# The .999... = 1 "debate"

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I'm amazed by how many people fail to accept repeated proofs of .999... = 1 utilizing multiple methodologies. Don't read the body of this blog, just read the comments:

There are some rather intelligent people making revprez-style bombasitc arguments towards the contrary. Why? What fundamental misunderstanding compels these people?

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imho the main reason is that people are usually used to think that the letters we use to write down numbers have a meaning of themselves. That´s what you are used to when you learn to calculate numbers (add them letter by letter, divide them letter-combination by letter-combination, ...). The abstraction that these letters are just a pictorial diagram for some abstract concept (and that two totally different paintings can both show a house, for example) is a way of thinking you usually do not need unless you are doing algebra.

Since I almost know this will happen: I doubt that any discussions about whether it is true or are welcome here. We have a very long thread about it (search function should reveal it) which was closed because some mod/admin (I think it was Dave) got fed up with the discussion.

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I closed that thread simply for the reason that nobody was listening to the arguments being put forward. This thread is somewhat different in nature

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That is really strange. Disagreeing is one thing, but why on Earth is there all that hostility? I've seen the same thing in other discussions of exactly the same thing. It's as if for some people, even if they've never thought about it before, the notion that 0.9_=1 is deeply, deeply insulting. I don't understand either this defensiveness or even the disagreement itself (how are the demonstrations less convincing than the "counter-arguments?"), but perhaps both can be explained by a mistrust and resentment of the "trickier" aspects of mathematics. As one who tutors college students in calculus, I've noticed that lots of people never really grok stuff like infinintesimals or limits, and so suspect that I'm trying to "put one over on them" when I do things based on them, like it can't possibly be "real" math. To say that 1, the most basic foundation that they've all known since before they could remember, is the same as some crazy infinitude, might well be the last straw.

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Why?

Because some people don't understand what a limit is, and that "..." means the limit where the number of 9s goes towards infinite. I talk from my own experience actually. I was one of those stupid lads who disagreed that 0.99...=1, saying that "no matter how many 9s you place behind the '.', the result will never be exact 1!".

I am less ignorant today.

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Okay I'll be part of the crowd that believes that those who believe .999....=1 are either smoking crack or are trying to screw with my head and I've seen too many "proofs" drawn up that try to prove to me that 1+1=3. In general all of this stuff makes my head hurt so I won't dive into the merits of the proof.

In regards to Sisyphus's question about the hostility, I think we could create a thread on just about anything and it would turn into hostilities. I've seen it happen way to often in way too many forums (I visit/participate in around 1 dozen different forums).

The simple fact of the matter is that people seem to mistake hostility and personal attacks with reasoned debate and I blame this on things like FOX News. I honestly think that reasoned constructive debating is a dying art form as is being able to accept and acknowledge valid points from opposing sides of an issue. There is no effort to find a middle ground or listen to other points of view. People simply want to ram their viewpoint down the throats of others. Look at the 9/11 thread that got closed a couple of hours ago.

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Okay I'll be part of the crowd that believes that those who believe .999....=1 are either smoking crack or are trying to screw with my head and I've seen too many "proofs" drawn up that try to prove to me that 1+1=3. In general all of this stuff makes my head hurt so I won't dive into the merits of the proof.

why? all of the 1+1=3 arguments are trivially wrong, and they all rely on dividing by zero: at some point one states xy=xz hence y=z, which is just not true if x=0.

you state that those who assert 0.999...=1 are wrong, yet you don't seemingly know the reason why it is true (and it is in base 10 decimal representations of real numbers). look at the proof, learn the definitions of all the terms (I'll bet you don't know what the real numbers are; this is not ignorance, it is just that the real nature of the real numbers is not taught to school kids any more than quantum mechanics is taught to 11 year olds).

The simple fact of the matter is that people seem to mistake hostility and personal attacks with reasoned debate and I blame this on things like FOX News. I honestly think that reasoned constructive debating is a dying art form as is being able to accept and acknowledge valid points from opposing sides of an issue. There is no effort to find a middle ground or listen to other points of view. People simply want to ram their viewpoint down the throats of others. Look at the 9/11 thread that got closed a couple of hours ago.

there is no middle ground. this is *not* about opinion or debate. there is no debate. it is not a viewpoint. it is a demonstrable, provable statement. what really is annoying is people arguing with conviction from a position of ignorance against a proof that they do not, will not, understand, and refusing to acknowledge the mathematics. instead they rely upon 'gut instinct' or something: I know you wrote a proof but i just don't believe you, I can't refute it, or point out where it is wrong, but it is, you're wrong (insert fingers in ears and go La La La, I can't hear you at this point).

0.999..=1 (decimals) is true, and is true for *exactly* the same reason that 1/2=2/4, and I bet you have no problem accepting that.

there are literally hundreds of explanations of this fact available all over the place. I must have run through the argument on this forum at least 20 times alone.

