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

[Billy T]

Post 106 re-states the puzzle about repeat lengths that Pete guessed a not fully correct answer to but something seems to be going on related to his answer, at least for base 10. See post 117.

I have not read much of "the divide pie into three equal pieces" discussion, but with just the compass that made the circle, swinging about the center, it is trivial to mark the circumference into 6 equal division - pair wise, with the center, they divide the pie into three equal parts.

 

[Motor Daddy]

Post 106 re-states the puzzle about repeat lengths that Pete guessed a not fully correct answer to but something seems to be going on related to his answer, at least for base 10. See post 117.

I have not read much of "the divide pie into three equal pieces" discussion, but with just the compass that made the circle, swinging about the center, it is trivial to mark the circumference into 6 equal division - pair wise, with the center, they divide the pie into three equal parts.

When you sum up the percentages of all three parts does it total 100% and nothing less???? If the circle has an area of 1 m^2, what is the area of each part?

 

[Undefined]

Hi Quantum Quack.

@undefined

I think what you are getting at is the paradoxical nature of applying infinity on a finite object. If I read you correctly you are I feel quite correct in stating as you have. If not then I apologize.

Example by way of problem:
take a house brick and divide it into an infinite number of slices so that all slices are equal in thickness.

Q: How thick are the slices?

Then :

Q:How many slices does it require to recompile the brick?

Now if the slice thickness is deemed to exist then the slices have a finite thickness.
How many finite thick slices are needed to recompile the brick?

Choices: 1] a finite number of slices or 2] an infinite number of slices?

If the slices are deemed to be "finite" infinitesimals, or given a fixed value, then the recompiling the brick is a finite function and not the same infinite function that was used to de-compile the brick.

Compare with using 2 dimensional slices instead.

What do you discover from the thought experiments?

The bottom line question is:
If you divide a house brick into an infinite number of slices :
Do the resultant slices exist in 3 dimensional space or not. If so, in what way and with what dimension of thickness are they?
To me this highlights the paradox associated with real world use of "Infinity"
1/infinity = 0 or does it equal 1/infinity

Its exactly the same question being asked of

1- 0.999... =
zero
or
1/infinity
when applied to the real 3 dimensional world.

if the answer is 0 then the brick vanishes.. non-existent. [and can not be recompiled as (0 x infinity) = 0]
if the answer is 1/infinity
what happens?
Does the brick [slices compiled] still exist?

see the paradox?
When:
1/infinity = 0
"When infinitely reducing a sphere, the sphere ceases to exist as a sphere" and it is a one way street only, for once the sphere is reduced infinitely it can not be recompiled from nihilo"

When:
1/infinity = 1/infinity
"When infinitely reducing a sphere, the sphere maintains form as a sphere" and it can be recompiled from 1/infinity" ~ yet this grants 1/infinity a finite value

Yes, it has to do with all these kinds of 'non-sequitur' treatments which have insidiously (and seemingly imperceptibly) crept into the maths formalisms/beliefs (as if they are not fundamentally non-sequiturs when looked at closely in discussions such as these). Excellent thread, QQ!

If you will read this part of my last post to Fednis48, you will see that in that bolded part I went straight to the (infinitesimal?) NON-dimensional 'point' (pun intended ). They all boil down to that essential (infinitesimal?) 'nothingness' which we abstractly 'treat' as a 'point' of 'no spatial dimensional extent'. Here is the relevant bits of that post...

Hi Fednis48.

...

According to Standard Model assumptions/postulates/theory, the Big Bang Scenario is essentially an UNIDENTIFIED IN REALITY some fundamental (infinitesimal?) 'nothing' state/thing which due to 'quantum fluctuation' of/in that 'nothing' became 'something'?

In my reality based perspective, it is an Energy-space arena 'something' that existed all along, but we will not go into that now.

Insofar as Standard Model goes, the space-time and all its mathematically modeled 'properties' ultimately depend on certain 'infinitesimal' things associated with 'pointlike' things which when treated in some 'mathematical space treatments' effectively represent the 'forces' and 'energy' and other dynamical entities which are used for the abstractions of reality into the mathematical formalism dependent 'models'.

So I now ask:

If 'pointlike' (ie, having no real or abstract dimensional extent) things are NOT ultimately THE closest thing (logically and effectively in reality context) TO 'infinitesimals of reality effectiveness' (regardless of mathematical formalism abstractions/modeling), then what are these pointlike 'nothings' that underlie all theorizing in 'from-nothing-to-something' models/treatments like that used for the conventional Big Bang Standard Model? Fair enough question, mate?



Again, the 'reality limitations' of mathematics AS IT STANDS NOW is not disputed, so it is a 'given' whenever such philosophical discussions OF it (and of other 'limited' systems of thought, especially including the physics system) are brought into 'review' like this via NON-prejudicial and 'uncommitted stance' discussions which are not yet FORMAL PRESENTATIONS as 'completed review'. So a certain leeway and patience needs to be allowed for developing all the various thoughts and observations which such philosophical discussions will inevitably elicit for further discussion in the various contexts they arise in. Yes?
...

