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cusiosity about fractions

Hi all,
This one is about converting a fraction to decimal notation and vice-versa:

consider the number 0.99999999999999...
Let's call it X.

So, if X = (0.999999...) then 10X = (9.9999999...)

10X - X = (9.999999...) - (0.9999999...)
9X = 9
X = 1

But X is 0.99999999..., not 1!!!
Why is that?

Please, keep writing us. Your posts are better than "a joke every day" by e-mail.

Have you taken your pills today?


I have judged this post malicious and have deactivated the John_Gabriel account. This is to notify others that replying further to this thread would be pointless.


I see that it is also fruitless to argue here. It also seems very likely that you are John Gabriel given your fairly unique writing style and your very unique ``logic.'' You also have just joined strangely about the time that john's account was disactivated. Thus I will assume you are john and will stop responding.


john_gabriel, your phihosophy is difficult! You should clearly say whether an "infinitely little" amount is a positive real number or 0. Or do you not operate within real numbers?


>Yes, 0.9999.... is less than 1. Much less. In fact it is infinitely >less than 1 but you can be sure it is less than 1.

Could you prove for us that 0.9999... is less than 1?

Hi Mr. john_gabriel,
The number S = 0.999... implies that it is representing in a base b>9.
Consider the n-th partial sum
S_n = 9(b^{-1}+ b^{-2}+ ... + b^{-n}). So,
b^{-1}S_n = 9(b^{-2} + ... + b^{-n} + b^{-(n+1)}). Substracting,
(1-b^{-1})S_n = 9b^{-1}(1-b^{-n}). Taking lim when
n--->\infty, we have S = 9/(b-1). Therefore your claim: "0.999... is less than 1" is false for b=10. S<1 if and only if b>10.


Firstly, John got cut off not because he was "direct" but because he lapsed into childish insults. This is a place for rational discourse, not emotional tantrums.

Secondly, your argument hinges on the claim that the "...", as used in combined with decimal notation, means "SUM" and not "LIMIT". I would argue that it is meant to mean "LIMIT". There is really no right answer here without appeal to how the decimal representation is defined. The definition I have been using, which corresponds to a particular textbook, has "..." as "LIMIT". Hence .999... is identical with 1 by definition.


> So you see, there is a BIG difference in calculating
> the LIMIT of the sum to infinity and calculating the
> ACTUAL sum to infinity. You cannot perform the latter.
> Unfortunately our intellect does not allow us to comprehend
> infinity, much less to actually calculate sums to infinity.

The ``sum'' is defined as a limit of partial sums (in this case since the terms are all positive you can take any definition of a sum over an infinite set, that is, you can sum in any order). Both ``limit'' and ``sum'' are definitions. The usual definition of real numbers is equivalence classes of cauchy sequences of rational numbers. Again this is a definition, there is nothing wrong or right with it. There are alternate definitions of real numbers, but are all equivalent. That is if you approach say the number 1 with two different sequences, say: (1,1,1,1,...) and (0.9,0.99,0.999,...) then these are equivalent cauchy sequences (the first is just 1 repeated over and over) as their terms get arbitrarily close to each other (for any $\epsilon > 0$, I can find an $N$ beyond which terms of the first sequence are withing $\epsilon$ of the the terms of the second sequence). Thus they define the same number. A decimal representation is just a cauchy sequence of rational numbers.

While $r^n$ is never zero, we are taking the limit. The definition of the limit is not that $r^n$ ever becomes zero, it's that it gets arbitrarily close to 0. This is not a fact or a theorem, this is the definition. You cannot show that it's wrong or right.

So we DEFINE that a sum to infinity means that the partial sums get arbitrarily close to a given number (real number, but in this case we can stay in the rationals since we're looking at the number 1).

It is true that we cannot ``think'' of infinite quantities, simply because you can never ``get'' to infinity. That was the mistake John Gabriel made by making up a number 1.00000 ... 1 (with the last 1 at infinity). The terminology is confusing, we never ``sum to infinity''. We just take limits of partial sums, that is we sum arbitrarily large finite subsets, and that's all we can do.

Similarly we cannot think of real numbers really. Real numbers can become very unintuitive very quickly. We can think of rational numbers. We wish to however think of real numbers because we wish limits to exist. We made up real number purely to satisfy our desire to have limits to all cauchy sequences. Real numbers don't come from any axioms, we define them. I think the definition of reals should be given earlier in education rather then leaving it to graduate level as it is currently done.

