# 1=0.999... infinities and box of chocolates..Phliosophy of Math...

Discussion in 'General Philosophy' started by Quantum Quack, Nov 2, 2013.

1. ### UndefinedBannedBanned

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1,695
ATTENTION MODERATORS/ADMIN. Reporting the troll has proved ineffective. Tach continues to dishonestly evade. The only recourse is to bring this to your attention in open forum: Here is the exchange so far...

Tach, you chose to dishonestly evade again. You have been banned previously for that. You have been reported in this instance also.

Please stop evading now, and answer properly and honestly OR admit you don't know what you meant when you suggested 'sharing' a dimensionless 'endpoint'. Thankyou.

Edit:

Tach: If you continue to evade instead of either answering the question OR admitting you don't know, then you are vulnerable to being banned for evading. How many infractions is that against you now?

3. ### TachBannedBanned

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5,265
Not at all, you are still stuck on solving the simple exercise that would teach you how to divide a circle in 3 equal parts. The fact that you cannot do it blocks your ability of learning that it can be done (actually, it has been done for the last 2000 years. Remedial class in 8-th grade geometry is highly recommended for you. Oh, and please stop trolling with your silly demand as to how to "divide a point".

5. ### UndefinedBannedBanned

Messages:
1,695
ATTENTION MODERATORS/ADMIN. Reporting the troll has proved ineffective. Tach continues to dishonestly evade. The only recourse is to bring this to your attention in open forum: Here is the exchange so far...

Tach, you chose to dishonestly evade again. You have been banned previously for that. You have been reported in this instance also.

You are asked to explain your suggestion to rpenner to 'share' a dimensionless 'endpoint'. The exercise in reality context, not in mathematical abstraction, is what the question asked requires you to explain about your 'suggestion'. Nothing else is involved.

Please stop evading now, and answer properly and honestly OR admit you don't know what you meant when you suggested 'sharing' a dimensionless 'endpoint'. Thankyou.

Edit:

Tach: If you continue to evade instead of either answering the question OR admitting you don't know, then you are vulnerable to being banned for evading. How many infractions is that against you now?

7. ### Quantum QuackLife's a tease...Valued Senior Member

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23,328
well said... IMO
It is indeed a common problem when some mix the two paradigms, mathematics and "real world". I have been and in some ways still are guilty of such an approach but am endeavoring to re-organize the way I think so as to accommodate it better.
Most of the angst seen on this board is associated with just this issue IMO.
Where some mathematicians believe they are expressing a real world view and some non-mathematicians believe they "should" express a real world view. aahgg! confusion reigns supreme!!

8. ### someguy1Registered Senior Member

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727
Math is math, physics is physics. It's as simple as that.

It's as if you and I were out to lunch and I suddenly said to you, "You haven't gone ten yards so you can't have a new first down."

You might say to me: "Yes, but we're eating lunch, NOT PLAYING FOOTBALL."

[American football, of course. I realize there's an international audience here and I don't mean to confuse anyone

]

It's the same thing in these .999... = 1 discussions. That's a basic, established fact in mathematics. But in the real world it's meaningless since there are no infinite sequences of real numbers in the physical world.

So if I say that .999... = 1 and someone says, Well, you can't really show me that in the real world, the answer is Duh, I'm not talking about the real world, I'm talking about mathematics.

It's true that math is incredibly useful in the real world, but math itself is an abstract mental construct.

ps I have no idea why I'm bothering to belabor this point, which I'm sure has already been made dozens of times in this thread and millions of times in similar threads all over the Internet.

9. ### Quantum QuackLife's a tease...Valued Senior Member

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23,328
What you are saying is quite correct however..
• What is the point of maths?
• What is it's objectives?
• Where is it applied when it's real world application is essential?

Most would conclude that physics as an umbrella field is the answer to the above.
• Why is the use or misuse of the "axiom of infinity" critical to the issue of 0.999...=1
• Why is the use or misuse of the "axiom of infinity" critical to the way physics looks at the universe?

If the use of the "axiom of infinity" is essential to the solution 0.999... = 1

Then is it not important to understand why it is therefore so essential that infinity be limited to terminating at zero and granted a quasi finite status?

10. ### someguy1Registered Senior Member

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727
You are absolutely right that the Axiom of Infinity is essential to the theory of infinite series. And that's why math differs from physics. Nobody's ever seen an infinite set in the real world and it's highly questionable as to whether one exists. We have no basis in experience for making such an axiom.

Rather, the Axiom of Infinity is based on the intuition that we all have of the counting numbers 1, 2, 3, 4, 5, ... The Axiom of Infinity basically says that there exists a set that models our intuition of the counting numbers ... bearing in mind that "set" is an undefined term! We don't even know what a set is, but we assume there's one that looks like the counting numbers.

