Thread: Concept of Quantity

To prove it, ask me what the value at **any** given index/digit is and I can give you an answer.

Lets say index number 5. With your logic
Sum the 8 and 3, we get 11, keep the second 1, add the last 1 to the last sum.
So you have: 4.22221
So how exactly do you get that any digit will be 2? It rather seems that there will always be a digit that will be 1....

Repeated summing is far from the definition of multiplication. How do you do 1/2 * 3/4 with repeated addition? Summing is a special case, not the rule.

1/2 * 3/4 = 1*3 / 2*4 = 3 / (4+4)

Code:

 .999...
+ .999...
 0.888...
 1.111...

Wrong, you add one additional 1. If you exclude that 1 you will get 8 as the last digit. In any case, the carrier will NOT apply to the sum that "created" the carrier, so with this logic you always have an 8.

float the decimal point to the right when multiplying by 10 in base 10 should hold even if you cannot fathom how to actually add the numbers on paper using the given representation.

And what is exactly the proof that we can do so for a decimal number with infinite digits? When you normally float you end up with one less digit, or the digit 0. Which is not the case here. So this is not exactly consistence, I don't find it really obvious in the sense that it doesn't need further proof or analysis.

I would actually say that all the above rather disprove the point rather than prove it.
Furthermore, if I told you "you have two decimal numbers, one has an integral part of 1 and the other of zero, which one is the bigger?" usually you would answer "the one with integral part 1". And this would be true for all decimal numbers with finite digits. Or actually all decimal number with the exception of trailing 9s. So where I am getting at is that the the infinite 9s doesn't follow anyway all the rules. Why do we assume that it just follows the rule of "multiplying by 10 you just float the digits"?

Originally Posted by User Name:

Wrong, you will have an 8 after the last 9. Is there a last 9?

You use the "last 9" to say I am wrong and you ask me if there is a "last 9"?
If you are implying that there is not last digit at all then how do you say that you the sum (1.999...) has infinite 9s? How do you know that all the digits are 9?

Originally Posted by C_ntua

You use the "last 9" to say I am wrong and you ask me if there is a "last 9"?

It's called a "rhetorical question."

Originally Posted by C_ntua

If you are implying that there is not last digit at all then how do you say that you the sum (1.999...) has infinite 9s? How do you know that all the digits are 9

It sums to 1.999... because there is no last digit. If there was a last, it would end in 8, but there isn't, so it doesn't.

EDIT: I didn't answer specifically. I know there the 9s continue in 1.999..., because the carry continues as long as the string of 9s in .999... The string of 9s is infinite, so the carry is also infinite. The idea of there being an 8 at the end is incorrect because of the axiom of real numbers that all digits in a decimal expansion must have a definite position. We can say the digit a_x = 9 for all a in 0.a_0a_1a_2a_n..., but, we have no way to specify the index of the 8 "at the end of infinity".

Originally Posted by whiteflags

This is not a repeated sum. As proof of the idea that multiplication is repeated sums, I would accept that you added up 1/2 a total of 3/4 times. I'm genuinely interested in you demonstrating this idea as you stated it -- that multiplication is repeated addition, for reals other than integers.

But let's do it your way anyway!

If you want to multiply a number like 0.9... You could imagine multiplying each term in a equivalent series:

9/10 * 10 + 9/100 * 10 + 9/1000 * 10 and continuing. I don't know where you get the idea that there would be a last 8, because all you do when you multiply a number by its base is shift places. It's plenty easy to think of doing this for every 9 in 0.9...

Well, I had in mind that that is the way you define multiplication, but the way you use it makes sense (of course) as well.

The point is there that you have one less decimal digit. So if you do that with finite number of digits , lets say N, you get N-1 9s. So you are implying that when N is infinite then N is the same as N-1. In a way having one more 9 is still infinite 9s. Which is true if you say that N->oo but then the number itself doesn't have a specific quantity, it has a quantity always growing to some limit (that is 1). So is it really a number or just some process? I mean, would you define an ever growing quantity as a number?

Putting what I said earlier more clearly maybe
1) A number is a specific quantity, something static. A process changes.
2) Infinity is either the biggest possible number or a process of a quantity going bigger and bigger.

Using infinity in the sense that it is the biggest possible number would result that

Code:

0.99... (N=inf digits)* 10 = 9.999....(X digits) where X = N-1 < inf

if you use infinity as a process going bigger and bigger then 0.99... itself always grows, so I wouldn't consider it a number.

The way we use infinity is normally as something growing bigger without an upper bound, thus from the beginning of this topic I see 0.999... as a process, not really a number.

Of course I use my own definition of number. So my conclusion is that we can see the argument as having two types of quantity
a) Static quantities with a specific valued
b) Growing quantities. They either grow without a limit, which then we call infinity and negative infinity, or they go towards a limit.

Lets get away with the lexical difference and just use both. In this case how would you define the equality between a numberA and a numberB??
The way it is usually defined is that they are equal if there distance is as small as possible. Then how do you define the order? To differentiate you say that one is bigger than the other if there difference is bigger than the smallest possible quantity NOT if its bigger than zero.

