0.999...= 1 | Hacker News Reader

0.999...= 1

en.wikipedia.org

218 pointsyurisagalov626 comments

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

If you tell a person that 3/6 = 1/2, they'll believe you - because they have been taught from an early age that fractions can have multiple "representations" for the same underlying amount.

People mistakenly believe that decimal numbers don't have multiple representations - which, in a way is correct. The bar or dot or ... are there to plug a gap, allowing more values to be represented accurately than plain-old decimal numbers allow for. It has the side effect of introducing multiple representations - and even with this limitation, it doesn't cover everything - Pi can't be represented with an accurate number, for example.

But it also exposes a limitation in humans: We cannot imagine infinity. Some of us can abstract it away in useful ways, but for the rest of the world everything has an end.

I wonder if there's anything I can do with my children to prevent them from being bound by this mental limitation?

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

I think this is a very insightful remark. People think that numerals _are_ numbers, and it's hard to explain why this is not the case, because we have no way to talk about specific numbers _except_ by using numerals. But many frequently-asked questions are based in a confusion between numbers and numerals. For example, many beginner questions on Math SE about irrational numbers are based in the mistaken belief that an irrational number is one whose decimal representation doesn't repeat. I've met many people who were just boggled by the idea that “10” might mean ●●, or ●●●● ●●●● ●●●● ●●●●, rather than ●●●●● ●●●●●. A particularly interesting example I remember is the guy who asked what were the digits that made up the number ∞. It's a number, so it must have digits, right? (https://math.stackexchange.com/q/709657/25554)

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.”)

Apparently the New Math (https://en.wikipedia.org/wiki/New_Math) tried to address this kind of issue quite explicitly, by drawing a consistent distinction between numbers and numerals (where a numeral is a symbol that names a number). Reportedly most American math students found this kind of distinction extremely hard to grasp when they were presented with this kind of issue in elementary school. Maybe it would have worked better when they were a bit older.

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 mistaken belief that an irrational number is one whose decimal representation doesn't repeat

...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?

Maybe I'm misunderstanding, but I think the issue with dates is strictly different. Dates are hard not because time is fundamentally hard, but because there is lots of complexity in human representation of time (different places at different times have had similar but different representations of time). But that's not inherent to time. Ignoring relativity, if everyone throughout time used something like seconds since Unix epoch (or some similarly arbitrary point in time), then writing programs about time would be much simpler.

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.

For example, many beginner questions on Math SE about irrational numbers are based in the mistaken belief that an irrational number is one whose decimal representation doesn't repeat.

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.

Proof: bring up removing time zones, such that there is only one time zone.

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.

> A particularly interesting example I remember is the guy who asked what were the digits that made up the number ∞. It's a number, so it must have digits, right?

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)

Reminds me of Terry Pratchett's description of the mathematical reasoning of camels in Pyramids:

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.

Possibly people are looking at two different symbols and asking "can you show me logically why those are equal." If they're given a definition of "equal" and they still object, that's a different problem.

I have this problem every time I play with group theory again. You get the axioms for a group, which say there is some identity but don't explicity require the identity to be unique. You can easily prove that the identity of a group is unique ... so long as you define "unique" to mean "if element e1 and element e2 are equal, then we say they are the same element."

You could count things differently and say the identity is "not unique", it would just lead to a lot of stupid and un-illuminating consequences.

I used to have the same annoyance about group and category theory; learning some about the type-theory-as-foundations work (particularly homotopy type theory and cubical type theory) has helped w/ this; in that setting, you have several distinct but well-developed notions of equality: propositional equality, which group theory and most math cares about, vs judgemental equality, which is the one that's "obviously true" by the rules of the system.

e.g. 5 = 5 is true under judgemental and propositional equality, whereas x + 2 = 2 + x is only true under propositional

I think a lot of people don't think of math in terms of definitions and proof. Math was just something they were taught as kids. And even if they've gotten into more advanced math, i think the 1 = .9... question hits their kindergarten brain and they just say "no" to it the same way they'd say "no" to someone singing the alphabet song in the wrong order.

This is a good point, but an even more basic issue is that the question "what is a number" is a matter of definition. There isn't a "correct" definition of numbers; only one that we've accepted as standard. The accepted definition of a "real number" is actually quite complicated [1], and it's certainly not easy to convey why this complexity is necessary. Other definitions are also possible [2], but nonstandard.

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.

[1] https://en.wikipedia.org/wiki/Dedekind_cut

[2] https://en.wikipedia.org/wiki/Hyperreal_number

Sorry, no.

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.

https://en.wikipedia.org/wiki/Surreal_number

Yes, but at the same time it is common for people to insist that 0.999… only "approaches" unity as if it were a series approaching a limit, rather than an unique number. Intuition is a funny thing.

Saying "a number isn't a limit" is true, but it's only really relevant if you're talking to someone who genuinely has no idea what limits are. In actual math the number 0.99... can be defined as the limit of a series.

great point. My thought on (1/3 = 0.3333...) * 3 = 1 = 0.999... was that it is intuitively obvious that the "problem" is that we use base-10 for decimals. There is nothing magic or unknowable about the quantity 1/3.

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.

I think the problem is that people are often only taught base-10 so they confuse numerals with numbers. If you learn base 2 and then base 16 and then base pi, you start to realize that numbers are something more abstract than whichever numeral system we use to represent them. Rightly or wrongly, the way I imagine integers now is an infinite set of different numerals (base infinity?) such that there is only ever 1 digit (I don't actually have concrete pictorials assigned to those numerals).

Base 12 is a better base overall. It is divisible by more numbers. That is why it is used in various monetary, time and measurement systems.

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.

I think the confusion comes when mathematicians say "a sheet of paper can be 0.333... (0.3_, ie 0.3 recurring) units long" or something because in experienced reality we can always choose a measure that's rational (in the maths sense). That 1/3 of a meter can be measured in one-third-meter units and be precise and easily written.

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 ...

Whatever base you use, it will only be useful for rationals. You can't use any base to significantly improve the representation of pi. In base 7, 3.1 would be a nice approximation, but you can't go beyond approximations.

You can't fix irrationals by changing base. But, there are other approaches.

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.

https://en.wikipedia.org/wiki/Continued_fraction

Decimals are easier to compare than fractions. I can't easily work out in my head which one is bigger, 457/790 or 580/924, but I can easily see that (approximately) 0.57848 is smaller than (approximately) 0.62771.

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?

"Players and painted stage took all my love, And not those things that they were emblems of."

The Circus Animal's Desertion, W. B. Yeats

Similarly, perhaps, many take for granted that the written version of a word, in current standard spelling is the word.

The real issue is that we don't define the real number system before we use it. The fact that 0.999... = 1 is a consequence of a formal definition of decimal numbers. We can create a new definition of decimal numbers that does not satisfy this equation and use it in place of our current one.

