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October 19th, 2016, 07:18 AM   #1
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Decimal Representation and Expansion

Definition: Property at infinity:
P$\displaystyle _{\infty}$ =$\displaystyle \lim_{n\rightarrow \infty}$ P$\displaystyle _{n}$

You can't define a property at infinity without P$\displaystyle _{n}$.

"Property at infinity" is convenient termijnology for the above definition; it does not mean a property evaluated at n=$\displaystyle \infty$, $\displaystyle \infty$ is not a number.

Examples:
1) Decimal representation of numbers in [0,1).
P$\displaystyle _{n}$=.499...9
P$\displaystyle _{\infty}$=.499.......

2) Decimal expansion of numbers in [0,1).
P$\displaystyle _{n}$ = $\displaystyle \sum_{i=1}^{n}\frac{a_{i}}{10^{i}}$
.499...9 never equals .500...0, therefore
.499.... never equals .500....

3) Analytic definition of Limit.
$\displaystyle \lim_{n\rightarrow \infty}$ .499.... = .500....

4) Cantor's diagonal argument

Notes
.4999...9 means 4 followed by 9's to n places, and is representation or expansion depending on context.
.499.... means 4 followed by an endless (infinite) string of 9's, and is representation or expansion depending on context.

Last edited by skipjack; October 19th, 2016 at 08:43 PM.
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October 19th, 2016, 07:30 AM   #2
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Another one? (groans)

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October 19th, 2016, 08:28 AM   #3
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At this stage all I can recommend is getting a book on real analysis and then study the hell out of it, something I intend to do once I get free time (which unfortunately seems to be never for me )
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October 19th, 2016, 10:30 AM   #4
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Quote:
Originally Posted by zylo View Post
You can't define a property at infinity without P$\displaystyle _{n}$.
One certainly can.

Quote:
Originally Posted by zylo View Post
"Property at infinity" is convenient terminology for the above definition
I think it is spectacularly bad terminology designed to promote confusion.

Quote:
Originally Posted by zylo View Post
therefore
.499.... never equals .500....
This is just as false as it was before you started trying to hide things behind your new terminology.

Quote:
Originally Posted by zylo View Post
Analytic definition of Limit.
$\displaystyle \lim_{n\rightarrow \infty}$ .499.... = .500....
That is not a definition of anything. The analytic definition of limit already exists.

Quote:
Originally Posted by zylo View Post
Cantor's diagonal argument
Cantor was certainly not using any of your terminology. He wasn't talking about any sort of limit either. He was talking about infinite sequences and enumerations.

Last edited by skipjack; October 20th, 2016 at 12:02 PM.
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October 19th, 2016, 02:45 PM   #5
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Another example of the beauty, clarity, and simplicity of OP:

What is $\displaystyle \infty^{2}$?

$\displaystyle \infty^{2}$= P$\displaystyle _{\infty}$ where P$\displaystyle _{n}$ = n^{2}

Without P$\displaystyle _{n}$, $\displaystyle \infty^{2}$ would be meaningless.
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October 19th, 2016, 03:02 PM   #6
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It is meaningless. By your definition of $P_\infty$ doesn't exist. Again, there is no mechanism for an increasing $n$ to tell you anything about any infinity other than via the standard definition of convergent sequences. Your sequence doesn't converge.

There are the ordinals that you could look at, where $\omega^2$ has some meaning.
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October 19th, 2016, 08:56 PM   #7
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Quote:
Originally Posted by v8archie View Post
By your definition of $P_\infty$ doesn't exist.
Isn't something omitted from that statement?
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October 20th, 2016, 03:37 AM   #8
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There a missing $\infty^2$ before the word "doesn't".

Last edited by v8archie; October 20th, 2016 at 03:40 AM.
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October 20th, 2016, 04:06 AM   #9
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If we take Zylo's word that $\infty^2$ is meaningful, then his definition tells us that $\infty^2$ is finite (because every $n^2$ is finite). Specifically, there exists some natural number $m$ that is greater than $\infty^2$ (because for every $n^2$ there exists some natural number $m \gt n$).

Moreover, we can say the same about $\infty$, so now "infinity" is finite and smaller than some natural number.
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October 20th, 2016, 08:21 AM   #10
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Reference Post#4 (with numbering added for convenience)

Quote:
Originally Posted by v8archie View Post
1) One certainly can.
2) I think it is spectacularly bad terminology designed to promote confusion.
3) This is just as false as it was before you started trying to hide things behind your new terminology.
4}That is not a definition of anything. The analytic definition of limit already exists.
5) Cantor was certainly not using any of your terminology. He wasn't talking about any sort of limit either. He was talking about infinite sequences and enumerations.
1) Please give a (meaningful) definition of a property at infinity without P$\displaystyle _{n}$.
2) What is your definition of a property at infinity? I was simply clarifying an ambiguous terminology sometimes used incorrectly.
3) Taken out of context.
4) I didn't give definition of Analytic limit. I gave definition of Analytic limit of decimal expansion.
5) You can refer to an infinite sequence. But discussing its properties without P$\displaystyle _{n}$ is meaningless.

Last edited by skipjack; October 20th, 2016 at 11:54 AM.
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