So you could argue that this is all made up nonsense, and it is. But all our models of nature work on this principle. Real analysis is the basis of all modern (post 1700's) physics for example (you need to solve differential equations there and that's very hard to do without real analysis). This blind leading the blind has worked remarkably well so far.


"I am a school teacher..."

"So, in conclusion, 1 is not equal to 0.99999...."

uhmmm... I sometimes wondered why students often misunderstand maths...

I have tried to stay out of this fray for the past few weeks because it is obvious that nothing anybody says will convince you that you are wrong. So instead I am taking the different approach of saying that you are right, although not in the way that you think.

> No matter how large n becomes, this term will become infinitely
> small but never actually zero.

> I believe the problem arises here because teachers and lecturers do
> not understand these basic concepts.

The problem here is actually that you do not understand the concept of the real number system. As the author of this definition on planetmath, I would like to make a few comments on the definition.

In the real numbers, BY DEFINITION, it is impossible for any number to be "infinitely small but never actually zero." Why? Because the real numbers are DEFINED to be equivalence classes of infinitesimally close (a.k.a. Cauchy) sequences.

So, for you to argue that a sequence is simultaneously infinitesimally small and nonzero is a contradiction of the definition of real numbers. In the real numbers, once something is infinitesimally small, the definition decrees BY FIAT that ANYTHING INFINITESIMALLY SMALL IS DEFINED TO BE EQUIVALENT TO ZERO even if it is not identical to zero. Indeed, the entire purpose of an equivalence class is to give mathematicians a technical means of saying that two things are to be considered equivalent even if they are not literally identical.

The conclusion is that 0.9999..., by your own admission, must be equal to 1 in the real numbers, because the two numbers are infinitesimally close, and the real numbers have the defining property that infinitesimally close numbers are considered equivalent.

Does that mean you are wrong? Well, yes and no. Your only error is that you used the wrong number system for asserting your claims. As I have just explained, the real numbers by definition do not allow infinitesimal separation. However there are other number systems that do allow infinitesimal separation. One example of such a system is the hyperreals, which I also authored on planetmath. Many of your arguments are acutally correct in the hyperreals, and in the hyperreals the number 0.9999... is actually infinitesimally less than 1.

So, to summarize: Your arguments fail in the real number system because the real numbers are specifically defined with the provision that infinitesimally close numbers, even if not identical, are considered to be equivalent for the purposes of that number system. Your arguments fare much better in other number systems such as the hyperreals for which this provision is not present.

0.9999... is not a valid decimal representation of a number because by definition a decimal representation has to have a finite number of decimals. There is no valid decimal representation of PI, because it is a transcendant number.

0.999... is a convenient way to write the more complicated
Sigma (i=1...infinity) (10^-i).
This quantity is equal to 1.

Haven't you heard about the candy bar?
If you return the paper from 10 bars, you get a free bar.
So, how much is a bar really worth?
One bar has the value of one bar, and additional 1/10th bar. But with this 1/10th of a bar, you get an additional value of 1/100 bar, and this gives an extra value of 1/1000 bar, and so on.

In the end, you get that each bar is worth 1.1111... bars.
Since 1.111... = 10/9, this means that 9 bars have the value of 10 bars. So, to get 10 bars, you only need to buy 9 bars.

You say: Give me 10 bars, I will only pay for 9 bars, and I will eat the 10th before I pay it, and afterwards I will give you 10 papers, and I will get a free bar (the 10th that I've already eaten)


I happened upon this thread and although I think John Gabriel is a
little too 'direct', his arguments are correct. Some of your readers
are studying for Phds and the arguments they are presenting hold no water. It is very short-sighted of you to simply cut off his voice when you find you cannot agree with him. Out of all the posts on this
topic including yours Drini, the only posts with any truth in them are John Gabriel's.

I am a school teacher and I would like to show you how I present these topics to my students. Of particular interest is the idea which John G mentioned, i.e. the difference between an actual sum and its limit.

The formula:

----- [A]
1 - r

which is derived from:
a(1 - r )
Sum to n-terms = Lim --------- [B]
n -> Infinity 1 - r

is not the SUM to infinity of a sequence/series with first term a
and common ratio r.