Let's imagine a new decimal number system with some vague notion of infinitesimal numbers. We lose some properties we enjoy in our current system but all of those properties still hold for numbers with no infinitesimal part. We can still use our every day numbers like nothing has changed yet we also have a notion to describe infinitesimal values. We can make statements like 1/3 is infinitesimally less than 0.333... and carry on like nothing else has changed.

Now let's sit someone down, start with the rational numbers, introduce Dedekind cuts to define the real numbers and prove that in the real number system that 0.999... is exactly equal to one. Let's also convince them that the real numbers are the unique complete ordered field and that each of these properties are indispensable. Then they will believe that 0.999... should be equal to 1.

> There is no proof that will ever satisfy a person dead-set against this.

Indeed. I've torn my hair out trying to convince smart people with PhDs in hard sciences and had to give up in frustration.

I usually find that the most success can be had by kicking the ball to them immediately and having them define what they actually mean when they say "0.999…". If we're going to debate whether that thing equals another thing, we better make sure we know what we're talking about. Inevitably, this either causes the dead-set person to give up, or give a myriad of definitions that are either meaningless, ill-defined, or causes them to realize that they don't actually know what "0.999…" means (or what they want it to mean). It is hard to have the patience to chase down the consequences of their ill-fated definitions, though.

Ask for a number between .9 repeated and 1

"There isn't one, 1 is the very next number right after 0.999... Checkmate atheists." (In all seriousness I don't think it's a very convincing argument for someone who doesn't buy the proofs -- it requires you to believe and have internalized the idea that there are an infinite number of reals between any two distinct reals, and therefore that any pair of reals with nothing between are the same number. Those seem like bigger logical leaps to me than the simple proofs for someone who hasn't thought about this stuff.)

How about, ask for an integer between 1 and 2. Can't think of one? Guess they're the same number then.

I don't understand how they're the same number.

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.

So does this mean that an infinitely small number is zero? As in 1/∞ ?

It's the smallest number bigger than 0.

Ask for a letter between G and H.

0.00...1

> It is hard to have the patience to chase down the consequences of their ill-fated definitions, though.

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.

> 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.

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.

(STATEMENT OF PERSONAL IGNORANCE [SOPI]: Anyone who actually understands this stuff please correct my mistakes below. Thanks.)

In the real numbers, which are not always simple or intuitive, 0.99... = 1. That's true and I seem to understand the proof.

But the real numbers aren't the only system that might be sitting behind "0.99..." and "1" when I write those symbols down and talk intuitively to people in my family. The reals are just the system we're taught first.

I believe there are other systems (I think the surreals are an example) that work just as well for everyday purposes, but where ( my understanding is that) there are numbers that differ from 1 by a value that approaches zero, yet those numbers are not equal to 1. (I've played with the surreals but only as a hobbyist.)

If you do calculus in these other numbers, I think physically meaningful problems will still yield the same answers. (For example Zeno's Paradoxes are still not an excuse for failure to attend school.) But it isn't a law of nature, I think, that all number systems that can hold 0.999... and 1 must make them equal.

The easiest way I know to explain it is fractions.

  1 / 3 = 0.33333....

  2 / 3 = 0.66666....

So what's 3 / 3?

Some people don't like that one. They might like this one better:

  1 / 11 = 0.0909090909...

What's 10 times that?

  10 * 0.0909090909... = 0.90909090...

So, let's do some addition and let the values zipper together because a nine will always line up with a zero:

  10 * 0.0909090909... + 0.0909090909... = 0.90909090... + 0.0909090909... = 0.9999999999...

However, 10 * 1 / 11 = 10 / 11. And 10 / 11 + 1 / 11 = 11 / 11. So 11 / 11 must be the same as 0.99999....

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

And 1.0 in base 11 is 1.0 in any base.

A number is just a number, it doesn't approach anything. A series can approach something, but a number can't. In any system where 0.99... is valid notation for a number, it doesn't approach anything.

Those number systems do exist, but I'm not sure it's right to say they work just as well for everyday purposes. They work only as long as you use them in a way that reduces to treating them as real numbers, either never computing an infintesimal in the first place or calculating 23 + 6ε and saying "oh that's basically just 23".

Even in surreal numbers, where there indeed exists numbers "infinitely close" to 1 but smaller than it, 0.9999.. would still be exactly 1. That's a quirk/feature of the decimal representation, not the underlying theory of numbers.

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...)

    0.999... + 0.000...1 = 1
    0.000...1 = 1/∞
    0.999... = 1 - 1/∞

1/∞ is zero or not?

One way to introduce the idea that a number represented as decimal digits can have multiple representations is to talk about the numbers 1 and 1.0 being exactly the same. And that 1.00 is the same as 1. Just like 0, 0.0, and 0.00 are the same number. Most people would agree at this point.

Then keep stretching the number of zeroes to 0.000... - which, again, is exactly the same as 0.

From there, it is not a huge stretch to be able to go from that 0.000... is another way to write 0, then 0.999... is another way to write 1.

The real question is what do you get if you add:

0.999 ... infinite number of 9s ... 9

and

0.000 ... infinite number of 0s ... 1

When I was a child I was convinced by a pretty simple conversation with my father:

Me: 0.9999... is not the same as 1

Him: Well if it's not the same is it more than 1 or less than 1?

Me: Less

Him: Okay then how much less is it?

At this point I started trying to do 1 - 0.999..., using the methods I'd been taught, and after a few iterations of "borrowing" the 1 I realized the answer was 0.000... which I was pretty convinced was equal to 0.

Hehe... smart man.

Another one is that

1 / 3 * 3 = 1

<==> 0.333... * 3 = 1

<==> 0.999... = 1

"yes but 1/3 does not equal .333... it's just an approximation since there's no perfect way to represent 1/3"

In base 3,

    1/10*10=1, 0.1*10=1

What the problem?

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

This works because you have defined what equality means.

We all wanna talk infinity because it sounds more exciting, but I think everybody gets "infinitely close to 1" pretty well intuitively. What they don't get is whether "infinitely close to 1" means "equal to 1". That could happen because these people are stupid.[note] But it could also happen if nobody has defined equality.

[note]for example even highly educated people maybe don't listen, which is functionally a lot like being stupid.

is there any difference between a black hole and nothing? (somewhat joking but I was thinking of a physical analogy of the limit approaching zero)

Why not 0.00...1?

That's a great approach

I didn't grok infinity until I started thinking in terms of verbs rather than nouns. As a static number, the concept of infinity makes no sense; but once reimagined as a process (start counting up from 1, and never stop), all apparent paradoxes disappear.

This is the inverse problem: it could just as easily be reframed as 0.000...0001 = 0. Defined as static nouns (does such a thing exist in nature?), it's seemingly paradoxical, and fascinatingly debatable in a "is a hot dog a sandwich" sort of way. But reframe it as a process (or as code), and all confusion disappears: for how many loops would you like to proceed? If you never stop, 0.99999... clearly approaches 1, without ever reaching it, and asking if they're the "same" is as academic as asking if the Ship of Theseus is the same ship, or if an electron is the same entity from one picosecond to the next.

