Is 1 = 0.999... ? Really?

To be perfectly honest with you, Ecmandu, I think that most people on this board think that you’re mentally challenged – and rightfully so.

This is why they generally don’t bother responding to your posts.

I’m defending your argument, but whatever.

If you can’t understand my last post, that is not an indictment of my articulation, but rather your mental abilities.

You’re not quite getting what (\sum_{i=1}^{n} \frac{9}{10^i} < 1) for n > 0 means. In the case of sums, valid values for n are typically integers greater or equal to whatever initial value is set for i. In other words, n must be an integer, and integers are only found on the number line.

You got this part right:

(\infty) > 0

But you’re making an implicit assumption:

(\infty) has a specific place on the number line.

^ This you got wrong.

(\infty) is not among the values you have at your disposal to choose from for n. IOW, the condition that n > 0 does not just mean “anything greater than 0”, it also means values that fit with sums of the form (\sum_{i=x}^{n} f(i)) (that is, integers, those symbols above the tick marks on the number line).

(\infty) > 0 remains true because > and < don’t signify specific points on the number line. They signify directions on the number line. “Greater than” means: in the positive direction. “Less than” means: in the negative direction. (\infty) need not have a place on the number line for (\infty) > 0 to be true. It just has to be to the right of 0.

Now we’re cooking! You’re finally asking the right questions.

I believe my explanation above should suffice. Valid values for n are any integer found on the number line greater that 0 (at least in the case where i = 1). Why? Because that’s what the condition n > 0 is asking for. (\infty) is not an integer found on the number line. Why? Because it is the endlessness of the number line itself, at least in the positive direction. In other words, (\sum_{i=1}^{\infty} \frac{9}{10^i}) is not a special case of (\sum_{i=1}^{n} \frac{9}{10^i} < 1) for any n > 0. It’s an entirely different statement. It says: don’t select a value for n. Just let the sum continue forever.

^ This may not satisfy your request for further clarification on what I need from you, but now your move is to probe deeper into my response. Is there something in the above you need even further clarification on? (wanna get into hyperreals? :smiley:)

Again, I think the above addresses this. The important point is that (\infty) fails my definition of quantity. A quantity is what points on the number line represent (at least in terms of cardinality). Since (\infty) has no point on the number line, it doesn’t represent a quantity.

You say tomato, I say tomato. (<-- That doesn’t quite come through in text, does it? :smiley: ).

It doesn’t matter what “we all know”. I’m not a part of “we”. You’re dealing with me in particular. If you want to convince me of your point, the onus falls on you to prove your point to me.

You mean like, oh I don’t know, asking you to prove that what applies to finite sets also applies to infinite sets? How 'bout proving that (\infty) is a valid value for n? Did I forget to ask for that?

It takes two to tango. I very well may not have been detailed enough in explaining to you what I need, but that’s your cue to ask for further clarification. These things may go through a few rounds of back and forth before both parties are clear on what the other needs in order to be convinced. Nothing wrong with that. It’s how it goes sometimes.

You know, this is a desperate cry. You typically see it when your opponent has nothing left. It really means: I’ve got nothing but my going against the grain. I know that I’m taking an unpopular position. I know that all the professionals in the field, all the experts, all the really smart people disagree with me. I know the majority of people who follow this topic are not on my side. I know all my arguments are falling on deaf ears, that my attempts at convincing others seems to be an exercise in futility… but you know what, at least I’m not a sheep, at least I’m not mindlessly conforming to the masses. I’m thinking for myself, I’m exercising my independence of thought. And that’s something to be proud of. I’ve exhausted everything else in the debate, might as well start using this.

For good measure, I will object and say that we can equally well say cow is larger than zero as that we can say infinity is larger than zero; neither are quantities. And yet both clearly more than nothing.

Infinity is not a quantity, because quantities are definite.

You’d have to explain why you’re limiting yourself to integers.

Not necessarily but it can be the case.

