## Is 1 = 0.999... ? Really?

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## Is it true that 1 = 0.999...? And Exactly Why or Why Not?

Yes, 1 = 0.999...
13
42%
No, 1 ≠ 0.999...
15
48%
Other
3
10%

Total votes : 31

### Re: Is 1 = 0.999... ? Really?

surreptitious75 wrote:Infinity cannot be defined as the smallest number greater than any integer because it is not a number as such

I am not saying "This is what infinity is". I am saying "If this is what infinity means, then this is what follows". My point being that if $$\infty$$ refers to a specific number that is greater than every integer, then $$\infty - \infty = 0$$ is true but $$\infty + 1 = \infty$$ is not; and if it refers to a non-specific number that is greater than every integer then $$\infty + 1 = \infty$$ is true but $$\infty - \infty = 0$$ is not.

Where two infinities are identical the answer is 0 when they are subtracted
And so for example the infinite set of primes - the infinite set of primes = 0

Yes, but in such a case, $$\infty + 1 = \infty$$ is not true.

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### Re: Is 1 = 0.999... ? Really?

$$e^{i \pi}=-1$$

Go figure

Is 0.999...=1 so strange in comparison?
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### Re: Is 1 = 0.999... ? Really?

phyllo wrote:Actually $$\infty$$ - $$\infty$$ = $$\infty$$

If $$\infty$$ refers to a non-specific number greater than every integer (in the same way that "A number greater than $$3$$" refers to a non-specific number since it can be $$4$$, $$5$$, $$100$$ or $$1000$$), then $$\infty + 1$$ equals to $$\infty$$ (because "A number greater than every integer + 1 = a number greater than every integer") but $$\infty - \infty$$ is indeterminate.

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### Re: Is 1 = 0.999... ? Really?

Could you make up your mind what it means instead of using multiple meanings?
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### Re: Is 1 = 0.999... ? Really?

Another question:

In regards to 1/3 necessarily having no way of being expressed as a finite sum of rationals, that depends on the base of the number system one is using. Suppose our number system was base 3. That means our number line would look like this:

0 1 2 10 11 12 20 21 22 100 101 ...

Here's what counting from 0 to 1 would look like if went by increments of two decimal places:

0.00 0.01 0.02 0.10 0.11 0.12 0.20 0.21 0.22 1.00

Note the part in bold. This is exactly one third the way to 1.00. Note that it does not require an infinite decimal expansion. It's just:

0.1

The quantity hasn't changed. It still represents a third of whatever. The only thing that's changed is the notation. We use a different notation to represent one third because we are using a different base.

^ This shows, I would hope, that the problem of an infinite decimal expansion, and therefore the problem of an infinite sum of rationals, is a superficial problem having to do only with the notation system in use. Use a different notation system and the problem goes away. It's not a problem with the quantity itself. That remains the same regardless of the notation system being used.

So a third is not somehow outside the categories of "finite" and "infinite" (what I dubbed "semi-finite" on Magnus's behalf), and it is not an irrational number, and it is not something that can never be complete, never quite attain it's limit, it's just, well, a third.

Now unfortunately, it gets a bit more complicated in the case of $$0.\dot9$$, and I'm not going to tackle that in this post, but I hope the above sheds some light on the difference between the notation and what the notation represents, and how sometimes the problem is only a problem for notation and not for the quantity the notation represents.
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### Re: Is 1 = 0.999... ? Really?

Ecmandu wrote:I know you’re trying to sound profound —- thing is, without time we couldn’t possibly have or understand this discussion on ANY level!

Not really. The notion of time is completely irrelevant to this thread.

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### Re: Is 1 = 0.999... ? Really?

Phyllo, did it ever occur to you that people’s intuition that finite numbers are not “infinite digits” (the most blatant contradiction in this whole discussion) means that there’s something wrong with how humans conceive of and understand math?

I mean, if you can really stand there and say “well it makes NO sense but it’s true”

What’s to stop me from saying “well it makes no sense but it’s NOT true”?

