## 0.999... = 1

• 3.4k
I'm becoming increasingly astonished that this thread continues.

Hang on a sec, you're not trying to avoid those old record long threads of yours getting beaten? :)
Yeah, the comments here (one sub-thread in particular) have run their course.
• 3.7k
Yeah, the comments here (one sub-thread in particular) have run their course

Oh no! And we are so close to half a millennium! Don't stop now. This may be a record for total nonsense. :scream:
• 23.7k
Hang on a sec, you're not trying to avoid those old record long threads of yours getting beaten? :)

There's no easy way to find longest threads that I am aware of; but one would have to go a long way to beat the Trump thread. My record on this forum is only 1.5k replies. On the old forum it was many times that.

I'd still like to have a thread reach over 100 replies from a single post, though. You could easily have done that here, had you not replied to your own OP.
• 4
The proof is correct
• 3.4k
Fixed in the PDF.
Quantification --- Forming Propositions from Predicates — Shunichi Toida et al, Old Dominion University
• 14.8k
In my book 0.9 + 0.1 = 1 and 1 - 0.1 = 0.9 and so 0.9 does not equal 1. There's a similar argument for 0.99 and 1 and so on. So far each element of 0.99999....., I have an argument that it does not equal 1. However, I see that your proof involves limits and I know that in that context words change their meanings. So I'm curious.

${1\over3} = 0.333...\\3 \times {1\over3} = 1\\3 \times 0.333... = 0.999...\\$

Let's not distract from supertasks by questioning very simple mathematical facts.
• 1.2k
3×13=1 and 3×0.333...=0.999...

That makes 0.999999..... = 1 just an illusion created by the notation you have decided to use. It is not a proof. In my opinion. You might have a different idea of what a proof is.

On the other hand, it does show that looking at a problem another way might show that the problem is an illusion. But that would be philosophy.
• 14.8k
That makes 0.999999..... = 1 just an illusion created by the notation you have decided to use. It is not a proof. In my opinion. You might have a different idea of what a proof is.

Well it's not a mathematically rigorous proof as it doesn't prove each of the three steps. A mathematically rigorous proof is much more complex, as seen with TonesInDeepFreeze's answer.

But it's a simple proof for those that accept each step individually. If you want a proof of these then that's a topic for another discussion, probably on a forum dedicated to maths.
• 2.1k
One could argue that the equality does not hold, or that 1/3 does not equal 0.333...

There is
a=0.999...,
10a=9.999...,
10a-a=9,
9a=9,
a=1 therefore 0.999...=1

Of course, one could (with undesired consequences) reject the first two steps of this proof.

There is also the sum of an infinite geometric progression of term a = 9*10^(-n):

0.999 = 0.9 + 0.09 + 0.009 +...
= 0.9/(1-0.1) = 0.9/0.9 = 1

Then again, one could reject that the equation for the sum applies. The equation of the infinite sum relies on the notion of limit, and it is the notion of limit that is at play on the 0.999... debate.

The argument that 0.999... only approximates 1 has grounding in formal mathematics. In the 1960's, a mathematician, Abraham Robinson, developed nonstandard analysis (Keisler, 1976). In contrast to standard analysis, which is what we normally teach in K-16 classrooms, nonstandard analysis posits the existence of infinitely small numbers (infinitesimals) and has no need for limits. In fact, until Balzano formalized the concept of limits, computing derivatives relied on the use of infinitesimals and related objects that Newton called "fluxions" (Burton, 2007). These initially shaky foundations for Calculus prompted the following whimsical remark from fellow Englishman, Bishop George Berkeley: "And what are these fluxions? ... May we not call them ghosts of departed quantities?" (p. 525). Robinson's work provided a solid foundation for infinitesimals that Newton lacked, by extending the field of real numbers to include an uncountably infinite collection of infinitesimals (Keisler, 1976). This foundation (nonstandard analysis) requires that we treat infinite numbers like real numbers that can be added and multiplied. Nonstandard analysis provides a sound basis for treating infinitesimals like real numbers and for rejecting equality of 0.999... and 1 (Katz & Katz, 2010). However, we will see that it also contradicts accepted concepts, such as the Archimedean property

• 1.2k
Then again, one could reject that the equation for the sum applies. The equation of the infinite sum relies on the notion of limit, and it is the notion of limit that is at play on the 0.999... debate.
I can see that point. I didn't look at the issue in the light of infinite series or take on board that it was a question of the sum of an infinite series. I apologize for the distraction.

