Is 1 = 0.999... ? Really?

If my calculations are right . . .

(0.000\dotso1 = \frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} \times \dots = (\frac{1}{10})^\infty = \frac{1}{10^\infty} = 10^{-\infty} )

(0.\dot9 + 10^{-\infty} = 1)

How might this be applicable to, say, an apple?

It might not be.

Let alone to one of us?

Almost forgot: :wink:

Well, I’ve been doing a bit more than just asserting. I gave some counter-examples as a means of proof by contradiction:

Not exactly proof that (\frac{\infty}{50}) = (\frac{\infty}{60}) = (\frac{\infty}{70}), but definitely proof that (\frac{\infty}{50}) > (\frac{\infty}{60}) > (\frac{\infty}{70}) can’t be true.

I also tried to explain why you have no right to switch between contexts when talking about infinity:

^ But given your lack of response to this, I think it went over your head.

^ That, and I did argue earlier in this thread that infinity is not a number, and therefore arithmetic doesn’t necessarily apply to it. I know you can take the symbol (\infty) and plug into mathematical equations and do algebra with it, which is probably what you’re getting at when you say you can do arithmetic with it, but (\infty) in that case plays the role of a variable, an unknown, not an actual infinite quantity; you can put whatever you want in it’s place and the same rules would apply: (\alpha)/60, (\phi)/60, :smiley:/60. (\infty) in this case doesn’t mean “the highest number possible”, it means “some undetermined quantity”… but it has to be a quantity, otherwise it doesn’t apply. The only sense I agreed with you that you can do arithmetic on (\infty) is if you treat (\infty) as a unit (where the quantity would be 1… like 1cm ← cm is the unit, 1 is the quantity). This is why I said you shouldn’t switched contexts willy-nilly–you’re switching from saying “1 of something symbolised by (\infty)” to “and infinity of something else (like train carts)”. When you do that, the math suddenly doesn’t work. (\infty) in this case is not simply defined as “an infinite number of things”, it’s defined as “some undetermined thing” (like a variable). But if you want to say you can do arithmetic on infinity (as an infinite number of things), I deny your right to do so.

:astonished:

That’s pretty damn basic. I’d even say definitional… as in, I never thought you would have asked for proof given that that’s essentially how we define “shorter”. If Max is shorter than Harry, that means the tip of Max’s head is below that of Harry’s. If neither has a tip at the top of their head (because they’re infinitely tall or whatever), what does it mean to say one shorter than the other?

Cool story, bro.

Look, this dumbing-down approach of “you’re just denying yourself/lacking the language for it” is sadly wasted on me - I know all too well how it is to deal with people whose vocabulary is all too conservative and restricted.
In fact, I love thinking outside the box - unreasonably so. I love the opportunity to do so.
I also love the rigorous rationalisation of exactly how you get outside the box, and the appreciation of every tiny step with all the exact minutae involved to get there, including the illegitimate ones.

I know what you think you’re explaining and exactly how you’re getting there - but with the apparent lack of telepathy at my disposal, I don’t seem to be able to get you to understand that I understand every single thing you’re doing and saying, whilst also rejecting it.

You tell me:
how do I communicate the fact that I don’t need apple analogies to understand what you’re saying, whilst still rejecting it?
Tell me how, because you’re just wasting time, keystrokes and computer bits for both of us.