Basically, what you’re saying is that (\infty) does not refer to a specific number in the same way that the statement “An integer greater than (10)” does not refer to a specific integer since the quantity that it represents can be (11) or (12) or (100). However, you have yet to explain why it’s invalid for (n) (the upper bound of summation) to be a non-specific number.

My claim is that (\sum_{i=1}^{n} \frac{9}{10^i} < 1) for any (n) that is greater than (0). It does not have to be an integer. The only condition is that it’s greater than (0). Do you think there is (n) greater than (0) such that the sum is not less than (1)? You obviously do (otherwise, you’d not say that (0.\dot9 = 1).)

(\infty) is not an integer. It’s a number greater than every integer.

And (\sum_{i=1}^{\infty} \frac{9}{10^i}) is not a completely different statement. The only difference between (\infty) and integers is 1) (\infty) is a bigger number and 2) it’s non-specific. The fact that it’s non-specific shouldn’t be a problem at all. And if you absolutely hate non-specific numbers, you can switch to some specific infinity.

It takes two to tango, yes.

Magnus wrote:

“Infinity is not an integer, it’s a number greater than every integer”

How many times do you have to be told ?

Infinity is not a number!

It is simple.

1/9= 0.111… Equation 1 (ordinary division by 9)

(0.111… )*9=0.999… Equation 2 (ordinary multiplication by 9)

(1/9)*9=1 Equation 3 (ordinary division and multiplication)

Substitute for (1/9) in Equation 3 with the equivalent (1/9) from Equation 1 :

(0.111…)*9=1 Equation 4

The left side of Equation 2 is exactly the same as the left side of Equation 4. Therefore whatever is on the right side of equation 2 is exactly equal to whatever is on the right side of Equation 4.

Therefore :

1=0.999…

QED

However you want to call it, infinity is something that is greater than every integer.

How do you call things that are greater than other things?
Cows?

The size of cows is greater than zero.

I’m writing this post for Carleas to link to (not because I’m reporting anyone here - that’s absurd!) but because I know he loves numbers and has participated . I love numbers too! Any of you can take a stab at it:

I have two number theory positions:

1.) series don’t converge
2.) there are no orders of infinity

1.) series don’t converge! If a series converges, this means that every counting number is an infinite sequence (proof through contradiction).

2.) there are no orders of infinity!

I use this order symbolically (not using numbers) to disprove this!

A.) rational number
B.) uncounted infinity
C.) different rational number
D.) different uncounted infinity

Etc…

This places all the “lower” infinities in 1:1 correspondence with the “highest order of infinity”, thus there are no orders of infinity.

Abstract concepts in mathematics not using numbers are fair game!

Thus my proof sits.

Magnus, for you personally, if you don’t understand that, I’m not crazy, and it’s not my fault.

I know Carleas doesn’t have the time for this thread, but I know he’s a math pervert (we all should know that) and I don’t think he’s seen these arguments before!

You moved out of the decimal system to make your argument, you switched back to whole numbers, which is the same as never using decimal expression in the first place.

No I didn’t.

Do I need to rewrite it for you?
It is simple.

1.00/9.00= 0.111… Equation 1 (ordinary division by 9)

(0.111… )*9.00=0.999… Equation 2 (ordinary multiplication by 9)

(1.00/9.00)*9.00=1.00 Equation 3 (ordinary division and multiplication)

Substitute for (1.00/9.00) in Equation 3 with the equivalent (1.00/9.00) from Equation 1 :

(0.111…)*9.00=1.00 Equation 4

The left side of Equation 2 is exactly the same as the left side of Equation 4. Therefore whatever is on the right side of equation 2 is exactly equal to whatever is on the right side of Equation 4.

Therefore :

1.00=0.999…

QED

Happy now?

Nope. You did. Step three like I said before is where you switched from decimals to whole numbers exclusively.

It’s not about making me happy, it’s about what’s true.

It’s only 5 steps with very basic math operations and you can’t follow the reasoning. #-o

Nice try. You may as well as typed the proof without using decimals at all (which is how standard Mathematicians argue it). At some point, you dropped the decimals and went to only whole numbers.

