Is 1 = 0.999... ? Really?

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Is it true that 1 = 0.999...? And Exactly Why or Why Not?

Yes, 1 = 0.999...
13
41%
No, 1 ≠ 0.999...
16
50%
Other
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Total votes : 32

Re: Is 1 = 0.999... ? Really?

Postby Ecmandu » Sun Jan 26, 2020 8:09 pm

Magnus Anderson wrote:It's not a procedure.

\(\infty\) represents a number greater than every number of the form \(n, n \in N\).

Similarly, \(\frac{1}{\infty}\) represents a number greater than \(0\) but less than every number of the form \(\frac{1}{n}, n \in N\).

\(0 < \frac{1}{\infty} < \cdots < \frac{1}{3} < \frac{1}{2} < \frac{1}{1} < 2 < 3 < \cdots < \infty\)

Also:
\(0.\dot9 + 0.\dot01 = 1\)

Note that \(0.\dot01\) or \(\frac{1}{10^\infty}\) is actually smaller than \(\frac{1}{\infty}\).


Ok, fine. I don’t buy this, but let’s say that what you’re saying is absolutely true.

Per the argument I leveled. That means every whole number is EXACTLY equal to the lowest possible “number” (your argument, not mine) that’s not zero.

My argument still stands. It’s absurd.
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Re: Is 1 = 0.999... ? Really?

Postby Magnus Anderson » Sun Jan 26, 2020 8:39 pm

How did you arrive at the conclusion that every whole number is exactly equal to the lowest number?

That's not even true for \(\frac{1}{\infty}\) let alone for whole numbers.
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Re: Is 1 = 0.999... ? Really?

Postby Ecmandu » Sun Jan 26, 2020 8:51 pm

Magnus Anderson wrote:How did you arrive at the conclusion that every whole number is exactly equal to the lowest number?

That's not even true for \(\frac{1}{\infty}\) let alone for whole numbers.


Really Magnus ?!

I’ll use your own post for it!

viewtopic.php?p=2758485#p2758485

You’re right. 1=0 is a constradiction.

My argument proves that when numbers converge at infinity (and in saying this, infinity is NOT A NUMBER!)

That 1=0.

Thus, infinities do not converge.

All you did was change infinity to “lowest possible ‘number’ that’s not equal to zero, which by my argument, makes every whole number equal to “the lowest possible number not equal to zero” which is still a contradiction.

That means that 1=2!! Contradiction !
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Re: Is 1 = 0.999... ? Really?

Postby Magnus Anderson » Sun Jan 26, 2020 8:56 pm

What argument, Ecmandu? Where is it?
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Re: Is 1 = 0.999... ? Really?

Postby Ecmandu » Sun Jan 26, 2020 9:00 pm

Magnus Anderson wrote:What argument, Ecmandu? Where is it?


You replied to the post yourself ! Honestly! This is getting absurd!

viewtopic.php?p=2758484#p2758484
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Re: Is 1 = 0.999... ? Really?

Postby Magnus Anderson » Sun Jan 26, 2020 9:57 pm

What exactly does that prove?
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Re: Is 1 = 0.999... ? Really?

Postby Ecmandu » Sun Jan 26, 2020 10:36 pm

Magnus Anderson wrote:What exactly does that prove?


Just what I said it does. Infinite series don’t converge.
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Re: Is 1 = 0.999... ? Really?

Postby gib » Mon Jan 27, 2020 3:40 am

Magnus Anderson wrote:Why would you break the pattern? (Assuming you know what I mean.) Obviously, if you break the pattern, you might not get a number that is less than \(1\).


To answer your question.

This started out with you asking:

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

To which I said: "Try it without integers. It doesn't work."

I was showing you it doesn't work. Of course, what you really meant was the non-integer \(\infty\).

Magnus Anderson wrote:In the case of \(\sum_{i=1}^{n}\), \(i\) starts with \(1\), increases by \(1\) and ends with \(n\). The number of terms is \(n\), so the sum stops (is complete) after \(n\) number of terms.

Yes.