By the way, if you had “infinite apples in front of you”, you’d be an apple and there would be nothing but apple everywhere.
You can’t do that “twice”. Everywhere is already taken.
If you had an infinite line of apples extending out “in front of you”, you’d not be able to see beyond a small number of them, you’d never be able to get to (\frac{1}2) way along to even divide it into 2, nor would you ever be able to divide all them into any finite number of apples. There would be a finite limit on apples going upwards, downwards, left, right, even backwards in this situation - just not forwards: the same as the natural numbers. Thus the line of apples is distinguished from an entire universe of apples by virtue of its finite constraints up, down, left, right, back, and in all other ways other than forwards. So any “size” of such a line of apples would be dictated by the size of the finite constraints, not the infinity that stretched before you. The infinity has only 1 type (quality), and no tokens (quantities): I am being very very very specific about the kind of ends that are going on here - don’t give me your shit about how I’m being broad.
You can imagine “apples in front of you” in any way you like, and the same principle applies - you cannot get to the “end” of that line and add any more.
But pay attention to the following:
You could even set up a line of apples directly underneath the first, and there would still be infinite apples. The fact that you started the line with 2 apples on top of each other doesn’t give the infinite line “more” or “less” apples. You changed a finite constraint. The infinity is just as unforgivingly endless in its monolithic single extension either way. There was no end before, and there’s still no end - quantity is just as meaningless either way.
You regard the change in the finite number of apples that start the infinity as a change in the infinity.
This is just wrong, I’m sorry.
It’s the finitude that governs the size. Any infinity has only one way of being infinite, and it defies all possible quantification, or operation using quantities.

You can “imagine” what it would be like “hey, let’s say it was the infinity was affected by changes to finite constraints” and go on from there, and you have your arguments.
Fine.
Do that.
It’s not legitimate, but let’s see what happens.

Hey, there you go. That works. Good job!

That’s not true. You have to understand that I am not using the word “infinite” the way that you do. In fact, apart from a handful of people, noone is.

That would be true if what I meant by “infinite number of apples” is “there is nothing in space other than apples”. But that’s not what I meant. The word “infinite” simply means “endless”. It does not mean “endless in every single way one can think of”. You can have an endless number of things within a portion of the universe that is not endless in certain regards. Of course, in order to accomodate all of the infinitely many things, the portion itself must have enough room and that means it must have infinite room, so the portion must be infinite in at least one direction but there is no requirement to be infinite in all directions.

Again, the word “infinite” does not mean “the property of being the only type of thing that exists”.

That’s irrelevant.

I’m afraid you’re confusing conceptual matters with empirical ones. Here in this thread, we’re discussing concepts. We’re not talking about whether it’s possible, nor how is it possible, to determine whether things that can be represented by our concepts exist. So if you’re claiming that there’s no infinite quantities in the universe, or that it’s impossible to establish whether infinite quantities exist, you’re wrong, to be sure, but most importantly, what you’re saying is irrelevant.

You can divide the infinite line of apples into two smaller equally-sized infinite lines of apples by removing every second apple from it and placing it elsewhere.

Of course you can.

The set of natural numbers does not have such limits. That’s because it’s a set.

The infinite line of apples does because it’s a physical object, not merely a set, and the fact that it’s infinite in only one direction (and not all) is completely irrelevant to the subject at hand. This is because we’re talking about THE QUANTITY OF APPLES, not about THE PHYSICAL LINE OF APPLES. The quantity of apples can only be infinite in ONE direction, and if it is infinite in this one direction, then it IS infinite.

If a hungry man walked past the infinite line of apples, grabbed one and ate it, that act alone would tell us that the infinite line of apples has one apple less than before.

You don’t have to go the “end” of the line in order to add more apples. You can add them elsewhere (just like with finite sets.)

You’re trying to apply to infinite quantities what applies only to finite quantities (which makes you guilty of your own accusations.)

I feel sorry for you too.

Yes, because you say so.

An infinite number of inches is equal to a smaller infinite number of 90 inch long segments and an even smaller infinite number of 93.75 inch long segments. There is no contradiction. You’re just doing to the math the wrong way.

The only reason you think that length implies a beginning and an end is because you’re used to working with line segments.

The other problem is that you’re confusing conceptual matters with empirical ones. The same mistake that Silhouette is making.

It does not.

Very convenient.

Why is infinity not a number?

It’s not an unknown. And the word “infinity” does not mean “the highest number possible”.

Meters, centimeters, inches, feet, etc are not undetermined things. (1m = 100cm) is not the same as (1 \times x = 100 \times y) where (x) and (y) are unknowns. Units (\neq) unknowns.

That’s what it means with respect to line segments. And if we assume that the concept of length applies only to line segments, it does not follow that rays don’t have something similar. Either way, you’ve got nothing.