I mean, if you’re going with a contradiction, be prepared to have all the nonsense in the world thrown at you as well.
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:
Ecmandu wrote:I know you’re trying to sound profound —- thing is, without time we couldn’t possibly have or understand this discussion on ANY level!

Not really. The notion of time is completely irrelevant to this thread.

Oh sure... Magnus is a 23105th dimensional being

Magnus, in this thread, I wonder if you even begin to hear yourself talk sometimes.
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### Re: Is 1 = 0.999... ? Really?

Phyllo, did it ever occur to you that people’s intuition that finite numbers are not “infinite digits” (the most blatant contradiction in this whole discussion) means that there’s something wrong with how humans conceive of and understand math?
Then find the flaws in the math instead of proposing some bizarre definitions of your own. Start by writing down the definition of "INFINITE number".

My own solution to the 0.999...=1 question does not depend on infinities at all. I didn't use an infinite series. I didn't use infinite sets. I used strictly multiplication and division and alternatively addition and division.
What’s to stop me from saying “well it makes no sense but it’s NOT true”?
You can say anything that you like.
I mean, if you’re going with a contradiction, be prepared to have all the nonsense in the world thrown at you as well.
That has already happened in this thread.
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### Re: Is 1 = 0.999... ? Really?

phyllo wrote:
Phyllo, did it ever occur to you that people’s intuition that finite numbers are not “infinite digits” (the most blatant contradiction in this whole discussion) means that there’s something wrong with how humans conceive of and understand math?
Then find the flaws in the math instead of proposing some bizarre definitions of your own. Start by writing down the definition of "INFINITE number".

My own solution to the 0.999...=1 question does not depend on infinities at all. I didn't use an infinite series. I didn't use infinite sets. I used strictly multiplication and division and alternatively addition and division.
What’s to stop me from saying “well it makes no sense but it’s NOT true”?
You can say anything that you like.
I mean, if you’re going with a contradiction, be prepared to have all the nonsense in the world thrown at you as well.
That has already happened in this thread.

Phyllo, you’re being disingenuous.

You know that when you divide 1 into decimals by 3 or 9 and then multiply the decimal by 3 or 9 that it NEVER equals 1!!

I know you know this.
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### Re: Is 1 = 0.999... ? Really?

You know that when you divide 1 into decimals by 3 or 9 and then multiply the decimal by 3 or 9 that it NEVER equals 1!!
No. That's exactly what I don't know because I think that I can do it.

Now, explain to me why I am wrong.
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### Re: Is 1 = 0.999... ? Really?

gib wrote:That's really careless. Dismissing the definition of "quantity"? Really? Well then, I guess we can put in whatever we want for n. How 'bout "cow"? Is "cow" > 0? If so, then I can prove that $$\sum_{i=1}^{cow} \frac{9}{10^i} < 1$$. But I think even you can appreciate that there is a limit to what we can substitute for n.

We cannot put in anything for $$n$$. There is an explicit condition: $$n$$ must be greater than $$0$$. It does not have to be something that satisfies Gib's narrow definition of the word "quantity", but at the same time, it cannot be something that cannot be said to be greater than $$0$$. If we cannot say that $$cow > 0$$ then we cannot say $$\sum_{i=1}^{cow} \frac{9}{10^i} < 1$$. Fortunately for us, we all agree that $$\infty > 0$$, so we can use it for $$n$$.

Your belief that using the word "infinity" in mathematical equations is the same as using the word "cow" is a mistaken one.

But you're missing a crucial step between 2) and 3): $$\infty$$ is a valid value for n. I know you feel intuitively that $$\infty$$ must be a valid value for n because $$\infty$$ is a quantity, so no need to prove it. But when you're arguing with someone who disagrees with you on that, you do need to prove it. You can't just run on intuition.

And in order for me to do so, you have to tell me what you think is a necessary and sufficient condition for something to be considered a valid value for $$n$$. And you may also need to explain why.

I think that $$n > 0$$ is a sufficient condition. You obviously don't. Why? You keep mentioning the word "quantity" without ever bothering to explain what you mean by it and why it's necessary for $$n$$ to fit your definition of the word "quantity".