> it could just as easily be reframed as 0.000...0001 = 0

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.

I find that "infinite as an endless process" concept intuitively very heplful as well. However, reading Gödel, Escher, Bach[1] showed me that there's another, more static logical interpretation of infinity which also comes handy.

In an infinite process, you can always take "one more step" to create the item after that. Let's assume there exists a "final" mathematical object that goes after every finite item in the generation process (i.e. it is higher than any item in the list, or smaller, or has happened after all of them)... This object doesn't really belong to the infinite generative sequence, it's an item outside all of them, and can't be reached by completing the sequence; it merely exist outside the process and happens to have the property of "dominating" all the items in it.

You can assume the existence in the same way you assume the existence of a number which is the square root of -1, or how you define triangles whose angles add up to more or less than 180 degrees. If you do that, this object "at the infinite" can be formally defined and treated axiomatically to find out what mathematical properties it possesses.

[1] https://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach

> As a static number, the concept of infinity makes no sense; but once reimagined as a process

Super insightful. That's the key right there.

The same concept can also be applied to the physical world. Things are not static, they are in constant flux, everything is a process in motion.

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

If you want to work with the real numbers intuitionistically (or constructively), you quickly find out that infinite decimal expansions are not what you want.

In classical mathematics all the usual definitions of real numbers (decimal, Cauchy sequences and Dedekin cuts) are equivalent. If you overthrow the Law of Excluded middle, these are all different.

Infinite decimal expansions are bad intuitionistilcally for several reasons. The first one which come to mind is that you cannot add numbers together. Imagine your numbers started 0.33333 and 0.66666. OK, so far it would seem that the sum would start 0.99999, but somewhere down the line one the firs number could contain two 4s, making a 1 carry all the way up and leaving one behind, so that it should in reality be 1.00000000001…

On the other hand there could also show up a 2 later, making it 0.999999998. Thus, you cannot decide weather the first decimals should be 1.00 or 0.99 without looking at infinitely many decimals. And the fact that 0.999… = 1.000 will not help you out, since 1.00000000001 ≠ 0.999999998.

Being able to define addition on decimal expansions is equivalent for constructivists to solving the halting problem. It cannot be done.

It turns out Cauchy sequences are better behaved, and (with a bit computational improvement) you can make a lot of things work out. See Bihshop's book, Foundations of Constructive Analysis, for details.

I wonder if there's anything I can do with my children to prevent them from being bound by this mental limitation?

I would try to explain to them that numbers are a framework for us to understand both the observable universe and abstract ideas, depending on what we're using them for.

Like you said, it's hard for people to understand that numbers have multiple representations and to grasp the implications of those representations. I think that if you can communicate that different representations can have the same meaning, accepting those representations when they come across them may be easier.

Or, if they're experienced enough with math, I think going through Euler's identity in addition to the link could help.

https://en.wikipedia.org/wiki/Euler%27s_identity

I feel like most people are not answering this, they are giving the proof in a different way.

Answering how to teach a child not to be bound to 100%'s is really hard. I personally would say just let them explore on their own, teaching a person that 0.99999 = 1 results in the same as if you taught them that 0.99999 != 1. You need to teach them that all sciences and maths are changing constantly, what might be a fact today could change tomorrow. You need to teach them that anything can become wrong or right as we progress and to be open about accepting new information, while being hesitant enough not to not succumb to false/fake information.

That's a very hard lesson to learn and an even harder one to practice. But one that I think a lot of people need to learn.

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

I find the algebraic way convinces most people:

x = 0.9999..

10x = 9.9999...

10x - x = 9

9x = 9

x = 1

I was prepared to blow my then 6th or 7th grade daughter's mind with this algebraic proof. I started by asking if 0.999... = 1, to which she said "no." I rephrased it and said it is equal, do you know why? She thought for a moment and said "1/9 is 0.111..., so 9/9 is 0.999... and 9/9 is 1." And I had to admit she had a far better solution than I did.

Because I wondered, it’s not a trick:

x = 0.444444...

10x = 4.444444...

10x - x = 4

9x = 4

x = 4/9 = 0.444444...

That's how I was taught in my Algebra class

black magic! I wonder if there are any programming languages that are able to handle this properly?

There isn’t a proof because it’s actually kind of arbitrary that 0.999... = 1. Fundamentally, this is true because we chose it to be true.

Now, there are good reasons we chose it to be true, and that’s what people usually use as proofs. If it’s not true then a bunch of mathematical expressions become more inconvenient. But there is no reason as such why 0.999... could not have been defined as something that was always < 1.

Fundamentally, 0.999... has no intrinsic meaning, and it’s value depends on the meaning we decide to give this representation.

You are downvoted, but you are actually correct. 0.999... != 1 can be true in nonstandard analysis. So if using standard analysis over nonstandard analysis is a convention, then ultimately 0.999... = 1 is a convention too.

(The Wikipedia article even reproduces that argument)

.999... and 1 exist on a continuous line. If they are different numbers, name a number between them.

I've had the same experience, even debating this topic with engineers. I think there are actually two hang-ups.

1. People have had it drilled into their heads that humans can't comprehend infinity. It was taken for granted by philosophers, that an "infinite regression" is a logical fallacy (e.g., used in a proof by Thomas Aquinas), and that tricks such as infinity and the infinitesimal were not rigorous. Mathematical infinity has been a settled matter for all practical purposes since the early 20th century AFAIK.

2. Related to the above, most people also believe that there is always a gap in any knowledge, and something hiding in that gap. Thus it's perfectly natural to believe that there's something hiding between 0.999... and 1, that we just haven't found yet. Knowing for certain that there is nothing between 0.999... and 1 is regarded as a kind of arrogance.

I think the way to approach this with children is to teach math as an abstract topic, that's not necessarily rooted in the objects of everyday life. For instance there's no physics experiment that can test the necessity of any math being carried beyond roughly the 15th decimal place. Yet we enjoy exploring it anyway.

> It was taken for granted by philosophers, that an "infinite regression" is a logical fallacy (e.g., used in a proof by Thomas Aquinas)...

Aquinas specifically objected to the notion of an essentially ordered infinite causal series. He had no objection to an accidentally ordered infinite causal series or other kinds of infinite series.

This distinction is extremely important for the purposes of understanding his proofs of God's existence, and people often unfairly reject his arguments because they conflate the two.

More reading here: http://edwardfeser.blogspot.com/2010/08/edwards-on-infinite-...

Couldn't you formulate a problem for extreme decimal-place accuracy based on the multiplication of errors in a physical process that's repeated in ways that multiply small errors into bigger ones?

I too was unable to convince my family but based on your comment I just thought of a new (to me) example I might have tried. It leverages the grasp of fractions that you mentioned people already have.

