but if you're using 0.9999... to represent it, you're using a real number representation, so you're bound by the definition)
> Real numbers are defined as an equivalence class such that if the differences of two infinite sequences of rationals tend toward zero, then they are equal. The difference between 0.999... and 1.000... clearly tends towards zero as it heads of to infinity, and so they are equal.
> If you want to argue that it doesn't then you have to come up with some other definition for numbers which have an infinite decimal expansion.
> (Technically, of course, 1 is a rational number, but if you're using 0.9999... to represent it, you're using a real number representation, so you're bound by the definition)
I'm not sure why you think I don't know the difference between rational and real numbers, but I assure you I do. What I said was I don't see how a proof involving the standard arithmetic operations found within the rational numbers, but not including any concepts of limits, completeness, etc. is invalid. Let me know if you still don't understand my point.
Trying to rearrange it and remove as many negatives as possible, I started with your statement:
> I don't see how a proof involving the standard arithmetic operations found within the rational numbers, but not including any concepts of limits, completeness, etc. is invalid.
I think what you mean is that any proof that does not use the concepts of limits and completeness is going to be invalid.
That seems clear to me, the reason being that one needs to define what one means by the sequence of symbols "0.9999...".
You can say "It's infinitely many 9s stretching off to the right", but that doesn't tell me what it means.
People seem to think it does, but when I dig deeper, they usually don't have any sense of what it means. And therein lies the problem (as I see it). People blithely write the glyphs, but don't have a concrete interpretation.
If someone really wants to understand it then I'll explain current mathematical thinking, including non-standard analysis and the surreals. But most people don't want to put in the work to understand how these issues have been resolved, and just want to argue from their intuition.
https://thatsmaths.com/2019/01/10/really-0-999999-is-equal-t...
I don't quite know how to formalize it, but I'm pretty certain that if these proofs logically worked (in the "theory of proofs sense"), then they should work in the surreals as well.
Anyway it's just intuition. My main point in this thread is that I don't really accept the proofs of this that don't use completeness as a step. Though I do suspect that proofs not making use of it are actually incorrect proofs in their own right. If I were curious enough I'd think back about formal proofs and models and all that jazz, but I probably already have spent more time in this thread than I should. :)
edit: The more I think about it I feel like someone actually explained to me this (i.e. why this proof is wrong using surreals as reasoning) a long time ago and I'm just remembering echos of it in my mind. Wish I could remember something more useful...or that I were a logician...
(Disclosure: I have no idea what Peano arithmetic is.)
Regardless, the point has been clarified, but discussing it further is kind of pointless.
edit: It just occurred to me that the original pedantic comment essentially breaks the following Hacker News guideline:
Please respond to the strongest plausible interpretation of what someone says, not a weaker one that's easier to criticize. Assume good faith.
I just found it a bit interesting considering it is my response complaining about the pedantry and not the pedantry that is receiving downvotes.
The rules are essentially the same.
Anyway it's still an artifact of representation. If rationals were represented as fractions, then it is unrepresentable. 1/1 X 10 == 10/1?
Sure you can provide a hand-wavy argument and try to give some intuition if you'd like. That doesn't make it any sort of logical proof though. I guess it depends on what you're after.
You can redefine the "0.9..." symbol to mean something else as much as you want, you can have it meaning pi if you like, but then you are just changing the subject on the most unhelpful way.
> You can redefine the "0.9..." symbol to mean something else as much as you want, you can have it meaning pi if you like, but then you are just changing the subject on the most unhelpful way.
I _know_ that.
I think I maybe should just bow out of this conversation. I'm apparently incapable of explaining myself in a way that is understandable to people here. I'll consider this my fault.
I'll just summarize: I don't think any "proof" that 0.9... = 1 that is only expressed in terms of arithmetic operations and does not make use of limits is legitimate. In other words I claim that a proof like "0.9... = x" means "9.9... = 10x" means "9 = 9x" is illegitimate. Instead of taking "0.9... = 1" on faith it takes "10 x 0.9... = 9.9..." and "9.9... - 0.9... = 1.0... = 1" on faith. There's no proof here. It's just shifting around symbols. Of course there are logical proofs, but they make use of limits/completeness/properties of real numbers explicitly.
Feel free to disagree...
It’s worth remembering that most people who understand and agree rarely leave a reply.
Well, I do disagree, not with the above statement, but the meaning of "0.9..." itself requires limits, so the discussion can never go anywhere if your assumptions do not include limits.
I think the point is not defining 0.9... to be 1, the point is that “...” means an infinite number of 9s. If you shift the decimal point by 1, then nothing changes, there are still an infinite number of 9s. If you shift the decimal point by 5 places, there are still an infinite number of 9s to the right. And here is the logical (induction) step: if you shift the decimal point by an infinite number of places, then there are still an infinite number of 9s to the right. This works for any repeating fraction, in groups of more than 1 repeating digit.
> I’ve never considered them right at all.
Do you mean you disagree with the result, or that you agree with the result but don’t believe the proof is really a proof?
I've heard this argument many times. I understand the intuitive reasoning. I just don't find it a proof. I mean with reasoning like this why can't you have an infinite amount of 9s and then just put a 7 after that? What's keeping you from doing that? It's just a hand-wavy argument with no rules of any kind of what are allowed.
> Do you mean you disagree with the result, or that you agree with the result but don’t believe the proof is really a proof?
I've never considered that specific "proof" a proof. When 0.9... is given a proper definition of limits and considered within the real numbers, then sure of course it's true and the proof is legitimate.
You can. The proof still works if you do that.
What you’re refusing to accept here is the definition of infinity.
> What you’re refusing to accept here is the definition of infinity.
I'm refusing to accept the definition of infinity? I have no idea what you mean by that. Would you make the same statement were you aware that I do in fact have a PhD in mathematics in the field of analysis? That I have in fact studied logic? Just as a hypothetical scenario.
I haven’t heard a reason yet why the logic of the proof doesn’t work, I’ve only heard that you don’t accept it. What is the reason it doesn’t work?
I wasn't appealing to authority. I never said that implied that I was correct. (Had I wanted to do that, I would have brought that up much earlier in this comment thread.) I was really just curious if you still believed that I was "failing to accept the definition of infinity" even if you knew that about me. Apparently you do.
> I haven’t heard a reason yet why the logic of the proof doesn’t work, I’ve only heard that you don’t accept it. What is the reason it doesn’t work?
Assuming that all my suspicions in that comment are correct and these proofs actually are invalid proofs (not the results which are true), then the question might become: does it matter if the proof of a fact is incorrect if the fact itself is correct? That is a philosophical question and I'm honestly not sure how I'd answer it...
