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

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

1. ### Billy TUse Sugar Cane Alcohol car FuelValued Senior Member

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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.

3. ### Motor DaddyValued Senior Member

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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?

5. ### UndefinedBannedBanned

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Hi Quantum Quack.

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...

Cheers, QQ. Excellent work, mate!

Last edited: Nov 9, 2013

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8. ### Billy TUse Sugar Cane Alcohol car FuelValued Senior Member

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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.

Last edited by a moderator: Nov 9, 2013
9. ### PeteIt's not rocket surgeryRegistered Senior Member

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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)

Last edited: Nov 9, 2013

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11. ### hansdaValued Senior Member

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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.

12. ### someguy1Registered Senior Member

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

13. ### rpennerFully WiredValued Senior Member

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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.

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.)

14. ### TachBannedBanned

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

15. ### rpennerFully WiredValued Senior Member

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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.

16. ### TachBannedBanned

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

17. ### someguy1Registered Senior Member

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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.

18. ### Billy TUse Sugar Cane Alcohol car FuelValued Senior Member

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can be a

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.

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19. ### someguy1Registered Senior Member

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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?

20. ### Billy TUse Sugar Cane Alcohol car FuelValued Senior Member

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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.

21. ### Fednis48Registered Senior Member

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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.

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

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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?

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

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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.