Is 1 = 0.999... ? Really?

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.

“If two calculations both result in ∞, that doesn’t mean that these outcomes are equal.” (“Als uit twee berekeningen allebei oneindig komt, dat betekent niet dat die uitkomsten gelijk zijn.”)

  • The Elder Milikowski, VLIW architect and physicist

“Thus ∞ is not a number.”

“Base point”? I don’t see that used anywhere. A bit of wikiing brought me to “radix point”, which I’ve never heard before but seems to be what I’m looking for.

I don’t think there’s a non-question-begging way to proceed from there.

(10 * \sum_{i=1}^{\infty}\frac{9}{10^i} = \sum_{i=1}^{\infty}\frac{90}{10^i} = \sum_{i=1}^{\infty}\frac{9}{10^{i-1}}), but I don’t see that leading to anything that’s not question-begging: if (\infty) is finite (what?), then the latter two sums ‘end’ before the sum on the left side.

But what’s the difference between (\infty) and (L)?
(\sum_{i=1}^{\infty}\frac{9}{10^i} \stackrel{?}{=} \sum_{i=1}^L\frac{9}{10^i})

What kind of number is (L)? It can’t be a natural number, an integer, a rational number, or a real number without redefining every basic operation.

And everything just seems to behave strangely around (L):
(\frac{L}{2} + \frac{L}{2} = L )
But multiplying both sides by 2, as we can usually do with an equation, yields
(L + L = 2L)
A contradiction by definition. But this is hard to escape: if we can put (L) into any equation, and if we can use any standard operator on it, ever, you get weird things. Like
( L - 1 + 1 = L + (-1) + 1 = L + 1 - 1 )
That should be permitted, and if it’s not then addition involving (L) doesn’t mean what addition usually means.

But, as someone not able to follow the mathemtics here with any real sophistication, how important, in regard to creating technology and accomplishing engineering feats, is it to get this right?

That’s what I can’t wrap my head around. Is the above in some respect the equivalent of this: “You will never reach point B from point A as you must always get half-way there, and half of the half, and half of that half, and so on.”

Intellectually, that seems to make sense. But if point A is my front door and point B is the mail box, I make it every time. And then back again.

So I’m wondering if, “for all practical purposes”, 1 = .999…has any real meaning in our lives. Such that, for example, if someone assumed they were not equal, and this turned out to be true, it would actually make their life different in some way.

But, here again, I am more than willing to concede this revolves entirely around my ignorance of the math. Analogous, perhaps, in a larger sense, to someone who does understand Einstein’s space-time continuum wondering how his or her life might be different if Einstein turned out to be wrong.

What kind of question is that? What exactly are you asking?

It’s neither of those.

Well, you multiplied (L) by (2). Not sure what you expected.

(L + 1 - 1) is fine.

The “result” isn’t useful in itself. It’s a curiosity and an exercise in problem solving and reasoning.

That particular problem produces a couple of interesting conclusions about mathematics and the universe.

Either infinite series converge - which is useful to know for solving other problems (and it would lay to rest some of the arguments in this thread). Or the universe is not infinitely divisible - at some point trying to go “half the distance” produces a quantum jump to the end point.

In itself, it means nothing in our lives.

Science turns out to be wrong all the time.

Life goes on in spite of that. But there are repercussions in wasted effort, money and lives.

For example, useless medical procedures and treatments that either do nothing or cause harm. Money wasted on medical research that is going in the wrong direction.

He’s asking exactly the same question I was asking, which you refuse to answer. You yourself tell us: L is not a nature, it’s not an integer, it’s not a rational, and it’s not a standard real. Yet you seem to be obstinately illusive when it comes to explanation what kind of number L is.

Can’t speak for Carleas, but I would expect that multiplying L by 2 is impossible. If you can’t even add 1 to it, how can you multiply it by 2. Or is multiplying it by 2 possible despite not being able to add 1. ← Now that would be bizarre indeed!

I think Carleas’s point is that there should be nothing wrong with dividing L by 2. And there should be nothing wrong with adding half L to half L. Combining those allows you to do: (\frac{L}{2} + \frac{L}{2} = L ).

Then there should be nothing wrong with multiplying half L by 2. Thus you should be able to do it to both fractions in the addition, which gives you: L + L.