> does it matter if the proof of a fact is incorrect if the fact itself is correct?
Your language isn't allowing for a notion of precision, or for multiple forms of correctness, and it's not considering audience, communication or level of expertise either. I don't think it's a question of correct vs incorrect, I think you're asking for more precision, and/or for a form that meets your own higher standard.
It does matter if a proof is wrong, if there is a step in the proof that can be shown to be false. But that's not the case here, what you want is additional definition.
BTW, reading the blog post you linked to on surreals, the "proof" looks to me to be more hand-wavy than Euler's proof that .9bar = 1. The proof begins by stating there are a finite number of 9s, in direct contradiction to the hypothesis. 10^-inf = 0, so from this blog post I don't yet see any reason why surreals clarify anything here, it feels like the opposite, it feels like obfuscation.
This could be an argument over representation and not the values of numbers. If you start by defining 0.9bar to be a different number than 1 for the specific reason that it's written down a different way, then fine. That's what the surreal "proof" tells me. Euler's proof is talking about the value of 0.9bar in the limit, not the representation. (Even if that's stated without rigorous definitions of limits.) The proof is saying the values of .9bar and 1 are the same. If the surreal number .9bar were strictly less than 1, that must mean there's another surreal number closer to 1, but there isn't, so I don't accept the surreal argument as valid logic, other than playing a semantic trick by saying 'look I defined they way we write a number to be meaningful, therefore 0.999... is by definition different than 1.'
By the way, that blog post claims "The set of real numbers contains no infinitesimals." Wikipedia claims: "the surreal number system is a totally ordered proper class containing the real numbers as well as infinite and infinitesimal numbers..."
> I don't necessarily understand your use of the word "invalid", when what it seems you mean is incomplete and/or too informal for your taste.
> [...]
> It does matter if a proof is wrong, if there is a step in the proof that can be shown to be false. But that's not the case here, what you want is additional definition.
I'm going to try to be more formal, but not entirely formal since (1) the details are almost never-ending and require a lot of formal logic and (2) I'm not 100% sure about the reasoning myself.
What I mean by a "proof" is a sequence of logical steps that start with axioms of your logical system. We are obviously not looking at things that formally, but I actually think it is still an important point. The rational numbers can be thought of as being defined by certain axioms of arithmetic. E.g. you have the natural numbers as well as the minimal extra values so that you can add, subtract, divide, multiply, etc.--in other words you have a field. Let's call these the arithmetic axioms. Then when you go to the real numbers you basically extend the rational numbers in such a way so that you have completeness and retain all the previous properties. So basically for real numbers you have the prior arithmetic axioms and you have the completeness axiom.
Next comes the question of what is actually to be proved. For that we _must_ make some sort of definition of what we mean by "0.9...". Let us define it as the limit of the sequence of partial sums (all of which are rational numbers) _if_ it exists. So to prove "0.9... = 1" means to prove that the limit exists and equals 1.
So lets say we believe we have a proof that the limit equals 1 using an argument like this:
0.9... = x => 9.9... = 10x => 9 = 9x => 1 = x => 0.9... = 1
This is not a formal symbolic proof, but there is one thing we can see immediately: this proof does not make use of the completeness axiom. Therefore it should logically be the result of a sequence of logical steps starting from what I earlier referred to as the arithmetic axioms. Now here is the point where the surreals come in. The point with the surreals is that they contain rational numbers and the arithmetic axioms still apply. (To be clear, I haven't thought this through 100%, but I am almost certain this is true modulo my hand-wavy reference to "arithmetic axioms".) That means that that same proof should work inside the surreal numbers to also prove that the limit of the partial sums is 1. But here is the key important point. Within the surreal numbers, the limit of the partial sums is _not_ 1. So what does this tell us? The original supposition that there exists a proof only making use of the arithmetic axioms cannot be true. So the proof must make use of the completeness axiom. Well the surreal numbers is _not_ complete so that axiom doesn't exist there and therefore the fact that the proof exists and works within the real numbers and not the surreal numbers is not a contradiction.
Okay this is a bunch of logic mumbo jumbo and no grade student should be expected to worry about things at this level, but we can still give a more simplified version of the proof that actually is in essence correct. Some people in this thread use the argument: "Well since we know the number must be less than or equal to 1 and we also know it must be bigger than any number smaller than 1, then it must be 1." That argument (while a priori assuming convergence) is at least implicitly using an intuitive idea of completeness. Is it logically formal and 100% rigorous? Of course not. But it is explicitly making use of a property of real numbers while the first proof is not. I think if students are to be taught anything about the real numbers, then they should get some general intuition for this property. The algebraic version of the proof is invalid (my claim, which hopefully is at least a little more well-supported given this comment here) and it doesn't give the intuition they should (hopefully) get anyway.
In any case, hopefully this does some to help clear up what I've meant throughout these posts.
P.S. Finally, in response to this:
> By the way, that blog post claims "The set of real numbers contains no infinitesimals." Wikipedia claims: "the surreal number system is a totally ordered proper class containing the real numbers as well as infinite and infinitesimal numbers..."
So what? Are you saying those statements are contradictory? If so, how? And if not, what are you saying?
How did Euler actually write his proof, do you know, or have a link? I’ve poked around online but can’t find it.
By the way I wouldn't hold it against him if he did write the proof that way. I'm pretty certain all the logic/model theory that comes into play came long after his death. The surreal numbers certainly did.
>0.9bar7 is a completely nonsensical number
For the same reason that 0.9...7 isn't a meaningful number, you cannot move the decimal 'an infinite number of times' and then after this, look at what number you have left and see it still has infinite 9s left. It's like you're trying to perform transfinite induction on the set of numbers generated by moving the decimal point. You can only move the point a countable number of times, so there is no sense in which the property can still be true after infinity many times.
Yes you absolutely can, for exactly the same reason. 0.9bar7 is nonsensical precisely because you can move the decimal to the right an infinite number of times, and still have an infinite number of 9s before the 7.
> You can only move the point a countable number of times
Not true, the implicit definition of “...”, the very statement that there are an “infinite” number of 9s, means exactly the opposite of what you claim, it means you can move the decimal an infinite number of times.
>And here is the logical (induction) step: if you shift the decimal point by an infinite number of places, then there are still an infinite number of 9s to the right
This is not how induction works. The induction shows that you can shift the decimal point any finite number of steps to the right and there will still be infinite 9's after it. If you want to show something is still true after infinity steps, you require transfinite induction, but this doesn't make sense because the '...' decimal representation only represents a countably infinte number of 9's.
This is the same reason 0.9bar7 doesn't make sense - because decimal representations only have countably many digits.
