Is 1 = 0.999... ? Really?

I think “convergence theory” is a load of crap.

“Convergence theory” is the theory that infinite sums equal (not approximately equal) to a whole number.!!

Magnus

Yes.

Well, I think it’s best to define “sequence” in order to answer that question. To me, a sequence is a series of objects with a defininte order. These objects can obviously be absract but they must be discrete (i.e. the objects have clearly defined identities, no ambiguity about where one object ends and another begins, no blurring of identities, etc.). For example, sections of a river would be difficult to describe as a sequence as there is no clear border between one section and another, and if you wanted to define each section as a collection of (H_2O) molecules, the molecules would blend in with and over take the molecules of the other sections, thereby making it unclear where one section ends and another begins (to the point where it can be unclear whether one section is always “before” another). On the other hand, carts in a train can much more easily be described as a sequence as they are definitely discrete and always maintain a specific order.

Each member of the sequence can be represented by a unique and unambiguous symbol, and the sequence itself can be represented by a linear arrangement of symbols (ex. (a_1), (a_2), (a_3), …) demonstrating the order each one comes in. This linear arrangement necessitates that there are no gaps in the sequence. So (a_1), (a_2), , (a_3), … is just bad notation, and more or less means the same as (a_1), (a_2), (a_3), … (or if you want to say the gap really is a member of the sequence, it should be represented with a symbol–like 0 or ‘null’–thereby making it at least an abstract object).

The last item in the sequence would just mean the symbol in the sequence taking the last position. So if you wrote out the sequence from left or right, it would be the rightmost item. In the sequence of the first 10 integers after 0:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10

… 10 is the last item in the sequence.

I don’t dismiss the possibility of infinite sequences, so I’m not saying sequences must have an end. If we were to extend the above sequence to include all integers after 0, we would express it as:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …

…and it wouldn’t have an end.

Infinite sequences can be infinite in both directions, as in the case of the sequence of all integers (not just the ones coming after 0), and in that case it would have neither a beginning nor an end.

Any time you talk about an item that comes after the end of an endless sequence, I have a problem with that. I hope you see the problem. If the sequence is endless, it doesn’t have an end. But if the last item comes after the end of the endless sequence, we are essentially saying it comes after something that doesn’t exist. Please tell me you see the problem here.

To talk about 1 coming after the end of the sequence (0.9, 0.99, 0.999, …) isn’t a problem because we’re not quite saying it’s the last item in the sequence, but that it comes after the end of the region on the number line covered by 0.9, 0.99, 0.999… which isn’t infinite. It actually has an end. But when you say 1 is the last item in the sequence, you’re no longer talking about a point on the number line but a position in the sequence. You’re saying the sequence is infinite (and therefore doesn’t have an end), yet 1 shows up at the end. ← There is a clear contradiction there which is why I have a problem with it (and why I brought up the two layered sequence model of parent/child).

Picking up from my definition above, what makes this kind of sequence improper (maybe) is just the problem noted above. It suggests something contradictory. It says the last item of the sequence comes after the end of an endless sub-sequence. The relevance is that this decides whether we can call (0.9, 0.99, 0.999, … , 1) a sequence or not.

Good. Then we have at least one way to think about it that we can agree upon.

That’s what I thought. But now here’s a challenge: what if we wanted to add 2 and 3 to the sequence above:

(0.9, 0.99, 0.999, … , 1, 2, 3)

Then it looks as though we are saying there is a number that comes just before 1.

If you were to ask me, I’d say we need a new notation. Maybe a bar:

(0.9, 0.99, 0.999, …| , 1, 2, 3)

The bar after the … indicates that there is no continuim between the endless sub-sequence represented by … and the sub-sequence immediately following it (no item immediately preceding 1).

This is the first time I’ve seen you use the term “non-standard reals”. Are these the kinds of strange numbers you have in mind? Can there be a largest non-standard real?

I’m not sure if you meant to lump them together with all other number types in this statement, but it sounds like you all of a sudden are saying there can be a largest number greater than all naturals, integers, rationals, and standard reals. So let me get this straight: you’re saying there is no largest natural, no largest integer, no largest rational, and no largest standard real, but there is a largest non-standard real which is greater than all natural, integers, rationals, and standard reals. Is that correct? Are you willing to explain to me what a non-standard real is?

Carleas

I’m not sure what Magnus’s objection to this step was, but I seem to recall something about a 0 appearing at the end of the string of 9s. Magnus is of the camp that believes you can have a last position of an endless sequence, which I would guess implies he thinks there’s a last 9. If you multiply (0.\dot9) by 10, therefore, there ends up being a 0 at the end of the sequence. This means the number you get is not (9 + 0.\dot9), and therefore the step that follows (where we replace (0.\dot9) with X) is unjustified.

