
0.999...= 1 - yurisagalov
https://en.wikipedia.org/wiki/0.999...
======
undecisive
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?

~~~
lmkg
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.

~~~
mjd
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](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.”)

~~~
blauditore
>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?

~~~
tom-thistime
Yeah, can we have an example of an irrational number whose decimal
representation repeats or terminates?

Or a rational number whose decimal representation doesn't repeat?

~~~
barrkel
I expect mjd is thinking of irrational bases. The number might still be
written as 10, in digits that look decimal.

~~~
samatman
Thank you! This thread is full of people insisting on something wrong because
they were taught incorrectly; an irrational number is defined in terms of
integer ratios for a reason.

It's not like those people haven't worked with an irrational base before,
either! Radians have an irrational base. When we talk about 2π radians, or
1/4π radians, that's exactly what we're doing.

------
knzhou
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!

~~~
CogitoCogito
> 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.

~~~
judofyr
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.

~~~
CogitoCogito
> 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.

~~~
marcosdumay
The GP's definition is the fundamental definition of the decimal notation.
It's exactly what the "0.9..." symbol means.

You can redefine the "0.9..." symbol to mean something else as much as you
want, you can have it meaning pi if you like, but then you are just changing
the subject on the most unhelpful way.

~~~
CogitoCogito
> The GP's definition is the fundamental definition of the decimal notation.
> It's exactly what the "0.9..." symbol means.

> You can redefine the "0.9..." symbol to mean something else as much as you
> want, you can have it meaning pi if you like, but then you are just changing
> the subject on the most unhelpful way.

I _know_ that.

I think I maybe should just bow out of this conversation. I'm apparently
incapable of explaining myself in a way that is understandable to people here.
I'll consider this my fault.

I'll just summarize: I don't think any "proof" that 0.9... = 1 that is only
expressed in terms of arithmetic operations and does not make use of limits is
legitimate. In other words I claim that a proof like "0.9... = x" means
"9.9... = 10x" means "9 = 9x" is illegitimate. Instead of taking "0.9... = 1"
on faith it takes "10 x 0.9... = 9.9..." and "9.9... - 0.9... = 1.0... = 1" on
faith. There's no proof here. It's just shifting around symbols. Of course
there are logical proofs, but they make use of limits/completeness/properties
of real numbers explicitly.

Feel free to disagree...

~~~
marcosdumay
Oh, ok. If I understand correctly, you mean that there isn't any standard
algorithm for handling a sum with infinite terms if you don't include limits.

Well, I do disagree, not with the above statement, but the meaning of "0.9..."
itself requires limits, so the discussion can never go anywhere if your
assumptions do not include limits.

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

[http://us.metamath.org/mpeuni/0.999....html](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.

~~~
317070
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.

~~~
_Donny
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

~~~
alanbernstein
There is this phrase, often used when describing the decimal expansion of pi -
"keeps going infinitely". This phrase is not exactly incorrect, but I wonder
if it misleads people into thinking that an "infinite decimal" is "a kind of
infinity", which it really isn't in any meaningful way.

~~~
klank
I think it absolutely gets confused.

Infinity, the number, is routinely confused with creating an onto function
mapping digits of pi to a set with a cardinality of the natural numbers. But
sadly most people don't have the mathematical maturity to understand the
difference when they encounter their first irrational number (normally pi).

------
jl2718
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.

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

~~~
dvt
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...](http://www.math.toronto.edu/ilia/Teaching/MAT378.2010/Limits%20and%20real%20numbers.pdf)

~~~
jturpin
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.

~~~
dvt
> the decimal is a geometric series with a limit, right..

Right, but that's not really the crux of the matter. Hint: look at how the
supremum is _defined_ [1]. The _definition_ of the supremum is how we end up
with 0.999... = 1.

[1] [https://math.stackexchange.com/questions/1977204/limit-of-
mo...](https://math.stackexchange.com/questions/1977204/limit-of-monotone-
increasing-sequence-is-supremum)

~~~
jl2718
I suppose my point is that you could turn the repeating decimal into an
infinite series, and a student might accept that, and you could define the
suprememum and they might agree that it is 1, but then you ask them if the
series is equal to the supremum, so they don't know what to do with the
series, so they turn it into a sequence. Now they ask whether the last item in
the sequence is equal to the supremum. Of course not! This is by definition.