Expanding upon:

The symbol (\infty) represents an infinite quantity; believe it or not, even when I’m doing arithmetic with it. It certainly does not play the role of an unknown because the quantity is known – it’s not unknown. And yes, you can use any symbol you want (e.g. you can use (\omega) if you’re into hyperreal numbers or (\text{infA}) if you prefer James’s notation) but that does not mean we’ve somehow changed what we’re representing. We’re representing the same thing: infinite quantity.

Saying “1 of something symbolized by (\infty)” is exactly the same as saying “an infinite number” (such as an infinite number of train carts) because (\infty) means “infinite number”.

And the math doesn’t stop working. Quite the contrary.

That’s exactly what (\infty) means. It means “an infinite number of things”. The fact that I’m treating it like a unit DOES NOT change that fact.

When I use the symbol (m) to represent one hundred centimeters, does it suddently stop representing one hundred centimeters? Of course not. THAT’S EXACTLY THE PURPOSE OF SYMBOLS.

(m) is not an unknown and neither is (\infty).

But I can see the spirit of Ecmandu has taken over . . .

Nah, unfortunately, that doesn’t work.

10-infinity is not a quantity that’s defined.

It’s undefined in this equation.

Think of it this way… walk up to anyone, even mathematicians and say, “dude! 10-infinity” or “dude! 10 minus the power of infinity!”

They’ll be like, “dude! what the fuck are you talking about?”

You’ll have to explain this to everyone because it makes no sense.

(0.000\dotso1) is equivalent to (\frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} \times \cdots). Represent it using Pi notation and I believe you’d be fine with it.

Similarly, (\infty) is equivalent to (1 + 1 + 1 + \cdots). Represent it using Sigma notation and I believe you’d be fine with it.

So, both (\infty) (or (\text{infA}) or (\omega)) and (0.000\dotso1) are “valid” numerical representations, in the sense that, Silhouette can now understand their meaning.

Doesn’t work because you don’t understand it?

You can replace (10^{-\infty}) with (\frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} \times \cdots). It’s the same quantity expressed differently. And it should have been obvious. (10^{-\infty}) is a compact representation and it’s akin to a hyperreal number (10^{-\omega}).

That’s why I gave you the quantum argument, infinity is actually x (x operator), x (x operator) etc…

It’s undefined.

1+1+1… is not undefined.

Now! Infinite sequences are different than infinity.

But when you substitute those “x’s”. It’s quantum.

To say that infinity is 1+1+1… is absurd.

Alright, so let’s look at it a different way.

When does the 1 in 1/10 ever occur? If it always occurs in 1:1 correspondence then that means that .111… equals .0…1.

Problem is, you have the carry from THE LAST DIGIT!!

Slight problem with that, there is no last digit.

Carrying doesn’t work from the first digit

Oh, by the way QED.

I already pm’d prom and told him he can collect his reward.

At this point the same errors are recurring indefinitely, like this new (10^{-\infty}) notation still treating infinity like a finite quantity.

As Ecmandu correctly pointed out, there is no last (10^{-1}) in what that’s supposed to denote, just the same as there is no last (1) in (0.\dot0{1})

Until something comes up that hasn’t already been disproven (\lim_{n\to\infty}10^n) times over, I’m gonna go ahead and leave the nonsense peddlers to it.

Does reading up on last minute math make you an expert as you go along, Sil/Twit? It seems not!

I await both your responses with baited breath! Show me Math! Teach me! I doubt you can, but I know you’ll try.

Actually, I know Magnus well enough that he’s going to take the last two posts very seriously … you’re embarrassing yourself here.

If you don’t like the notation (merely because you don’t understand it, I must say), you may consider its equivalent which is (\frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} \times \cdots). That’s a “valid” numerical representation, isn’t it?

I’ve already stated it but you conveniently ignored it opting instead to focus on what you don’t understand.

It seems like you’re one of those rare people who can understand Ecmandu :slight_smile:

There is no last (1) in (0.\dot01). The product (\frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} \times \cdots) is infinite. The end is merely in the symbol.