If I argued that "cow" > 0, and therefore the formula applies to cows as well, you would insist I prove that "cow" is a valid value for n, wouldn't you?

No. I'd ask you what you mean by the word "cow" such that you can say $$cow > 0$$.

Same onus falls on you to prove that $$\infty$$ is a valid value for n.

It's not the same onus. We all know that $$\infty > 0$$.

But so far, all I've seen from you is re-assertion after re-assertion that $$\infty$$ is a number--no proof--which tells me you believe it on intuition, not logic.

It should tell you that you're not doing a good job at explaining what needs to be done (and why) in order for you to accept the position that I'm putting forward.

But what that means is: grab any number on the number line. $$\infty$$ is not on the number line. It's a direction in which the number line extends.

That's an extremely narrow view. Basically, you're limiting yourself to what is widely-accepted (or at the very least, to what is familiar to you.)

What makes you think you cannot place $$\infty$$ on the number line? The fact that it is not a widely accepted position?

There is a class of people known as parrots. Parrots are people who adopt other people's conclusions without knowing how to arrive at them on their own. Since they are not independent thinkers, they can never accept a belief unless their trusted authority approves of it.

How do you convince a parrot? By convincing their trusted authority.

If this is the only way that I can convince you, then we have nothing to discuss.

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### Re: Is 1 = 0.999... ? Really?

gib wrote:In regards to 1/3 necessarily having no way of being expressed as a finite sum of rationals

Who said that?

$$\frac{1}{9} + \frac{1}{9} + \frac{1}{9}$$ is a finite sum of rationals that is equal to $$\frac{1}{3}$$.

And it's a well-known fact that there is a base-3 representation of $$\frac{1}{3}$$ (which is, as you say, $$0.1$$). This has been discussed in this very thread $$60$$ pages ago (or more than $$3$$ years ago, if you will.) This is not what's being disputed.

What's being disputed is the claim that $$\frac{1}{3}$$, or $$0.1$$ in ternary numeral system, is the same number as $$0.\dot3$$.

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### Re: Is 1 = 0.999... ? Really?

phyllo wrote:
You know that when you divide 1 into decimals by 3 or 9 and then multiply the decimal by 3 or 9 that it NEVER equals 1!!
No. That's exactly what I don't know because I think that I can do it.

Now, explain to me why I am wrong.

This post is incomprehensible.

I’ll explain it to you very simply:

1/9 = 0.111...

0.111... * 9 = 0.999... NOT! 1

When I divide 1/4 and do 0.25 * 4 it equals 1.

I’m going to be perfectly blunt with you ...

If you see no serious philosophical difference between the two, you are being dense (foremost) and disingenuous.

If you claim they are equal, then your claim is that every natural number is an infinite digit expansion
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### Re: Is 1 = 0.999... ? Really?

To be perfectly honest with you, Ecmandu, I think that most people on this board think that you're mentally challenged -- and rightfully so.

This is why they generally don't bother responding to your posts.

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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:To be perfectly honest with you, Ecmandu, I think that most people on this board think that you're mentally challenged -- and rightfully so.

This is why they generally don't bother responding to your posts.

I’m defending your argument, but whatever.

If you can’t understand my last post, that is not an indictment of my articulation, but rather your mental abilities.
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:We cannot put in anything for $$n$$. There is an explicit condition: $$n$$ must be greater than $$0$$. It does not have to be something that satisfies Gib's narrow definition of the word "quantity", but at the same time, it cannot be something that cannot be said to be greater than $$0$$. If we cannot say that $$cow > 0$$ then we cannot say $$\sum_{i=1}^{cow} \frac{9}{10^i} < 1$$. Fortunately for us, we all agree that $$\infty > 0$$, so we can use it for $$n$$.

You're not quite getting what $$\sum_{i=1}^{n} \frac{9}{10^i} < 1$$ for n > 0 means. In the case of sums, valid values for n are typically integers greater or equal to whatever initial value is set for i. In other words, n must be an integer, and integers are only found on the number line.

You got this part right:

$$\infty$$ > 0

But you're making an implicit assumption:

$$\infty$$ has a specific place on the number line.