Cheers, QQ. Excellent work, mate!

 

[rpenner]

I asked for proof that 1/p where p is a prime has repeat length of p-1back in post 108

If q is prime and GCF(b,q) = 1 then $$b^{q-1} \; \textrm{mod} \; q \; = \; 1$$ (Fermat's little theorem) so q divides evenly $$b^{q-1} - 1$$ and so $$\frac{1}{q}$$ repeats in some factor of q-1 digits in base b.

But usually we describe the period as the smallest repeating segment.

http://en.wikipedia.org/wiki/Repeating_decimal#Fractions_with_prime_denominators

 

[Billy T]

If q is prime and GCF(b,q) = 1 then $$b^{q-1} \; \textrm{mod} \; q \; = \; 1$$ (Fermat's little theorem) so q divides evenly $$b^{q-1} - 1$$ and so $$\frac{1}{q}$$ repeats in some factor of q-1 digits in base b.

But usually we describe the period as the smallest repeating segment.

http://en.wikipedia.org/wiki/Repeating_decimal#Fractions_with_prime_denominators

Thanks. As I had noted in post 117 that 1/11 & 1/13 do repeat in block of 10 & 12 length digit stings, but their shortest repeat length is 2 & 6 respectively, so what Pete needs to show* for the base 10 part of his conger, in post 109, is that the 996 long repeat string is NOT a block of two 498 repeat strings (or three blocks of 332 length strings , or four of 249, or 6 of 166, or even 12 of the 83 length repeat string you found it to be in several different bases.)

Thanks also for the link - a fascinating read, which happens to include the method of my short proof that 1 = 0.99999... too. I "discovered" it easily as I have know for more than five decades how to do basically that same trick to evaluate the usual geomantic series (but not my more general, but still convergent, version with all the coefficients bounded, that includes negative and non-integer coefficients).

The link also confirms my guess in post 117 that yes: "When the prime, p, is a factor of the base, that may be a special case?"

* Until Pete does that, I'm not sure his answer (997) to the original puzzle, is correct.

PS - I cannot restrain myself from noting once again how impressed I am with your math skills and knowledge. If You don't already have a Ph.D. in math, some clever university should grant you one. - Just on the gamble that they can later claim that one of their alumni discovered the important theorem known by his name.

 

[Pete]

Thanks, rpenner, my guess was based on poorly understood cribbing from Wikipedia. I still don't understand multiplicative order.
But, by cheating with Wolfram Alpha, I now think that the answer to (edit) Billy's original questions are:

The longest repeating period in the decimal representation of a fraction with numerator and denominator being integers < 1000 is for:
n/983, with period of 982.

The longest repeating period in any base for a fraction with numerator and denominator being integers < 1000 (that's 1000 base 10) is for:
n/997, with period of 996 in bases 7, 11, 17, 21, and many more. (In base 10, the period is 166)

 

[Billy T]

... I now think that the answer to QQ's original questions are: ...

I don't think QQ has been part of this interesting (and educational for all but rpenner) discussion. Here is where the puzzle originated (at end of a 3 November 2013 post): http://www.sciforums.com/showthread.php?136842-1-is-0-9999999999999&p=3127851&viewfull=1#post3127851

 

[hansda]

It's impossible to split an object into 3 equal pieces! At least one of the three pieces has to be different than the other two, they are NOT equal!

You can divide a circle into three equal parts. Draw three radius from the center of a circle, which are 120 degrees apart. This will divide the circle into three equal parts.

 

[someguy1]

You can divide a circle into three equal parts. Draw three radius from the center of a circle, which are 120 degrees apart. This will divide the circle into three equal parts.

That's harder than it looks. How do you account for each of the points on the boundary?

 

[rpenner]

PS - I cannot restrain myself from noting once again how impressed I am with your math skills and knowledge. If You don't already have a Ph.D. in math, some clever university should grant you one. - Just on the gamble that they can later claim that one of their alumni discovered the important theorem known by his name.

While I frequently overestimate the skills other people should have, I really think most any Mathematics Baccalaureate should be able to do these simple things.

Thanks, rpenner, my guess was based on poorly understood cribbing from Wikipedia. I still don't understand multiplicative order.

If you look at my post with [post=3126937]division algorithm 1.[/post] One of the properties I proved was that $$s_k$$ was an integer between 0 and q-1. You can think of $$s_k$$ as the "state" of the division algorithm. It is the only piece of information of the previous step which is used in the computation of the present step. Since the state is finite and discrete, it must evolve into a cycle of states that repeat. (Repeated 0's or 1's for example.)
We have the map where the next value of $$s_k$$ is just the remainder after we multiply by b and divide by q. $$s_{k+1} = b s_{k} \, \textrm{mod} \, q$$.

Now if you think of powers of integers modulo q, its clear that the sequence $$n, \, n^2, \, n^3, \, n^4, \dots$$ (mod q) must also evolve into a cycle of states that repeat. $$n^{k+1} \equiv n \times n^{k} \; ( \textrm{mod} \, q ) $$. So these problems are the same.