This should rather say that it's because decimal representations have digits indexed by the natural numbers I guess, rather than by any larger countable ordinal
You can justify the idea by defining a decimal representation of a number x as a vector x_2, x_1, x_0, x_{-1}, x_{-2} ..., with x_n ∈ {0, 1, ..., 9}. Negative indexes are digits after the comma, positive indexes before the comma. You can recover the original number simply using
x = \sum_{n=-∞}^{n=∞} 10^n x_{n} (1)
(this sum always converges as long as x_n = 0 when n > N, for some big enough N ∈ ℕ).
For example, the number three is represented by x_0 = 1, x_n = 0 otherwise. 0.9... is defined as x_n = 0 for n >= 0, 9 for n < 0. Now, by using limits, we could recover the original number using the formula above. However, we can do it without them. For that, we define just enough operations for the proof.
1. If z = x - y and for all x_n, y_n we have that x_n >= y_n, then z_n = x_n - y_n for all n. 2. If z = 10x, then z_n = x_{n-1}.
For the first operation, in order to be rigorous, we need to ensure that if z_n = x_n - y_n in the same conditions, then z = x - y. The proof of this part just consists of plugging the recovery formula (1): z = \sum 10^n z_n = \sum 10^n (x_n - y_n) = (\sum 10^n x_n) - (\sum 10^n y_n) = x - y. We can perform all those operations as we are guaranteed that (1) always converges.
Now, let y = 0.9... defined as in the example above (y_n = 0 for n >= 0, 9 otherwise), and let x = 10y (therefore x_n = 0 for n > 0, 9 otherwise). Now, define z_n = x_n - y_n, so that z_n = 9 for n = 1, 0 otherwise, which yields z = 9. As we demonstrated above, this implies that z = x - y, therefore 9 = 10y - y => y = 1, so 1 = 0.9... .
PS: I don't think one can make a proof without at least using some bits of limits to be able to switch between decimal representation as a vector and the number itself. However I don't think it's a problem, because you need the same bits to be able to talk of "0.9..." as a well-defined number.
Another proof by contradiction I've heard of this is that if 0.9... != 1.0... then there exists a number in between the 2. What is it?
You say
> you're basically just defining 0.9... as a symbol to be some number x which has the property that 10x - x = 0.
Okay, well what is an alternate definition that makes more sense intuitively?
0.333...., for example, is one that seems pretty intuitive. We can get to .333... by iterated long division of 1 by 3.
3 | 1
0 (3 * 0 = 0) => 0.
10 (add zero)
9 (3 * 3 = 9) => 0.3
10 (add zero)
9 (3 * 3 = 9) => 0.33
...
So does this trick always work? If we have a repeated decimal, can we always multiply by 10 ^ (length of repeated sequence), subtract off, and get the value of that repeated decimal? If so, then it is reasonable to say that 0.999.... is equal to 1.
We can't really go in the forward direction without cheating (that is, going from 1 -> .999...); the best we can do is to modify long division to allow us to do it:
3 | 3
0 (3 * 0 = 0) => 0.
30 (add zero)
27 (3 * 9 = 27) => 0.9
30 (add zero)
27 (3 * 9 = 27) => 0.99
...
Obviously this isn't in Peano arithmetic exactly, but I think it holds together. If we allow repeated decimals in general to be valid representations of rational numbers, then we have to accept 0.999.... is equal to 1.
x = 0.8...
10x = 8.8... = 8 + x
9x = 8
8/9 = 0.8...
We do longhand division of 9 by 9, but start by putting a 0 in the first place.
0.99
9 | 9.0000
0
9 0
8 1
90
I should note that when I learned about rational & irrational numbers in elementary school (I think third or fourth grade), we used a "bar" notation where we'd put a bar over the last digits in a decimal expression that repeated forever (i.e. it corresponded exactly to a geometric series with r = (1 / 10)^k where k is the number of digits under the bar, though we didn't know about that at the time). Our teachers explained that the difference between a rational and irrational number was that there would be no pattern you could ever find in an irrational number that would allow us to use the bar, which is surprisingly accurate for grade school arithmetic.
http://us.metamath.org/mpeuni/0.999....html
Unlike typical math proofs, which hint at the underlying steps, every step in this proof only uses precisely an axiom or previously-proven theorem, and you can click on the step to see it. The same is true for all the other theorems. In the end it only depends on predicate logic and ZFC set theory. All the proofs have been verified by 5 different verifiers, written by 5 different people in 5 different programming languages.
You can't make people believe, but you can provide very strong evidence.
And at the point where students see this, the whole concept of real numbers and infinity is usually ill-defined. I actually understand the scepsis for this theorem and where it comes from. The proof relies on the existence of a supremum, which is non-trivial.
I am not very good at mathematics, so I never questioned my professors when they said that "You cannot treat infinites as regular numbers".
Perhaps due to that statement, I did not really pursue these kinds of equations. For instance, I do not really see how the algebraic argument on the Wiki is any different from:
2 * inf = inf
inf + inf = inf (subtract inf from both sides)
inf = 0Infinity is very slippery, and there are several divergent fields of math that depend on particular definitions of it.
Infinity, the number, is routinely confused with creating an onto function mapping digits of pi to a set with a cardinality of the natural numbers. But sadly most people don't have the mathematical maturity to understand the difference when they encounter their first irrational number (normally pi).
The only interesting step is step 32, which is just an application of http://us.metamath.org/mpeuni/geoisum1c.html, whose only interesting step is step 21 which is just an application of http://us.metamath.org/mpeuni/geoisum1.html.
They key steps for that are http://us.metamath.org/mpeuni/geolim2.html and http://us.metamath.org/mpeuni/isumclim.html , which is indeed the crux of the issue
Rigorous proofs like this are for mathematicians and computers, I doubt they help anyone believe who doesn't believe.
I'm not sure how best to help someone who doesn't believe, but it could take arguments with stronger intuition, or just allowing the person to demonstrate with their own proof. It probably depends on the person, and why they don't believe it.
This is illustrative of what I see as a fundamental problem in mathematics education: nobody ever teaches the rules. In this case, the rules of simple arithmetic hit a dead end for mathematicians, so they invented a new rule that allowed them to go further without breaking any old rules. This is generally acceptable in proofs, although it can have significant implications, such as two mutually exclusive but otherwise acceptable rules causing a divergence in fields of study.
When I was taught this, it was like, “Look how smart I am for applying this obtusely-stated limit rule that you were never told about.” This is how you keep people out of math. The point of teaching it is to make it easy, not hard.
Stating that 0.9999... = 1 without exposing these new tools meant to grapple with concepts that physically cannot be grappled with is a huge mistake.