^ This you got wrong.

$$\infty$$ is not among the values you have at your disposal to choose from for n. IOW, the condition that n > 0 does not just mean "anything greater than 0", it also means values that fit with sums of the form $$\sum_{i=x}^{n} f(i)$$ (that is, integers, those symbols above the tick marks on the number line).

$$\infty$$ > 0 remains true because > and < don't signify specific points on the number line. They signify directions on the number line. "Greater than" means: in the positive direction. "Less than" means: in the negative direction. $$\infty$$ need not have a place on the number line for $$\infty$$ > 0 to be true. It just has to be to the right of 0.

Magnus Anderson wrote:
But you're missing a crucial step between 2) and 3): $$\infty$$ is a valid value for n. I know you feel intuitively that $$\infty$$ must be a valid value for n because $$\infty$$ is a quantity, so no need to prove it. But when you're arguing with someone who disagrees with you on that, you do need to prove it. You can't just run on intuition.

And in order for me to do so, you have to tell me what you think is a necessary and sufficient condition for something to be considered a valid value for $$n$$. And you may also need to explain why.

Now we're cooking! You're finally asking the right questions.

I believe my explanation above should suffice. Valid values for n are any integer found on the number line greater that 0 (at least in the case where i = 1). Why? Because that's what the condition n > 0 is asking for. $$\infty$$ is not an integer found on the number line. Why? Because it is the endlessness of the number line itself, at least in the positive direction. In other words, $$\sum_{i=1}^{\infty} \frac{9}{10^i}$$ is not a special case of $$\sum_{i=1}^{n} \frac{9}{10^i} < 1$$ for any n > 0. It's an entirely different statement. It says: don't select a value for n. Just let the sum continue forever.

^ This may not satisfy your request for further clarification on what I need from you, but now your move is to probe deeper into my response. Is there something in the above you need even further clarification on? (wanna get into hyperreals? )

Magnus Anderson wrote:I think that $$n > 0$$ is a sufficient condition. You obviously don't. Why? You keep mentioning the word "quantity" without ever bothering to explain what you mean by it and why it's necessary for $$n$$ to fit your definition of the word "quantity".

Again, I think the above addresses this. The important point is that $$\infty$$ fails my definition of quantity. A quantity is what points on the number line represent (at least in terms of cardinality). Since $$\infty$$ has no point on the number line, it doesn't represent a quantity.

Magnus Anderson wrote:
If I argued that "cow" > 0, and therefore the formula applies to cows as well, you would insist I prove that "cow" is a valid value for n, wouldn't you?

No. I'd ask you what you mean by the word "cow" such that you can say $$cow > 0$$.

You say tomato, I say tomato. (<-- That doesn't quite come through in text, does it? ).

Magnus Anderson wrote:
Same onus falls on you to prove that $$\infty$$ is a valid value for n.

It's not the same onus. We all know that $$\infty > 0$$.

It doesn't matter what "we all know". I'm not a part of "we". You're dealing with me in particular. If you want to convince me of your point, the onus falls on you to prove your point to me.

Magnus Anderson wrote:
But so far, all I've seen from you is re-assertion after re-assertion that $$\infty$$ is a number--no proof--which tells me you believe it on intuition, not logic.

It should tell you that you're not doing a good job at explaining what needs to be done (and why) in order for you to accept the position that I'm putting forward.

You mean like, oh I don't know, asking you to prove that what applies to finite sets also applies to infinite sets? How 'bout proving that $$\infty$$ is a valid value for n? Did I forget to ask for that?

It takes two to tango. I very well may not have been detailed enough in explaining to you what I need, but that's your cue to ask for further clarification. These things may go through a few rounds of back and forth before both parties are clear on what the other needs in order to be convinced. Nothing wrong with that. It's how it goes sometimes.

Magnus Anderson wrote:
But what that means is: grab any number on the number line. $$\infty$$ is not on the number line. It's a direction in which the number line extends.

That's an extremely narrow view. Basically, you're limiting yourself to what is widely-accepted (or at the very least, to what is familiar to you.)

