Irrational Pi Defrocked

[Buffy]

  Robust said:

I'm not saying that the closed continuum cannot have an irrational area. I'm saying that the closed continuum must be described by a whole number or ending decimal - which the Lindemann pi does not allow.

I think you need to define your terms a little better here: What's the difference between "the area of a closed continuum" and "the description of a closed continuum"? And while you're at it, what is a "closed continuum"?

 

Cheers,

Buffy

[Qfwfq]

Quite. What's the meaning of "a closed continuum must be described by a whole number or ending decimal" and why doesn't the Lindemann pi allow it? How do or don't you "describe" a closed continuum by a number?

 

Buffy:

 

The meaning of 'closed continuum' is clear enough, in 2-D, any area delimited at finite distances by curves, in 3-D, any volume delimited at finite distances by superficies, in 4-D, any 4-volume delimited at finite distances by 3-superficies...

[Buffy]

  Qfwfq said:

Buffy: The meaning of 'closed continuum' is clear enough, in 2-D, any area delimited at finite distances by curves, in 3-D, any volume delimited at finite distances by superficies, in 4-D, any 4-volume delimited at finite distances by 3-superficies...

Yah, I know ("and in n-d any n-volume delimited at finite distances by n-1-superfices"...I love analytic geometry and topology!), but I'd still like to hear Robust's definition....

 

Cheers,

Buffy

[Qfwfq]

Only he and the Almighty know what words he would use to define it, I was hoping to avoid that, ;) but I think the idea in his head would be the same. It takes a bit of getting used to, before understanding what he says.

[Qfwfq]

Oh, yes!!!

 

If you love topology, I'm even allowed to say "Any compact and connected subspace."!

[Robust]

  Buffy said:

Yah, I know ("and in n-d any n-volume delimited at finite distances by n-1-superfices"...I love analytic geometry and topology!), but I'd still like to hear Robust's definition....

 

Cheers,

Buffy

Do we truly need to go into some kind of academic frothing as how to describe a closed continuum? Wouldn't a simple circle, square or some such suffice? What I.m contesting is that it can be described by otherthan a whole number or ending decimal. - such as the inscribed square of a circle defining it's 4 quadrants. If it by an ending decimal, then why not the area of the circle itself?

[Buffy]

  Robust said:

Do we truly need to go into some kind of academic frothing as how to describe a closed continuum? Wouldn't a simple circle, square or some such suffice? What I.m contesting is that it can be described by otherthan a whole number or ending decimal. - such as the inscribed square of a circle defining it's 4 quadrants. If it by an ending decimal, then why not the area of the circle itself?

Its not actually academic frothing. If every circle's area is representable either a whole number or a finite number of decimal places, it would be impossible to construct circles of certain radii. Heres a reductio ad absurdum argument. Lets start with saying the limit is one decimal place: then there's no circle with an area between 1.0 and 1.1. Okay, we can have numbers with two decimal places, then there's no circle with an area between 1.00 and 1.01. Okay we can have numbers with three decimal places, then we can't have a circle with an area between 1.000 and 1.001. I can keep doing this forever, describing a circle that cannot exist for some limit of decimal places, which we know is not possible, so therefore we must conclude that there are indeed circles for which the area is describable *only* in terms of an infinite number of decimal places. Now the reason why its important to understand what the description of a closed continuum is is that if you're going to be an engineer or a physisict rather than a mathematitian, you might argue that there are no relevant physical circles beyond enough decimal places to represent a number smaller than the Planck Distance, and thus you could say that the area of all known *physical representations* of circles can be described with no more than 23*(unit of measure)*(integral circle radius) decimal places. What we've been arguing here though is math, where there are indeed no physical limits.

 

Cheers,

Buffy

[MortenS]

But what do you mean by an ending decimal? A finite number of decimal places? Or something else?

Does this mean that you cannot divide a 1 m^2 square in three equally sized rectangles, as they would all have an area of 1/3 m^2, and 1/3 as you know, has an infinite number of decimal places, and therefore no ending decimal?

 

Or does it mean that a rectancle can not have an area of sqrt(2) m^2, sine sqrt(2) is an irrational number, with no ending decimal?

 

Or does it just apply to the circle, which cannot have an area of pi, if pi is an irrational and transcendent number?

 

It might just be me misunderstanding what you are trying to get across, but what I understand from your phrase "a whole number or an ending decimal", is numbers similar to these:

 

16

16,2

16,678545

 

but not numbers like these:

 

0,333333333333...

or

3,141592...

 

since these have an infinite number of decimal places, and therefore not an ending decimal.

[Robust]

  MortenS said:

But what do you mean by an ending decimal? A finite number of decimal places? Or something else?

Does this mean that you cannot divide a 1 m^2 square in three equally sized rectangles, as they would all have an area of 1/3 m^2, and 1/3 as you know, has an infinite number of decimal places, and therefore no ending decimal?

 

Or does it mean that a rectancle can not have an area of sqrt(2) m^2, sine sqrt(2) is an irrational number, with no ending decimal?

 

Or does it just apply to the circle, which cannot have an area of pi, if pi is an irrational and transcendent number?

 

It might just be me misunderstanding what you are trying to get across, but what I understand from your phrase "a whole number or an ending decimal", is numbers similar to these:

 

 

 

 

 

 

 

16

16,2

16,678545

 

but not numbers like these:

 

0,333333333333...

or

3,141592...

 

since these have an infinite number of decimal places, and therefore not an ending decimal.

I mean a finite number of decimal places

[C1ay]

  Robust said:

I mean a finite number of decimal places

How many then? A googol? A googolplex? What limit will make you happy because mathematically there is no limit?

[Buffy]

  Robust said:

I mean a finite number of decimal places

You may have missed it cuz its now on the previous page. Thoughts on my post #160 above?

 

Cheers,

Buffy

[MortenS]

  Robust said:

I mean a finite number of decimal places

 

So, can you divide a 1 m^2 square in three equally large rectangles, and tell me what the area of each of the resulting rectangles is? How many decimal places do the areas have?

[Qfwfq]

  MortenS said:

So, can you divide a 1 m^2 square in three equally large rectangles, and tell me what the area of each of the resulting rectangles is? How many decimal places do the areas have?

In base 3, they would each be 0.1 m^2.

 

:Alien:

[Qfwfq]

  Robust said:

Do we truly need to go into some kind of academic frothing as how to describe a closed continuum??

Can't Buffy and I have a bit of fun?

 

  Robust said:

If it by an ending decimal, then why not the area of the circle itself?

If you mean the area, I've already shown you that a closed continuum may have an area of root 2 over 6, and you agreed that's an irrational number.

[Robust]

  Qfwfq said:

Can't Buffy and I have a bit of fun?

 

If you mean the area, I've already shown you that a closed continuum may have an area of root 2 over 6, and you agreed that's an irrational number.

What is the radius of your circle?

[Qfwfq]

  Robust said:

What is the radius of your circle?

:Alien: Which one?

[C1ay]

  Robust said:

What is the radius of your circle?

How about a radius equal to the square root of any prime number?