> the supremum of an increasing sequence is equal to the limit
-- this is not misinformation (and to anyone familiar with some introductory analysis, correct[1]). Of course, calling it "dogma" is a bit inflammatory, but not technically wrong. It's kind of of a made-up rule to help us work with infinities (particularly in ℝ -- but it happens all the time in set theory, as well).
But to agree with GP, touting it as "intuitive" or "mind-blowing" is indeed silly.
[1] http://www.math.toronto.edu/ilia/Teaching/MAT378.2010/Limits...
Right, but that's not really the crux of the matter. Hint: look at how the supremum is defined[1]. The definition of the supremum is how we end up with 0.999... = 1.
[1] https://math.stackexchange.com/questions/1977204/limit-of-mo...
And now you realize that you and the student have been operating by different rules. Their rules of equality are based on symbolic equality, so you actually have to relax the rules a bit to make limit equality work. And then, more importantly, you have to show that all the other rules are still intact. Actually, in this case, they aren't. Symbolic equality involving infinity is now horribly broken, and you have to express all equality in terms of limits to maintain consistency. Explore this further and you keep finding more inconsistencies that have to be settled by new rules that define new areas of mathematics.
So who is right? The natural world appears to be much more permissive than limit equality, preferring epsilon-equality. Symbolic equality is the only purely self-consistent system, but you can't do much with it. It's also possible that the natural world works with symbolic rules (quantum) but the complexity is great enough to resemble epsilon equality (continuum).
So, .999... == 1 by tautology. It's not some brilliant mathematical insight. The interesting part is the consequence of defining it as so.
1/3 = 0.333..
3 * 1/3 = 3 * 0.333..
3/3 = 0.999..
1 = 0.999.. x = 0.9999.....
10x = 9.9999.....
(10x -x) = 9x = (9.9999.... - 0.9999....) = 9
x = 9/9 = 1 x = 0.9999...
10x = 9.999...
10x = 9 + 0.999...
10x = 9 + x
9x = 9
x = 1
x = 0.9999...
2x = 1.9999...
2x - x = 1
x = 1
"It's much more intuitive that 10 * 0.9999... = 9.9999... than that 2 * 0.9999...= 1.9999..."
x = 0.999...
to
2x = 1.999..
If you see an 8 it’s because you didn’t carry the 1. Keep going.
Another way to think about it is that you have n digits, and I’m using the n and n + 1 digit at the same time. But since n goes to infinity, +1 hardly matters.
Write
0.9
+ 0.9
—————-
1.8
Where people keep getting tripped up is thinking you can stop when you get tired, or die, or when the universe ends in heat death. You don’t get to stop. You never get to stop. It’s nines all the way down.
0.9 = 1 - 0.1
0.99 = 1 - 0.01
0.999 = 1 - 0.001
0.9999 = 1 - 0.0001
0.99999... = 1 - 0.00000... with a 1 at the end of the infinite series of 0...uhh
subtracting infinities is dangerous, you can achieve any result from it
That subtraction is just as valid as saying 0.333... + 0.333... = 0.666..., or that 1/3 + 1/3 = 2/3.
I also figure it's a bit more intuitive for pupils to just try out calculating the decimal representation of 1/3 and seeing that it'll just keep going forever.
as in 1/3 does not have a decimal representation. you can only approximate it but never reach it.
A definition of a third that most people agree with is that if we multiplied that value by 3, we should get 1. Let's check the right hand side: 3 * (0.33 + 1/100(1/3)) = 0.99 + 1/100 * 1 = 0.99 + 0.01 = 1. Great!
What other expressions for a 1/3 can we come up with? If you agreed with the previous statement, then you must surely also agree that 1/3 = 0.333 + 1/1000(1/3).
Inductively, we should be able to come up with a general formula that 1/3 = bar(3, n) + 1/pow(10, n)(1/3), where bar(3, n) = sum i = 1 to n 3/pow(10, i). We can check that bar(3, 2) = 0.3 + 0.03 = 0.33, and that our first example fits this formula. Intuitively, this formula is giving us a way to represent 1/3 in terms of n decimal places of accuracy and a recursive term.
The question is now, what happens when we run that formula with n to infinity? An infinite level of accuracy! That expression is equal to 0.333... as we have defined.
The right term, 1/pow(10, n)(1/3), goes to 0, so we can discard that. The left hand side, is a geometric series with 1/10 as the power, and a scalar multiple of 3. Using a closed sum formula for that [1], we can see that the left hand side goes towards 1/3. (Apply the formula from Wikipedia, but remember our index starts off at 1, not 0.)
In the end, we have found that 0.333... = n->infty bar(3, n) + 1/pow(10, n)(1/3) = 1/3
I think it's mostly a matter of definition, since mathematicians consider sums of infinite series equal to their limit (if it's finite), i guess for many practical reasons. If you accept this, then 0.999... = 1. If you don't, then 0.999... can't be assigned a value (but converges to 1), which may be the intuitive understanding of infinite series for some.
I disagree. Any middle school student can calculate 1/3 to be 0.33333... using long division, but there's no immediately obvious way to go from 1 (or 1/1) to 0.9999...
I can just do it backwards - is 1/3 equal to 0.33333...?
1 / 3 = 0.33333... 3 * 0.33333... = 0.99999... and my child brain "knows" that 1 != 0.99999...
In my child brain this proves that 1 / 3 is not equal to 0.33333..., it's just an approximation.
So I agree with larschdk, those problems are equivalent and one can't be used to prove the other ...
...the same way That Chuck Norris can count to infinity... twice!
And how will I smart middle-schooler know that the result of running the long division algorithm is exactly 1/3, rather than some approximation.
Yes, but there aren’t good arguments for either of them, and that’s the point. The difference is that you have probably already learned how to divide 1 by 3 and have thus convinced yourself that 1/3 does indeed equal 0.333 repeating. It’s not so simple to come to the conclusion that 0.999 repeating equals 1 from simple long division that you would encounter in grade school.
1/3 = lim(N -> oo) 0.3{N} (3 is N times repeated)
When you are calculating a limit, you can never just plug in the value for N (say if N is in the denominator and the limit goes to 0). Why should you be able to do this when N is infinity?
At least this is my personal justification why I find non-standard reals interesting. They also justify the nice calculation method where you can cancel out 'dx'es from fractions.
It's only with limits and proper formalism that I was reconciled with maths that frankly were just tending towards approaching an equality with bullshit.
1/3 ⅓
Ans * 3 1Me: Is 9.999... the same as 10, or is it just really close to 10?
Kid: Really close. It never gets all the way there.
Me: Well then how close? What do you get when you subtract 9.999... from 10?
Kid: (pause) An infinite number of zeroes. . .and then a one. . .wait, you can't do that.
Me: Right. You just have an infinite number of zeroes. Which is zero.