I’m putting words in his mouth, obviously, but if this is correct, he might not agree with your definition since it insists that for every 9, there is a 9 that follows, and he would say this is true except for the infinitieth 9.

Is .111… = 0 ?

More to the point [mine], is 1 apple = 0.999…apple?

Yes, for all intents and purposes. But why do you consider it more to the point? Oh. That’s right… cause it is [yours].

ecmandu, silhouette, andy and gib; you’re fired.

Biggs and mowk will take it from here.

My point on this thread has revolved around those like me who admittedly don’t have the capacity to really grasp the math here.

Yet it is things like math and science and actual empirical/material facts which, to me, make-up the bulk of the either/or world. So, it would seem that eventually in a discussion of this sort one or the other assessment and conclusion would finally prevail.

And yet the exchanges themselves revolve almost entirely around words discussing numbers. So, it seems reasonable to me to wonder how the points being made would translate in regard to an actual physical thing like an apple.

And, in emphasizing that this point is mine, I am only trying to reinforce the fact that I am not trying to suggest that it should be yours or anyone else’s.

In part because I don’t have a solid background in math and in part because it seems strange that, since mathematics is not something relevant to the manner in which I construe the meaning of dasein, there are still such fierce disagreements about this issue.

It makes me more uncertain in regard to the either/or world as well. In other words, how much of that might be dependent on a subjective frame of mind? I mean look at interactions in the quantum world.

Hope that helped.

I’d be happy to see that.

You know, the point I’ve raised about “convergent series” really hasn’t been addressed here.

0.999…
Divided by 1/3 equals

0.333… * 3

Which divided by 1/3 again equals

0.111 *9

Eventually this sequence will equal:

0.000…

Which means that 0.999… equals zero at convergence

Now take a number like 1:

It’s 1/2 + 1/2 if divided by 2

1/2 + 1/2 divided by 2

Is 1/4+1/4+1/4+1/4…

This means that at convergence:

0.000… = 1 !!!

How’s the math of convergence theory look to you?

To me it looks like bullshit!

In order for 0.999… to equal 1, EVERY number must equal zero!!!

Make sense to you?

Me neither!

Where does the boundary of an apple begin or end?

It’s as real or imagined as 1 or 0.999…

I expect a hefty severance and at least two good reference.

True. On the subatomic level in particular.

How does one imagine the missing piece of 0.999… of an apple. Anything you imagine, it must be even less than that. Hyperreal enthusiasts would tell you there is an infinitesimal piece of apple missing–if that’s not enough to say the whole apple, for all practical purposes, is there, then going passed that infinitesimal (because infinitesimal can be divided) should put the debate to rest.

I don’t know that Magnus has taken that position, but I think that is something like the usual idea when people object to that line in the proof. But that doesn’t really make sense. Why would we add a zero, as opposed to another 9?

I also think a more rigorous construction of (0.\dot9) would resolve the issue. Suppose we create (0.\dot9) by taking (3 * 0.\dot3), and we construct (0.\dot3) by dividing 3 into 1. Assuming long division is a reliable algorithm, we get an infinite string of threes (similar to my recursive definition of (0.\dot9)).

If we use a construction like that, we have an alternative way of asking whether (10 * 0.\dot9 = 9 + 0.\dot9), because we can multiply 10 elsewhere in the construction (by the associative property) and get to the same repeating long division. I don’t think that construction would actually work, but maybe we could find one (if not, we’re just not talking about the same thing, so there’s no actual disagreement about what equals or doesn’t equal 1).

I see this question of whether one can create a coherent mathematics that has a largest number as a red herring. Even if there is a largest number, (0.\dot9) still equals 1.

Because that’s what happens when you multiply something by 10: 10 x 123 = 1230.

Obviously, this isn’t the rule, it’s just a convenient short-cut for deriving the result. The rule isn’t: shift all the digits to the left and fill the last place with a 0. The rule is: add the multiplicand to itself a number of times equal to the multiplier. So with 10 x 123, you would add 123 to itself 10 times. You start with 123, then get 246, then 369, etc. until we get 1230. Nowhere in that process are we shifting digits to the left. It’s just convenient that, when multiplying by 10, we so happen to get a result that can be derived by a much simpler short-cut: shift the digits to the left and tack on a 0.

This short-cut applies to all numbers with finite decimal expansions, but because it is not the actual rule of multiplication, we cannot necessarily say it carries over to numbers with infinite decimal expansion. We’d have to derive a thorough explanation for why we get this short-cut when multiplying numbers with finite decimal expansions by 10, and then see if that explanation carries over to numbers with infinite decimal expansions.