So what is it then? It is the LIMIT of the sum IF AND ONLY IF
we assume that as n goes to infinity in the term

r where r = 1/10


------ [C]

then we can say that the smallest value [C] can have is ZERO
and consequently the largest value [B] can have is [A].

r is never zero.

No matter how large n becomes, this term will become
infinitely small but never actually zero. By the same
token, [B] is never the same as [A] but [A] is the
largest value that [B] can become.

So you see, there is a BIG difference in calculating
the LIMIT of the sum to infinity and calculating the
ACTUAL sum to infinity. You cannot perform the latter.
Unfortunately our intellect does not allow us to comprehend
infinity, much less to actually calculate sums to infinity.

I believe the problem arises here because teachers
and lecturers do not understand these basic concepts.
The irony is that we have these people teaching courses
on real analysis, etc. This is an exact example
of the blind leading the blind. On this particular subject I
can't help but agree with John Gabriel.

So, in conclusion, 1 is not equal to 0.99999....

it is exactly because of people like you that John Gabriel rsponded the way he did. So what do you have to say that's worth anything? Or are u so ignorant that this is where your contribution ends? Uhmmmm.

Also, don't confuse the finite sums:
0.9 = 9/10
0.99 = 9/10 + 9/100
0.999 = 9/10 + 9/100 + 9/1000
0.99...9 with a million nines
those indeed aren't equal to 1

but when taking the infinite sum, it indeed becomes equal to one

it's like the known sum
1/2 + 1/4 + 1/8 + 1/16 + .... = 1
G -----> H G
p \ /_ ----- ~ f(G)
\ / f ker f
G/ker f

I just hope these forums don't turn into a sci.math-like.

Let's keep things mature. I agree with jirka.
When discussion switches from exchanging questions and arguemtns to
dismissing/mocking , you know one side has gone childish and there's no point on continuing the dialogue.

These kids have always been around (usenet, irc, all math forum-like places I've been) and the only way to deal with them is ignoring.

Quoting: " It is very sad that the world is in the grip of these so called Academic fools who are mostly pedophiles, perverts and arrogant fools"

What place do that have in a serious and mature talk about mathematics?
So *please, please* DO NOT get carried. Just ignore these kids.
G -----> H G
p \ /_ ----- ~ f(G)
\ / f ker f
G/ker f

john_gabriel's picture

Hullo Jussi,

Yes, 0.9999.... is less than 1. Much less. In fact it is infinitely less than 1 but you can be sure it is less than 1.

See, the problem here arises because of the way we choose to represent numbers as sums of the powers of a certain radix. Whatever the quantity 0.99999.... really is, it can probably be represented 'finitely' in another base. Let's look at an example:

1/4 in base 10 is represented as 0.25

For example, in base 7 it is represented as:


This is caused as a result of trying to represent a finite quantity in terms of an infinite sum. We assume that at infinity, such a finite sum will have a limit.

e.g. 1/7 + 5/7^2 + 1/7^3 + .... etc is NOT 1/4 !!!

It is an infinite sum which you are assuming to be equal to 1/4. This is incorrect.

Do you know why the decimal (radix 10) system was invented? Maybe you should research this information before you make up your mind for sure.

If Thiago had thought:

1 x 10/10 rather than 1 x 0.999999...... he would not be scratching his hard head along with all the other hard heads as they are doing right now.....


0.99999999... = 9/10 + 9/100 + 9/1000 +...

= 9/10 \times (1 + 1/10 + 1/100 + ...)

= 9/10 \times (\sum_{i=0}^\infty 1/10^i)

= 9/10 \times (1/(1-1/10))

= 9/10 \times (1/(9/10))

= 9/10 \times 10/9 = 1.


john_gabriel's picture

I am afraid the logic is not correct.

0.9999... IS NOT EQUAL TO 1.
0.9999... has no rational (finite) representation amongst the real numbers. In fact, 0.99999... is an undefined quantity if it's not an approximation. It is simply the sum of:

9*[1/10 + 1/100 + 1/1000 + ....] Provided the radix is 10.

Arithmetic as we have defined it, works only on FINITE values. We only talk about 'limits' of infinite sums approaching some finite value.
If operations worked as well as you claim, then this anomaly would not exist as stated by the original author, i.e. that .9999999... is equal to 1, would it?