Kid: (pause) Oh, that's mind-blowing.
why not? why can't an infinitely small number exist?
1/9 = 0.111...
2/9 = 0.222...
3/9 = 0.333...
...
8/9 = 0.888...
9/9 = 0.999...
123/999 = 0.123123123...
999/999 = 0.999999999...
Once you accept this mapping, it's trivial to treat 0.999... as 9/9 (or 99/99, or 999/999, etc). Which can be simplified to 1.
I didn't have a good enough answer for him, so I had to look it up and found this page. I tried to explain it to him but since I'm a terrible teacher and he's only 5, it was hard for me to convince him. Luckily he has many years before it matters!
0.666... is equal to 0.666...7
If someone says 0.666…7 then you have a couple different ways you can take the discussion. You can say, “No! Real numbers don’t work like that!” or you can talk about what number systems would look like if you can do that.
It turns out that there’s a lot to learn from the alternative number systems, including formulations of calculus without limits that are easier to understand from an intuitive perspective, yet equally rigorous. The field is called “nonstandard analysis”. It’s not taught in college.
You might consider "real" numbers (in plain English) to mean physically measurable quantities, but there are plenty of numbers that we can write out because they're infinitely long, but trivially "made" (such as π, which just requires grabbing a compass and drawing a circle)
The bit you should be wondering about is why 0.666...6 and 0.666...7 are the same number: infinities cause digits written on paper (or a computer screen) to look like a kind of number that they're not. The two fractions (numbers in ℚ) 0.6666 and 0.66667 are 0.00001 apart, but the two reals (numbers in ℝ) 0.666...6 and 0.666...7 are 0.000...1 apart. That looks like a tiny tiny fraction, but it's not a fraction, it's an infinite number of zeroes, and thanks to that, this number, while it looks like a fraction, is just a silly way to write zero.
So thanks to infinities, the most-definitely-not-a-fraction number that we write as 0.666... is the same as the most-definitely-not-a-fraction number 0.666...(some numbers here). The difference between the two is zero.
Infinities are fun. And difficult. But also fun.
Ill-defined. Try again.
The blog post that (I think) started off this chain of posts, at https://blog.plover.com/misc/half-baked.html , has the most concise and lucid explanation I've found, so I'm just going to straight-up copy it: 0.666...7 is an “an object... said to ‘have order type ω+1’, and is completely legitimate.”
It's not very useful – it's exactly equal to 0.666... ! – but it's legitimate and well-defined.
My absolute favorite construction of objects like this is Conway's surreal numbers. These things appear perfectly naturally in the surreal numbers, and are completely well-defined, if (again) not very useful.
---
1. we assert that 0.666...7 is a sequence of digits.
2. we assert that each digit in a sequence can be assigned an integer index corresponding to its position in the sequence. I.e. we can defined each digit's index as "the number of digits that precede this digit".
3. from (1) and (2) it follows that the index for 7 must be an integer.
4. the ellipses represents an infinite number of digits (infinitely repeating the repeated digit pattern preceding it).
5. from (2) and (4) it follows that the index for 7 must be the integer value "infinity", because it has an infinite number of digits preceding it.
6. (3) and (5) cannot both be true, because infinity is not an integer.
7. from (6) it follows that (1), and/or (2), and/or (4) must be false
8. (4) is, by definition, true.
9. from (7) and (8) it follows that (1) and/or (2) must be false.
10. (1) is our fundamental assertion. If (1) was false then there is wouldn't even be a sequence of digits for us to reason about. So (1) is true.
11. from (6), (8), and (10) it follows that (2) must be false for there to be no contradiction.
QED
---
Now, certainly, for finite length numbers the assumption that each digit in a sequence has an integer index holds true, but it turns out we have mathematical notation that lets us write down numbers for which that property does not hold.
Infinities are fun. And difficult. But also fun.
A sequence in the mathematical sense is a function whose domain is the natural numbers. Please define that function for the creature you're working with here. Otherwise you're trying to prove things about an object with no definition. You will end up in trouble.
As it currently stands, assumption (1) is similar in nature to me saying "gnarfgnarf is an imaginary number". It's completely meaningles unless I define what I mean by gnarfgnarf.
So: what do you mean by 0.666…7?
Same here: we have a number written as 0.666...7 using conventional mathematical notation. The comment that is being replied to asserts that this can be treated as a sequence, and so we start the proof with that definition: "0.666...7 is a sequence", and now we're done. You, as reader of the proof, have been informed that those nine symbols, in that order, for the rest the proof, represent a sequence. Not "a specific sequence", but "any sequence", and it must follow all the rules that sequences follow.
We then show that simply by being "a sequence", due to the properties of sequences, we get a contradiction. Our first assertion is the definition for the purpose of this proof, and is sufficient.
This would make me very proud.
https://stats.stackexchange.com/questions/218821/round-to-ev...
and depending on career choices, it might never matter at all.
You’ll see various proofs involving real numbers that must account for the fact that 0.999…=1.0. There are, of course, many different ways to construct real numbers, and often it’s very convenient to construct them as infinite sequences of digits after the decimal. For example, this construction makes the diagonalization argument easier. However, you must take care in your diagonalization argument not to construct a different decimal representation of a number already in your list!
Mathematically, mathematicians prove that there is a unique number that this process goes to, (and not, say, two distinct numbers), and define the notation to represent this unique number.
Arguably the sign symbol ruins it for whole numbers as well, as +0 and -0 could be equally valid representations of the number 0. We just conventionally don't allow -0 as a representation. There are other number representations that don't have this problem.
But for the sake of argument, let's just define numbers as sequences of digits with a mixed in period somewhere:
MyNumber := {
a = (a_1, a_2, ...) -- list of digits a_i = 0 .. 9; a_1 != 0.
e -- exponent (integer)
s -- sign (+/- 1)
}
We can go on and define addition, subtraction, multiplication and division in the familiar way.
Problems arise only when we try to establish desireable properties, e.g.
(1/3) * 3 = 1
Does NOT hold here, since 0.9999... is a difference sequence than 1.000....
So yes, you can define these number systems, and you will have 0.999... != 1. But working with them will be pretty awkward, since a lot of familiar arithmetic breaks down.
a_i = 9 for i \in \IZ and i < 0
a_i = 0 for i \in \IZ and i >= 0
If any two real numbers are not equal, then you can take the average and get a third number that is half way between them. Conversely, if the average of two numbers is equal to either of the numbers, then the two numbers are equal. (this isn't a proof, just a way to convince yourself of this)
What's the average of .9999... and 1?