Once you grant the Axiom of Infinity then you can model the counting numbers within set theory, and then you can model the integers, rationals, and reals; and then you can develop the theory of infinite series, and then you can prove that .999... = 1.

It's purely an exercise in fictional, abstract math.

But physicists and engineers use infinite series all the time. Digital signal processing, which underlies all computer communications including the Internet, relies on Fourier series. It's perhaps interesting that Cantor was actually studying the discontinuities of trigonometric series when he was led to the discovery of set theory.

How can it be that starting from an axiom that's obviously false about the real world, leads us to a theory that's so useful in the real world? That is a legitimate mystery.

11. ### Quantum QuackLife's a tease...Valued Senior Member

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23,328
Why do say that?

Zeno of Elea had no problems describing the use of infinity in the real world, as did Archimedes . I would argue, quite successfully too, I might add, that infinity does indeed exist as a reality but have never seen arbitrary limits placed up on it in reality, as is the case with 0.999... = 1.
Does infinity define the math or does math define the infinite?

Only have to look at a sphere or a ball and infinity is in view.

The human mind, due to it's inherent fears of chaos, needs to be able to manage the infinite and thus a finite method was developed. IMO

For example:

Zeno of Elea and no doubt other well known ancient per-Aristotle Greeks easily recognized that you could expand something in to an infinite volume and also contract something infinitely as well.

ie. 0.999... = 1
or 9999.... = ? [no decimal place]
Allegation:
Zeno recognized a paradox existed and since then maths has been trying to resolve it. They did that by placing an arbitrary limit on infinity.. and I think I hear an ancient Greek voice from the distance, in Elea, yelling "δεν είναι δίκαιο" or "Not fair"..."You cheated!" when contemporary math has Achilles beating [or calling a draw] the tortoise to his own position...thus claiming the paradox was never a paradox.
and Zeno was not talking about Mathematics he was talking about an observation he and others could witness in reality.
"Probably when the ancient Greeks raced each other they probably joked about how no one could beat any one to their own position in a race - ["I always win and you always loose" type kidding around] and Zeno just formalized it into his famous paradox.

Last edited: Nov 12, 2013
12. ### Quantum QuackLife's a tease...Valued Senior Member

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23,328
seriously,
if 0.999... =1
then what does 9999... = ? [no decimal places]

ironically 999... = 1(0) could be a rather profound answer... [1 being "whole", complete, totality, universal sum etc]

"the infinite oneness of it all...." [chuckle]

13. ### rpennerFully WiredValued Senior Member

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4,833
0.999... = 1 because we are talking about the real numbers and
$0.999... \equiv 0.9 + 0.09 + 0.009 + \dots \equiv \sum_{k=1}^{\infty} \frac{9}{10^k} \equiv \lim_{n\to\infty} \sum_{k=1}^{n} \frac{9}{10^k} = \lim_{n\to\infty} \left( 1 - 10^{-n} \right) = 1 - \lim_{n\to\infty} 10^{-n} = 1 - 0 = 1$
where $\equiv$ denotes equivalence by basic definition and $=$ denotes application of the axioms of the real numbers.

...999 is however not a real number because
$...999 \equiv 9 + 90 + 900 + \dots \equiv \sum_{k=0}^{\infty} 9 \times 10^k \equiv \lim_{n\to\infty} \sum_{k=1}^{n} 9 \times 10^k$
because this limit does not exist in the real number system. It is not bounded in magnitude.

There is a math system where ...999 has meaning, which is called the p-adic numbers. Wikipedia, Wolfram Mathworld
The p-adic numbers are useful in a different way than the reals -- both extend the rationals but only the reals keep the same idea of magnitude as we mean in geometry. In addition, each different prime p gives you a different p-adic number system. In fact in a p-adic expansion of a number the further you move to the left, the less significant your digits become.

For example the difference (p-adic metric) between 100 and 2 in the base 7 p-adic numbers is $\left| 100 - 2 \right|_7 = \left| 202_7 - 2_7 \right|_7 \left| 200_7 \right|_7 = \frac{1}{49}$.

And this leads to a number system where $...013201320132_7 = \frac{-3}{100}$ because $...013201320132_7 \times 202_7 + 3 = ...026402640264_7 + ...640264026400_7 + 3_7 = ...000000000000_7$.