So, 1 x 0.999999... IS NOT EQUAL TO 1.
Just like 3 x 0.3333... IS NOT EQUAL TO 1 either.

In fact NOT every real number has an infinite number of digits in its decimal representation - I cannot understand how you arrived at this. Consider a 1/4 = 0.25 - it appears to have a very finite representation to me. No?

Regarding zeroes after each real number: I think you may have missed something in your primary school education - were you not informed that trailing zeroes after the last significant digit are in fact redundant? Don't take this up wrongly, but I think fractions should be retaught at high school when students are a little wiser.

This is clearly evident since the radix representation can be finite or infinite as the sum of the powers of its base.

> (we say it tends to one)
I think you mean zero, actually :)

I'm not sure there no "infinite representation", as you say. What do you mean by that? Surely in real life you will find only approximated values on your statistics tables. But then it would make no sense to speak about the number 0.999999..., you simply reject to use an infinite number of digits. If you do that, you'd better throw away the decimal representation of numbers, since you wouldn't be able to write 1/3=0.33333....
When you use such a number you mean that you are specifying a sequence of "simpler" numbers (0.3, 0.33, 0,333 ...) which will approximate 1/3. Now, in effect, if you take the sequence (0.9, 0.99, 0.999 ...) you will see that the number you are approximating is 1. In fact if you proceed enough in the sequence the error you make (the difference between 1 and the member of the sequence)becomes as small as you want.
I don't understand why you only want to deal with numbers with a finite quantity of digits to perform operations. These operations are perfectly correct, even if giving a rigorous justification would require a little analysis. But the analysis involved would not be deep: the matter is just that you haven't a rigorous definition of 0.99999.... This really is the sum of an infinite series, and you have to prove thet you can do some operatios on these.

The logic is correct, 0.9999... = 1. Operations work just as well on values which have as you say ``infinite'' representation. We really only deal with finite number of digits (approximations), but then all arithmetic operations are continuous, and so taking the limit and then doing the operation or doing the operation and then taking the limit gives you the same thing. So you can really do the operation on a finite number of digits and then take a limit and you still see that
X = 1. And of course 1 x 0.999999... = 1. Just like 3 x 0.3333... = 0.99999... = 1.

I'm not sure what your ``infinite point of view'' really is. X is a real number and by definition of the decimal representation, it is a series of approximations. In fact every real number has an infinite number of digits in its decimal representation, just that we usually don't write the infinite number of zeros, but 1 = 1.00000...

In fact any number for which the decimal representation terminates (ends in infinitely many zeros) can be represented in two different ways. We usually use the one representation which ends in all zeros, rather then the one which ends in all 9's.


> Regarding zeroes after each real number: I think you may have missed
> something in your primary school education
> I think fractions should be
> retaught at high school when students are a little wiser.

How true. Ironically, the target of this remark is not who you

Normally I'd reply pointing out your errors, but because I did a little search on the internet for your name and found some other discussion on some other boards that you were in, I know this would be futile. You would just say that I don't know what I'm talking about, you would assert to be the only person capable of this knowledge and obviously all of the other mathematitians are completely wrong. Oh well, what can I say.

Those that care about rigorous treatment of this should read a good book on real analysis such as Royden or Rudin.


Hi john_gabriel! If 0.9999... is not equal to 1, do you think that it is less than 1? How much less? About some 10^{-200} or what?


john_gabriel's picture


I think you have summed it up pretty well: most mathematicians are absolute fools. Real Analysis is mostly convoluted garbage - just look at the source and you will find out why this is true. There are some good things in real analysis but have you stopped to think that this garbage was not even around when the greatest mathematical concepts and ideas were born? I somehow don't think you have.

So many books have been written on real analysis - mostly by insecure professors who don't have a clue what they are talking about (and don't really care) and then poor, unsuspecting souls just suck up the unproven non-sense because they do not understand it. Ignorance is bliss - how true.

Cantor, Riemann and Weierstrass: What did these fools do exactly? And where were they when the true Foundations of Mathematics were laid? What is true Mathematical rigour? Let's start with Cantor:

Set theory: Since the time it was spawned by Cantor, nothing significant has come of it. Millions of 'papers' have been published containing a few facts but mostly mixed with absolute rubbish of the utmost junk. Almost every Tom, Dick and Harry lecturer and professor has on is website his interests in Banach spaces, topology, metric spaces and set theory which is what most choose since it is the least understood by most.