Let C be the countable product of the set with ten elements, i.e. {0, 1, 2, ..., 9}. The space C naturally has the topology of a Cantor set (compact, totally disconnected, etc). Furthermore, for example, in this space the tuples (1, 9, 9, 9, ...) and (2, 0, 0, 0, ...) are distinct elements.
The space C can also be described in terms of a directed graph, where there is a single root with ten outward directed edges, and each child node then has ten outward directed edges, etc. C can be thought of as the space of infinite paths on this graph.
A continuous and surjective map from C to the unit interval [0, 1] can be constructed from a measure on these paths. For any suitable measure, this map is finite-to-one, meaning at most finitely many elements of C are mapped to a single element in the interval. For example there is a map which sends (1, 9, 9, ...) and (2, 0, 0,....) to the element "0.2".
The point is that all decimal expansions of elements of [0, 1] can be described like this, and we can instead think of the unit interval not as being composed of numbers _instrinsically_, but more like some kind of mathematical object that _admits_ decimal expansions. The unit interval itself can be described in other ways mathematically, and is not necessarily tied to being represented as real numbers. Hope this helps someone!
0.999... = 1 - 1/∞
We talk about infinity all the time in mathematics, teachers use the concept to introduce calculus in a way that people can more easily understand, but using infinity directly is almost universally banned within classrooms.
Nonstandard analysis is a much more intuitive way of understanding calculus, it's the whole "infinite number of infinitely small pieces" concept, but you're allowed to write it down too.
0.333... = 1/3 - 1/∞
Which implies 3/∞ = 1/∞
0.999... implies a number infinitesimally smaller than 1. You wouldn't use 0.999... in a hyperreal system because you can represent it directly.
I shouldn't have mixed different systems and claimed they're mathematically equivalent, you've proven that doesn't work.
Computers agree: never trust precision to floats
I understand and accept this is wrong. However, somewhere in my brain I still believe it. Sort of like +0 and -0, which are also different in my head.
Mathematically you are wrong.
It is indeed true that if there was an entity that could reason beyond infinity 1-0.999... would be greater than 0.
This seems like a common thread in the comments on this article, and I don't quite understand it.
Humans can reason about infinity just fine. We have a hard time picturing it, but it's overall a pretty simple concept: It never ends.
So there's no such thing as being able to reason "beyond infinity", because beyond infinity doesn't exist. Its very existence is precluded by the definition of infinity.
What is the difference between them?
The problem here is our language for mathematics. Just like you have to accept the silent "k" on the word "knife", even when it doesn't make sense, in math, you have to understand that rational numbers can't always be expressed accurately as decimals.
There are approaches to mathematics that avoid infinite constructions, and a "strict finitist" would not assign 0.999... a meaning.
The stunning success of limit based mathematics makes finitism a fringe philosophy.
Remember, class, for every epsilon there is a delta.
https://www.youtube.com/watch?v=WabHm1QWVCA
I mention him because I would think he sympathizes with those who have concern over the meaning of this kind of notation.
I actually have some sympathies with his contention that real numbers (limit points of infinite series) are somehow a different animal than rational numbers. But it might be easier for me to go there because practically all numbers on computers that we work with are rational, floating point values. On the other hand, it seems like a philosophical distinction in the end because you can fully order them both on a number line.
I guess I shouldn't phrase it as "you can fully order it". :D Zermelo's theorem at that point right?
> but presumably they lie somewhere regardless of my inability to do it on a TM.
Why?
> Zermelo's theorem at that point right?
It is declared by fiat in standard set theory that infinite sets can be well ordered. This is no real mathematical justification. The real justification is social: that it is convenient for mathematicians to not care about the ontology of these nasty infinite objects so long as results are mostly reasonable for objects that mathematicians actually care about. You don't get into too much trouble pretending the reals are nice so long as you don't look too hard.
console.log(0.1 + 0.2)
// 0.30000000000000004
A mathematician might say that this shows that you do not really have accurate floating point values and arithmetic in your computer, but instead something close to it.
Racket starts with arbitrary precision rationals.
However as a representation of physical world, there is a caveat. What we understand is physical world appears and behaves discretely, because at planck scale (approx. 10^-35) the distances seem to behave discretely.
Although common people don't know/ understand planck scale, they do grasp this concept intuitively. What they are really saying is that in physical world there's some small interval (more precisely, about[1 - 10^-35, 1]) which can't be subdivided further, based on our current knowledge.
Same thing applies to planck time (approx. 5 * 10^-43) too.
So people are arguing two different things - the pure maths concept, or the real world interpretation.
The Planck length might or might not be a physical limit of the universe. We don't have any specific proof that it's the smallest, just that we will not be able to observe any of that size or smaller. To look at something, we need to use light, and we must use a wavelength smaller than the details we wish to resolve. For something of the Planck length or smaller, this ultimately results in a photon that would have more energy in that area than can exist without a black hole forming... so one does, which then prevents us from measuring it, much less anything smaller.
Space and time might very well be discrete and not continuous - certainly the Loop Quantum Gravity folks would agree there. But there are widely supported theories that take both sides.
(I tend to lean towards them being discrete, but I would hesitate to call myself even an amateur hobbyist when it comes to theoretical physics...)
0.99
7 | 7.00000
0
7 0
6 3
70The notation "0.999..." looks non-threatening, which tricks people into believing that they understand what it means. We could make "0.999... = 1" look scarier by writing it as [n ↦ 1 - 10^(-n)] = [n ↦ 1], where [n ↦ a_n] denotes the equivalence class of a Cauchy sequence of rational numbers. These statements mean the same thing, but with the scarier notation much fewer people would mistakenly believe that they understand what it says.
I would expect mathematics majors to learn what 0.999... means during their undergraduate university courses. But then there's still the question of why mathematicians chose to define it that way. To really understand that, you need to be able to come up with alternative definitions and to investigate the consequences of those definitions. And for most undergraduates, it might still take a few years to build that level of mathematical maturity.
For anyone who is not a math major, I certainly don't want to discourage any curiosity about this subject. Just don't be discouraged if you feel you can't fully understand what's going on. Understanding what 0.999... means and why mathematicians chose to define it that way is quite subtle.
I’ll keep trying to understand it.
The problem isn't that you can't come up with axioms to convince people you have a proof - the problem is with people not understanding that 0.99999.... is not a number - it's one representation of an abstract entity called a number.
The problem is, the maths required to actually define the concept of a number is fairly complicated, so it's hard to explain to someone why all of these axioms make sense in the first place.
Generally the proofs of .9...=1 rely on the fact there is no number that exists that can be between .9.. and 1 and therefore .9... is equal to 1.
.9... is the least upper bounds of the set. My question is if .9... was removed from the set what would be the new least upper bounds. Another way of asking the question is if we define it in this context doesn't any set bounded by a real number have a least upper bounds and aren't all real numbers equal to each other?