In the case of \(\sum_{i=1}^{\infty}\), \(i\) starts with \(1\), increases by \(1\) and does not end. The fact that \(i\) does not end tells us that the number of terms is \(\infty\). This means the sum stops (is complete) after an infinite number of terms. Which is to say it doesn't stop. (I assume you're one of the people in this thread who have no problem with the concept of "actual or completed infinity".)

I have no problem with saying things like "suppose there is an infinite number of items." But I don't think you can "build up" to infinity.

\(\sum_{i=1}^{\infty}\) does not mean that \(i\) ends with \(\infty\). Agreed. In other words, \(\sum_{i=1}^{\infty} i\) is not equal to something like \(1 + 2 + 3 + \cdots + \infty\). (Note that such a sum would have more than \(\infty\) terms.)

The value of \(i\) is always a natural number. So if we are asking a question such as "What's the value of the sum of terms whose index is \(i, 1 \leq i \leq x\)?" then \(x\) cannot be anything other than a natural number since the range of \(x\), in such a case, must be the range of \(i\) Well, x must be the end of the range of i-- and this means that \(x\) can't be \(\infty\). But if we're asking a question such as "What's the value of the sum after \(x\) number of terms?" then the range of \(x\) goes from \(1\) to the number of terms of the sum. If the number of terms is infinite, then \(x\) can be \(\infty\). And it is precisely this question that we're asking.


I don't get the distinction between these two cases. Sounds like the exact same case just worded differently. In the one case you're saying x can equal \(\infty\), in the other that it can't.

Magnus Anderson wrote:What is the value of the sum \(\sum_{i=1}^{\infty} \frac{9}{10^i}\) after an infinite number of terms?

You're saying it's \(1\), I am saying it's less than \(1\).

My argument (which is basically James's argument) is that the pattern of this sum prohibits its value after \(x\) number of terms to be equal to \(1\) for any \(x > 0\). (You can limit the value of \(x\) to numbers that have no fractional component, if you want.) Since \(\infty\) is greater than \(0\), it applies to \(\infty\) as well.

I don't see anything new in this argument that I haven't already addressed.

By what logic does the value of this sum become \(1\) after an infinite number of terms?


This logic:

\(x = 0.\dot9\)
\(10x = 9.\dot9\)
\(10x = 9 + 0.\dot9\)
\(10x = 9 + x\)
\(9x = 9\)
\(x = 1\)
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Re: Is 1 = 0.999... ? Really?

Postby surreptitious75 » Mon Jan 27, 2020 4:14 am

Magnus Anderson wrote:
In the case of \(\sum_{i=1}^{\infty}\) \(i\) starts with \(1\) increases by \(1\) and does not end. The fact that \(i\) does not end tells us that the number of terms is \(\infty\). This means the sum stops ( is complete ) after an infinite number of terms. ( I assume you re one of the people in this thread who have no problem with the concept of actual or completed infinity )

A sum cannot stop after an infinite number of terms because if it could it would be finite so the concept of completed infinity is entirely fallacious
And so your first sentence and third sentence contradict each other because if \(1\) increases by \(1\) and does not end then logically the sum cannot stop
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Re: Is 1 = 0.999... ? Really?

Postby surreptitious75 » Mon Jan 27, 2020 4:31 am

I0x = 9.999......
I0x = 9 + .999...
I0x = 9 + x
I0x = 9 + I
I0x = I0
x = I
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Re: Is 1 = 0.999... ? Really?

Postby Magnus Anderson » Mon Jan 27, 2020 9:38 am

surreptitious75 wrote:A sum cannot stop after an infinite number of terms because if it could it would be finite so the concept of completed infinity is entirely fallacious


Not really.
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Re: Is 1 = 0.999... ? Really?

Postby Magnus Anderson » Tue Jan 28, 2020 12:31 am

gib wrote:This logic:

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


I addressed this "proof" around 20 pages ago and I can restate what's wrong with it but I think it's pointless since you don't agree that we can do arithmetic with infinite quantities.

Basically, you don't agree that adding a green apple to an infinite line of red apples increases the number of apples in the line. Instead, you prefer to contradict yourself by saying that the number of apples remains the same.
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Re: Is 1 = 0.999... ? Really?