Weierstrass(not so sure he was the founder..): The so-called epsilon-delta proofs are a complete waste of time. There is no rigour anywhere except perhaps what was in his hard head.

Riemann: He supposedly proved the fundamental theorem of calculus. Non-sense, all he did was show that it was bounded. And it is unfortunate they use terms like Riemann Sum and Integral when these should be called Archimedian sum/integral.

LaGrange: Introduced the mean value theorem which up to this day was not proved conclusively or in terms a lay man can understand. If you ask anyone, they will say it requires deep properties of real numbers to prove - this is of course rubbish once again. Check out my average tangent theorem - it proves the mean value theorem in very simple math which even a high school student should be able to understand.

Believing R.A books is exactly what I would not recommend. Rather I would advise someone to study these with an open mind, remembering they were written by some soul who probably needed to obtain a PHd.

It is very sad that the world is in the grip of these so called Academic fools who are mostly pedophiles, perverts and arrogant fools
who continue to impede the progress of true Mathematics. I have seen hundreds of papers which are not even worth the paper they were published on. How sad that most people cannot begin to think for themselves!

And I'll say this again, all the other 'mathematicians' are wrong and I am right. If you can find something wrong, argue the point but don't write an email which shows you don't have the answers and then heap discredit upon me which is what you have seen on other trashy sites. Why did all those critics shut their mouths? Because they realized they could not disprove my claims. So they opted to bring disrepute and slander rather than try to understand. Now for the record: There are Phds out there who agree with me but don't have the courage to say so - for obvious reasons. I have confidential emails which I cannot publish. So, you may want to rethink your ideas - perhaps you might learn something.

I think you have missed the point Jiri: I believe John was suggesting a number such as 0.000... 1 exists only at infinity. It cannot be represented in any form or fashion. The example he gave showed how you can explain Thiago's confusion, i.e. Thiago was performing 'illegal' arithmetic when he took the difference of 10X and X. It made sense to me.

> all equivalent. That is if you approach say the number 1 with two
> different sequences, say: (1,1,1,1,...) and (0.9,0.99,0.999,...)
> then these are equivalent cauchy sequences (the first is just 1
> repeated over and over) as their terms get arbitrarily close to
> each other

What you have just stated in the above paragraph Jiri, proves my point. As for the rest of your email, I have no idea what you are talking about - it appears to be all confused. e.g. contrary to what you say, you cannot assume that 'sum' has just any definition. 'Sum' is well-defined just like 'limit'. In actual fact, when you take the limit to infinity of the sequence 0.999.... (and Thiago was not using this as a sequence but rather as a number), you are showing that it is bounded above by 1 which clearly implies that 0.9999.... is LESS than ONE. Finally, I believe you provoked John Gabriel by asking him questions like "Have you taken your pills today?" It is unfortunate that you are allowed to get away with this, whilst he is denied further access to this forum. But you are not the only one guilty - the other comments I have seen also have zero value. I often ask myself why on earth do people say anything when they have nothing of any worth to add to the discussion?.... And ironically, your readers are the losers, not John Gabriel.

Akrowne: Regarding the decimal system or any radix system, these are well-defined and designed with the idea that as you encounter more positive significant digits to the left of the radix, the number increases. Going the other way (no matter how far), the number decreases. This is not left up to 'real analysis' or chance of any sort. It is built into the design of the system which would have been rejected a long time ago had your suggestion been true.

The reason Differential Equations work in physics is because of verification through experimentation. DEs are very hard to understand (and pdes even harder) because almost no one has a deep and solid understanding of the same.
I think I'll stay with the topic at hand as I am no expert at DEs.

john_gabriel's picture

Let me begin by saying you are all pathetic fools, but I am going to grace you with a proof. Thing is, will you be able to understand it?!

Firstly, if 0.9999.... were equal to 1, we would have a serious problem in that unique representation would no longer be possible in the decimal system. Therefore, 0.9999... and 1 are two different numbers. What this implies is that we can find another
number such that the sum of 0.9999... and this number is equal to 1.
It is safe to assume without any loss of generality that 1 is greater
than 0.9999.... and thus such a difference is possible. Why? Because they are both positive and 1 is a significant digit before the radix point in the decimal system.

