Re: BigDecimal and trigonometrics

Roedy Green wrote: 
On Wed, 16 Nov 2005 22:44:27 GMT, Jeffrey Schwab
<jeff@xxxxxxxxxxxxxxxx> wrote, quoted or indirectly quoted someone who
said :


I'm not familiar with Lebesgue, or "measure theory." If you have any good links on them, I'd be interested in learning.


You are dancing around an area of advanced mathematics that you might
find fascinating. 

Things you might google for are Aleph, countable, uncountable, 1-1
mapping, advanced probability theory, Lebesgue measure theory, the
different kinds of infinity, Georg Kantor,


ITYM Cantor.

transfinite numbers. 

I'm familiar with all of those things except Lebesgue and "measure theory."

rationals are like thin strips of celery fibre in a thick soup of
irrationals. Even though you can always find an irrational between two
rationals and a rational between two irrationals, in a very strong
sense, irrationals are vastly more numerous, and most definitely can't
be enumerated by a set of strings.


Right on.  "Celery fibre" is an interesting analogy. :)

How do you quantify that? You do it by integrating over all the
rationals or over all the irrationals.  The set of rationals is called
a set of measure 0, because when you do that, their contribution comes
out 0.  Just how you do those integrals is called Lebesgue measure
theory.

Not sure I follow, but I'll investigate. The proof I've seen before is this one:

http://en.wikipedia.org/wiki/Cantor's_diagonal_argument

The way you usually tackle the domain of knowledge is with first
calculation of finite probablities and combinatorics, then
probability, then the various types of infinity, then probability
theory over these various infinite sets.

Probability theory as I studied it in school was approached a very different way. We never saw irrational outcomes. It seems strange now, but this never struck me as odd. Anyway, probability is usually based on the randomness of some physical process, which means some random point is being chosen from a continuum of space or time; but whether the infinity of the continuum is the same as the infinity of irrationals is provably unprovable. Btw, have you read this?

http://www.amazon.com/gp/product/019514743X/104-2580866-1835953?v=glance&n=283155&v=glance

It's very good.

You then wander around in Markovian processes (the things that turned
me on most since they are so much like finite state automata.) and the
astounding 0-1 law.

I don't know about that either, although I gather this is the same Markov of "Markov chain" fame. Looks like I've got some reading to do!

This is stuff I did not learn until after I had my BSc back when I was
studying it, so the books you may find won't necessarily be that
accessible. Mathematicians tend to go for brevity, and elegance, as if
they were constructing puzzles.  They are not big on handing you any
sort of intuitive understanding.

Some great books about Mathematics were not written by mathematicians. My personal favorite was written by a dentist:

http://www.wwnorton.com/catalog/fall96/math.htm

I've had it for three or four years now, and I'm still not half-way through it.
.

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