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mathematical proofs are one thing, matt grime. Conceptualization is quite another.

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the real nature of the real numbers is not taught to school kids any more than quantum mechanics is taught to 11 year olds)
Should it be? You'd have to explain some types of non-real numbers to explain it properly I doubt most kids could handle that.

mathematical proofs are one thing, matt grime. Conceptualization is quite another.
On that note, I'd argue that there are no proofs for $0.\ddot{9}=1$ just as there aren't any for $5=5$, it just is
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You don't have to explain 'non-real' numbers at all. all you have to do is teach them that real numbers are *not* real. They do not exist in any physical sense. What exists in the physical world are things we wish to measure. Whole numbers do that wonderfully for many things. Fractions even begin to capture descriptions of quantity in quite a subtle way. Now, we can show that if the equation x^2=2 has a solution at all, whatever it is is not a rational number, so where do we go from here?

It would be stupid to go into details about analysis to school kids, so the best we could do is the common white lie: we use decimals to represent things, and one of the *rules* of the representation is that we *declare* the symbols 0.99... and 1 to be the same number, just as we declare 1/2 and 2/4 to represent the same number. The reason for this is simple: it makes the decimal representations of real numbers consistent. If they weren't declared to be the same thing then we have all kinds of problems. In particular, and this is exact reason why they must be the same representation of a real number in base ten, 1/n would not tend to zero.

Remember, there is no reason to suppose that 1/2 and 2/4 must represent the same thing unless we say what it is they're representing and what properties these symbols must have. Of course we want these symbols to be representative of the addition and multiplication of rational numbers, which is accurately summed up as a/b=c/d if and only if ad=bc

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@ Matt grime - I'm not really a math person, but this is how I understand it.

.999... is an infinite number. If you wanted to start the number in full, you would never stop saying it.

Because nothing in the physical world can really be infinite, we can only approximate infinate numbers by giving them finite values, as the infinite number approaches that finite value.

.999... repeating approaches 1, as you take the limit. It never actually reaches there, but for the purposes of coming up with a meaningful definition in our physical reality for .999..., we can call it 1.

Is this correct?

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mathematical proofs are one thing, matt grime. Conceptualization is quite another.

They aren't arguing against the concept if they refuse to learn the definitions, they are arguing against their perceptions of the concepts (which is merely the archimidean principle that 1/n tends to zero; here's why the archimidean principle must be true in any reasonable place where we want to do analysis.

1/n is decreasing and bounded below, we want these things to converge (otherwise we can't do analysis at all), so it must tend to x for some x. But 1/n^2 must tend to x^2 if we are to have a reasonable analysis, and it must tend to x because it is also a subsequence, hence x=x^2 so x=0 or 1, it can't be 1, so it must be zero *if we are going to do analysis* and the real numbers are where we *do* choose to do analysis, they are constructed (and that is what is missing) as a place to do analysis, and they are a useful structure for measuring things like lengths.

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@ Matt grime - I'm not really a math person' date=' but this is how I understand it.

.999... is an infinite number. If you wanted to start the number in full, you would never stop saying it.[/quote']

no I cannot agreewith that: it is not an 'infinite number' unless you define an infinite number for me, since that is not a label that I am aware of

Because nothing in the physical world can really be infinite, we can only approximate infinate numbers by giving them finite values, as the infinite number approaches that finite value.

this has nothing to do with maths, though

.999... repeating approaches 1, as you take the limit.

No! 0.999... is a number, it is not approaching anything anymore than 1 or 1/2 or 0.25 is approaching anything. It *is* a limit.

It never actually reaches there, but for the purposes of coming up with a meaningful definition in our physical reality for .999..., we can call it 1.

Is this correct?

0.999... is just a symbol, it doesn't approach, or attempt to reach anything. It is easiest to describe it as the limit of the following sequence:

0.9, 0.99, 0.999, 0.9999, etc

and since the limit of that must be 1 in the real numbers these symbols must represent the same real number.

Remember the real of real number is just a silly hang up from several centuries ago when people didn't understand how to rigorously deal with things like x^2=-1. The real does not indicate we are talking about the behaviour of real physical things.

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look at the proof, learn the definitions of all the terms (I'll bet you don't know what the real numbers are; this is not ignorance, it is just that the real nature of the real numbers is not taught to school kids any more than quantum mechanics is taught to 11 year olds).

To me these types of statements are efforts to downgrade the validity of others by questioning their intelligence or education and by using grandiose jargon designed to make one's self look like a learned expert who should not be questioned. This may not be the case here; however, it is a classic "debating" stunt that is used constantly in forums and on TV news programs.