Everyone knows that 1/3 = .333... and it can be pretty easily shown that 1/3 + 1/3 = 2/3 = .666...

So I would ask them that since .333... + .333... = .666... does it make sense that .333... + .333... + .333... = .999...? And since .333... = 1/3 isn't .333... + .333... + .333... = .999... the same as saying 1/3 + 1/3 + 1/3 = 3/3? And since 3/3 = 1 and 3/3 = .999... it makes sense that 1 = 3/3 = .999...

This might work on your kids but in my experience recalcitrant people will either act bored as if they don't care or will try to claim that somehow they understood it all along.

What's interesting is that people pretty quickly become comfortable with the idea that 1/3 = 0.333…

So using that as a foothold, we can express 1/3 + 1/3 + 1/3 as 0.333… + 0.333… + 0.333… and it should be pretty easy to digest. At once we can see that in this little zone we've defined, 1 and 0.999… mean the same thing.

Not a rigorous proof, and one or two people will probably bring up whataboutisms like "that's just because the calculator can't do stuff!" but it should at least be proof of comfort for most people.

This is a really good point. Maybe the problem is how we define equality. What's the test for when two numbers are equal?

People accept that 1/3 = 0.333333... The same people don't always seem to accept that 3*0.33333... = 1. Well, how are we defining "equals"? If we can give that definition in black and white, I think that may help.

Maybe try expressing it in the form of money? Let's say a gallon of gas is 99 cents with infinitely repeating 9's. 0.99999999999 cents. You're still going to end up paying a dollar a gallon for it because eventually it's going to get rounded off. No gas store operator is going to try to cut a penny for you and give you a fraction of a penny. They could argue that the fraction of a penny becomes infinitely small and that giving you a shaving off the side of a copper penny would be infinitely too large.

Now don't mind me while I open up a store where every price tag ends in 0.99...repeating and have a poor college student at the checkout lane with a penny shaver to calm down any rowdy customers he or she can't explain away.

What set in stone the equality for me was learning about limits and series, because 0.999... is essentially a funny way to represent a serie.

Before that, despite accepting the proofs that were given to me, there was always something in the back of the brain telling me "mmmm there is something wrong in that". The only thing close to that was a reasoning like the following:

1 divided by 3 = 1/3 = 0.333..., but then 3 * 0.333... = 0.999... so 1 = 0.999...

This comment in the wikipedia page nails it down:

"The lower primate in us still resists, saying: .999~ doesn't really represent a number, then, but a process. To find a number we have to halt the process, at which point the .999~ = 1 thing falls apart. Nonsense."

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

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

Is this a mental limitation, or is it a simple defense mechanism against diving into rabbit-holes of thought with no end and no real productivity? It seems much easier to come to the conclusion that .99999... and 1 are different numbers, is it really worth the effort to consider otherwise?

We create these abstractions to simplify our thought- and analyzing or over-analyzing these simplifications can have the opposite effect.

Infinity will forever be an abstraction, as there is no infinite physical actions you can carry out.

Which means what infinite actions actually constitue is purely by definition, as you cannot experimentally verify it. And that's why under some definition it makes sense to say 1+2+3+4+5+... = -1/12

This is essentially just a matter of limits, without which the world wouldn't make any sense.

So you must explain that if you move your hand closer to an object, technically you are halving the distance infinitely many times, but if 0.999... != 1 then your hand would never touch anything.

> We cannot imagine infinity

Now try imagining that some infinities are bigger than others: https://en.wikipedia.org/wiki/Aleph_number

This is one of my pet peeves in maths.

Although I do understand the concepts presented, the notion of "greater" makes no sense when applied to something without boundaries.

Yet it's used all the time.

People mistakenly believe that decimal numbers don't have multiple representations - which, in a way is correct.

It is correct if you take the limit, people usually do not.

Not only can numbers have more than one representation, but they can also have zero!

Looking at you, irrational numbers.

Irrationals have a unique decimal representation in the mathematical sense: given a definition such as $x^2 = 2 $, any digit of the decimal expansion of $x$ can be determined.

just tell them to write out the complete infinite sequence of 9's

when they are done they will have undesrtood.

chuckle

Yep. Agree 100%. It is like the blue dress.

I think the problem is the repeating function. Infinite things are non-intuitive and should be presented differently.

Even here on HN you still see people confused about "convergence" and "identity". 0.999... doesn't CONVERGE, it literally is 1.

I suspect this persists even with students that have had second year college calculus that discusses convergent series and sums.

Fine. Define 0.999... as the limit of the series sum(n=1 ... N)(10^-n), as N-> infinity. This is standard high school calculus. "Number" and "series" and "limit" and "convergence" don't all mean the same thing. However this number is defined as the limit of a convergent series. So the question really is meaningful. (One clue that this question is meaningful is the amount of space introductory calculus textbooks use to address it.)

Because I can still ask, in black and white, what law of "equality" do I use to establish that my limit equals 1? (It does, if I import the definition of "equality" from the real numbers. That's what they do in calculus class. )

> There is no proof that will ever satisfy a person dead-set against this.

Yes there is. There is a proof that uses only fundamentals of first year university analysis. When you see

0.99999....

this can be written as an infinite sum

\sum_{i=0}^\infty 0.9 x 10^{-i}.

Truncate the sum and set

S_n = \sum_{i=0}^n 0.9 x 10^{-i}

and now simply use the rules of arithmetic progressions to get the limit out:

0.1 S_n = \sum_{i=0}^n 0.9 x 10^{-i-1}

S_n - 0.1 S_n = 0.9 - 0.9 x 10^{-n-1}

0.9 S_n = 0.9 ( 1 - 0.1^{n+1} )

S_n = 1 - 0.1^{n+1}

Now let n tend to infinity to find the limit, which is 1.

You don't need to imagine infinity to do any of this.

Personally I've always thought "proofs" using "arithmetic" are right, but kind of stated backwards.

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!

The way '...' is used here is perfectly consistent with being defined as a geometric series where the ratio between successive elements is 1/10 and the start term is the final digit. Geometric series always converge when the absolute value of the ratio of less than 1.

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.

> Personally I've always thought "proofs" using "arithmetic" are right, but kind of stated backwards.

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.

You only need to define 0.9… as 9/10 + 9/100 + 9/1000 + …. Without knowing how that series converges you can then use the two expressions mentioned to conclude that it has to be equivalent to 1.

> You only need to define 0.9… as 9/10 + 9/100 + 9/1000 + …. Without knowing how that series converges you can then use the two expressions mentioned to conclude that it has to be equivalent to 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’re basically just a priori defining 0.9... to be 1.

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 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 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.

Limits and completeness are convenient shorthands here. But let's start from a more basic perspective.

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
          ...