Postby Magnus Anderson » Tue Jan 28, 2020 1:25 am

It has been claimed that it's a contradiction in terms to say that an infinite sequence has a beginning and an end.

An infinite sequence has no end, so you cannot say that it has an end.

Well, you actually can, provided that the first occurrence of the word "end" and the second occurrence of the word "end" mean two different things (i.e. provided that they refer to two different ends.)

Don't be fooled by homonyms.

\(S = (e_1, e_2, e_3, \dotso, e_L)\) is one such sequence. It's an infinite sequence with a beginning and an end.

Note that a sequence with no repetitions is no more than a relation between the set of positions and the set of elements.

\(S = (e_1, e_2, e_3, \dotso, e_L)\) is a relation between the set of positions \(P = \{1, 2, 3, \dotso, \infty\}\) and the set of elements \(E = \{e_1, e_2, e_3, \dotso, e_L\}\).

Note that sets have no order. This means that, when visually representing a set, you can place its elements anywhere you want. This means that \(P = \{1, 2, 3, \dotso, \infty\} = \{\infty, 1, 2, 3, \dotso\}\). The same applies to \(E\). By moving the last element of the set to the beginning of the set, there are no longer any elements after the ellipsis, so there is less to complain about (:

Let's represent the sequence as a set of pairs \((\text{position}, \text{element})\). \(S = (e_1, e_2, e_3, \dotso, e_L) = \{(\infty, e_L), (1, e_1), (2, e_2), (3, e_3), \dotso\}\). And voila! There is nothing beyond the ellipsis anymore, so absolutely nothing to complain about (((:

"The last position in the sequence" refers to the largest number in the set of positions \(P\). Either there is such a number or there is not. In the case of our sequence, there is such a number and it is \(\infty\).

This is not the same as "The number of elements in the set of positions \(P\)". This is an entirely different thing. In the case of our sequence, the number of positions is infinite (i.e. there is no end to the number of elements.) It's also not the same as "The last element in the set of positions \(P\)". No such thing exists, not because the set is infinite, but because sets have no order.
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Re: Is 1 = 0.999... ? Really?

Postby gib » Tue Jan 28, 2020 8:45 am

Magnus Anderson wrote:I addressed this "proof" around 20 pages ago and I can restate what's wrong with it but I think it's pointless since you don't agree that we can do arithmetic with infinite quantities.

Basically, you don't agree that adding a green apple to an infinite line of red apples increases the number of apples in the line. Instead, you prefer to contradict yourself by saying that the number of apples remains the same.


Aw, what a sad way to go out. We were so close to a break through. I don't know why you didn't want to get into hyperreals. I think that's where you had your best shot and where I think you *might* have had a point. But I guess frustration got the better of you. Sayonara chico.
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Re: Is 1 = 0.999... ? Really?

Postby Magnus Anderson » Tue Jan 28, 2020 9:41 am

Talking about hyperreals is both unnecessary and pointless. How can you accept hyperreals if "you don't agree that adding a green apple to an infinite line of red apples increases the number of apples in the line"?
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Re: Is 1 = 0.999... ? Really?

Postby gib » Tue Jan 28, 2020 8:46 pm

Magnus Anderson wrote:Talking about hyperreals is both unnecessary and pointless. How can you accept hyperreals if "you don't agree that adding a green apple to an infinite line of red apples increases the number of apples in the line"?


I don't accept hyperreals. But I'm willing to entertain them conditionally. Under the condition that you can have numbers greater than infinity (or numbers that are infinitely small), then hyperreals become not only a possibility but a necessity. We could then go on to debate the logic of hyperreals, argue about what can and can't be said about them.
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Re: Is 1 = 0.999... ? Really?

Postby Magnus Anderson » Wed Jan 29, 2020 12:34 am

You need to address this argument:

Max wrote:Suppose there is an infinite line of people somewhere in the universe and that YOU are one of the people waiting in it.

Suppose now that I take you by your hand, remove you from the line and place you somewhere outside of it.

The line is the same as before except that you're no longer part of it. Noone joined the line, noone left the line -- except for you.

