# Concept of Quantity

This is a discussion on Concept of Quantity within the General Discussions forums, part of the Community Boards category; But you cannot do that here whiteflags. It is exactly this closeness between an infinitesimal and 0, or 0.999... and ...

1. But you cannot do that here whiteflags. It is exactly this closeness between an infinitesimal and 0, or 0.999... and 1 that we are discussing. It may be true for certain types of operations where limits are used to facilitate calculations. But it is wrong to propose a proof that 0.999... is 1 by using a method in which we simplify 0.999... to 1. Or an infinitesimal to 0. What use can that be to anyone?

Here is something a lot more interesting. Why, I think, a certain level of rigor and a more critic attitude to seemingly established concepts serves us better:
Is 0.999... = 1?

I don't propose to understand all that is being discussed. But clearly here the author understands that under certain scenarios 0.999... may be equal to 1, if we allow ourselves to introduce a margin of error into our calculations. But as an universal truth, we cannot. We are then forced to introduce other concepts such as, and I quote:

Clearly 0.9* = 1 + 0¯, so 0¯ is a sort of negative infinitesimal. On the other hand, you can't solve the equation 0.9* + X = 1 because, in cut D, the sum of a traditional real with any real is a traditional real.
The acceptance that we musty introduce new concepts such as 0¯ or 1¯ and that our mathematical operations are powerless to deal with these quantities (real numbers) are fundamental to accept certain proofs into our lives without that meaning we should defend them as universal in the domain of Real numbers.

In other words: A little more humility serves better anyone pretending they can prove 0.999... is equal to 1.

2. Originally Posted by Mario F.
But you cannot do that here whiteflags. It is exactly this closeness between an infinitesimal and 0, or 0.999... and 1 that we are discussing. It may be true for certain types of operations where limits are used to facilitate calculations. But it is wrong to propose a proof that 0.999... is 1 by using a method in which we simplify 0.999... to 1. Or an infinitesimal to 0. What use can that be to anyone?

Here is something a lot more interesting. Why, I think, a certain level of rigor and a more critic attitude to seemingly established concepts serves us better:
Is 0.999... = 1?

I don't propose to understand all that is being discussed. But clearly here the author understands that under certain scenarios 0.999... may be equal to 1, if we allow ourselves to introduce a margin of error into our calculations. But as an universal truth, we cannot. We are then forced to introduce other concepts such as, and I quote:

The acceptance that we musty introduce new concepts such as 0¯ or 1¯ and that our mathematical operations are powerless to deal with these quantities (real numbers) are fundamental to accept certain proofs into our lives without that meaning we should defend them as universal in the domain of Real numbers.

In other words: A little more humility serves better anyone pretending they can prove 0.999... is equal to 1.
We didn't 'round' anything. It was an exact answer. You have been corrected with your "point" in my proof that was wrong. Look it up, it is known that:
lim[x -> infinity] 1/x = 0
That doesn't say that a very tiny thing is 0. It means that as x goes to infinity, 1/x will go to 0 and will get extremely close and even keep getting closer as x grows even bigger.
But in my proof I never used 1/infinity = 0. Just the limit.

3. Originally Posted by Mario F.
How can I possibly name a rational, when the interval for which you require me to do so is bounded by irrationals? Explain to me how can I name a rational in [0.(9), 1[ and, more importantly even, how my physical inability to do so does in any way mean that 0.(9) is the same as 1. Do you deny irrationals? Why don't you ask me for an irrational when you know perfectly well that's the only number representation I can give you?

As for the axiom you introduced, I'm afraid at this point you are possibly taking advantage of my basic knowledge of mathematics. I'm pretty sure that it may be valid for some sort of calculations where the infinite string of decimals required by 0.(9) does not need to be factored in. But it cannot be valid here where the proposition 0.(9) == 1 is being discussed to an absolute certainty. And because we are discussing the infinite nature of rational numbers, you cannot honestly say 0.(9) equals one without introducing an error margin.

So, it's my turn to play cat and mouse with you: Name me a rational representation of 0.(9) where it becomes equal to 1 and the error margin is 0. Dishonest question, I know. But you made me one too.