Now for the proof:

Let X = 0.999999......

Let Y = 1/Inf where Inf = 10^n (as n -> Infinity)

Now Y is an infinitesmal number, say

0.00000.... 1 (the 1 appears at infinity)

So, 1 = X + Y

Must be true since there is a number such that X = 1 - Y

X = 0.99999.... + 1/Inf (1)
10X = 9.99999.... + 10/Inf (2)

(2) - (1) as per Thiago's problem:

9X = 9 + (10/Inf - 1/Inf)

Now we can really do nothing with (10/Inf - 1/Inf) except perhaps to simplify it to 9/Inf. So, this implies:

X = 1 + 1/Inf

which is what we started out with in (1).

You are confused when you misinterpret the limit of the sum to infinity of the terms 9/10 + 9/100 + 9/1000 to be the same as the representation of 1. This is NOT TRUE. What the limit says is that this sum is upper bounded by 1, it does not say it is EQUAL to 1.
This is what it means when we say the limit equals ...
There is a big difference! You cannot compute sums to infinity!!! Your brains are not capable of comprehending infinity.

If this does not make sense, you should be looking into other
subjects more suited to your abilities (if you have any).

Oh, and Jerka, maybe it's you who hasn't taken your pills eh?

Excuse me Mr. john_gabriel, but you don't know clearly the concept of limit of a sequence. 0.999... is exactly ONE in the decimal system.

You wrote:
X:=0.999... (1)
X=0.999...+1/INF (2)
X=1+1/INF (3)

(1) AND (2) implied: 1/INF=0 (4)
(3) AND (4) implied: X=1 (5)
(1) AND (5) implied: 1=0.999... (6)

Hunor Peto"

Hi john_gabriel, can you prove also that 4arctan(1) is less than pi (the perimeter of circle divided by radius)?

Sorry, of course I mean that pi is the perimeter of circle divided by diameter.

> Firstly, if 0.9999.... were equal to 1, we would have a serious
> problem in that unique representation would no longer be possible
> in the decimal system.

Whatever made you think the decimal representation of number was unique?

There are many constructions of the real number system. One of them is uses Equivilence classes of Cauchy sequences of rational numbers. Go read about and it might shed some light on your misunderstanding.

If you do not accept the validity of infinite decimal expansions how do you deal with things like root(2), or worse still e, and pi?

You wrote:
* X = 0.99999.... + 1/Inf (1)
* 10X = 9.99999.... + 10/Inf (2)
*(2) - (1) as per Thiago's problem:
* 9X = 9 + (10/Inf - 1/Inf)
*Now we can really do nothing with (10/Inf - 1/Inf) except perhaps to simplify it to 9/Inf. So, this implies:
* X = 1 + 1/Inf
*which is what we started out with in (1).
From (1) and your last equation, by substracting we get

1-0.999999... =0 <-> 1 = 0.999999...

Thank you for the proof,

> Your brains are not capable of comprehending infinity.

Obviously yours is since you just put the digit 1 in the place at infinity. In any case, I'm still not sure if you are a 10 year old trying to provoke people into giving you attention, or if you are a burned out college dropout angry at the professors who failed him. It is very obvious from your posts that you don't care about anybody proving or disproving your claims. No matter what anybody says you'll just call him a fool (or worse) and then claim that you are the only person who understands anything. Then you say something even more outragous to get more attention from people. So I do believe that you know that what you are saying is utter nonsense and that you are saying that on purpose to get reaction out of people. Though I think that most people are just using you to get a good laugh. I am certainly getting a good laugh out of this. As much fun as it would be to argue with you I know it's a pointless waste of time since it's the argument and not the result that you obviously care about.

So in closing, I think we'd all be wiser to not respond to john's outragous claims. It is not going anywhere and it's just encouraging him more.

john_gabriel's picture

It really is all very simple:

The Decimal system was designed as follows:



1 = 1 . 0 0 ...
0.999... 0 . 9 9 ...

Look carefully at the above Pedro! What do you see? 1 appears
before the radix point for the representation of 1 and we have
a ZERO under UNITS for 0.9999.... - both are positive numbers.
So which is beeeeger ?! Must be 1, no?

