Is 1 = 0.999... ? Really?

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.

Phyllo,

Engineers only use pi to 6 decimal places.

That’s not what we’re talking about here.

I’m talking about how to solve problems without getting all fucked up.

Something like 1=0.999… is just one example.

You’re preeching to the choire.

How 'bout the “base”?

Magnus likes to use (\sum_{i=1}^{\infty}\frac{9}{10^i}).

Yeah, I’m not sure why Magnus brought it up, nor did he care to explain what he meant. He did say it was a “special” kind of number. Not a natural, not an integer, not a rational, and not a standard real. He dropped the term “non-standard real” at one point, but he has yet to confirm this is what he meant. We’ve also been discussing hyperreals (infinitely large and infinitely small) and he might have this in mind, but if so, I’ll dispute it as I see no reason to assume there is a largest hyperreal number (hell, they’re already larger than infinity, what would stop them beyond that?).

I don’t think it’s a matter of being able to pin it down, I think it’s a matter of unanimity. As far as I’m concerned, the matter was pinned down the minute someone posted this proof:

X = (0.\dot9)
10X = (9.\dot9)
10X = 9 + (0.\dot9)
10X = 9 + X
9X = 9
X = 1

I didn’t.

(L + 1) refers to a number larger than the largest number which is a contradiction in terms. On the other hand, (L - 1) is perfectly fine (i.e. no contradiction.)

And there’s no need to change the definition of addition.