EDIT: BTW, I took a long and closer look at the axioms you linked me to. And nowhere I can see where anything even resembling your proposed proof that 0.(9) == 1 is being discussed, particularly where it relates to this statement of yours: "The fact that the distance(amount you must add) between them is an infinitesimal means they are equal". This is a very strong thing to say. Nowhere it is being discussed there and never before I heard such a statement. It's particularly confusing, because those axioms reach a point where it is established that "Between any two distinct real numbers there is a rational number. (In fact, there are infinitely many such rational numbers.)". This pertains also to Cantor's proof that shown there are more irrationals in the Real set of numbers, than there are Rationals (another issue of differently sized infinities).

The question I pose you is then, given there's an infinite amount of rational numbers between any two real numbers, how does one represent one such rational number between irrational limits? Our inability to name such a number really means there isn't any? If there isn't (and then 0.(9) equals 1), how come there's in fact an infinite number of rationals between these two limits?
Irrational limits? You're making up concepts and arguing completely from intuition.

How does the fact that .999... is irrational [, you incorrectly assume,] affect the fact that that if .999... != 1, you could name a rational between them? pi is irrational, yet you can name a rational in [pi/2, pi], or can you? It wouldn't surprise me.

I'm not taking advantage of your ignorance, you're just too stubborn(or otherwise) to understand the plain theorems I show you. Ever heard of the Dunningâ€“Kruger effect - Wikipedia, the free encyclopedia, you need to read it.

4. Originally Posted by brewbuck
Where is this "infinity" thing coming from? The fact that the number of digits is infinite is irrelevant.
Um, certainly not. There is no equality without an infinite number of digits. That's where the infinity comes from. See my last post, the one you quoted was before I really understood what was going on here (ok ok, I didn't read the whole thread, busted!). As the number of digits approaches infinity, the value of the expression approaches 1. I prefer that understanding. It is the same equality, but the wording makes it easier to understand I think.

The problem is that people often use infinity as if it were a number (I think this is the basis of Mario's beef with some of the explanations). There are some number of nines, call it "n". "n" is not infinity (that doesn't make sense), but as "n" approaches infinity the value of the expression approaches 1.

5. Originally Posted by Perspective
Um, certainly not. There is no equality without an infinite number of digits. That's where the infinity comes from. See my last post, the one you quoted was before I really understood what was going on here (ok ok, I didn't read the whole thread, busted!). As the number of digits approaches infinity, the value of the expression approaches 1. I prefer that understanding. It is the same equality, but the wording makes it easier to understand I think.

The problem is that people often use infinity as if it were a number (I think this is the basis of Mario's beef with some of the explanations). There are some number of nines, call it "n". "n" is not infinity (that doesn't make sense), but as "n" approaches infinity the value of the expression approaches 1.
Yes you're absolutely right here, and that's kind of the point: if the number of nines approaches infinity then the number approaches 1.
However, if you write "0.9999..." that means an infinite amount of nines, meaning that according to the limits it approached 1.

6. ...while never reaching it.

Is that hard to say?

7. Originally Posted by Mario F.
...while never reaching it.

Is that hard to say?
No, we never reach it because we can never reach infinity. Assuming we did reach infinity (as 0.99999.... does), we did reach one.
Really, we're talking limits 101 here... Look up limits.
And if you still want to argue start off with showing me what's wrong in my last proof. Because the last time you showed something "wrong" you only showed your lack of understanding of limits.

Read Limit of a function - Wikipedia, the free encyclopedia

8. I thought I made it clear to you that you cannot prove 0.999... equals 1 by saying that an infinitesimal equals 0. We are no better off from when we started.

Your use of limits, what it clearly reveals, is that you are the one not understanding their meaning and purpose. When you write lim(x -> oo) ( 1*10^-x ) = 0 you are reducing a irrational to an integer. And as I said before, while this may be acceptable under the context of limits usage, it cannot be used as the basis of a proof that 9.999... equals 1, because that is the very essence of what we are discussing here; the nature of a number as it approaches infinity.