There is
a=0.999...,
10a=9.999...,
10a-a=9,
9a=9,
a=1 therefore 0.999...=1
That's very neat.
• 1.2k
a=0.999...,
10a=9.999...,
10a-a=9,
9a=9,
a=1 therefore 0.999...=1

But this doesn't seem to work with other similar sequences, such as 0.333... or 0.444... or 0.1212....
What have I got wrong?
• 2.1k
a=0.3333333
10a=3.3333333
9a=3
a=3/9
a=1/3
1/3 = 0.33333

b = 0.12121212
100b=12.12121212
99b=12
b=12/99
12/99=0.121212

• 2.3k
If 0.9999 = 1 than it seems that 0.000...0001 should be equal to 0.

But now I think we'll all agree that as you divide a number by smaller and smaller numbers your output gets ever greater.

So:
1/0.1 = 10
1/0.0001 = 10,000, etc.

But since 0.00...001 = 0 and 1 / 0.000...001 = ∞ then 1/0= ∞.

Now it is also true that 4/0=∞ and 9.7181=∞. And with a little more leg work I shall demonstrate that all numbers are actually equal to each other. Multiplicity is mere illusion, a result of the Fall and Adam's sin.

:razz: :nerd: :up: :100: :heart: :strong: :strong: :strong:
• 3.2k
Let's not distract from supertasks by questioning very simple mathematical facts.

Oh I see what happened. @Ludwig brought up the old .999... = 1 chestnut in the staircase thread, and it apparently got moved over here to revivify this four year old thread.

Ludwig, let me put to you a question.

Suppose that .999... is not 1. If they are different numbers, then there must be a third number strictly between them. What is it?

Put another way, and echoing the point you made in the other thread, suppose I have the sequence

.9, .99, .999, .9999, .99999, ...

What number can possibly get between ALL the terms of that sequence, and the number 1?

Well maybe it's .95. No, that's smaller than .99.

Ok maybe it's .995. No, that's smaller than .999.

Ok then maybe it's .9995. No, that's smaller than .9999.

You see how this works? You can't find any number to stick in between ALL of the elements of the sequence, and 1.

Since you can't find a number between them. the limit of the sequence is 1. Or putting it another way: .999... = 1.
• 1.2k
Oh I see what happened. Ludwig brought up the old .999... = 1 chestnut in the staircase thread, and it apparently got moved over here to revivify this four year old thread.
It probably saves time and energy. Actually, you mentioned it and I got curious. I'm afraid I innocently asked a question and set off a land-mine.

What number can possibly get between ALL the terms of that sequence, and the number 1?
Well, if I've understood how this works, there is a number that gets between each element of the sequence - the next element in the sequence - and is there is no last element of the sequence. So there is no answer to your question.

However, it is also true that 1 is the sum of the infinite series 0.999... - and therefore the limit.
But an infinite series never reaches its limit. To put it another way, "=" in this context (an infinite series) does not mean what it usually means.
• 1.2k

Thank you very much for those.

If I've understood, your argument shows what the sum of the infinite series is.

Right?
• 1.2k
Now it is also true that 4/0=∞ and 9.7181=∞. And with a little more leg work I shall demonstrate that all numbers are actually equal to each other. Multiplicity is mere illusion, a result of the Fall and Adam's sin.
• 1.2k
I'm becoming increasingly astonished that this thread continues.
It's probably just that everyone who joins needs to be taken through it. Each person has to learn everything for themselves.
• 3.2k
It probably saves time and energy. Actually, you mentioned it and I got curious. I'm afraid I innocently asked a question and set off a land-mine.

It's typically a land mine of ignorance and confusion. Mathematically there is no question whatsoever. .999... = 1 is a theorem of ZF once the appropriate definitions of the real numbers, limits, and infinite series are made. It's like looking at a chess position and saying yes or no, is this a legally reachable position according to the rules. .999... = 1 is a legally reachable position in ZF.

Well, if I've understood how this works, there is a number that gets between each element of the sequence - the next element in the sequence - and is there is no last element of the sequence. So there is no answer to your question.

There is an answer. The answer is that there is no number greater than all the terms of the sequence, and less than 1.

However, it is also true that 1 is the sum of the infinite series 0.999... - and therefore the limit.

But an infinite series never reaches its limit.

We had this conversation. 1 is the limit of the sequence .9, .99, .999, ... "reaching" is just something people say to confuse themselves.

To put it another way, "=" in this context (an infinite series) does not mean what it usually means.

It means exactly what it usually means. The limit of .9, .99, .999, ... is 1. Or equals 1.

I think I'm a little bit puzzled that you have this confusion after I've explained it in the other thread.

The limit is equal to 1, in exactly the same sense that 1 + 1 equals 2.
• 1.2k
There is an answer. The answer is that there is no number greater than all the terms of the sequence, and less than 1.
Yes. I assume you mean all the terms of the infinite sequence?

I think I'm a little bit puzzled that you have this confusion after I've explained it in the other thread.
And I'm puzzled why you think I'm disagreeing with you.

