Accomplishing the same task through a different set of rules

[Acme]

It also goes along to proving that .9999... is equal to 1.

 

We know that 1/3 equals .333.. and 3*1/3= 1

 

There is also other proofs of this, such as 1/9 = .1111... and 1/9 * 9 = 9/9 and .999.....

 

Therefore, a similarity between those two proofs is 1/3^n * 3^n = 1.

 

EDIT: Breaking this down, 1/n^s * n^s = 1.

That particular problem is base specific and not fundamentally related to number. That is, it is an artifact of numeration. Since computers ultimately use binary -i.e. base two-, can you re-write your evaluation in that base?

[Unity+]

That particular problem is base specific and not fundamentally related to number. That is, it is an artifact of numeration. Since computers ultimately use binary -i.e. base two-, can you re-write your evaluation in that base?

I can rewrite it in binary, yes.

 

So, 1/3 = .333... which, in binary, is .0101010101...

 

.0101010101... + .0101010101... = .10101010...(.666...).

 

.0101010101...+ .10101010... = .111111111...(.999... = 1).

[Acme]

 

That particular problem is base specific and not fundamentally related to number. That is, it is an artifact of numeration. Since computers ultimately use binary -i.e. base two-, can you re-write your evaluation in that base?

I can rewrite it in binary, yes.

 

So, 1/3 = .333... which, in binary, is .0101010101...

 

.0101010101... + .0101010101... = .10101010...(.666...).

 

.0101010101...+ .10101010... = .111111111...(.999... = 1).

 

D'oh! Touche! Binary was a poor choice of base on my part. My argument still stands if you try base q where q is the KomornikLoreti* constant with approximate decimal value 1.787231650... En garde!

 

*http://en.wikipedia.org/wiki/KomornikLoreti_constant

[Unity+]

D'oh! Touche! Binary was a poor choice of base on my part. My argument still stands if you try base q where q is the KomornikLoreti* constant with approximate decimal value 1.787231650... En garde!

 

*http://en.wikipedia.org/wiki/KomornikLoreti_constant

The reason why properties, with slight changes, mainly remain the same because even with a change of base the actions stay similar and seem similar in many cases. You can correct me on this though.

[Acme]

The reason why properties, with slight changes, mainly remain the same because even with a change of base the actions stay similar and seem similar in many cases. You can correct me on this though.

I don't understand that statement. Anyway, the whole .9999=1 thing doesn't work in the base I last gave. That was my only point in posting it.

[Unity+]

I don't understand that statement. Anyway, the whole .9999=1 thing doesn't work in the base I last gave. That was my only point in posting it.

Can you show me the example in work?

[Acme]

Can you show me the example in work?

I can give you a reference, yes.

 

source: .999... @ Wiki >> http://en.wikipedia.org/wiki/0.999...#In_alternative_number_systems

...Generally, for almost all q between 1 and 2, there are uncountably many base-q expansions of 1. On the other hand, there are still uncountably many q (including all natural numbers greater than 1) for which there is only one base-q expansion of 1, other than the trivial 1.000.... This result was first obtained by Paul Erdős, Miklos Horváth, and István Joó around 1990. In 1998 Vilmos Komornik and Paola Loreti determined the smallest such base, the KomornikLoreti constant q = 1.787231650.... In this base, 1 = 0.11010011001011010010110011010011...; the digits are given by the ThueMorse sequence, which does not repeat.[26] ...

[Unity+]

I can give you a reference, yes.

 

source: .999... @ Wiki >> http://en.wikipedia.org/wiki/0.999...#In_alternative_number_systems

I think this improves my point because that means there are multiple ways to represent 1 in other bases.

[Acme]

I think this improves my point because that means there are multiple ways to represent 1 in other bases.

But again, just because you have a specific example/case, you cannot then logically conclude all examples/cases follow [the rule]. The all business is how I read/interpret your initial comment in the OP, where you said;

So, it got me thinking that these tasks being done in a different set of restrictions can be generalized by a function that if you have found the process of completing a task within one set of restrictions then you can find the process of completing a task with any set of restrictions.