But then you get a result which, by definition, you shouldn’t get: L + L = 2L.

Think of it this way: I have L cats and L dogs–that’s possible, right?–but then I have 2L pets.

There are numbers that function something like the way you’re describing (L), but you’re avoiding identifying (L) as any of those. It clearly doesn’t function like other numbers (as you’ve acknowledged), but you’re really only providing an accounting of how it does function when it’s function allegedly prevents what we’re trying to do.

You’re claiming that a thing exists which makes an equality false, and there are two responses to that: 1) the thing doesn’t exist, or 2) the thing doesn’t make the equality false. To explore either, we need a rigorous definition of the thing you’re positing.

How can that be? L+1 is undefined! You can’t subtract 1 from an undefined quantity and get a defined quantity.

Okay, if everyone sophisticated in math here can agree, it’s good to know.

And the part “the human condition” plays in that universe?

It’s the extent to which these “solved problems” ramify on human interactions at the juncture of science, philosophy and theology, that most interest me. Given the distinctions that I make between the either/or and the is/ought worlds.

Given the extent that you or any one of us here can possibly know this.

Meaning, of course, there is a path that one can take to be right. And, in regard to the either/or world, it always intrigues me when confronted with seeming antinomies…from the existence of something instead of nothing, to quandaries embedded in the determinism/free will debate, to the evolution of lifeless matter into living matter into self-conscious matter, to all of the fascinating speculations about sim worlds and dream worlds and solipsism and a matrix reality.

And, in a way that both intrigues and baffles me, this.

Thinking more about (L), I have some additional objections.

The proof that there are infinite primes goes like this: let’s ask if (p_n), the (n)th prime, is the largest prime. In commonly accepted math, we can see that it can’t be the largest because ((p_1 * p_2 * p_3 * … * p_n)+1) must be divisible by a prime number not included in the first (n) prime numbers, and thus greater than (p_n).

What’s problematic about this in your system is that the response to this proof is that at some point, (\prod_{i=1}^{n}p_n > L), i.e. the product of the first (n) primes is greater than the greatest number. What’s strange about that is that this is a normal multiplication, it’s not adding something to (L) in a way that’s clearly undefined, and yet we’re saying it’s running into the ceiling. But if (L) isn’t specified, how can we know if some product is too big?

It seems that (L) must be unreachable by operations on the real numbers, since any operation multiplication or addition on the real numbers produces a real number, and L is greater than all real numbers. And consequently, no operation on (L) can produce a real number, because then reversing the operation would produce (L) (so if (L - 1 = r, r\in\mathbb{R}) then (r+1 = L).

I’d say that, in that case, both (0.\dot9) and (9.\dot9) have the same number of decimal places, even if (L) exists: they both have (L) nines.

Consider: at the point of lack of differentiabiliry of the hypothetical -mathematical absolute -proximal , the functional difference is no longer either .9999999999999-99999=1 OR .9999999999999-9999 not = 1, since it is a non function.
There principles of mathematics demand that both =1 and (-=1) to be functional equivalents.

That is the whole point of Leibnitz’ supposition.
If we further enquire by the validity of the principle, (-&+), we fall into the same mathematical trap.
So I would vote not either yes, or no on this .

Note: I think voting has been timely expired. This type of census should temporally be left open.

I suppose it has something to do with the fact that you can subtract (1) from (1) to get (0)?

Not sure what you mean when you say that you expect that “Multiplying (L) by (2) is impossible”.

The point is that it’s a logical contradiction to speak of a number that is twice the size the largest number. It’s not impossible to speak of such a number (indeed, it’s not impossible to contradict yourself) but it’s a logical contradiction to do so.

I can say that Socrates is a man and then later on say that he’s a woman. Nothing impossible about that. But it’s a logical contradiction to do so. Either Socrates is a man or he’s a woman. You can’t have it both ways. Either there is no number larger than (L) or there is a number larger than (L). You can’t have it both ways.

(2L) is a reference to a number larger than the largest number. Nothing impossible about saying such a thing. But it’s a logical contradiction to do so.

It is not my argument that (9 + 0.\dot9 \neq 9.\dot9). Rather, my argument is that (10 \times 0.\dot9 \neq 9.\dot9).