These types of comments in threads aren't designed to support one's stance by helping others to understand one's position better; rather they are designed to discredit the stance of others. It is these types of arguments that spark hostile and personal exchanges that ruin good debates/discussions.

there is no middle ground. this is *not* about opinion or debate. there is no debate. it is not a viewpoint. it is a demonstrable, provable statement.

If you believe that .9999....=1 then prove it in a fashion that others can understand and accept even if they were educated in the U.S. educational system and hasn't had to deal with proofs for twenty years.

what really is annoying is people arguing with conviction from a position of ignorance against a proof that they do not, will not, understand, and refusing to acknowledge the mathematics. instead they rely upon 'gut instinct' or something: I know you wrote a proof but i just don't believe you, I can't refute it, or point out where it is wrong, but it is, you're wrong (insert fingers in ears and go La La La, I can't hear you at this point).

Again these types of arguments go back to my first comment in this reply. Insulting others is what drives threads down and it is a typical tactic use to discredit others in so many discussions. It is also a very commonly used tactic by those who can't really argue for their position.

In a discussion that is either true or false like .9999....=1, one shouldn't need to insult others or use confusing logic to explain why something is or isn't.

From a practical stance, .9999.... would close enough to equal one; however, given the limits of my U.S. education, the best I understand about numbers is that no matter how far out one carries .9999.... it will always be .00001... short of 1. Put another way we would all agree that 1=1.0000.... and 1.0000.... !=0.9999.... Again the difference would always be 0.00001.... (or however you want to note it)

In reality, I don't see how it matters if .9999....=1 or not; the point of my post was to show why threads like this get drawn down into nasty exchanges on a regular basis.

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KLB: the 9's continue for ever and ever so the 1 never shows up so the difference is 0.000... if the 1 never arises, therefore 0.999... = 1

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The problem arises that .9 repeated has an infinate number of nines associated with it.

But, doesn't that mean that the distance will be infinately small? It doesn't mean that they are equal.

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It's not nasty: I did explain that the necessary tools to understand what the proof is are beyond the majority of highschool (and now college) courses (going with the US model of terminology).

If you aren't going to look up these things that's fine by me. There's no real need to learn abuot them, any more than you need to learn about special relativity to drive a car (or even use GPS).

That's all ok.

But do please, do not tell people who are experts that they are wrong because they don't prove it in a way you can understand. It does embody difficult concepts that take more time to learn then is reasonable to expect. That's the trouble with maths: it's hard. However, if you read what I've written here in other posts all you need to realize is

1. decimal strings of digits are just representations of things that mathematicians call real numbers.

2. it is necessary in that definition to declare that the strings 0.999.. and 1.000... are different representations of the same real number.

It is just a convention if you will.

If you want to understand why it is that the real numbers are the thing that we have come (after many centuries of playing around with ideas) to use as our preferred mathematical system of measurement then that could take a long time.

It is important to understand the culture of mathematics: things are just definitions; this is the definition we have come to use.

Let me illustrate perhaps some of the justifications for you:

1/n, as n tends to infinty, well, we'd want that to tend to zero (read the other stuff in the previous few posts too to get an idea of where this came from)

square root of two isn't a rational numbers. so lets suppose that i add in a symbol X and define it to be larger than all the positive rationals whose square is less than 2 and less than all those who square is large than 2, can i continue this construction? What do I get? what's the smallest space with all these 'gaps' filled in? what about taking limits? What happens if i put in all limits? But i'll want unique limits, right? I need the real numbers for this.

you say that 0.999.. differs from one cos it will always be 0.00001... short of 1.

but it isn't. your string 0.0001 has a finite number of ones. adding that to even 0.999999999 produces something larger than 1, so adding it to 0.999.... which is surely larger than 0.99999999 must also produce an answer greater than 1.

and that has to be true for any *finite* number of 0s then a 1.

so you're verging towards the normal argument of 'well it's an infinite number of zeroes then a 1' but that does not make sense: decimal numbers do not have an infinite number of digits and then some more. So, what number are you adding to 0.999... to get 1?

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To me these types of statements are efforts to downgrade the validity of others by questioning their intelligence or education and by using grandiose jargon designed to make one's self look like a learned expert who should not be questioned.
The fact is, in order to take part in a debate (which this isn't) then you need to know some of the basic terminology and theories.
These types of comments in threads aren't designed to support one's stance by helping others to understand one's position better;
But there is no stance, so what is the problem?
rather they are designed to discredit the stance of others. It is these types of arguments that spark hostile and personal exchanges that ruin good debates/discussions.
There is no debate or discussion to be ruined.
If you believe that .9999....=1 then prove it in a fashion that others can understand and accept even if they were educated in the U.S. educational system and hasn't had to deal with proofs for twenty years.
Okay, that's easy enough.

$\frac{9}{9}=0.\ddot{9}$ and $\frac{\alpha}{\alpha}=1$ so $0.\ddot{9}=1$.