Again, not to get subtracted, my thought is that when you use a number as both a) and b) I describe above, that we usually do, then you imply that a quantity very very small is the same as zero. In other words, when you measure a quantity you can just measure its limit, since it might change. So the limit and the quantity itself cannot be distinguished thus we use them the same way. More or less that is the background of all these proofs in my opinion. But this also implies that the equality sign the meaning of two quantities being equal, means something a little bit different, as I phrase above.

Originally Posted by C_ntua

And what is exactly the proof that we can do so for a decimal number with infinite digits? When you normally float you end up with one less digit, or the digit 0. Which is not the case here. So this is not exactly consistence, I don't find it really obvious in the sense that it doesn't need further proof or analysis.

There are inconsistencies in the Real Numbers system that have been studied by others. I have not looked yours up in detail. I'm merely picking your quote for truth.

The Real numbers are dubbed by some as self-inconsistent. I'm unsure as to the real meaning of that wording. I suspect it tries to refer to the fact that these inconsistencies are what keeps the number system from falling apart when dealing with infinity. Basically the number system falls apart when dealing with these quantities. However, the rules and axioms that have been established, while inconsistent, keep the number system consistent with the classical approach to calculus that we know as Limits.

Meanwhile, non classical approaches exist for calculus, and for the study of Real numbers, that support the notion of a precise definition for an infinitesimal that is non zero. And this changes everything. What some here seem to keep implying is that this does not happen in R, when it does! While hyperreals do extend the number space, other methods employed by non-standard analysis don't. That is the case of IST. It's not an extension of R. It's an extension to ZFC, the set of axioms governing pretty much all of modern mathematical thinking.

So what can be said at best is that 0.999... equals 1 in classical calculus employing Limits. But definitely not in R where other approaches demonstrate them to be different. This is not a limitation of the number system. This is merely a limitation of the methods employed and their mathematical foundations. And it's also one of the foundations of my claim that 0.999... equals 1 can't be proved.

In fact the importance of an infinitesimal has lead to a sort of friction between renowned investigators. So I find it particularly curious that by the time we reach page 14 of this debate, we are still hearing some students of maths in here fully convinced they can provide proofs on subjects that are at a very minimum minimum arguable and far from being universally accepted. Or that may be subject to change in the future.

Originally Posted by C_ntua

Furthermore, if I told you "you have two decimal numbers, one has an integral part of 1 and the other of zero, which one is the bigger?" usually you would answer "the one with integral part 1". And this would be true for all decimal numbers with finite digits. Or actually all decimal number with the exception of trailing 9s. So where I am getting at is that the the infinite 9s doesn't follow anyway all the rules. Why do we assume that it just follows the rule of "multiplying by 10 you just float the digits"?

I can only conclude because they are stuck with classical calculus as their one and only tool to handle real numbers. They see the inconsistency, but they trust their axioms. That's the type of indoctrination that happens at our schools (I apologize if I sound rude or pretentious. That's not what I mean).

In fact, on many places, exposing these inconsistencies or daring to reject established axioms (regardless of still studying them, employing them and showing good knowledge of them) is grounds for a negative note. I wouldn't be surprised if some teachers would flunk you for even daring mentioning non-standard analysis in their classroom.

Originally Posted by User Name:

What approach demonstrates them to be different? (That hasn't been shot down yet.)

Haven't I been clear enough already? In non-standard analysis, an infinitesimal is a quantity different than 0. In R, it is defined as a number which absolute value is smaller than any number in the form of 1/n, where n is any natural.

Here there's a clearly a distance between 0.999... and 1.

Limitations are limitations, you can't bypass them because you don't like them or because they are unintuitive. You can redefine them, but by doing so you can't necessarily say statements true of your redefinition are true of the reals.

Oh yes I can! If I extend the existing axioms without changing the number space. What do you think the Internal Set Theory is? I've mentioned this to you several times already. How many more times do I need to say it again? Or are you going to confuse it with hyperreals again?

For how long will you keep believing that non-standard analysis is dependent exclusively on extending R?

Originally Posted by User Name:

I haven't confused IST with anything. I've purposely ignored it, just as you've ignored the 50+ standard analysis proofs I've given you.

I haven't ignored them. I've consistently told you you weren't successfully providing a proof.

Originally Posted by User Name:

IST is, by definition, an nonstandard extension(it adds axioms, what else could it be?), and thereby just as irrelevant as the other tangential counterexample you proposed, the hyperreals.

To my knowledge there's no such thing as a "nonstandard extension" in mathematical theory. If there is I'd like you to tell me what that is. IST is a conservative extension.

If however you can't see the difference between extending a number space and extending the axioms governing a number space, or that you can't understand the fact different methodologies have been applied to studying R, or even that IST has already had practical uses in R in fields like hydrodynamics and harmonics, there's nothing I can do for you. You simply refuse to accept the fact there is indeed other methods to calculus besides limits.