And now you realize that you and the student have been operating by different
rules. Their rules of equality are based on symbolic equality, so you actually
have to relax the rules a bit to make limit equality work. And then, more
importantly, you have to show that all the other rules are still intact.
Actually, in this case, they aren't. Symbolic equality involving infinity is
now horribly broken, and you have to express all equality in terms of limits
to maintain consistency. Explore this further and you keep finding more
inconsistencies that have to be settled by new rules that define new areas of
mathematics.

So who is right? The natural world appears to be much more permissive than
limit equality, preferring epsilon-equality. Symbolic equality is the only
purely self-consistent system, but you can't do much with it. It's also
possible that the natural world works with symbolic rules (quantum) but the
complexity is great enough to resemble epsilon equality (continuum).

So, .999... == 1 by tautology. It's not some brilliant mathematical insight.
The interesting part is the consequence of defining it as so.

------
ginko
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..

~~~
larschdk
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.

~~~
notRobot
> 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...

~~~
ant6n
> Any middle school student can calculate 1/3 to be 0.33333... using long
> division, ...

...the same way That Chuck Norris can count to infinity... twice!

~~~
mywittyname
I smart middle-schooler is absolutely capable of understanding that dividing
1/3 results in an infinitely repeating sequence of 0.33333... Even without
understanding the concept of infinity, they will quickly realize that there's
no reason to believe the problem will stop adding a 3 to the end of the result
with each iteration.

~~~
ant6n
> I smart middle-schooler is absolutely capable of understanding that dividing
> 1/3 results in an infinitely repeating sequence of 0.33333..

And how will I smart middle-schooler know that the result of running the long
division algorithm is exactly 1/3, rather than some approximation.

------
ping_pong
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!

~~~
rudolph9
> 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.

~~~
boxy310
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.

------
klodolph
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!

~~~
rini17
I never understood the fixation on diagonalization. Why can't ever exist
another way for mapping any set to countables?

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

------
bytedude
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.

~~~
username90
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?

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

[https://en.wikipedia.org/wiki/Nonstandard_analysis](https://en.wikipedia.org/wiki/Nonstandard_analysis)

------
orthoxerox
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.

~~~
jcfields
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.

------
steerablesafe
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.

~~~
virgilp
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.

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

------
heinrichhartman
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](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.

~~~
astrobe_
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.

~~~
heinrichhartman
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.

------
ltbarcly3
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?

~~~
saagarjha
0.999…5 obviously.

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

~~~
saagarjha
Position 999…, duh.

------
sleepyams
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!

------
cjfd
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.

------
calibas
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.

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

0.333... = 1/3 - 1/∞

Which implies 3/∞ = 1/∞

~~~
calibas
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.

~~~
__s
Agreed, I think the main lesson from this topic is that decimal numbers are a
pain. Stick to integers & symbolic operations. You can spit out a decimal
approximation at the end

Computers agree: never trust precision to floats

------
russellbeattie
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.

~~~
thdrdt
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.

~~~
cthalupa
>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.

------
JJMcJ
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.

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

[https://www.youtube.com/watch?v=WabHm1QWVCA](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.

~~~
rrmm
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.

~~~
danharaj
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.

~~~
rrmm
Certainly not non-computable ones, but presumably they lie somewhere
regardless of my inability to do it on a TM. Which presumably gives rise to
all the weirdness uncountable infinities give you.

I guess I shouldn't phrase it as "you can fully order it". :D Zermelo's
theorem at that point right?

~~~
danharaj
Even computable reals do not have computable ordering.

> but presumably they lie somewhere regardless of my inability to do it on a
> TM.

Why?

> Zermelo's theorem at that point right?

It is declared by fiat in standard set theory that infinite sets can be well
ordered. This is no real mathematical justification. The real justification is
social: that it is convenient for mathematicians to not care about the
ontology of these nasty infinite objects so long as results are mostly
reasonable for objects that mathematicians actually care about. You don't get
into too much trouble pretending the reals are nice so long as you don't look
too hard.