Nothing difficult about this. However, we have a lot of fools in
high places who are trying to teach the masses but are themselves
horribly confused. This is a problem.

I know it is likely that this question will create more dicussion, but here we go:

As some of you are saying, an infinitesimally small number would be zero (equal to, or equivalent, or whatever). So, the number 0.999..... would just be another form to REPRESENT the number 1 (so they would be equivalent, with the same value?)

If so, I wonder: this implies that we can't have arbitrarily small or arbitrarily large numbers? I mean, how small should be considered non-zero or zero? One can always make some number a little smaller but non-zero. So, (1 - 0.9999.....) could be as small as I wanted, but never zero.
Or, by making a number arbitrarily small, it would not be "infinitesimally" small anymore, but finite and well defined?

(Though in practice we just disconsider those very small quantities since most devices in the real world do not have precisions better than 0.01%... So, anything less than 0.0001 is simply be treated as zero without any problem, and we still can live with that).


In a common explicit construction of the hyperreals, where hyperreals are defined as sequences of reals, the sequence

.9, .99, .999, ...

is strictly less than the sequence

1, 1, 1, 1,...

i.e. in this number system, it is true that .999... is less than 1. The difference between the two numbers is the sequence

.1, .01, .001,...

which is an infinitesimal number (it is nonzero, yet smaller than every real number (i.e. sequence a, a, a,..., where a is real) and it has the property that no matter how many times you add it to itself, you never exceed 1, this is another defining property of an infinitesimal.

The distinction is whether to interpret the sequence

.9, .99, .999,...

as a hyperreal number or identify it with its real number limit.

Hi Thiago,

10^N is arbitrarily large as N grows (we say it tends to infinity),

10^{-N} is arbitrarily small as N grows (we say it tends to one).
And 0.9999...=\sum_{i=1}^{\infty} 9/(10^i) =1.


You can make numbers arbitrarily small or arbitrarily large, that's the idea because there is no ``smallest'' number (no infinitesmal). That is if x and y are two different real numbers such that |x-y| is positive (non-zero), then you can find a z in between such that |x-z| < |x-y|
and such that x is not equal to z, that is |x-z| > 0. That is the reason for how we construct the reals. We want a complete ``metric space'', and we want |x-y| to be the ``metric'', meaning ``distance''. So we want |x-y| = 0 if and only if x = y. That's why since |1-0.999...| cannot be any positive number, then |1-0.999...| = 0, and that's why 0.999... is just another representation of 1. (this is all kind of handwavy because I'm talking about contructing reals on the right and the metric (the distance) is defined as a real number. This is just motivation, not the construction or definition:) So if we are to talk about distance between numbers as we are used to and talk about limits as we are used to, then we need to also have 1 be the same as 0.999..., that is, we cannot have (well we don't want to have) some number smaller then 1 such that no number is between that number and 1 (which 0.999... would be), so we really want to have arbitrarily small numbers.

The reason why you want to treat small things as zero in the real world is that you have some error introduced. Obviously if you cannot represent a number in the computer (such as pi) you have an approximation and errors will arise. But it's not as simple as saying small numbers are zero, they are not. Small numbers are very non-zero. For example when you are taking a derivative you have a quotient where both numerator and denominator go to zero, and the closer you get to zero (the smaller the numbers) You need to first do some analysis on your problem to figure out what the error is. For example if your system is very unstable then a small error will translate to a lot. Take for example the numbers you gave, 0.0001 and let's take the function x^10000. Then let's look:

1^10000 = 1
1.0001^10000 = 2.71814...

So a very small change in the input gives a large error (this is just an example, but similar problems occur when trying to numerically solve differential equations for example). That's why trying to do analysis on your calculator won't work, analysis is about arbitrarily small quantities and they make a lot of difference. They only make little difference in very well behaved systems. So it is important to know when 0.0001 can be treated as zero. Best to always keep in mind that it is not zero, but just very close.


Dear Mr. thiago,
1=0.999... by definition, simply because the concept of limit of an infinite sequence is a DEFINITION universally accepted in math. And it works!!

> Hi all,
> This one is about converting a fraction to decimal notation
> and vice-versa:
> consider the number 0.99999999999999...
> Let's call it X.
> So, if X = (0.999999...) then 10X = (9.9999999...)
> 10X - X = (9.999999...) - (0.9999999...)
> 9X = 9
> X = 1
> But X is 0.99999999..., not 1!!!
> Why is that?