What makes you think you cannot place $$\infty$$ on the number line? The fact that it is not a widely accepted position?

There is a class of people known as parrots. Parrots are people who adopt other people's conclusions without knowing how to arrive at them on their own. Since they are not independent thinkers, they can never accept a belief unless their trusted authority approves of it.

How do you convince a parrot? By convincing their trusted authority.

If this is the only way that I can convince you, then we have nothing to discuss.

You know, this is a desperate cry. You typically see it when your opponent has nothing left. It really means: I've got nothing but my going against the grain. I know that I'm taking an unpopular position. I know that all the professionals in the field, all the experts, all the really smart people disagree with me. I know the majority of people who follow this topic are not on my side. I know all my arguments are falling on deaf ears, that my attempts at convincing others seems to be an exercise in futility... but you know what, at least I'm not a sheep, at least I'm not mindlessly conforming to the masses. I'm thinking for myself, I'm exercising my independence of thought. And that's something to be proud of. I've exhausted everything else in the debate, might as well start using this.
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### Re: Is 1 = 0.999... ? Really?

gib wrote:
Magnus Anderson wrote:We cannot put in anything for $$n$$. There is an explicit condition: $$n$$ must be greater than $$0$$. It does not have to be something that satisfies Gib's narrow definition of the word "quantity", but at the same time, it cannot be something that cannot be said to be greater than $$0$$. If we cannot say that $$cow > 0$$ then we cannot say $$\sum_{i=1}^{cow} \frac{9}{10^i} < 1$$. Fortunately for us, we all agree that $$\infty > 0$$, so we can use it for $$n$$.

For good measure, I will object and say that we can equally well say cow is larger than zero as that we can say infinity is larger than zero; neither are quantities. And yet both clearly more than nothing.

Infinity is not a quantity, because quantities are definite.
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### Re: Is 1 = 0.999... ? Really?

gib wrote:You're not quite getting what $$\sum_{i=1}^{n} \frac{9}{10^i} < 1$$ for n > 0 means. In the case of sums, valid values for n are typically integers greater or equal to whatever initial value is set for i. In other words, n must be an integer, and integers are only found on the number line.

You'd have to explain why you're limiting yourself to integers.

$$\infty$$ has a specific place on the number line.

Not necessarily but it can be the case.

$$\infty$$ > 0 remains true because > and < don't signify specific points on the number line. They signify directions on the number line. "Greater than" means: in the positive direction. "Less than" means: in the negative direction. $$\infty$$ need not have a place on the number line for $$\infty$$ > 0 to be true. It just has to be to right of 0.

Basically, what you're saying is that $$\infty$$ does not refer to a specific number in the same way that the statement "An integer greater than $$10$$" does not refer to a specific integer since the quantity that it represents can be $$11$$ or $$12$$ or $$100$$. However, you have yet to explain why it's invalid for $$n$$ (the upper bound of summation) to be a non-specific number.

I believe my explanation above should suffice. Valid values for n are any integer found on the number line greater that 0 (at least in the case where i = 1). Why? Because that's what the condition n > 0 is asking for.

My claim is that $$\sum_{i=1}^{n} \frac{9}{10^i} < 1$$ for any $$n$$ that is greater than $$0$$. It does not have to be an integer. The only condition is that it's greater than $$0$$. Do you think there is $$n$$ greater than $$0$$ such that the sum is not less than $$1$$? You obviously do (otherwise, you'd not say that $$0.\dot9 = 1$$.)

$$\infty$$ is not an integer found on the number line. Why? Because it is the endlessness of the number line itself, at least in the positive direction. In other words, $$\sum_{i=1}^{\infty} \frac{9}{10^i}$$ is not a special case of $$\sum_{i=1}^{n} \frac{9}{10^i} < 1$$ for any n > 0. It's an entirely different statement. It says: don't select a value for n. Just let the sum continue forever.

$$\infty$$ is not an integer. It's a number greater than every integer.

And $$\sum_{i=1}^{\infty} \frac{9}{10^i}$$ is not a completely different statement. The only difference between $$\infty$$ and integers is 1) $$\infty$$ is a bigger number and 2) it's non-specific. The fact that it's non-specific shouldn't be a problem at all. And if you absolutely hate non-specific numbers, you can switch to some specific infinity.