You should refocus that stubbornness you attribute to me. As I said before a little more humility about math's shortcomings will serve you better. The only thing you proved is that limits aren't the best way to prove that 0.999... equals to 1. The only thing you shown was a formulaic simplification in the form of lim(x -> oo) ( 1*10^-x ) = 0. If you don't understand that the limits rule that validates this expression are a simplification and not a pure mathematical law, you failed to understand limits.

I also gave you a link that I think shows to you the actual requirements for a proof where 0.999... equals 1. It involves the creation of new concepts. That I believe is the way to do it.

What you don't want to understand is that I'm actually quite comfortable with the thought that 0.999... equals 1. It's convenient and for all purposes, almost true. It's so infinitely close to the truth, that it stops becoming an issue in about any use we want to make out of it. But that does not mean it is. And it is exactly the fact that it isn't, that no mathematical proof can exist for it. In fact, what saddens me somewhat is that almost all mathematical proofs ever devised to address this issue use the target interlocutor lack of knowledge as a means to establish "proof". It's the case of the 1/3 proof, or the 10x-x proof. It's low. Instead I'm more comfortable with this thought:

0.999... is infinitely close to 1. For all purposes it should be considered 1. But it isn't.

9. Originally Posted by Mario F.
I thought I made it clear to you that you cannot prove 0.999... equals 1 by saying that an infinitesimal equals 0. We are no better off from when we started.

Your use of limits, what it clearly reveals, is that you are the one not understanding their meaning and purpose. When you write lim(x -> oo) ( 1*10^-x ) = 0 you are reducing a irrational to an integer. And as I said before, while this may be acceptable under the context of limits usage, it cannot be used as the basis of a proof that 9.999... equals 1, because that is the very essence of what we are discussing here; the nature of a number as it approaches infinity.

You should refocus that stubbornness you attribute to me. As I said before a little more humility about math's shortcomings will serve you better. The only thing you proved is that limits aren't the best way to prove that 0.999... equals to 1. The only thing you shown was a formulaic simplification in the form of lim(x -> oo) ( 1*10^-x ) = 0. If you don't understand that the limits rule that validates this expression are a simplification and not a pure mathematical law, you failed to understand limits.
Your stubbornness is impressive. How you can not only reject perfectly valid proofs and established definitions but also well known facts by well known mathematicians.

The lim(x -> infinity) 1/x = 0. That's not a simplification. No, that's a definition of what "lim" does in our world. Identically, lim(x -> infinity) (1 * 10^-x ) = 0. Again, not a simplification. Definition.
The only remaining thing you'll have to see for my proof is that:
Code:
`0.9999.... = lim(x -> infinity) 9*10^-1 + 9*10^-2 + 9*10^-3 + ... + 9*10^-x`
But again, this is the definition of 0.9999...!

So in fact, we're using the definition of 0.9999... AND the definition of a limit. The only step in between is to subtract this limit from 1.

But okay, if you fail to accept that either one is defined as the other, then it's clear why we disagree on this. If you agree with both definitions, then you have to accept that 0.9999.... = 1.

The one you seem to disagree with is the definition of a limit. Read the definition here:
Limit -- from Wolfram MathWorld
It shows easily that lim(x -> infinity) 1/x = 0. Not APPROXIMATELY, no it's actually 0. No, there is no x with which the result is 0, but that's the limit: it never gets there, but it comes closer and closer.
Hence lim(x -> infinity) 1/x = 0.

The same with the definition of "0.9999...". We simply keep adding nines. We never actually get to the infinity but we get closer and closer. Yes, that's a limit indeed.

10. It's clear we are going around in circles here, repeating the same arguments. I'll leave it at that. Sorry.

If anything -- I believe you have that advantage -- I do you a dare: Play devil's advocate with your math teacher. Challenge your own proof with him, using the arguments I've used here. That is, if my arguments mean anything to you. If they don't... oh well. Let's call it a day and celebrate Mario's infinite stubbornness.

11. Originally Posted by Mario F.
It's clear we are going around in circles here, repeating the same arguments. I'll leave it at that. Sorry.