Or, if you're willing to learn what numbers actually are:

$0.\ddot{9}-1=0$ should be sufficient.

It is also a very commonly used tactic by those who can't really argue for their position.
Of course he can't argue it, what is there to argue about?
the best I understand about numbers is that no matter how far out one carries .9999.... it will always be .00001... short of 1.
Do you understand what the ... means? It means that the 9s continue ifinitely.
Put another way we would all agree that 1=1.0000.... and 1.0000.... !=0.9999....
No, I wouldn't agree.
Again the difference would always be 0.00001.... (or however you want to note it)
The difference would be $0.\ddot{0000}$

In reality, I don't see how it matters if .9999....=1 or not
If it dosen't then 50% + 50% wouldn't be equal to 100%, all statistical analyis ever done, ever, would be invalid.
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The problem arises that .9 repeated has an infinate number of nines associated with it.

But' date=' doesn't that mean that the distance will be infinately small? It doesn't mean that they are equal.[/quote']

but that's the whole point of the real numbers: there is no such thing in them as infintely small but non-zero: there is no real number less than all strictly positive real numbers but greater than zero. (that is the surreal or hyper real numbers). the real numbers were invented precisely to put on a rigorous footing things like limits, it is a choice we want and a natural one.

that's because as I keep saying these are the real numbers where by construction these things do not exist, and the decimals are *just* representations of the real numbers which are a purely abstract concept.

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so you're verging towards the normal argument of 'well it's an infinite number of zeroes then a 1' but that does not make sense: decimal numbers do not have an infinite number of digits and then some more. So, what number are you adding to 0.999... to get 1?

that does show you can't add anything to .999... to make it equal to 1.

But does that mean .99... is = 1? I'm not sure that it does.

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KLB: the 9's continue for ever and ever so the 1 never shows up so the difference is 0.000... if the 1 never arises, therefore 0.999... = 1

Okay how about if it were written like this?

1.00000.... minus 0.9999... equals 1^-n where n equals infinity.

The question is what is the purpose in trying to prove this one way or another? Does it really matter? For human uses, no matter how percise we needed to make a calculation, eventually we would determine the number of significant digits and round off which would mean that for our purposes 0.999... would equal 1.

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heh there was a post on forums for a game i play where the topic got up to 173 posts in 2 days debating this.

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that does show you can't add anything to .999... to make it equal to 1.

But does that mean .99... is = 1? I'm not sure that it does.

you agree 0.999... ought to be a real number, and you agree that 1 is a real number and that addition and subtraction are defined on real numbers, right? so i'll let e=1-0.999..., that's a real number.

Now, what is 0.999..+e?

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Okay how about if it were written like this?

1.00000.... minus 0.9999... equals 1^-n where n equals infinity.

that is not a real number. infinity is not a number. and 1 to the -n where n is any integer is 1.

for our purposes 0.999... would equal 1.

they are equal owing to the properties that we declare they must have to be *representatives of real numbers*, it is nothing to do with rounding or measuring or phyisical niceties like that.

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This thread is moving faster than I can keep up, so my posts will be lagging a few posts behind the post they are referring to.

But do please, do not tell people who are experts that they are wrong because they don't prove it in a way you can understand. It does embody difficult concepts that take more time to learn then is reasonable to expect.

The point of my first post wasn't to support or disprove .9999.... = 1. The point was to try and answer the question why threads get so nasty so fast.

The worst mistake any one can make is to not question an "expert". The moment one declares that they are an expert, one must begin to question. Case in point, the Pope is supposedly an expert on God. If we aren't to question experts, then we shouldn't have question Popes when they told us that God created the world in six days. Supposed experts told us that there was definitive proof that there were WMD in Iraq yet none were ever found.

The reasoning and proofs of experts in the math and sciences have been turned on their head throughout history. We can only try to get to the truth by questioning.

Too often threads turn into personal attacks because "experts" take offense at being questioned. If an expert is truly an expert then they should have the confidence not to take the questioning of their knowledge personally and try to explain something in a way those questioning can understand. Otherwise it looks like arrogance, which doesn't help dispel bad information. Only a fool does not question.

In the sciences, "experts" must get over the arrogance of saying take the time to learn to those who question and doubt and instead learn how to explain things in a manner that the common person can understand. Unless this is done, the debate over things like evolution vs. creationism will go on for all eternity and we will be forever having to battle to keep the various forms of creationism out of the science classroom.

At the very least without questioning, one does not learn. Who knows, today I might learn more than I really cared to learn about numbers.

Two hours ago I, if someone would have told me that 0.9999...=1.000... I would have told them they were nuts. Now while, I may not fully see how .9999....=1, I do see how 1.0000.... - 0.9999.... = 0.0000.... Still, best I can grasp is that the difference between 1.0000.... and 0.9999..... is infinitesimally small, but it is not 0.

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