If you say that the number of people waiting in that line is the same as before, it either means that I didn't really took you out of that line (that you're still there) or that I did but that someone else joined it. Both are contradictions.

You never addressed this argument.
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Re: Is 1 = 0.999... ? Really?

Postby WendyDarling » Wed Jan 29, 2020 12:37 am

Magnus Anderson wrote:Talking about hyperreals is both unnecessary and pointless. How can you accept hyperreals if "you don't agree that adding a green apple to an infinite line of red apples increases the number of apples in the line"?


Infinites can be measured?
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Re: Is 1 = 0.999... ? Really?

Postby Ecmandu » Wed Jan 29, 2020 12:54 am

Magnus Anderson wrote:You need to address this argument:

Max wrote:Suppose there is an infinite line of people somewhere in the universe and that YOU are one of the people waiting in it.

Suppose now that I take you by your hand, remove you from the line and place you somewhere outside of it.

The line is the same as before except that you're no longer part of it. Noone joined the line, noone left the line -- except for you.

If you say that the number of people waiting in that line is the same as before, it either means that I didn't really took you out of that line (that you're still there) or that I did but that someone else joined it. Both are contradictions.

You never addressed this argument.


I’ll address it. You’re assuming a “completed infinity”, that’s minus 1 or plus 1 adds or subtracts from it.

A COMPLETED infinity (contradiction, oxymoron!!)
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Re: Is 1 = 0.999... ? Really?

Postby Magnus Anderson » Wed Jan 29, 2020 1:01 am

Max wrote:1. You have an infinite line of green apples in front of you.
2. You add one red apple at the beginning of the line. The line is otherwise unchanged.
3. If the number of apples is the same as before, it follows that you didn't add the red apple (contradicts premise #2) or that you did add the red apple but that some other apple was removed (also contradicts premise #2.)

Where's the flaw?


Your response was that the argument is wrong because I'm assuming that adding an apple to an infinite set of apples makes the set larger.

gib wrote:The flaw is in #3--your assumption that adding an apple to an infinite set of apples makes the set larger.


Note that you did not say that I'm wrong because adding an apple to an infinite set of apples does not make the set larger (that would be a pretty bold statement.) No, you said that I'm assuming that adding an apple to an infinite set of apples makes the set larger. That's not pointing out a flaw, that's you not being to tell whether my conclusion logically follows or not. "I don't see how it follows" is not pointing out a flaw. It's merely an expression of ignorance. "It does not follow because of this and that", on the other hand, is pointing out a flaw.

And the reason my conclusion follows is because by definition the operation of addition is the operation of increasing the quantity of things. What do you think the word "add" means?

You might want to argue that it is a contradiction in terms to say that the size of an infinite set has been increased. But this isn't true because the word "infinite" does not mean "the largest number". Indeed, if that's what the word meant, then \(\infty + 1 = \infty\) would be just as wrong as \(\infty + 1 > \infty\). But that's not what the word means. And that's precisely what the word must mean in order for there to be a contradiction. To increase some number is to create a larger number, and if you're increasing the largest number, then you're creating a number greater than the largest number -- which is a contradiction because by definition the largest number is a number greater than every other number i.e. there is no other number greater than it.

The word "infinite" is merely a number greater than every integer. And there isn't one such number. There's an infinity of them, the largest number being merely one of them.
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Re: Is 1 = 0.999... ? Really?

Postby Magnus Anderson » Wed Jan 29, 2020 1:04 am

Ecmandu wrote:I’ll address it. You’re assuming a “completed infinity”, that’s minus 1 or plus 1 adds or subtracts from it.

A COMPLETED infinity (contradiction, oxymoron!!)


Completed infinity is not an oxymoron. You're merely confused by homonyms.

And so far, you've done nothing to show that it's an oxymoron.

Try this: define what infinity is and then define what completed infinity is.
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Re: Is 1 = 0.999... ? Really?