~~~
rrmm
I mean that's Wildberger's whole point isn't it?

~~~
danharaj
Yes, I'm just trying to emphasize that there is real serious mathematics
behind his point, it's not just a matter of philosophical taste.

------
sv_h1b
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.

------
sebringj
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.

~~~
cthalupa
>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...)

------
gigatexal
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.

~~~
MrManatee
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.

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

I’ll keep trying to understand it.

~~~
gigatexal
It has to do with infinities (infinite sums) which might explain why it’s so
interesting

------
edanm
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.

------
fluganator
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!

------
jdashg
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.

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

------
rs23296008n1
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?

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

~~~
tomtomtom1
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?

~~~
atq2119
The crux of the matter is that you have to define what things like "0.3333..."
mean in the first place. Any reasonable definition of it as a representation
of a real number is going to lead to it being equal to 1/3.

If you want to redefine it explicitly as _not_ a real number, you can do that,
and maybe even get to some amusing math that way, but you're no longer talking
the same language as the rest of the world.

~~~
tomtomtom1
>"but you're no longer talking the same language as the rest of the world"

yes, in the standard real numbers 1 = 0.999.., but people have dealt with
numbers like "pi" and "sqrt(2)" before the standard real numbers were defined.

Hence the question, if we define such a system such as 0.333... != 1/3\. what
are the consequences?

by 0.3333... I mean a countably infinite sequence of 3s.

~~~
atq2119
I think an important distinction is that in those "old days", people were
largely working in what we now know to be _subsets_ of real numbers, and the
same conclusion applies there.

If you want to go to _supersets_ of real numbers, you may be interested in
[https://en.wikipedia.org/wiki/Surreal_number](https://en.wikipedia.org/wiki/Surreal_number)

------
j-pb
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](https://en.m.wikipedia.org/wiki/Surreal_number)

------
jefftk
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...](https://mathwithbaddrawings.com/2013/08/13/the-kaufman-decimals/)

~~~
cthalupa
>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.

~~~
ubercow13
That is, until Cantor showed otherwise

~~~
cthalupa
Are you saying that Cantor showed you can have 0.9..4?

I'm not even remotely an expert here, so I might certainly be wrong, but I
don't understand how Cantor's theorems show an ability to stick a finite
number and stop on the end of an infinite number.

~~~
ubercow13
Yes, one thing Cantor showed is that it is somehow meaningful to define
numbers like infinity+1, where there are an ordered infinity of elements
followed by one more element. Sets like this are called ordinals

[https://en.wikipedia.org/wiki/Ordinal_number](https://en.wikipedia.org/wiki/Ordinal_number)

So you could, if you like, define 0.9...4 to be a bunch of digits indexed by
the ordinal ω+1. However the thing you have now defined isn't really a
representation of a real number any more, unless you just ignore all the bits
after the ... I guess.

~~~
cthalupa
Huh, interesting. Looks like I've got some more reading to do.

Thank you!

------
shrimpx
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.

------
vfinn
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.

~~~
kosievdmerwe
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.

~~~
vfinn
"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...

~~~
kosievdmerwe
That statement is based on:

> First hopefully we can agree that if we have two real numbers x and y that
> are not equal then we have a number z, such that x < z < y. The easiest
> example is z = (x + y)/2.

It's one of the properties of the reals and rationals that if two numbers
aren't equal, then there are infinitely many numbers between them. It doesn't
work with the integers, 2 and 3 aren't equal, but there's no integer x, such
that 2 < x < 3.

So if we say 0.999... != 1 then that means 0.999... < 1 and then that means
(0.999... + 1) / 2 is a number different from 1 and different 0.999... but
that lies between them.

But this is modern mathematics. In the past mathematicians dealt with
"infinitesimals", especially in the early days of calculus, but I think they
were discarded because they were confusing and also not necessary in favor of
limits. This is where I think where some of your confusion is coming from.
Infinitesimals don't exist in the real (and therefore rational) number space,
but the concept exists for other "weirder" number systems.

According to the wikipedia page "This repeating decimal represents the
smallest number no less than every decimal number in the sequence (0.9, 0.99,
0.999, ...)"