[Unity+]

But again, just because you have a specific example/case, you cannot then logically conclude all examples/cases follow [the rule]. The all business is how I read/interpret your initial comment in the OP, where you said;

But that is why this would be considered a conjecture.

[Acme]

But that is why this would be considered a conjecture.

I suppose I could give you wide latitude there, but usually a conjecture is based on evidence. My choice of term would be a musing.

 

We think, therefore we argue. ~ Unattested

[Unity+]

I suppose I could give you wide latitude there, but usually a conjecture is based on evidence. My choice of term would be a musing.

 

We think, therefore we argue. ~ Unattested

A conjecture would be that a proposition is presented that has examples of it being true(the examples being evidence). For example, the 3x+1 problem has examples of it being true. This proposition has examples of it being true in some form(whether directly or indirectly).But, it could just be a musing.

[Acme]

A conjecture would be that a proposition is presented that has examples of it being true(the examples being evidence). For example, the 3x+1 problem has examples of it being true. This proposition has examples of it being true in some form(whether directly or indirectly).But, it could just be a musing.

I love to muse; I just temper my expectations. I did have to read up on the 3x+1 problem as it wasn't familiar to me and it put me in mind of my musing over the all-perfect-numbers-are-even conjecture. Now if we just sum the reciprocals of the modulus of..........

[Unity+]

I love to muse; I just temper my expectations. I did have to read up on the 3x+1 problem as it wasn't familiar to me and it put me in mind of my musing over the all-perfect-numbers-are-even conjecture. Now if we just sum the reciprocals of the modulus of..........

Well, I am trying to make a concept, mathematically, of this idea, and here is what I got.

 

Given an algorithm A there exists the Big-O(which is the representation of efficiency of a given algorithm, simply put http://en.wikipedia.org/wiki/Big_O_notation).

 

[math]A(x)\in O(g(x))[/math]

 

There are many possibilities that could occur within this concept, which is either that the other algorithms that branch off from algorithm A have Big-O's that add up to the Big-O of algorithm A or are the product of, or both?

 

Either that, whatever combination of functions of Big-O to get the function of the original Big-O.

 

[math]O(\delta) = \sum_{n=1}^{k}O(\delta _{n})=O(\delta _{1})+O(\delta _{2})+...O(\delta _{n})[/math]

 

[math]O(\delta) = \prod_{n=1}^{k}O(\delta _{n})=O(\delta _{1})\times O(\delta _{2})\times ...O(\delta _{n})[/math]

 

[math]O(\delta) = \sum_{n=1}^{k}O(\delta_{n} )\pm \prod_{n=1}^{k}O(\delta _{n})[/math]

 

Of course, this assumes that this conjecture is correct, which is that there is a conservation of information(?) when dealing with the break down of algorithms to their simpler form.

Edited May 11, 2014 by Unity+

[Acme]

Well, I trying to make a concept, mathematically, of this idea, and here is what I got.

 

Given an algorithm A there exists the Big-O(which is the representation of efficiency of a given algorithm, simply put http://en.wikipedia.org/wiki/Big_O_notation).

 

[math]A(x)\in O(g(x))[/math]

 

There are many possibilities that could occur within this concept, which is either that the other algorithms that branch off from algorithm A have Big-O's that add up to the Big-O of algorithm A or are the product of, or both?

...

I did study Big O notation a couple decades ago, but I didn't pursue programming and so it's never been of much use or interest to me. This is not to say that I don't recognize its value of course. When I need code for my mathematic probing I get someone to write it for me and then just use it and give the programmer the benefit of the doubt that the code is as efficient as they can make it. Big numbers mean big time and I'm content to hurry up & wait. Imagine Fermat's or his contemporaries' amazement at the computing power we all take for granted on our desktops. Further imagine their chagrin at the ratio of porn to math use. Oy vey ist mir.