The limit is equal to 1, in exactly the same sense that 1 + 1 equals 2.
So it is. But what is the element of the sequence immediately preceding 1?
• 3.2k
Yes. I assume you mean all the terms of the infinite sequence?

Yes. "Sequence" and "infinite sequence" are basically synonymous, since finite sequences aren't of interest in this context.

And I'm puzzled why you think I'm disagreeing with you.

It's late, time for bed. I don't think you're disagreeing, but possibly misunderstanding.

So it is. But what is the element of the sequence immediately preceding 1?
[/quote]

There is none. Why do you think there is one or should be one? That's why I think you're misunderstanding. There's no element of the a sequence immediately preceding the limit point.
• 1.2k

For me, this issue has a wider context.

This may be a step too far. But there are many people who turn up on this forum - and elsewhere - who deeply believe that nothing is true and everything is probable.

The usual basis for this is traditional (since Descartes) scepticism, and one usually tries to meet it by arguing about that.

But what if they have been introduced to probability theory and infinity? Suddenly, there is a mathematical proof.
Sometimes probability = 1 and 1 = 0.9999... So everything is probability,

I think this is a mistake, because it neglects context. But it is new angle on the mistake.

I'm basing this on an assumption that both theses are correct - in their context.

There is none. Why do you think there is one or should be one? That's why I think you're misunderstanding. There's no element of the a sequence immediately preceding the limit point.
I think it follows that "0.999...." does not equal 1.

Sadly, my best time for philosophy is first thing in the morning...
• 2.1k
If I've understood, your argument shows what the sum of the infinite series is.

It shows how we get the fraction representation of repeating decimals.
• 1.2k
It shows how we get the fraction representation of repeating decimals.
OK. I wondered if it worked a bit more widely than that. I don't think that it would work for sqrt2, since Aristotle could prove that it was "incommensurable" without involving decimals. What about π? I was taught that it was 22/7 or 3.14....?
• 2.1k
Neither pi or sqrt(2) or e don't have repeating decimals. They are both irrational numbers — by definition numbers that cannot be represented by fractions.
• 1.2k

Thanks.
• 3.2k
For me, this issue has a wider context.

This may be a step too far. But there are many people who turn up on this forum - and elsewhere - who deeply believe that nothing is true and everything is probable.

I never gave any thought to the relation of truth and probability. Probability is just a number we assign to an event. Before you roll a die there is no truth to its outcome, it hasn't happened yet. You do know with probability 1 that it will turn up 1, 2, 3, 4, 5, or 6. And that if you roll it a million times, about 1/6 of the time it will turn up each number. I don't know how you relate that to truth.

The usual basis for this is traditional (since Descartes) scepticism, and one usually tries to meet it by arguing about that.

But what if they have been introduced to probability theory and infinity? Suddenly, there is a mathematical proof.
Sometimes probability = 1 and 1 = 0.9999... So everything is probability,

I don't understand what you're saying. I have 1 apple in the fridge, and there is 1 president of the United States, but there is no deep philosophy there. 1 is a number that has many applications.

You are still (I think) applying some mysticism to .999... = 1 but there really isn't any. It's a theorem of ZF. If you wanted .999... to be 47 you could make up a system in which that's a theorem.

1 is a probability and 1 is the number of stars in our solar system. I simply do not see the point you're making.

I think this is a mistake, because it neglects context. But it is new angle on the mistake.

Well if I have a hammer I can use it to pound a nail or go out into the parking lot and smash everyone's windows. A hammer is just a tool with many distinct and unrelated uses; and if you think of it that way, the number 1 is also a tool with many uses.

The hardware store owner has no use for hammers, to him it's the buyer who supplies the use. Likewise the pure mathematician has no use or application for the number 1; he just makes sure all the numbers are nice and shiny and logically constructed, for others to use.

Any of this make sense? I don't get what you are trying to say.

I'm basing this on an assumption that both theses are correct - in their context.

Hammers and numbers. Tools for most people, objects of interest in and of themselves to hardware store owners and mathematicians, respectively. I am baffled at where you are going with this.

I think it follows that "0.999...." does not equal 1.

In what system of rules? In ZF? You are wrong. In the "point-9-repeating equals 42" system? You're right. What underlying assumptions are you making?

If you assume the axioms of ZF, then .999... = 1 can not be challenged or disputed, any more than you can argue with how the knight moves in chess. But if you make up a chess variant in which the knight goes, say, three steps vertical or horizontal and two step diagonal, then that's how the knight moves in his alternative variant.

Make sense?

Now, if you would like to chat about why .999... = 1 in ZF, I am trained to know this. I had it beaten into me by professors at some of our finest universities. But if you prefer the "point-9-repeating equals 42" system, I'm perfectly happy to work with that as well.

