Thread: Concept of Quantity

Originally Posted by Yarin

Will there be? You stopped calculating at the digit in question, you can't do that! In order to answer what a certain digit is, you must calculate any remaining digits that will effect the outcome. In this case, any digit beyond the '.' will be a 2 *every time*.
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That's a strange question, the very text "999..." means "9s then continue on forever".

I don't think we can calculate something that goes forever now can we?
You cannot use it as something that implies time which has nothing to do with a specific quantity, static, irrelevant of time. Unless you have an actual way of calculating what you want. I can very well accept that, as long as you use a logic that is correct. Saying that 0.(3) + 0.(1) = 0.(4) because *all* digits are 1+3=4, that I can accept. Won't be my preferred expression, but still it make sense for me. Saying though that you do something forever, or there is a shift of infinite digits etc etc should not make sense by definition of infinity.

Problem is, 0.000....0009 = 0 ;-)
I'm guessing you'd like to say that 1 - 0.000....0009 = 0.999..., which, is true, but, so is this:

0.999... + 0.999... = 1.999...
1.999... - 0.999... = 1

why is 0.000...0009 = 0?

What I like to say is that all this lacks some definition, which more or less is an infinitesimal which is non-zero. Then we can define better decimals with infinite digits as a whole and have no paradoxes.

If we use infinitesimal I would claim that automatically by definition you have some error in your measurement, since you claim that you have a quantity that you cannot distinguish from 0. Then we can define two equalities. One that is absolute and one with having an error a quantity that is an infinitesimal.

Then we can think how we can actually distinguish two quantities of having or not an error. The same logic quantum physics use. There is no real sense declaring something as the "exactly X" if you have no way of measuring exactly anything. It is like asking a blind to speak about colors.

So I would personally conclude that equality has an error, always, if you use infinitesimals. So saying that 0.000...000999... is equal to zero or it -> 0 is the same thing.

Originally Posted by C_ntua

why is 0.000...0009 = 0?

because 0.0...9, by it's very definition, has an infinite number of zeros, every single digit to the right of the decimal point is a 0

It's pretty much the same thought behind an infinitesimal being zero with traditional calculus. Simply, you cannot have some entity beyond infinity. So an infinite number of zeroes followed by yet another numeral different than zero makes no sense in this methodology. An infinite quantity by definition excludes the possibility of a "beyond" or a "at the end of".

Of course, this is open for debate since the Compactness Theorem proved that we can have infinitesimals (and by extension such numbers as described here). But still 0.000...9 being equal to zero is pretty much valid in traditional calculus, I suppose.

I'm just pointing out that calculus does not use infintesimals. Newton's original formulation used them, but they were abandoned in favor of limits when calculus was formalized. (Probably for some of the same reasons discussed in this thread)

Indeed.

edit: it's another one of those interesting inconsistencies (limitations?) of Real numbers with the current calculus methodologies. On this case it seems that at infinity numbers break apart. You no longer seem to have a continuum. What comes after 0.888...?

Originally Posted by ಠ_ಠ

because 0.0...9, by it's very definition, has an infinite number of zeros, every single digit to the right of the decimal point is a 0

But 9 is not a 0.

It's pretty much the same thought behind an infinitesimal being zero with traditional calculus. Simply, you cannot have some entity beyond infinity. So an infinite number of zeroes followed by yet another numeral different than zero makes no sense in this methodology. An infinite quantity by definition excludes the possibility of a "beyond" or a "at the end of"

Don't disagree, but then the "adding another 9" or "there is always another 9" on 0.999... and any other number make no sense either.

So in the end the definition of infinite decimal digits is still missing. For me is the same thing as trying to figure out if you can have infinite apples in a basket with infinite volume. Doesn't really make that sense. But here we are saying that you can have infinite number of digits in an infinite possible "space" for those digits.

In a way a better definition is needed for decimals with infinite digits or a more complete one.

Naturally I expect no less than a number like 0.000...9 to exist is some way or another. I think anyone in their sane mind does (including our friends in here). In fact, if we adjust the language a bit, it can be more easily perceived, I suppose: "it's a number that has all 0s for decimals with a 9 at an infinite distance".

There's however very little practical use for it and I expect that to be what Sang-drax meant earlier. The reason infinitesimals were dropped from traditional calculus is probably because we don't find that many uses for them anyways -- or at least didn't at the time -- and their introduction posed serious problems then (it's not up until the last 40 or 50 years that a serious effort was given to reintroduce the infinitesimal in R with valid axioms). Limits were (or are) otherwise good enough for almost anything concerning mathematical analysis. So we stick to them until a new boundary of knowledge is required to again move the field of calculus (limits were not the first, and for sure won't be the last).