You mean like, oh I don't know, asking you to prove that what applies to finite sets also applies to infinite sets? How 'bout proving that $$\infty$$ is a valid value for n? Did I forget to ask for that?

It takes two to tango. I very well may not have been detailed enough in explaining to you what I need, but that's your cue to ask for further clarification. These things may go through a few rounds of back and forth before both parties are clear on what the other needs in order to be convinced. Nothing wrong with that. It's how it goes sometimes.

It takes two to tango, yes.

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### Re: Is 1 = 0.999... ? Really?

Magnus wrote:

“Infinity is not an integer, it’s a number greater than every integer”

How many times do you have to be told ?

Infinity is not a number!
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### Re: Is 1 = 0.999... ? Really?

I’ll explain it to you very simply:

1/9 = 0.111...

0.111... * 9 = 0.999... NOT! 1
It is simple.

1/9= 0.111... Equation 1 (ordinary division by 9)

(0.111... )*9=0.999... Equation 2 (ordinary multiplication by 9)

(1/9)*9=1 Equation 3 (ordinary division and multiplication)

Substitute for (1/9) in Equation 3 with the equivalent (1/9) from Equation 1 :

(0.111...)*9=1 Equation 4

The left side of Equation 2 is exactly the same as the left side of Equation 4. Therefore whatever is on the right side of equation 2 is exactly equal to whatever is on the right side of Equation 4.

Therefore :

1=0.999...

QED
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### Re: Is 1 = 0.999... ? Really?

Ecmandu wrote:How many times do you have to be told ?

Infinity is not a number!

However you want to call it, infinity is something that is greater than every integer.

How do you call things that are greater than other things?
Cows?

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### Re: Is 1 = 0.999... ? Really?

Fixed Cross wrote:For good measure, I will object and say that we can equally well say cow is larger than zero as that we can say infinity is larger than zero; neither are quantities. And yet both clearly more than nothing.

Infinity is not a quantity, because quantities are definite.

The size of cows is greater than zero.

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### Re: Is 1 = 0.999... ? Really?

I’m writing this post for Carleas to link to (not because I’m reporting anyone here - that’s absurd!) but because I know he loves numbers and has participated . I love numbers too! Any of you can take a stab at it:

I have two number theory positions:

1.) series don’t converge
2.) there are no orders of infinity

1.) series don’t converge! If a series converges, this means that every counting number is an infinite sequence (proof through contradiction).

2.) there are no orders of infinity!

I use this order symbolically (not using numbers) to disprove this!

A.) rational number
B.) uncounted infinity
C.) different rational number
D.) different uncounted infinity

Etc...

This places all the “lower” infinities in 1:1 correspondence with the “highest order of infinity”, thus there are no orders of infinity.

Abstract concepts in mathematics not using numbers are fair game!

Thus my proof sits.

Magnus, for you personally, if you don’t understand that, I’m not crazy, and it’s not my fault.

I know Carleas doesn’t have the time for this thread, but I know he’s a math pervert (we all should know that) and I don’t think he’s seen these arguments before!
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### Re: Is 1 = 0.999... ? Really?

phyllo wrote:
I’ll explain it to you very simply:

1/9 = 0.111...

0.111... * 9 = 0.999... NOT! 1
It is simple.

1/9= 0.111... Equation 1 (ordinary division by 9)

(0.111... )*9=0.999... Equation 2 (ordinary multiplication by 9)

(1/9)*9=1 Equation 3 (ordinary division and multiplication)

Substitute for (1/9) in Equation 3 with the equivalent (1/9) from Equation 1 :

(0.111...)*9=1 Equation 4

The left side of Equation 2 is exactly the same as the left side of Equation 4. Therefore whatever is on the right side of equation 2 is exactly equal to whatever is on the right side of Equation 4.

Therefore :

1=0.999...

QED

You moved out of the decimal system to make your argument, you switched back to whole numbers, which is the same as never using decimal expression in the first place.
Ecmandu
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