If anything -- I believe you have that advantage -- I do you a dare: Play devil's advocate with your math teacher. Challenge your own proof with him, using the arguments I've used here. That is, if my arguments mean anything to you. If they don't... oh well. Let's call it a day and celebrate Mario's infinite stubbornness.
Well, actually, in my proof there is ONE thing you can argue about, but it's not what you pointed out. As I said, "lim[x -> infinity] 1/x = 0". That's an established fact that results from the definition of limits. That's what us humans agreed was the definition of limits. So arguing that is arguing whether we made the right choice to call it that.

That leaves us with the only actually arguable step in my proof, which is to say that:
Code:
`0.9999... = lim[x -> infinity] (9*10^-1 + 9*10^-2 + 9*10^-3 + ... + 9*10^-x)`
Which is actually saying that "0.9999..." is the number it goes to if you add an infinite number of nines.
Really, you should have pointed THAT out in stead of the last bit. Because that's the only debatable part, and it's about what the definition of "..." really is.

However, I've yet to see a definition that supports your side of discussion. But if you don't even want to accept MathWorld as source to show that your hypothetical definition is not the usual one:
Repeating Decimal -- from Wolfram MathWorld
Then indeed there's very little left to argue about. (Note the "1/3 = 0.333...", and note the existence of "approximately equals" symbols and note their common usage on mathworld and their absense here).
By not believing this you claim yourself to be a better mathematician than most on this forum and, worse, a better mathematician than the people at wolfram and wikipedia (as it all would have been fixed by now if it wasn't right). It's quite a claim...

12. Originally Posted by EVOEx
By not believing this you claim yourself to be a better mathematician than most on this forum and, worse, a better mathematician than the people at wolfram and wikipedia
... why don't you go all the way and say I'm claiming I'm the best mathematician in the whole wide world? *sigh* I've been quite patient with you...
Anyways, what I'm claiming instead is that I know perfectly well what all the best mathematicians in the world mean when they write something like lim(x -> oo) ( 1*10^-x ) = 0. And it's not that an infinitesimal equals 0. Rather, that it approaches 0 towards infinity.

The fact that you know this very well, but still insist in using this as a proof, is at the center our our little debate here. Not the hypothetical claims of grandeur that you are fantasizing over me.

The equality sign does not have here the same meaning as in traditional algebra. It means, "approaches" and not "equals". So when you then pick that "0" up and transport it to traditional algebra to produce the final part of your proof, you treat the "0" you reached at with the help of limits as if you had arrived at it through "equals". And that is your mistake. Not mine. That is the distinction I'm making, and you are not.

So, for the very last time:

lim(x -> oo) ( 1*10^-x ) = 0 is true. But it doesn't get us any closer to prove that 0.999... equals 1. Only on the domain of limits. Not as a proof of identity.

13. Originally Posted by Mario F.
The equality sign does not have here the same meaning as in traditional algebra. It means, "approaches" and not "equals".
Here's where i object! The equality sign never means "approaches", there's a different sign for that. The exaggeration of (x->infinity) is that x becomes that much high that everything collapses and lim(x->infinity)(1*10^-x) equals zero exactly. But anyway, infinity doesn't exist, so what's the point?

14. No, it actually means "approaches". See the definition of limits.

What I find somewhat sad is that EVOEx proof as a an actual merit of its own. In a very simple way he manages to show why 0.999... should be treated as 1; "Here it is, why 0.999... should be treated as one. For any infinite requirement of rigor, for any type of calculus, we can treat 0.999... as 1".

But it does not prove, neither it could ever prove that 0.999... (an unaccountable number) IS 1 a (countable number). And I wished that the whole debate around 0.999... where the identity of both number representations is debated, this distinction was made immediately obvious: That we can safely treat them as being the same, but we aren't proving they are. We are simply proving they can be used as the same.

15. Originally Posted by Mario F.
But it does not prove, neither it could ever prove that 0.999... (an unaccountable number) IS 1 a (countable number).
What exactly is an "unaccountable number" and a "countable number"?