And we can verify the reverse by doing the same trick above; 0.333... = 10 * 0.333... - 3 => 3 = 9 * 0.333... => 0.333... = 3 / 9 = 1 / 3.

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
         ...

And so on.

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.

> 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.

Peano arithmetic only covers nonnegative whole numbers, so one will never exist.

> 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"?

Yes that proof depends upon the representation in text of rational numbers (a dot and a series of digits). Try it in hexadecimal - it becomes opaque nonsense. Without some mathematical basis for 0.9... X 10 being something, there's a dangerous dependency on the representation that makes many folks uneasy.

I don't see how it would be significantly different in other bases. In hexadecimal it'd be 0x0.ffff.. * 0x10 = 0xf.fff..

The rules are essentially the same.

Here's one stupid, simple proof that doesn't require any predefinition.

We do longhand division of 9 by 9, but start by putting a 0 in the first place.

Hence, 9 / 9 = 0.999 = 1.

What's very neat is that the algebraic argument also works for other sequences like 0.888...

x = 0.8...

10x = 8.8... = 8 + x

9x = 8

8/9 = 0.8...

And I thought mathematicians would reject normal arithmetic operation over the domain of `N...` elements. They're always so ultra rigorous to classify what is or is not, what is defined, the domain .. but then they let an infinite sequence be treated as any finite number.

> but then they let an infinite sequence be treated as any finite number

Technically it’s an infinite series (sum of an infinite sequence), which is a finite number in certain cases like this one.

certain cases, so define this special one but frankly it doesn't make my brain happy (but who am I)

> 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.

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.

if we say that infinitesimals exist. that 1/3 != 0.33.. and 1 != 0.9999... and the probability of possible events is never 0.

what are the properties that we would lose?

If we say that infinitesimals exist, it still happens that 1 = 0.999…. It just happens that 0.999 ≠ 1 - 𝛚.

0.999… = 1 is a property of the way we write some rational numbers, not of the number system itself.

wouldn't 0.999.. be equal to 1 - 10^w since it's only a countably finite series of nines.

I have never seen a number system with infinitesimals where the addition wasn't updated to ignore smaller classes if they come with larger ones.

That is, for any number system I've seen, 1 = 1 + dx, and infinity = infinity + 100.

The obvious one seems terrible enough, that division is no longer the inverse of multiplication: (1/3)*3 != 1

Nonstandard analysis exists (with infinitesimal and infinite numbers) , but 1/3 and 9/9 is the same there. The problem is that the numbers 0.333... and 0.999... don't really exist.

Completeness is one of the most important properties of real numbers. Basically, you will have to completely throw away real analysis.

1/3 does equal 0.3.. though.

A formally rigorous proof of this (in Metamath) is here:

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.

It depends on more than just ZFC, also on the definitions of the real/complex numbers. The crux of the proof is that 0.99999... is being constructed within the real/complex numbers, and in that system it is equal to 1.

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 think this is spot on, at least for me personally.

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 = 0

It's true that it depends on the definitions of real/complex numbers. Many other things turn out to be provable from ZFC. A discussion about this, from the viewpoint of Metamath, is here: http://us.metamath.org/mpeuni/mmcomplex.html

Very cool page!

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

I wonder if this particular example does the opposite. It's too rigorous for a normal human. It seems humorously rigorous to state things like 10 is not equal to 0, and 10 is a real number and 1 is less than 10, not to mention the number of seemingly redundant repetitions showing how each of the numbers discussed is a complex number, separately an individually. Does defining 1+9=10 make the proof more believable?

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.

That is cool! I've been thinking about building a site that lets you explore a big graph of math proofs for a long time, since doing my pure math degree. Glad to see someone else has done something like it.

The proof relies on the assertion that the supremum of an increasing sequence is equal to the limit. This is mathematical dogma, and should be introduced as such. Once that is accepted, it becomes obvious.

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.

This is in large part due to the difficulty with reasoning about infinite representations. You do have to add axioms to your system to be able to reason about 0.9999...

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.

And this I think is the real issue. When someone says that 0.999... = 1.0, what they are saying is that this is true given a number of assumptions that we are taking for granted that would not be obvious to a non-mathematician. There's a lot of math hiding in those '...'.

What? 0.999... = 1 is not dogma. Please don't spread misinformation. And at least read the link before commenting on something.

Did you read what @jl2718 posted? Namely:

> 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...

I do think it is technically wrong to call it dogma - the decimal is a geometric series with a limit, right? And limits have an unambiguous definition, it's the smallest value that the series approaches but never exceeds as it tends to infinity. I think the part that is admittedly weird is that the notation "0.999..." refers to the limit as the series tends to infinity, and it kind of hides that fact from you. Even just writing the geometric series down and plopping "infinity" as the value for x would be wrong, as it's the limit that is equal to 1 as x tends to infinity. So there's arguably more hidden notation than the ellipses implies, but nothing is pulled out of a hat here or defined for definitions sake.

I remember being doubtful when being presented with this in middle school, but after being shown this as fractions makes it obvious:

      1/3 =     0.333..
  3 * 1/3 = 3 * 0.333..
      3/3 =     0.999..
        1 =     0.999..

I don't mean to troll you, but if you were doubtful that 0.999... = 1, then you should also be doubtful that 0.333.. = 1/3. Any argument that 0.999... is not quite 1 can also be used to argue that 0.333... is not quite 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.

> if you were doubtful that 0.999... = 1, then you should also be doubtful that 0.333.. = 1/3

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...

> Any argument that 0.999... is not quite 1 can also be used to argue that 0.333... is not quite 1/3.

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.

See my other reply along the same lines. I was thinking about adding this caveat to my original message as well, but I think understanding that decimal numbers with infinite digits exist and that 0.999.. = 1 are separate things. The second being less intuitive.

I agree. The argument ignores the rule that infinity is a point which can never be reached; it can only be approached. So repeating 9s infinitely many times will still not reach 1, it will only approach it.

Another secondary school 'proof'

  x = 0.9999.....
  10x = 9.9999.....
  (10x -x) = 9x = (9.9999.... - 0.9999....) = 9
  x = 9/9 = 1

or,

      x = 0.9999...
    10x = 9.999...
    10x = 9 + 0.999...
    10x = 9 + x
     9x = 9
      x = 1

Presented slightly more clearly

https://en.wikipedia.org/wiki/0.999...#Algebraic_arguments

The difficult part is to convince that (9.9999.... - 0.9999....) = 9. Someone else mentioned about Zeno's paradox involving Achilles and the tortoise. Since you multiplied x by 10 to get 9.9999...., you know that it's always going to be 1 decimal place slower in pace to reach convergence compared to 0.9999.... What you might get instead is (9.9999.... - 0.9999....) = 8.9999....