Sadly, my best time for philosophy is first thing in the morning...

I've always been a night person. I generally post in the evenings US left coast time.
• 6.1k
I don't know how you relate that to truth.

Just an aside. You probably know this stuff. But others might not. This is not a rigorous presentation.

When you talk about the probability of something, that needs to be defined as an event. Which is a particular kind of mathematical object. It does not tend to be the kind of mathematical object that a formula in a mathematical argument is. Eg the probability that it will be raining in 2 hours given that it is raining now makes sense. The probability that 2+2=4 doesn't make too much sense.

However. If a statement A is provable from a statement B and concerns a quantity
*
(in some amenable sense I won't specify)
, the probability of A given B is 1. As an example, what's the probability of X+1=4 given that X=3? Probability 1.

Another fact like this is that if A and B are mutually contradictory, the probability that A occurs and B occurs is 0. That also works with entailment. Like the probability that X=3 given that X+1=2 is 0, since X+1=2 implies X=1, and there's "no way"
**
(in some amenable sense I won't specify)
for X to be 3 given that assumption.

The same holds for statements
***
(in that same amenable senseI haven't specified)
you can derive from B using classical logic and algebra and set operations. eg if the probability that X=3 is 0.3, what's the probability that (X=3 or X!=3)? 1, since those are exhaustive possibilities. The latter does have a connection to truth, as if you end up asking for the probability of something which must be true, its probability is 1.

For folks like Fishfry, I'm sure you can make the amenable sense I've not specified precise. Logical, algebra and set operations which can be represented as measurable functions on the sample space work like the above. "no way" corresponds to the phrase "excepting sets of measure zero". Which is the same principle that stops you from asking "What's the probability that clouds fly given that x=2?", as there's no way of unifying both of those types of things into a cromulent category of event.

The latter also blocks a more expansive connection to truth. Since the kind of things that humans do while reasoning from premises typically aren't representable as measurable functions. Maths objects themselves also have plenty of construction rules that behave nothing like a probability - like the ability to conjure up an object by defining it and derive a theorem about it, there's just nothing underneath all maths that would take take a probability concept which would usefully reflect its structures I believe.
• 3.2k
Just an aside. You probably know this stuff. But others might not. This is not a rigorous presentation.

Actually I never formally studied any probability theory. I've seen measure theory but not the fine points of probability.

When you talk about the probability of something, that needs to be defined as an event. Which is a particular kind of mathematical object. It does not tend to be the kind of mathematical object that a formula in a mathematical argument is. Eg the probability that it will be raining in 2 hours given that it is raining now makes sense. The probability that 2+2=4 doesn't make too much sense.

However. If a statement A is provable from a statement B and concerns a quantity *, the probability of A given B is 1. As an example, what's the probability of X+1=4 given that X=3? Probability 1.

Another fact like this is that if A and B are mutually contradictory, the probability that A occurs and B occurs is 0. That also works with entailment. Like the probability that X=3 given that X+1=2 is 0, since X+1=2 implies X=1, and there's "no way" ** for X to be 3 given that assumption.

I've never seen probabilities assigned to mathematical facts like that. Not sure what it means.

The same holds for statements *** you can derive from B using classical logic and algebra and set operations. eg if the probability that X=3 is 0.3, what's the probability that (X=3 or X!=3)? 1, since those are exhaustive possibilities. The latter does have a connection to truth, as if you end up asking for the probability of something which must be true, its probability is 1.

Same remark. Don't follow this at all. If you pick a random real in the unit interval, the probability that it's between 0 and 1/3 is 1/3. That I understand, from measure theory. But I don't follow assigning probabilities to equations at all.

For folks like Fishfry, I'm sure you can make the amenable sense I've not specified precise. Logical, algebra and set operations which can be represented as measurable functions on the sample space work like the above. "no way" corresponds to the phrase "excepting sets of measure zero". Which is the same principle that stops you from asking "What's the probability that clouds fly given that x=2?", as there's no way of unifying both of those types of things into a cromulent category of event.

You are giving me too much credit. I have no idea how to assign a probability to an algebraic statement. I've never seen that.

The latter also blocks a more expansive connection to truth. Since the kind of things that humans do while reasoning from premises typically aren't representable as measurable functions. Maths objects themselves also have plenty of construction rules that behave nothing like a probability - like the ability to conjure up an object by defining it and derive a theorem about it, there's just nothing underneath all maths that would take take a probability concept which would usefully reflect its structures I believe.

I don't know if reasoning from premises is amenable to probabilities. I may have missed much of what you said in this post.
• 6.1k
I've never seen probabilities assigned to mathematical facts like that. Not sure what it means.

P(X=1|X+1=2). Where X is a random variable. That'll give you probability 1.
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