For example $\pm \sqrt{2} = ...266421216213_7, \; \mp \sqrt{2} = ...400245450454_7$ in base 7 p-adic numbers.
You can kind of see how this works in ordinary arithmetic modulo 7:
$\begin{array}{rcrcrcr} 3_7^2 & = & 1\underline{2}_7 & \quad & 4_7^2 & = & 2\underline{2}_7 \\ 13_7^2 & = & 20\underline{2}_7 & \quad & 54_7^2 & = & 430\underline{2}_7 \\ 213_7^2 & = & 4600\underline{2}_7 & \quad & 454_7^2 & = & 32000\underline{2}_7 \\ \vdots & & \vdots & \quad & \vdots & & \vdots \\ 266421216213_7^2 & = & 116446165651000000000000\underline{2}_7 & \quad & 400245450454_7^2 & = & 22030343022200000000000\underline{2}_7 \\ \vdots & & \vdots & \quad & \vdots & & \vdots \end{array}$

But while ...999 makes a kind of sense because displayed digits all have fixed meanings, the 999... doesn't begin to have meaning since the meaning of the first 9 is undefined.

14. ### Quantum QuackLife's a tease...Valued Senior Member

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23,328
So whilst 0.9(9) is limited, 9(9)... isn't. [if I understand you correctly]

thanks rpenner very interesting....

15. ### Quantum QuackLife's a tease...Valued Senior Member

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23,328
In a sense the idea of asymmetrical use of the limit makes sense.
let us presume that no limits are involved for a moment.

If we take a sphere and reduce it infinitely we must assume that the reduction is contained with in the sphere [IN] and is infinite yet it leads to a single unresolvable point.

However if we expand the sphere outwards [out] it must lead to encompass all possible points [ not a single point ] and is like wise unresolvable.
so infinity in this context is asymmetrical...
Physically limited [Inwards towards 1 point] and physically unlimited [Outwards from zero] towards encompassing all possible points.

This is possibly one of the many reasons the limits were developed the way they were...? Perhaps?

IMO very interesting indeed!

16. ### hansdaValued Senior Member

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2,424
You can consider the area of a circle as pi*r^2, where r is the radius. You can divide the circle geometrically into three equal parts and there will be no remainder left.

17. ### hansdaValued Senior Member

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2,424
Going by your logic, you cannot divide a circle into two or four equal parts also, because in these cases also the centre point will remain in the origin; just like the case for dividing the circle into three equal parts.

18. ### Quantum QuackLife's a tease...Valued Senior Member

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23,328
deleted as trivial

19. ### Quantum QuackLife's a tease...Valued Senior Member

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Just to help understand how science perceives infinity the following video was posted to the other thread.
The video is about a fictional infinite hotel called Hilbert's Hotel.

Complaint:
the claim that the hotel manager can suddenly just free up rooms in a fully occupied infinite room hotel is totally unsupported in the video.
I wonder how this could be possible?
If all infinite rooms are occupied then where do the free rooms come from?
Are we not simply making a false claim that the hotel was infinitely occupied to begin with? [in the video]

20. ### PeteIt's not rocket surgeryRegistered Senior Member

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10,167
The video says "Hilbert's answer is just to make each guest shift along one."
Is there any guest that can't move to the next room?
So now room number 1 is free, right?
They're the same rooms as before, but we free up rooms by changing the relationship of the infinite passengers to the infinite rooms.
No. What rooms were vacant?

21. ### Quantum QuackLife's a tease...Valued Senior Member

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how can you just shift one passenger to another room when they are all occupied infinitely?
On what logical grounds can you say infinity some how implies an open ended situation when you have already claimed all infinite rooms are full?

one possible slant:

The manager says to the incoming guests, "You can't be looking for a room, as you already have one...after all are you not a part of the infinite guests to begin with? So what room do you already have?

the rational being explored is:
Infinity may be "never ending" but it is always a complete series [ with in a given scenario ].
In this case we have an HOTEL with infinite rooms and infinite guests. [ a "complete" yet "never ending" scenario ]
or;
Since when has an infinite number of guests not included all possible guests?

22. ### Quantum QuackLife's a tease...Valued Senior Member

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23,328
We can have an infinite number of rooms but a less than infinite number of guest....

"We just move them along one..." implies to me that whilst the rooms are infinite the guests aren't.

(R1,G1) + (R2,G2) + (R3,G3) ....
whereby the rooms and guest are directly associated infinitely.
All possible rooms and guests are accounted for. [in this single HOTEL]

23. ### UndefinedBannedBanned

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1,695
Hence the question in all such exercises offered so far to 'sector/halve' MD's real disc:

"How does one 'share' a central point that is 'dimensionless'; or a middle line that has 'no dimension other than length' ?"

Can you or anyone else answer that for me? I would be very interested to see how you and others would do this 'sharing' process in reality. Thanks.