Yes. That proof always works with only whole numbers.

Try it without integers. It doesn’t work.

Let’s take an example. Let’s say n = 2.5.

First, we’d have to define how one even does sums with non-integers, how to increment the value of i. Obviously you start with the initial value (1 in the cases we’ve been dealing with), but what’s the next number? Do you go:

1, 2, 2.5?

Do you go:

1.0, 1.5, 2.0, 2.5?

Do you go:

1.0, 1.1, 1.2 … 2.4, 2.5?

Do you stop at 2 (because you can’t jump by a whole integer amount from 2 to 2.5)? Do you stop at 3 (because maybe n = 2.5 means don’t stop as long as you’re under 2.5)?

But for argument’s sake, let’s say you go: 1, 2, 2.5. Then the sum is:

(\frac{9}{10^{1}} + \frac{9}{10^2} + \frac{9}{10^{2.5}} = \frac{9}{10} + \frac{9}{100} + \frac{9}{316.227766} = 0.9 + 0.09 + 0.02846 = 1.01846)

I’m not just saying (\infty) does not refer to a specific number, I’m saying it doesn’t refer to any number. It can’t. You’ve heard me say it before and I’ll say it again. Infinity means endless. It refers to a property of sets. It’s not that we’re not being specific about which number it refers to, it’s that we (or I) are not referring to something that could be a number.

Like I said, (\sum_{i=1}^{\infty} \frac{9}{10^i}) is not a special case of (\sum_{i=1}^{n} \frac{9}{10^i}). It’s a case that lies outside all specific cases of (\sum_{i=1}^{n} \frac{9}{10^i}) (because (\infty) is not a valid value for n). (\sum_{i=1}^{\infty} \frac{9}{10^i}) is saying: don’t pick a value for n. Let the sum run forever.

So I agree that there is no n such that (\sum_{i=1}^{n} \frac{9}{10^i} \geq 1), but (\infty) is not a valid option for n. Plugging (\infty) into the upper bound of the sum isn’t picking a value for n. It’s refusing to pick a value for n. Only in that case does the sum equal 1.

You’re so close to biting, Magnus. Wanna talk about hyperreals?

Show how it fails.

Show the cases where it doesn’t work.

Apologies in advance for the drive-by, I haven’t followed this thread but Ecmandu asked for my response.

This seems question begging of the subject of this thread: if .999… = 1, then every counting number is an infinite sequence (.9 + .09 + .009 + .0009 + …)

I don’t follow this proof. Are you rejecting the idea that each rational number represents a different infinity? Can you cast the proof in terms of Hilbert’s Hotel?

It’s totally on topic for the thread (thanks for stopping by btw!)

Think about what this means. It means that 1 (a definitely finite number)

IS!!! ( bolding important )

An infinite sequence (which is a disproof through contradiction of definitions)

As for the second part. I’m sorry it didn’t make sense.

All I’m trying to say is that ALL infinities can be paired in 1:1 correspondence. The argument that they cannot, is put forth that you cannot enumerate all infinities in one single list.

So I CHEATED =)

I just list them (symbolically (not numerically)) as “uncounted numbers”

Math uses abstract symbols constantly that people complain about!! But, from my perspective, my CHEAT is fair game!

As far as Hilbert hotel is concerned, this is already a 1: 1 correspondence, you’re not doing “dimensional flooding” in that thought experiment. “ Dimensional flooding” is when every number in a single list has an infinite amount of numbers that cannot be put in each slot of the list.

Carleas,

I also wanted to add to my last post.

Lots of people think number theory is dry, boring and useless.

Little do they know, I’m actually making a theological argument.

If there are no orders of infinity, then that means that no being is greater than another being (including hypothetical “god”)

I think everyone in this thread can all agree that they are “math perverts”. Math always has philosophic implications, in fact, it is the purest language we have for discussions.

That’s what I wanted to add. Again, thanks for joining the thread.