Thanks!
To me, 0.9999 indicates a directional limit, which can't necessarily be evaluated and substituted separately from its context.
What gets broken? What consequences do we hit?
1/3 * 3 could still be equal to one. but 1/3 != 0.33333... that is, 1/3 is not representable in base 10. Which makes way more sense.
I wonder if taking 0.9999.. != 1, that is 0.0000...1 exists would allow us to reslove, the fact that some possible events have probability 0?
If you want to redefine it explicitly as not a real number, you can do that, and maybe even get to some amusing math that way, but you're no longer talking the same language as the rest of the world.
yes, in the standard real numbers 1 = 0.999.., but people have dealt with numbers like "pi" and "sqrt(2)" before the standard real numbers were defined.
Hence the question, if we define such a system such as 0.333... != 1/3. what are the consequences?
by 0.3333... I mean a countably infinite sequence of 3s.
If you want to go to supersets of real numbers, you may be interested in https://en.wikipedia.org/wiki/Surreal_number
But the issue is that this is easy to verify experimentally via (in this case infinitely) long division that you can do by hand. So it’s hard to convince people of this.
why would it terminate at countable infinity?
You'd be asserting that if I eat the three parts I have not eaten the whole pizza.
I'm unconvinced.
This is one of the many (equivalent) ways the real numbers are defined to begin with, https://en.wikipedia.org/wiki/Construction_of_the_real_numbe...
There are lots of other ways to define sets with operations, but they won't be anything at all like the normal numbers you are used to.
However, I was a bit fast and loose, the sequence .9, .99, .999, .9999 also gets closer and closer to 2 but it doesn't converge to 2, I should have said if you have a metric || and some number X such that for any d, there exists an N such that |X -An| < d for all n > N, then the sequence A converges to X. But I wasn't trying to write a proof.
https://en.wikipedia.org/wiki/(%CE%B5,_%CE%B4)-definition_of...
Well, you fundamentally can't. If the 9s go on for forever then you never reach a point where you can add the 4. The definition of infinity precludes anything after infinity, because it never ends so you can never get there.
I'm not even remotely an expert here, so I might certainly be wrong, but I don't understand how Cantor's theorems show an ability to stick a finite number and stop on the end of an infinite number.
https://en.wikipedia.org/wiki/Ordinal_number
So you could, if you like, define 0.9...4 to be a bunch of digits indexed by the ordinal ω+1. However the thing you have now defined isn't really a representation of a real number any more, unless you just ignore all the bits after the ... I guess.
Thank you!
Base case: Given 𝑎⁰ = 0, 𝑎⁰ ≠ 1.
Inductive case: Given 𝑎ⁿ⁺¹ = 1 - (1 - 𝑎ⁿ) / 10, 𝑎ⁿ ≠ 1 ⇒ 𝑎ⁿ⁺¹ ≠ 1
My intuitive way to see this is that infinitely repeating decimals are an abomination that breaks the nice property of decimal notation where each number contains a single representation (without zeroes at the beginning or at the end of the decimal). Fractions are the one true way to represent rational numbers.
I feel the easiest "proof" is a proof by contradiction.
First hopefully we can agree that if we have two real numbers x and y that are not equal then we have a number z, such that x < z < y. The easiest example is z = (x + y)/2.
If 0.999...!= 1 then there must exist a number A, such that 0.999... < A < 1.
Now since 0 < A < 1 (as 0 < 0.999...) it should be easy to see that A's decimal expansion is of the form 0.abcdef... . Since, A != 0.999... one of the digits in the decimal expansion of A has to be something other than a 9.
For instance, we might have A = 0.99998999... .
However, this would mean A < 0.999... as all digits other than 9 are smaller than 9 [1], but this contradicts our initial assumption that 0.999... < A and since we're dealing with strict inequalities both can't be true at the same time! Thus no such A should exists and thus 0.999... = 1.
Now this isn't a rigorous proof, the thing that makes me most uncomfortable is the bit that I state A < 0.999..., but I'm uncomfortable because A might have multiple decimal expansions and I don't know how the algorithm in [1] interacts with that, however, if someone quibbles about that bit of the proof for that reason I feel they should have already accepted that 0.999... = 1 via another rigorous proof.
[1] If this is not clear think about how you would compare two decimal expansions to see if one is smaller than the other. You go through every digit until you find one that is different between the two numbers and then you compare those.
Thanks for the reply, but I don't know how you can make the above claim, because to me the question of what we mean by 0.999... is intertwined with the claim itself. I mean to me it seems to be a matter of how you interpret the approach to infinity. I don't see why there has to be A in between, if you interpret 0.999... as the "biggest possible number below 1", as then there would be also a "difference of the smallest possible amount" between those numbers approaching 0, but not quite getting there. But then again, if it's by some fundamental definition (limit) that 0.999... = 1, then ok.
I'm slightly embarrassed I don't know more mathematics, but I'm trying to learn some more...
> First hopefully we can agree that if we have two real numbers x and y that are not equal then we have a number z, such that x < z < y. The easiest example is z = (x + y)/2.
It's one of the properties of the reals and rationals that if two numbers aren't equal, then there are infinitely many numbers between them. It doesn't work with the integers, 2 and 3 aren't equal, but there's no integer x, such that 2 < x < 3.
So if we say 0.999... != 1 then that means 0.999... < 1 and then that means (0.999... + 1) / 2 is a number different from 1 and different 0.999... but that lies between them.
But this is modern mathematics. In the past mathematicians dealt with "infinitesimals", especially in the early days of calculus, but I think they were discarded because they were confusing and also not necessary in favor of limits. This is where I think where some of your confusion is coming from. Infinitesimals don't exist in the real (and therefore rational) number space, but the concept exists for other "weirder" number systems.
According to the wikipedia page "This repeating decimal represents the smallest number no less than every decimal number in the sequence (0.9, 0.99, 0.999, ...)"
The clearest resolution to "if you interpret 0.999... as the "biggest possible number below 1"" is that in the reals and rational this concept doesn't exist. There's no biggest number smaller than x and ditto for smallest number bigger than y. (The distinction from the wikipedia definition is the difference between < and <=, <= exists, but < doesn't)
It's similar to saying you can't divide by 0. Sure in some cases you can define it, but doing so causes many issues and costs you so much that it's not worth it.[1] Another example is 1 not being prime, there's no reason for it not to be prime, but it's just much more convenient to say arbitrarily that it's not prime.[2]
The other thing that might confuse you is a proof by contradiction, I know it certainly confused me the first few times I saw it. I'm happy to help you if this is tripping you up too.
> I'm slightly embarrassed I don't know more mathematics, but I'm trying to learn some more...