Originally Posted by User Name:

You can correctly say(assuming it's true, I've not taken the time to study 2 irrelevant subjects) "Under IST, 1 != .999..." But the fact it's true under IST doesn't have any bearing on R.

Yes, I believe you can correctly say that you haven't studied these subjects. Because your conclusion that this has no bearing on R, demonstrates that.

Originally Posted by Mario F.

Oh yes I can! If I extend the existing axioms without changing the number space. What do you think the Internal Set Theory is? I've mentioned this to you several times already. How many more times do I need to say it again? Or are you going to confuse it with hyperreals again?

You have been reading too much on Wikipedia. No one is using those constructs for anything serious.

What's going on here? You started out by asking in post #30

"Isn't it self-evident that an irrational number composed entirely 9s is smaller than its closest higher integer?"

Then a lot of proofs were presented that in fact 0.999... = 1. Then you brought up non-standard analysis.

How many do you think were talking about alternative sets of axioms?

Originally Posted by Sang-drax

What's going on here? You started out by asking in post #30

"Isn't it self-evident that an irrational number composed entirely 9s is smaller than its closest higher integer?"

Then a lot of proofs were presented that in fact 0.999... = 1. Then you brought up non-standard analysis.

How many do you think were talking about alternative sets of axioms?

Actually I've brought non-standard analysis early on in this discussion, not just now. I've been mentioning IST for quite some time.

But to answer your question directly, I no longer question the fact 0.999... equals 1. I've said this repeatedly. This may be true of R and of limits. This has been as much a place of debate for me as of learning and I have had no problem in changing my views as I did.

I just presented on that post you quote what I believe is a corollary of my position: 0.999... equals 1 in classical calculus employing Limits. But this can't be proved. Definitely not in R where other approaches demonstrate them to be different. Quite frankly people may thwart this as they want, but the whole issue of 0.999...l equaling 1 is dependent on how a certain method or axiom understand the figure of an infinitesimal. And an effort to introduce infinitesimals into standard analysis has already been made, which does not affect the number space. If you think this isn't worth of mentioning when I see these "proofs" pretending to prove something for R, I don't know what to say. If that were true, non-standard analysis wouldn't be possible, because at least for IST, ZFC axioms all remain valid. So, proving in ZFC Reals that 0.999... equals 1, would mean that IST (a valid and accepted extension) should have not been formulated, when it fact it was developed exactly to provide a means to make exactly these type of distinctions between real numbers because it is clearly understood as a limitation of classical calculus.

As for you dismissal of non-standard analysis and real-life applications of IST, I find them curious. I presented already other links to them (and no, they weren't wikipedia) many pages behind, so forgive me if I don't go search for them or do another research project on my search engine. But more curious is not so much that you negate any real-life uses, but that you actually seem to pretend this has an effect on the validity of these axioms or that they shouldn't be worth of mention. Particularly because, since your peers in this thread are so deeply interested in being specific about what constitutes a proof on R, IST doesn't change this number space in any way and it is considered a valid set of axioms.

Originally Posted by Mario F.

Basically the number system falls apart when dealing with these quantities.

That is exactly what I am trying to say, well honestly my phrasing is bad and I am naturally really bad expressing myself, but still the infinity is the issue. And for me the way the system falls apart is the difference between something being static and something actually defined as growing. You can of course use both but then you have to keep in mind that when you say "0" you mean "0+-e" where "e" can be seen as some quantity as small as possible. The reason you have this error is because the quantity you are measuring can be something that is growing, as infinity does. You can see that it is growing towards a limit, 0, but you cannot prove in any possible way that it isn't 0+e or 0-e.

If you don't like the 0+e then you can write it as 0.000.....00001. If you don't like the fact that when you say 1 you mean "1+-e" you can see 1 as 0.999.... or 1 or 1.000...0001. In this case, yes, 0.999... is equal to 1. It is its lower possible limit.

In classical calculus you see numbers as specific quantities where you can order them and even if the have the slightest distance (even e->0) then they are different numbers. Using this terminology and trying to prove that 0.999... = 1 is contradictory for me. You are simply expanding the meaning of numbers or your system as a whole and that expansion has some immediate logical results.

Note that using the "e" is just a way to see it. You can see it differently if you want by changing the meaning of "=". Thus you can say that x=1 even if x is not 1. You see in classical calculus if I tell you x=1 then you can say that x IS 1. Here it is not true, because x can be 1+-e where e->oo or if you prefer x = 0.999... or x = 1.000...0001.

Or lets use "0.999.... = 1" as something proven, since most agree with this. Then I would go and say that every equation you get with a variable you would actually give 3 possible results. One a quantity that is specific and two with the meaning of limits to that quantity.

All of the above are in the same wavelength, the phrasing is different, the logic maybe different, but in all cases I believe that when you use infinity in any way, the infinitesimal has to be defined and it is used either you want it or not. The same way you talk about something going bigger and bigger you talk about something going smaller and smaller. And generalizing, something that goes towards a limit. Lets name that number? OK, but then all the numbers just got a some special qualities. Or, you just re-define the signs of order (=, <, >).