Postby Ecmandu » Wed Jan 29, 2020 1:05 am

Magnus Anderson wrote:
Max wrote:1. You have an infinite line of green apples in front of you.
2. You add one red apple at the beginning of the line. The line is otherwise unchanged.
3. If the number of apples is the same as before, it follows that you didn't add the red apple (contradicts premise #2) or that you did add the red apple but that some other apple was removed (also contradicts premise #2.)

Where's the flaw?


Your response was that the argument is wrong because I'm assuming that adding an apple to an infinite set of apples makes the set larger.

gib wrote:The flaw is in #3--your assumption that adding an apple to an infinite set of apples makes the set larger.


Note that you did not say that I'm wrong because adding an apple to an infinite set of apples does not make the set larger (that would be a pretty bold statement.) No, you said that I'm assuming that adding an apple to an infinite set of apples makes the set larger. That's not pointing out a flaw, that's you not being to tell whether my conclusion logically follows or not. "I don't see how it follows" is not pointing out a flaw. It's merely an expression of ignorance. "It does not follow because of this and that", on the other hand, is pointing out a flaw.

And the reason my conclusion follows is because by definition the operation of addition is the operation of increasing the quantity of things. What do you think the word "add" means?

You might want to argue that it is a contradiction in terms to say that the size of an infinite set has been increased. But this isn't true because the word "infinite" does not mean "the largest number". Indeed, if that's what the word meant, then \(\infty + 1 = \infty\) would be just as wrong as \(\infty + 1 > \infty\). But that's not what the word means. And that's precisely what the word must mean in order for there to be a contradiction. To increase some number is to create a larger number, and if you're increasing the largest number, then you're creating a number greater than the largest number -- which is a contradiction because by definition the largest number is a number greater than every other number i.e. there is no other number greater than it.

The word "infinite" is merely a number greater than every integer. And there isn't one such number. There's an infinity of them, the largest number being merely one of them.


Magnus!!! For the nth-millionth time!!! Infinity is not a number!!!
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Re: Is 1 = 0.999... ? Really?

Postby Ecmandu » Wed Jan 29, 2020 1:08 am

Magnus Anderson wrote:
Ecmandu wrote:I’ll address it. You’re assuming a “completed infinity”, that’s minus 1 or plus 1 adds or subtracts from it.

A COMPLETED infinity (contradiction, oxymoron!!)


Completed infinity is not an oxymoron. You're merely confused by homonyms.

And so far, you've done nothing to show that it's an oxymoron.

Try this: define what infinity is and then define what completed infinity is.


Sure, infinity is that which begins but never ends.

Completed (anything) is that which ends. (That which you can add or subtract from). (Finite)
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Re: Is 1 = 0.999... ? Really?

Postby Magnus Anderson » Wed Jan 29, 2020 1:38 am

Ecmandu wrote:Sure, infinity is that which begins but never ends.

Completed (anything) is that which ends. (That which you can add or subtract from). (Finite)


That's a simple-minded understanding of these terms, no doubt responsible for your confusion.

Try tackling this post of mine.

Note that \(S = (e_1, e_2, e_3, \dotso, e_L)\) is an example of so-called completed infinity.

Show me the contradiction.
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Re: Is 1 = 0.999... ? Really?

Postby Magnus Anderson » Wed Jan 29, 2020 1:55 am

Properly speaking, infinity is a number greater than every integer.

When you say that a set has an infinite number of elements, what you're saying is that the number of its elements is greater than every integer.

Note that sets have no beginning and no ends defined. There is no first element, no last element, no beginning and no end of any sort. And yet, they can be said to be infinite. What this tells us is that the concept of infinity has little to do with notions such as beginnings and ends. It really is just a number greater than every integer.

You can even take an infinite set and define where it beings (i.e. which one of its elements is the first element) and where it ends (i.e. which one of its elements is the last element) thereby turning it into some sort of sequence that nonetheless remains infinite (since it still has an infinite number of members.)

I can take a set of natural numbers \(N = \{1, 2, 3, \dotso\}\) and turn it into a sequence with a beginning and an end like so \(N = (2, 3, 4, \dotso, 1) = \{(\infty - 1, 1), (1, 2), (2, 3), (3, 4), \dotso\}\). The resulting sequence, despite having a beginning and end, is still an infinite sequence.
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