I like this one better:

    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

> 10x -x

subtracting infinities is dangerous, you can achieve any result from it

https://www.youtube.com/watch?v=-EtHF5ND3_s

Well, if I could choose I wouldn't personally accept 1/3 = 0.333... . But rather, I'd say it equals a limit:

    1/3 = lim(N -> oo) 0.3{N}     (3 is N times repeated)

Especially I would distinguish between infinitely many threes, and N threes, where N goes to infinity. In the first case, you would still be missing an infinitisimal amount, in the latter case you have the usual situation and the sequence has the least upper bound of 1/3.

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.

But you absolutely can evaluate that limit as N goes to infinity and correctly conclude that 1/3 does equal 0.333 repeating.

I remember a conversation I had with my daughter in the car when she was starting out with algebra...

Me: 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.

> An infinite number of zeroes. . .and then a one. . .wait, you can't do that.

why not? why can't an infinitely small number exist?

I personally like using fractions of 9.

  1/9 = 0.111...
  2/9 = 0.222...
  3/9 = 0.333...
  ...
  8/9 = 0.888...
  9/9 = 0.999...

What's neat is that this trick works for any repeating decimal, with any number of digits in the repeating part. For instance:

  123/999 = 0.123123123...
  999/999 = 0.999999999...

Multiply or divide by powers of 10 as necessary to shift the decimal point, and add the non-repeating part.

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.

Nice to see a few different proofs/intuitions here. Not being a big fan of symbol manipulation, I always felt partial to the proof/intuition that you couldn't find another number between the two :-)

Well, here you reduced 1=.9999... to 1/3=0.333... What if I don’t believe that second equation.

What if we recursively defined 1/3? This will allow us to ignore the infinite 0.333... for a second. As an example, let 1/3 = 0.33 + 1/100(1/3).

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

[1] https://en.wikipedia.org/wiki/Geometric_series#Sum

As in the 0.333... will stop at some point? That would still mean that 3 time 0.333... with a LOT of 3s end up being being 1.

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.

This also happens to be the test for whether your calculator is any good.

My high-quality scientific calculator makes this a bit uninteresting:

  1/3            ⅓
  Ans * 3        1

My 5 year old stumped me with this, and I had to look it up. He asked me why 1/3 + 1/3 + 1/3 = 1, since it's equal to 0.333... + 0.333... + 0.333... which is 0.999... How can that possibly equal 1.000...? And is 0.66... equal to 0.67000...?

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!

> He asked me why 1/3 + 1/3 + 1/3 = 1, since it's equal to 0.333... + 0.333... + 0.333... which is 0.999... How can that possibly equal 1.000...? And is 0.66... equal to 0.67000...?

This would make me very proud.

Yes, it's quite clever. An equivalent proof is dividing 0.999... by 9 using long division, which comes out to 0.111... which is equal to 1/9. Now use fraction notation and it simplifies to 9/9 = 1. Not quite as robust as the limit-based proofs but it's a quick answer and gets to the heart of the issue of repeating notation not capturing the whole picture.

Is this problem simpler than we want it to be? Meaning 1/3 is a concept stating there is 1 part of 3 total. If you have 3 total parts, added then it is a whole. Trying to shoe-horn it into the decimal system, similarly to try to represent pie as a clean number into the decimal system etc. Isn't the issue representing the number in one for and another, not the actual logic of the issue? idk

Why this is an issue, when 0.9999... is exactly 1?

5 year old is curious or asking such question is mind blowing.

> Luckily he has many years before it matters!

and depending on career choices, it might never matter at all.

I asked my math teacher this when I was a kid. He told me to accept that's the way it is so I did.

Kinda like why 5 rounds up instead of down

No .666666 is not equal to .6700000

0.666... is equal to 0.666...7

Not sure if you’re joking, but 0.666...7 is not a real number. Can you define it?

An interesting consequence of this in proofs.

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!

I never understood the fixation on diagonalization. Why can't ever exist another way for mapping any set to countables?

Diagnolization is a pretty deep argument about fixpoints, Godels incompleteness argument is essentially a diagnolization. So why wouldn't there be fascination?

The point is that diagonalization works whatever map you have come up with: no matter how you construct your list of the reals, you can come up with another real not on the list.

Flame wars over this used to be common on the internet. People intuitively have the notion that the left side approaches 1, but never actually equals it. They see it as a process instead of a fixed value. Maybe the notation is to blame.

The intuition is right, and the mathematical definition relies on the intuition. It's just that people haven't been exposed to the actual definition when it comes to real numbers.

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.

Repetition can easily be seen as a process, which would indeed approach 1. But I think the idea of infinite repetition is very hard to get.

The intuition that there is something in between isn't really wrong, it make sense and they work, otherwise physicists wouldn't be able to work with them. So that intuition is correct, it is mathematicians who just don't understand it fully yet. Maybe fully formalizing this is what unlocks the final piece keeping us from creating a unified theory in physics?

Nonstandard analysis is a rigorous framework for working with infinitesimals (and infinitely large numbers).

https://en.wikipedia.org/wiki/Nonstandard_analysis

I remember WarCraft 3 official forums being torn apart by this, with probably thousands of comments in the thread. Blizzard even had to post their official stance on the issue, but that didn't calm those who insisted 0.999... was 1 minus epsilon and not exactly 1.

I'm glad someone else remembers this. To this day, whenever I see 0.999... = 1, I think of the Battle.net forums inexplicably flooded with threads about it for what felt like ages.

Maybe the major source of confusion is that our decimal representation for whole numbers is supposed to be unique. Then when we extend it to rationals and reals this property fails at rationals in the form of a/10^n.

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.

I think the source of confusion is that people can't cope with recurring numbers. When someone says .999...=1 the listener assumes that the 0.999... stops at some point, and if that happens it'll always be below 1 because they can imagine adding another 9. Essentially, people actually ignore the "..." because that's the hard part.

Right - I also find it easier to say that really, "1" is just a different/shorthand notation for 0.(9) It's not "two different, but equal numbers" - it's two different notations for the same number. Like how you can write same number in different ways in different bases - this is just writing the same number, in the "infinite number of decimals" vs "natural" way.

Arguably 1 is just as an infinite number of decimals as 0.(9) . 1 is just short for 1.(0) .

I don’t know, I don’t recall students having much trouble accepting that 1 = 1.0 = 1.000.

0.9999 = 1 is a consequence of the way we define rational and real numbers and limits. There are alternative definitions of numbers where this equality does not hold: Non Standard Analysis https://en.wikipedia.org/wiki/Nonstandard_analysis being the most famous one.

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)
    }

Each such sequence corresponds to the (classical) real number: s * \sum_i a_i * 10^{i + e}.

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.

1 = 0.9... is the consequence of purposely ambiguous and questionable notation. That's an old teachers' trick to make students talk and listen about mathematics.

This has nothing to do with notation. It's perfectly possible to define infinite sequences without using dots. In particular if they are constant. In the above case:

    a_i = 9 for i \in \IZ and i < 0
    a_i = 0 for i \in \IZ and i >= 0

Where \IZ are the integers.