This is only your presumption. What you've discovered is a
proof of X = 1, which tells you that your presumption was
wrong. X is in fact equal to 1. Many people rediscover this
for themselves.

Perhaps, what is most shocking to most people is that the
decimal representation of real numbers is not unique. But that
is just life, and there is no point in loosing sleep over it.

john_gabriel's picture

Hi Thiago,

I believe your logic fails when you assume 10X-X=9

Your difference is only an approximation since 0.999... is an
'infinitely represented' quantity. You need to follow "finite rules" for subtraction and arithmetic in general when dealing with 'infinitely represented' values, i.e. assign a cut-off point in the radix representation. Example:


And in this case, X=0.999999, not 1. This is an interesting problem because it cannot be evaluated at 'infinity', it can only be approximated for some 'cut-off' point. By your assumption one might say:

1 x 0.999999..... = 1 which is in fact not true.

However, the more digits you take after the radix in 0.99999..., the closer your product becomes to 1. So, I think the answer to your question might be:

Logic fails when you take the difference (assuming that you could in actual fact have formed the product to begin with, i.e. 10X)
One might also argue that from an 'infinite' point of view, there is no difference between 1 and 0.9999....

Another interesting and related topic here is the use of mathematical and statistical tables - these are all approximations and I am amused when I see questions in textbooks asking the reader to find an 'EXACT VALUE' when there is really no such thing given a quantity is 'infinitely represented'.

dear thiago,

I assume you have been reading the replies you get to your innocent looking question. thanks to mr john gabriel this has turned into an annoyingly amusing issue. apart from his extremely inappropriate rhetoric, some of his remarks have merit while his delusions of actually proving them don't.

the thing is:
like most mathematicians, we can treat properties and definitions as extensional to the object they nominate. another alternative is from a bit more philosophical point of view that they can be intrinsically more important than the object they nominate. if you like to ponder on a doubt then i thought at least the doubt can be a bit better formulated.

According to me it's simple.
Take the average of 1 and .999..., and you will find that
it's .999... And so if the average of two numbers is one
of the numbers then they have to be the same.

john_gabriel's picture

In the following publication, I have systematically debunked all the so-called ”proofs” that 0.999… is the same as 1:

As for thiago’s question, there is a problem with the 10x ”proof”. For starters, it was never a valid proof.

Example: x = 0.999… 10x = 10(0.999…) 9x = 9(0.999…) x = 0.999…

If you input anything into the algorithm, you should get the same output. Now, only ignorant academics (and there are many on planet Math!), would do something as stupid as input a non-number (0.999…) into the algorithm. Actually they are capable of doing a lot more stupid things!

A little closer examination will reveal that in their false proof, they have already assumed that x=1. How so?

Well, add 9x to the LHS of first line and 9 to the RHS:

10x = 9.999…

Observe that 9x=9 implies that x=1, long before you arrive at the so-called conclusion. However, modern academics who are not much smarter than baboons, would easily miss this detail.

Of there is no valid construction of the real numbers and 0.999… or any other non-terminating representationPlanetmathPlanetmath is the result of a dysfunctional mind.

I have no doubt the pathetic cowards who run this site will delete my comments before anyone can learn something worthwhile.

jeremyboden's picture

If 0.999… ≠1 then what is the value of (1-0.999...)10.999...(1-0.999...)?

john_gabriel's picture

In order to say anything about the difference between 0.999… and another number, you would have to prove that 0.999… is indeed a number. It’s not a number of ¡any kind¿. Well, don’t even try to tell me it is a ¡real number¿ because no such thing exists. I have also debunked the myth that ANY valid construction of the real numbers exist.

Not a single ”proof” exists that 0.999… is equal to anything but itself.

Please don’t mention non-terminating representationsPlanetmathPlanetmath - these are junk math that has nothing to do with Euclid’s alogorithm. In fact, any concept that includes infinity is ill-defined.

Before you respond with all sorts of rants, just remember that I am a mathematician and know what you ”believe”. I understand it perfectly well but am telling you ”I” don’t agree with it.

Please DO NOT remove the link to my site:

Thank you in advance.

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