No problem, we all start knowing nothing :)
[1] https://www.youtube.com/watch?v=BRRolKTlF6Q [2] https://www.youtube.com/watch?v=IQofiPqhJ_s
Is 0.999... = 1? Yes, because we define decimal numbers to behave that way. Why do we define them to behave that way? That's the real question.
The result of 1 minus 0.999... is 0.000 with zeroes that go to infinity. And I think its easier to reason that 0.000 with repeating zeroes forever is in fact equal to zero.
For most purposes hyperreals are too much machinery when learning and manipulating limits would be simpler.
That would be adding a special rule for purely cosmetic reasons. It's typically not done in mathematics, where concise rules are usually more cherished than special-casing things.
How much faster than 99.999..% the speed of light would you need to go to get as fast as the speed of light?
If your answer is 0.00..1% the speed of light, this answer is nonsensical because infinity never ends, and you never ever have the ability to add that 1 at the end. So, the only answer can be 0.00.., and 0.00.. = 0, so 99.99.. has to = 1.
Also, I’d like to understand how if we can say 99.999... is = 100, wouldn’t we also be able to say 99.9999888999... = 99.9999..., and therefore also just 100? And so on?
You've just introduced a novel symbol into the discussion. What is the definition of 99.9999888999...? Before anyone can answer what it equals, you must define it. Recall that if the digits repeat, we already have a definition, so that case is fine. Your case is not à priori well-defined.
Assume x is the largest number smaller than 1.
(x + 1)/2 is a number larger than x but smaller than 1. Our assumption that x is the largest number smaller than 1 must be wrong. QED.
0.9999... < 1
It is because the "limit" in
0.999... = lim[eps->0] 1-eps
UPDATE: So instead of interpreting the expression as:
(lim[eps->0] 1-eps) < 1
lim[eps->0] ((1-eps) < 1)
What is this supposed to even mean?
If you accept both of those two (which you should, because they're correct), then since equality is transitive, 0.999... = 1. Therefore it is not less than 1, it is equal to 1.
0.9999... < 1
lim[eps->0]( 1 - eps < 1)
1 - 0 < 1
lim[eps->0] (x - eps)
x - 0
You're simply wrong here. You can read that wikipedia article if you don't trust me, or you can watch any of the thousands of youtube videos of mathematicians explaining this to you.
This is one of those things that is hard to do. Once you have an idea in your head that you're sure is right, it is incredibly difficult to dislodge it. It takes an enormous amount of humility and intellectual flexibility. But it is healthy and good to do it, every time you do it, you become a better person for it.
It isn't a valid argument for 0.999... != 1 but an interesting one nevertheless.
Assuming you don't mean some special notion of limit, I would guess that by `(1-eps) < 1` you mean the function from the reals to `Y = {false, true}` that is defined as sending `x` to `true` if `1-x` is strictly less than `1`, and `false` otherwise. Let's call this function `f`. I assume you're endowing `Y` with the discrete metric?
If so, `f` does indeed have a limit from above that is `false` and a limit from below that is `true`. Where do you wanna go from here?
Edit: Corrected stupid wrong assertion about limit from below, d'oh.
https://www.wolframalpha.com/input/?i=Limit%5BSign%5B1-x-1%5...
This shows that the limit does exist from both sides (but is different from both sides).
That's the other way around. From above you get true, and from below you get false.
Note that 0.999... represents the limit from above for 1-eps. Hence the result is true.
Yeah, my bad.
> Note that 0.999... represents the limit from above for 1-eps. Hence the result is true.
I mean you can define 0.999… like that if you want. You'll get 1. So what? Your detour via `f` provided nothing.
Given ε = 1/∞ then: ε = 0
Am I wrong in thinking this way? It seems as though there's no way to actually truly prove that an infinite series converging towards zero actually hits zero (from a constructivist pov)
Also go read Generatingfunctionology and Concrete Mathematics
1 - sum{k=1}^{\ifnty} 10^-k is one example.
I think you'd have to dismiss them all to make your claim (to do it well, that is).
Any non-empty set of real numbers with an upper bound has a least upper bound.
We can't prove your statement
Given ε = 1/∞ then: ε = 0
because it is not well-defined, but we can prove this:
If ε ≥ 0 and, for every natural number n, ε < 1/n, the n ε = 0.
For suppose there exists an ε which is a counter-example, i.e. ε > 0 and ε < 1/n for every natural number n. Then the set
S = { x : x a real number, x > 0 and x < 1/n for every natural n},
is non-empty, and has an upper bound (e.g. 1). So it has a least upper bound, say y. In particular, y is an upper bound, so 2y is not in S. It is > 0, so there must exist a natural number N for which 2y >= 1/N. But then y/2 > 1/4N, and 4N is also a natural number. So for any element x of S, x < 1/4N < y/2; thus y/2 is an upper bound which is less than y.
This is a contradiction, so the claim is proved.
At least one can simply prove that 0.999... = 1 without much hard work. Maybe less controversial than the following:
1 + 2 + 3 + ... [somehow] = -1/12 {{Riemann's zeta(-1)?}}
1 + 2 + 4 + 8 + 16 + ... [somehow] = -1
Have seen that.
Another one by 3b1b on that topic: https://youtube.com/watch?v=sD0NjbwqlYw
How about we prove that an infinite number of 9s is impossible?
Assume that we have a finite number of 9s. Add a 9. The result is not infinite. Add another 9. The result is still not infinite. We can repeat this process for an infinite amount of time and still not have an infinite number of nines.
Any process that can not be completed in a finite amount of time can not complete and can not have a valid result based on that completion. Any process that can not be completed in an infinite amount of time is also bogus, but is in a sense even more bogus.
Added: Note that this is different than the case where we are asked to contemplate infinity with respect to continuous functions. By defining the number of 9s as a discrete (integer) value it opens things up to a discrete argument. These pointless navel gazing exercises always end up as a war of what everyone things things are defined as.
Well, no, you cant. How about I prove it to you:
Assume that we have a finite number of processes. Repeat the process. The number of processes is still not infinite.
/s
First you need to define what you even mean by this statement. The rest of what you wrote makes no sense either.
Isn't this somewhat ill-defined? You may have a finite number of 9s at any one point when adding more 9s, but there is no point "at infinity" where you can stop and look at how many 9s you've added because by definition there are still more 9s to add.
It's like trying to prove infinity is impossible:
> Assume that we have a finite number. Add 1 to that number. The result is not infinite. Add another 1. The result is still not infinite. We can repeat this process for an infinite amount of time and the number is still not infinity.
Sure, individual numbers "on the way" to infinity are not infinite, but that doesn't necessarily disprove the existence of infinity.