This is 'more intuitive' if you think about it this way:

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?

0.999…5 obviously.

But at which position that "5" is? Tell the number please.

Okay, now multiply that by 2.

Or: if they're different, what is their difference?

0.000...1 obviously

There is a nice characterization of decimal expansions in terms of paths on a graph:

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!

Ultimately this is more the definition of R than that it is a theorem. One can also work with sets of numbers in which the completeness axiom does not hold. E.g., sets of numbers in which one also has infinitesimals.

And this is why I prefer hyperreals.

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.

I think what's important here is that if you're making that claim,

0.333... = 1/3 - 1/∞

Which implies 3/∞ = 1/∞

Apologies, I should have explained it differently.

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.

I'll just chime in with my completely ignorant theory that 1 - 0.999... = the infinitely smallest number, but is still, in my mind, regardless of any logic, reason, or educated calculations, greater than 0.

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.

So 0 and that number are two different numbers.

What is the difference between them?

One is positive and the other is negative. I thought that was pretty obvious!

Well, if you perform the same calculation in base12, then you'll get a whole number representation, because 12 / 3 is 4. Thus, in base 12, 1 / 3 = 0.4

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.

Just to be clear, as I said, I totally accept the correct answer. I was just trying to humorously point out the non-intuitive nature of it.

Philosophically you are right.

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.

>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.

From the perspective of IEEE-754, they’re pretty close ;)

Usually the concept of a limit, which assigns a meaning to 0.999..., isn't studied until calculus.

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.

Professor N.J. Wildberger is probably among the most well known "ultrafinitist" on YouTube.

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.

Wildberger is great. His lectures that he teaches at UNSW (i think) are interesting, and he usually keeps a clear dividing line between std math and his own predilections. It threads the line between being a kook and legitimate published mathematician very finely.

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.

If I give you two representatives of real numbers, say turing machines that write out on their tape the binary digits of those real numbers, in general you will not be able to order them.

But on computers, you get things such as

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.

0.999...=1 is true in the mathematical sense, period.

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 thing that helps me "understand" it is that the universe has finite sizes of things like the Planck length for example being a theoretical thing at the smallest distance I would imagine. Now imagine it going smaller than the Planck length (finite) in terms of the difference of .9 repeating and 1 since infinitely small differences can do that. Essentially there is no way to tell the difference between .9 repeating and 1 then from a practical or theoretical perspective of measurement. So not imagining infinity lets us at least imagine smaller than the smallest measurable thing.

>The universe has finite sizes of things like the Planck length for example being a theoretical thing at the smallest distance I would imagine

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...)

I hate to say it but I still don't believe this, it just goes against all intuition that I have, but people much smarter than I have proven it so I take it on faith for doing things like calculus etc just my lizard brain won't let me accept something that looks like less than 1 being 1 the same way that the limit of 1/x as x goes to infinity is zero but it doesn't seem like ti should be. The number gets infinitesimally small but it's still some non-zero number -- I dunno this is probably proving my ignorance it's just what it is.

Don't feel too bad.

The 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.

Thank you. I had totally expected to get dunked on and find your empathy refreshing.

I’ll keep trying to understand it.

Maybe try this: do longhand division of 7 by 7 or 9 by 9 or whatever, but instead of putting 1 in the first place, start with 0.

        0.99
    7 | 7.00000
        0
        7 0
        6 3
          70

I follow that but then I could see it just continues to .9999 forever never converting to 1 it’s always that tiny fraction less than 1...

I think if you're trying to "prove" this using axioms, you've already lost.

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.

Can someone help me out here with least upper bounds?

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!

I think this is a notation and definition problem. To me, it behaves differently in `Y = 1 / X`, which distinguishes quite strongly between `X = 1 - 0.9999` and `X = 0.9999 - 1`! If 0.9999 ought be exactly equivalent to 1.0, there ought to be no difference between `1 / (1 - 0.9999)` and `1 / (0.9999 - 1)`.

To me, 0.9999 indicates a directional limit, which can't necessarily be evaluated and substituted separately from its context.

Luckily your taste doesn't factor into whether it's true or not :-)

I'm actually curious what impact it would have on various proofs if 0.999... wasn't accepted as 1.

What gets broken? What consequences do we hit?

Arithmetic breaks, as multiplication is no longer the inverse of division. (For example, 1/3 * 3 = 0.999… would no longer work.)

why?

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?

So, lets take .9, .99, .999 and so on. If a sequence of rational numbers converges, it converges to a real number. What number does .9, .99, .999, .9999 (and so on) converge to? Which is to say, is there a number that it gets closer and closer to at every step? Clearly it gets closer and closer to 1 at every step, so the sequence converges to 1.

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.

I don’t think this reasoning is correct, because the limit of an expression does not have to be the same as the value at the point the limit is being taken. For example, if your sequence is “sin(1/x) * x“, then it’ll slowly appear to converge to one as x approaches infinity, but it cannot reach it. So there’s really no relevant conclusion you can make.

You can multiply 1/3 by 3 and not get 1.

I have a pizza. I divide it into three parts.

You'd be asserting that if I eat the three parts I have not eaten the whole pizza.

I'm unconvinced.

I'm not a mathematician, but I would guess that the Surreal numbers developed by John Conway, do contain values that start with an infinite sequence of 0.9_ but "end" with something that makes them != 1.

https://en.m.wikipedia.org/wiki/Surreal_number

What if you have 0.9̅4? Can we say 0.9̅5 > 0.9̅4 > 0.9̅3? More on what happens if you allow this: https://mathwithbaddrawings.com/2013/08/13/the-kaufman-decim...

That's fun but it's not clear how interesting those numbers are compared to real numbers, which have turned out to be pretty interesting over the years

>What if you have 0.9̅4?

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.

In this alternate number system you can think of it as a tuple: (.9̅, 4). Which is larger than (.9̅, 3) and smaller than (.9̅, 5).

That is, until Cantor showed otherwise

A dumb consequence of the axiom of choice? The reals are like a membrane with no atomic pieces. You can move in either direction infinitely and you can zoom in infinitely without reaching any “Planck unit” so to speak. So what does it even mean to pick out a “real number”? To me anything built on this concept is nonsense.

Sorry for my naivety, but why one couldn't prove by induction that adding 9s never close the gap, or let's say, that by definition the operation is such, that it never closes the gap. If you can always halve the pie, then you can continue eating forever. To me it would be much easier to accept that (1/3)*3 is not 1.

Other people have pointed out that induction never makes the jump from a finite number of 9s to an infinite number of 9s.

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.

"If 0.999...!= 1 then there must exist a number A, such that 0.999... < A < 1."

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...

Let's try a naïve inductive proof of 0.999… ≠ 1.