The clearest resolution to "if you interpret 0.999... as the "biggest possible
number below 1"" is that in the reals and rational this concept doesn't exist.
There's no biggest number smaller than x and ditto for smallest number bigger
than y. (The distinction from the wikipedia definition is the difference
between < and <=, <= exists, but < doesn't)

It's similar to saying you can't divide by 0. Sure in some cases you can
define it, but doing so causes many issues and costs you so much that it's not
worth it.[1] Another example is 1 not being prime, there's no reason for it
not to be prime, but it's just much more convenient to say arbitrarily that
it's not prime.[2]

The other thing that might confuse you is a proof by contradiction, I know it
certainly confused me the first few times I saw it. I'm happy to help you if
this is tripping you up too.

> I'm slightly embarrassed I don't know more mathematics, but I'm trying to
> learn some more...

No problem, we all start knowing nothing :)

[1]
[https://www.youtube.com/watch?v=BRRolKTlF6Q](https://www.youtube.com/watch?v=BRRolKTlF6Q)
[2]
[https://www.youtube.com/watch?v=IQofiPqhJ_s](https://www.youtube.com/watch?v=IQofiPqhJ_s)

~~~
vfinn
Thanks :). I think the resolution to my problem is moving away from
infinitesimals. I did learn some calculus, linear algebra, number theory etc.
in university, but I did it quite superficially, since it didn't feel that
relevant to me at that time. Now I feel different.

------
zests
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.

------
fourseventy
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.

------
j7ake
This can be solved by using base 12 rather than base 10 to do the
calculation...

~~~
makotoNagano
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

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

~~~
makotoNagano
We haven't even started the Pi Tau argument yet

------
2OEH8eoCRo0
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?

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

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

~~~
maxnoe
Or any other language using IEEE 754 64 bit floats

~~~
juanmacuevas
I see... Nice!

------
berkeleynerd
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.

~~~
jhanschoo
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.

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

~~~
gspr
> 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.

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

------
heavenlyblue
So is then ‘0.(0...)1 = 0’?

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

~~~
cthalupa
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.

~~~
novacole
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?

~~~
gspr
> Also, I’d like to understand how if we can say 99.999... is = 100, wouldn’t
> we also be able to say 99.9999888999... = 99.9999..., and therefore also
> just 100? And so on?

You've just introduced a novel symbol into the discussion. What is the
definition of 99.9999888999...? Before anyone can answer what it equals, you
must _define_ it. Recall that if the digits repeat, we already have a
definition, so that case is fine. Your case is not à priori well-defined.

------
clevbrown
Given infinity that doesn’t make sense.

------
grensley
What is the largest number smaller than 1?

~~~
iiv
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.

------
amelius
How about the expression:

    
    
        0.9999... < 1
    

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

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

~~~
amelius
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.

~~~
gspr
> which is indeed false, you can also interpret the expression as: >
> lim[eps->0] ((1-eps) < 1)

Assuming you don't mean some special notion of limit, I would guess that by
`(1-eps) < 1` you mean the function from the reals to `Y = {false, true}` that
is defined as sending `x` to `true` if `1-x` is strictly less than `1`, and
`false` otherwise. Let's call this function `f`. I assume you're endowing `Y`
with the discrete metric?

If so, `f` does indeed have a limit _from above_ that is `false` and a limit
_from below_ that is `true`. Where do you wanna go from here?

 __Edit: __Corrected stupid wrong assertion about limit from below, d 'oh.

~~~
amelius
Have a look here:

[https://www.wolframalpha.com/input/?i=Limit%5BSign%5B1-x-1%5...](https://www.wolframalpha.com/input/?i=Limit%5BSign%5B1-x-1%5D%2C+x-%3E0%5D)

This shows that the limit does exist from both sides (but is different from
both sides).

~~~
gspr
Oops, sorry, that was a stupid blunder. Thanks for the correction.

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

------
ristos
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)

~~~
karatinversion
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.

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

------
smlckz
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](https://en.wikipedia.org/wiki/Zeno's_paradoxes)

~~~
frogpelt
Your two examples can be debunked though.

See here:
[https://www.youtube.com/watch?v=YuIIjLr6vUA](https://www.youtube.com/watch?v=YuIIjLr6vUA)

~~~
smlckz
ah, Mathologer video.

Have seen that.

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

------
upofadown
> ...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.

~~~
eof
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.

~~~
upofadown
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.