Of note however the fact that even with infinitesimals, a number like 0.000...9 still poses serious problems. That is, non-standard analysis doesn't fix all the issues with R either. I cannot find anywhere how a number like that can be treated in R with the current non-standard methods. Maybe I'm missing something, but I suspect there isn't. And if indeed there isn't, guess what's the candidate to "take its place", much the same way 1 took 0.999...'s in traditional calculus?

EDIT: I suppose this is all the holy grail of mathematical analysis. In a world were non-standard methods took hold of calculus we would gain the infinitesimal and all this "nonsense" of 0.999... being equal to 1 would be a thing of the past. Still, we would be no closer to deal with the problem of dealing with a rigorous approach to Reals. 0.000...9 still satisfies the non-standard analysis definition of an infinitesimal. So we would still be "gobbling up" numbers and all sorts of "proofs" would come up by everyone claiming that 0.000...9 = 0.000...1 (or something in those lines). In a distant future we may one day come up with a formulation that indeed once and for all closes the issue of rigorously representing R in all its glory. But that is for now a pipe dream. My only beef?... that we call "proofs" to these stepping stones on that long road, knowing full well we didn't reach the end of it.

Originally Posted by Mario F.

Naturally I expect no less than a number like 0.000...9 to exist is some way or another. I think anyone in their sane mind does (including our friends in here). In fact, if we adjust the language a bit, it can be more easily perceived, I suppose: "it's a number that has all 0s for decimals with a 9 at an infinite distance".

nope, if you were to ask me "how many how many numbers to the right of the decimal point are 0" I would answer "all of them". The very definition of this number forbids the existence of the 9

Originally Posted by ಠ_ಠ

nope, if you were to ask me "how many how many numbers to the right of the decimal point are 0" I would answer "all of them". The very definition of this number forbids the existence of the 9

Yes, but you can also answer "infinite 0s and then one 9".

Might not make sense to you but that is never enough. Unless we use some more proper definitions about decimals with infinite digits, which we don't really.

You have to answer what is the quantity of digits.
If that quantity is defined as infinite then you can add one more digit as you can do to any quantity.
If you define it as the upper bound of a quantity, that you can never reach, then you cannot have infinite digits.
If you define it as at the upper bound of a quantity, which you can reach (the maximum) then you are OK. But if you have one less digit, you no longer have the maximum number of digits, but exactly one less than the maximum.

I believe infinite digits is just used carelessly. I would suggest that a proper definition to be given to it prior any attempt for proof so we can be on the same page.

Originally Posted by whiteflags

I think that you're missing a critical point here. 0.0...9 is not a number.
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We would have to surpass infinite places in order for the place where 9 is to exist

I don't think you read the rest of the post.
You have to define what "infinite places are" in order to use that argument. I gave three possibilities of how you can see infinity. There can be more, but you have to define it first, otherwise we will keep just making circles.

Originally Posted by Clairvoyant1332

I think that in order for a numerical representation to be considered a number, you need to be able to define any given digit
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In the case of 0.0...09, you can't define what digit the 9 occurs at therefore it isn't a number

I clearly defined it as "it is one digit after infinite digits".
Not to sound crazy, my point is that if you cannot add a something to infinity, you cannot subtract either, thus you cannot "shift" the digits of 0.999...
True?

Originally Posted by C_ntua

I clearly defined it as "it is one digit after infinite digits".
Not to sound crazy, my point is that if you cannot add a something to infinity, you cannot subtract either, thus you cannot "shift" the digits of 0.999...
True?

you cannot shift the last digit of .999..., which is why 10*0.999... - 0.999... = 9, and not 9.0...9 (which I would argue is also = 9, if it could exist)

Originally Posted by C_ntua

I clearly defined it as "it is one digit after infinite digits".
Not to sound crazy, my point is that if you cannot add a something to infinity, you cannot subtract either, thus you cannot "shift" the digits of 0.999...
True?

It's different, because you can say every digit is a 9. You cannot, however, say which digit of 0.000...9 is nine. You can say after infinity, but after infinity doesn't exist. For example, you can write .999... as a sum of digits to powers of 10, but you cannot write 0.000...9 as a similar sum. Or can you? A try wouldn't hurt, I guess.