So instead of asking whether (10 * 0.\dot9 = 9.\dot9), we ask whether (10 * (3 * 0.\dot3) = 10 * (3 * \frac{1}{3}) = 9.\dot9)?

I doubt that would convince the skeptics. Magnus has already ruled out the fact that (\frac{1}{3} = 0.\dot3).

It’s a tangent. This thread is full of tangents. Personally, I’m okay with that as I’m not actually trying to get to the bottom of the main question–does (0.\dot9) really equal 1?–I just enjoy a good debate regardless of where it leads.

(But just to address your point, if there is a largest number, it would fundamentally change the way we understand numbers (it would for me at least), and this could affect the way we understand the question of does (0.\dot9) = 1.)

Does the “=” sign represents the requirement of a balanced equation. Seems it is tipping ever so .0111… to one side. Or does .0111…= 0? Seems like a smallest number question as well.

The short-cut isn’t necessarily wrong, multiplication by 10 in base 10 shifts the decimal point to the right. But (123 = 123.0), so we aren’t adding a zero, the zero was already there. That’s not the case for (0.\dot9) (if there are (L) decimal places, there can’t be any number in the (L+1)th place, because (L+1) is undefined).

I also suspect that we can prove the short-cut as a general theorem, since it’s the case that for any number in base x, multiplying by x shifts the decimal point one place to the right. (Tangent: is there a base-agnostic word for the ‘decimal’ point?)

I was thinking (10 * (3 * 0.\dot3) = (3 * \frac{10}{3}) = 9.\dot9), because (\frac{10}{3}) results in the same repeating decimals in the same way.

I agree this isn’t the construction we need to convince skeptics, but I’m not sure what construction the skeptics are using to define (0.\dot9). Again, without that construction, I’m not sure that we all mean the same thing when we say (0.\dot9).

Absolutely. Numbers aren’t closed on addition? A number where addition is defined for only half the number line (all negative numbers, but no positive numbers)? An end to all decimal expansions, such that there is uncertainty about what happens to any infinite expansion when multiplied by 10? Does that make a whole class of rational numbers that aren’t closed on multiplication? Is multiplication by (L) defined?

My impression is that the existence of (L) is ad hoc, unnecessary, and has a lot of unintended consequences. I don’t believe it can be proven from other standard axioms, so we’d be adding it as an additional axiom, and it isn’t clear why.

Ah, so the red herring is actually sport fishing! Fair enough.

The limit of the sequence is (0.0000… * \infty). My calculus is rusty, but I think that means the sequence doesn’t converge.

So, even here someone or something the equivalent of God is necessary. In other words, an omniscient point of view who/that knows everything that can possibly be known about both apples and math.

In the interim, mere mortals such as ourselves carry on as best we can. I’m just curious as to why those here who obviously do have an enormous amount of understanding with respect to math still can’t pin this down conclusively.

This tells me something about reality [human or otherwise] that doesn’t quite seem to sink in with others. For better or worse.

No. You can find logical flaws without omniscience or knowing everything.

We can know that 6 times 7 does not equal 43 even when we don’t know the true answer. That’s because the product must be an even number. We know something about it.

That’s because a consistent point of view is not being maintained by some posters. Sometimes they say that infinity is a number and sometimes it’s not. Sometimes they say mathematical division works and sometimes it doesn’t. Sometimes they talk about numbers and sometimes they talk about sets.

It’s like trying to argue against a square-circle. They see a circle when it suits them and a square when it suits them. They don’t recognize that you can’t have both at the same time.

Demonstrating that inconsistency is difficult.

You know my pitch here. Whether in regard to living matter evolving into apples, the definitive understanding of all things mathematical or human minds capable of discussing either one, we are all embedded in that same profoundly mysterious and problematic gap between what we think we know about anything and all that there is to know about everything.

Besides, the discussion revolves not around whether 6 X 7 = 43 – who here would get into a fierce debate about that – but 1 either equaling or not equalling 0.999…

Okay, but as long as the exchanges revolve around words talking about numbers pertaining only to more words still, it’s hard for folks like me to grasp the relevance of the debate as it might be applicable to, say, technology and engineering. Again, it’s actual use value and exchange value in human relationships.

A square circle? Isn’t the whole point of the expression “squaring the circle” to suggest something impossible?

The point is that you don’t need to know everything in order to come to some reasonable and true conclusions.

If you want to engineer something, then you can’t be flip flopping around.

Take a stand and see where it goes. That’s the lesson to be taken away from this argument.