  Base case: Given 𝑎⁰ = 0, 𝑎⁰ ≠ 1.
  Inductive case: Given 𝑎ⁿ⁺¹ = 1 - (1 - 𝑎ⁿ) / 10, 𝑎ⁿ ≠ 1 ⇒ 𝑎ⁿ⁺¹ ≠ 1

This proves that for every 𝑎ⁿ = 0.999…9, there's an 𝑎ⁿ⁺¹ that's a 9 larger and still different than 1, which is similar to your halving the pie example. However, you can see that it always happens that 𝑎ⁿ⁺¹ > 𝑎ⁿ, so the "last" infinite 0.999… is not part of the sequence of the inductive case.

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.

It's because the rigorous mathematical definition is more subtle. It's defined as the smallest real number such that repeating this process (putting more 9's in the end) can't result in a number that's greater than it. So as long as you have a number that's smaller than 1, there's a gap there, and repeating the process of adding 9's will eventually give you a number that lies in that gap.

Yeah, it makes sense if the whole foundation of mathematics is laid out in such way that the above has to be true (kinda circular, but still), and this would mean that what I'm talking about is actually not mathematics but something else :).

By induction, you can prove that no _finite length_ sequence of 9s after the decimal reach one. It has nothing to say about the infinite limit, though.

Induction can be non-intuitive to people who are troubled by this problem.

Induction has nothing to do with this. It's about a taking the limit of a process.

Is 1 a prime number? No, because we define it not to be. Why do we define it not to be a prime number? That's the real question.

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 best way to think about it is that 1 - 0.999... = 0.000...

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.

This can be solved by using base 12 rather than base 10 to do the calculation...

Firm believer that adopting base 12 would have had a ripple affect on society, preventing many trials tribulations and wars. Pity we only have base 10 and donald trump

base 12? Them are fighting words...I call WAR!

Yes it does but it seems like a less correct way of writing it. Like you could represent the number 10 as 10/1 (ten over one) but why would you? Why would you represent 1 as .9 repeated?

Basically an infinite series of 9's just means they're all maxed so the .999... = 1 makes sense to me (kind of anyway).

# on Python (3.7.4) 1 == 0.99999999999999994448884876874217297882 # but 1 != 0.99999999999999994448884876874217297881

Or any other language using IEEE 754 64 bit floats

I see... Nice!

Perhaps the natural discomfort many face when confronted by this challenging formulation instead indicates that a limitation of the real number system has been perceived? I would encourage those who have this reaction to study hyperreal and other alternative systems as mentioned in the article. If this clicks for them they may help lead us in new direction mathematically and advance the state of the art.

This comes from the fact that most people don't learn limits. Once the definition of the decimal representation is understood to be in terms of a limiting process, the meaning becomes clear.

For most purposes hyperreals are too much machinery when learning and manipulating limits would be simpler.

Why do we accept .999... as a valid notation. Why not only allow 1 to denote this concept?

> Why do we accept .999... as a valid notation. Why not only allow 1 to denote this concept?

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.

I'd have figured "approaches 1 from the left" would be more accurate.

So is then ‘0.(0...)1 = 0’?

So 99.999..% of the speed of light is just the speed of light?

Yes.

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.

Hmm, This reminds me a lot like Zenos paradox.

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?

Given infinity that doesn’t make sense.

What is the largest number smaller than 1?

0, if we're talking natural numbers or integers. There isn't one if we're talking real numbers. Simple proof:

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.

How about the expression:

    0.9999... < 1

And consider that if a < b then a != b.

Both are false. 0.99999... is not less than 1. It is the same as 1.

Yes, because someone defined it that way.

It is because the "limit" in

    0.999... = lim[eps->0] 1-eps

is implicit and defined as being applied before anything else. But you might as well define that implicit limit as applying over the entire expression.

UPDATE: So instead of interpreting the expression as:

    (lim[eps->0] 1-eps) < 1

which is indeed false, you can also interpret the expression as:

    lim[eps->0] ((1-eps) < 1)

which is true (assuming that -> denotes a limit from above). Note that here the "lim" has been taken out and acts over the entire expression.

It's false.

"In other words, "0.999..." and "1" represent the same number." - Ok, I'm done with this world.

I don't think that 0.999... = 1 is actually provable. I think this and all of calculus is actually axiomatic, which has the following axiom:

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)

That's not an axiom we take; the study of real numbers takes some algebraic and order axioms, and a completeness axiom:

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.

Why is your proof for every natural number n and not infinity (ω)? also isn't the law of excluded middle axiomatic?

A mathematician would quibble with your notation, but you're basically right. The fact that 0.999... = 1 depends on the fact that 0 is the only number smaller than 1/n for every integer n. This isn't exactly an arbitrary axiom, it encodes some of our natural intuitions about what a number is, but you can construct a system known as the "hyperreal numbers" where it isn't true and 0.999... doesn't converge.

Well, that wiki page has several proofs using various approaches. There's also a section which addresses common objections.

I think you'd have to dismiss them all to make your claim (to do it well, that is).

Afaict all of these rely on the axiom of Archimedes which is what I'm talking about, some of the proofs explicitly stating it and others implicitly. Unless I missed something.

I think the mistake in your thinking is that infinity is a value or reachable destination. It's a concept and it behaves differently than a number.

Also go read Generatingfunctionology and Concrete Mathematics

That's the point I'm trying to make actually. 1/infinity is an infinite series, which would be a computation that takes infinite time to compute. Saying that 0.9999...=1 is saying that the infinite series is the same as that concrete value, which don't actually know and can't prove.

Reminds me of the paradoxes of Zeno [1], especially the paradox of Achilles and the tortoise.

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

As well as the weird prime product (Product of 1/(1-(p^-2)) for p prime) and the sum of x^-2 from x=1 to [sigh] being equal to (pi^2)/6 are some example of infinite beauty of mathematics that I remember.

[1]: https://en.wikipedia.org/wiki/Zeno's_paradoxes

It's less controversial than those because it's actually true. The latter statements are not true with the traditional definition of a sum.

1 + 2 + 3 + ... does not equal -1/12, except if you redefine what you mean by +. https://www.youtube.com/watch?v=YuIIjLr6vUA

Your two examples can be debunked though.

See here: https://www.youtube.com/watch?v=YuIIjLr6vUA

ah, Mathologer video.

Have seen that.

Another one by 3b1b on that topic: https://youtube.com/watch?v=sD0NjbwqlYw

> ...infinitely many 9s...

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.

> We can repeat this process for an infinite amount of time and still not have an infinite number of nines.

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.

> We can repeat this process for an infinite amount of time

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.

I tried this route as a counter to Cantor's diagonal argument, and got chastised by my then professor. I hope you have better luck, as I was never able to convince myself otherwise.

By that point you were already working in a universe where infinity was in some sense real. The universe of mathematics is different than the universe we live in.

> How about we prove that an infinite number of 9s is impossible?

First you need to define what you even mean by this statement. The rest of what you wrote makes no sense either.

Reply on news.ycombinator.com