And the multiplicative order of n modulo q, $$\textrm{ord}_q ( n )$$, is (when n and q are co-prime) exactly the smallest exponent k such that $$n^k \equiv 1 \; ( \textrm{mod} \, q ) $$. (Because all elements of a finite algebraic group generate a finite cyclic group, the concept of the order of an element is part of group theory.)

 

[Tach]

You can divide a circle into three equal parts. Draw three radius from the center of a circle, which are 120 degrees apart. This will divide the circle into three equal parts.

This is one way of doing it, not the most elegant. How would you do it with a ruler and a compass only?

 

[rpenner]

That's harder than it looks. How do you account for each of the points on the boundary?

Those points have measure zero, so they really don't count, but a simple procedure of assigning each point to it's clockwise-adjacent arc will split the circle into N congruent parts.

 

[Tach]

Those points have measure zero, so they really don't count, but a simple procedure of assigning each point to it's clockwise-adjacent arc will split the circle into N congruent parts.

"Sharing" the endpoints by congruent arcs accomplishes the same result.

 

[someguy1]

Those points have measure zero, so they really don't count, but a simple procedure of assigning each point to it's clockwise-adjacent arc will split the circle into N congruent parts.

That's exactly the -- no pun intended -- point.

In math, points have measure zero so they can be ignored.

In the real world, you have to assign each molecule/atom/quark to one part or the other. In the physical world you can't even cut an object exactly in half, let alone thirds.

 

[Billy T]

can be a

... In the physical world you can't even cut an object exactly in half, let alone thirds.

That is true, but mathematics is about concepts, a closed tautology, so the "physical world's" limitations are of no concern. That is the domain of physics, not math.

Certainly relationships developed in math theory can be applied in the real world, and achieved in the real world accurately enough to make them very useful. For example if you want to fence off an area, say to keep cattle out while newly planted pasture gets established, the shape of the fence for less cost per unit area protected should approximate a circle. etc.

 

[someguy1]

That is true, but mathematics is about concepts, a closed tautology, so the "physical world's" limitations are of no concern. That is the domain of physics, not math.

Yes, I fully agree. And like I said I have not read all of the deep thinking in this thread. That's why I'm so confused about using a folded piece of paper as proof of anything. Of course the square root of 2 is not physically realizable in the real world. Neither is the number 2 when used as a length. You can't show me an object of exactly length 2. So why are people going on about a physical piece of paper?

 

[Billy T]

... So why are people going on about a physical piece of paper?

Most likely to make understanding the concepts more easily. We do this all the time. E.g. I speak of going fishing at sun rise, but know better that it does not. etc. Being a little inaccurate, often helps with communication.

 

[Fednis48]

That is true, but mathematics is about concepts, a closed tautology, so the "physical world's" limitations are of no concern. That is the domain of physics, not math.

Certainly relationships developed in math theory can be applied in the real world, and achieved in the real world accurately enough to make them very useful. For example if you want to fence off an area, say to keep cattle out while newly planted pasture gets established, the shape of the fence for less cost per unit area protected should approximate a circle. etc.

Thank you for stating clearly what I've been trying to get to with Undefined. Fractions - like any other mathematical constructs - can only reflect the real world up to a finite degree of precision. It's important to realize, though, that this is an entirely different issue from the nature of infinitesimals.

 

[Quantum Quack]

It appears upon reading most of the preceding discussion that the main reason for angst on this forum is that is the untrained philosopher/critical thinker protesting about how mathematics c/o physics may not reflect real world situations. So often argument seems to be generated by:

1] Cross purpose and loss of contextual application by the philosophically minded.
2] The failure of mathematicians and physicist to remind themselves and the philosopher of the real world / abstract world context.

So often unnecessarily angst is generated IMO..

For example :
0.999... = 1

Is it a "real world" case or a purely mathematical case?
How does it relate to the "real" world?

[QQ calls upon philosophical license]

 

[Quantum Quack]

I don't think QQ has been part of this interesting (and educational for all but rpenner) discussion. Here is where the puzzle originated (at end of a 3 November 2013 post): http://www.sciforums.com/showthread.php?136842-1-is-0-9999999999999&p=3127851&viewfull=1#post3127851

I may not have been participating but certainly I have been spectating this fascinating discourse and learning all the time.

Relating to my previous post, I am sure rpenner, of all people, given his significant skills, knows that the context of math abstraction vs real world physics requires constant vigilance.
His specialized skills would be rendered ineffective if he and others didn't.
"Grounding the "genius" within all of us is not that easy to do sometimes."

as to the quote from Pete about answering the questions I raised in the OP:

The main thing IMO that differentiates Philosophy and Mathematics is that in philosophy there are no answers, only questions. [ stemming from the premise that: "no matter how smart and knowledgeable we think we are we will always die ignorant."]

So generally speaking the "philosopher within" , asks the questions and the "scientist within" strives to find the answers.