BigDecimal and trigonometrics

[Roedy Green]

Yes, that's true. But can you please point me to an irrational number
that cannot be derived by a sequence of mathematical operations on
rational numbers?

It has already been proved there are more irrational numbers that can
be put into 1-1 corresponding with integers, hence Strings. "Most" (in
the Lebesgue sense) irrationals cannot be described by a String. You
are dabbling in measure theory, which is pretty far removed from
computer science which is the study of finite numbers.

 

[Googmeister]

The time until George Bush's death in microseconds from noon UTC
today.

Nice one.

Choose a random number uniformly between 0.0 and 1.0.
It will almost surely be uncomputable.

 

[Jeffrey Schwab]

It has already been proved there are more irrational numbers that can
be put into 1-1 corresponding with integers, hence Strings.

Right, we agree on that. I'm just saying that a program needing to
represent irrational numbers with "infinite" precision can do so by
representing them in an abstract, symbolic way.

"Most" (in
the Lebesgue sense) irrationals cannot be described by a String. You
are dabbling in measure theory, which is pretty far removed from
computer science which is the study of finite numbers.

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

Computer science is NOT strictly the study of finite numbers. That's
like saying Astronomy is the study of telescopes. Finite numbers are
just some of the fundamental tools of computer science.

 

[Jeffrey Schwab]

No, but I think I can prove their existence

Yes, of course they exist. Name one.

if I can prove that all
sequences of mathematical operations on rational numbers is countable (since
the there are uncountably many irrational numbers).

Every sequence of mathematical operations on rational numbers can be
represented by some ASCII string (e.g. "1+1")

You can order them by using Java's standard string sorting algorithm.
Associate the first such legal string with the integer 1.
Associate the second such legal string with the integer 2.
And so on.

You now have a 1 to 1 mapping between sequences of mathematical
operations on rational numbers and the set of natural numbers, thus showing
that there are only countably many sequences of mathematical operations on
rational numbers.

Note that I'm assuming the ASCII representation is finite, which I think
is true as long as the number of operators and the number of arguments to
each operator is finite in the sequence (and as long as each operator and
each term can be represented by a finite number of characters, which is true
for rational numbers).

Nicely done!!! So there certainly are unrepresentable irrational
numbers. Let me "clarify" my position: Any irrational number that can
be clearly represented in a traditional mathematical formula or proof,
can also be represented in a computer program.

 

[Jeffrey Schwab]

Nice one.

Choose a random number uniformly between 0.0 and 1.0.
It will almost surely be uncomputable.

Nice job. I think you've actually got me there. We could certainly
use symbols to stand for these numbers, but I don't think the symbols
could really be said to "represent" the irrationals; for example, I
don't think symbolic representations could be used to order such random
numbers. Just for giggles, here's a way to assign symbols to truly
random numbers:

In order to be observed, such numbers must be generated by some process,
at some finite rate. (It is understood that other irrational numbers
exist, but it should also be understood that only finite number may ever
be observed by people or computers, since there is a finite number of
people and computers, each having finite faculties of observation.) We
can then refer to the first number generated by the process as R1, the
second as R2, etc. Randoms generated at the same instant may be refered
to as RnA, RnB, etc., where n is an ordinal integer.

 

[Roedy Green]

Any irrational number that can
be clearly represented in a traditional mathematical formula or proof,
can also be represented in a computer program.

That is like saying that every describable mathematical problem has a
solution that can be represented in Java, clear grandiosity. And
clearly false -- look at the halting problems.

You may be attempting to say something much weaker -- that anything
that can be expressed in mathematical notation could be formulated in
a Java String, e.g. as a TeX or PostScript program to typeset it.

 

[Roedy Green]

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, transfinite numbers.

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.

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.

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.

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.

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.

 

[IchBin]

[snip]

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

http://en.wikipedia.org/wiki/Measure_theory

--


Thanks in Advance...
IchBin, Pocono Lake, Pa, USA
http://weconsultants.servebeer.com/JHackerAppManager
__________________________________________________________________________

'If there is one, Knowledge is the "Fountain of Youth"'
-William E. Taylor, Regular Guy (1952-)

 

[Oliver Wong]

Any irrational number that can
be clearly represented in a traditional mathematical formula or proof,
can also be represented in a computer program.

[snip]

You may be attempting to say something much weaker -- that anything
that can be expressed in mathematical notation could be formulated in
a Java String, e.g. as a TeX or PostScript program to typeset it.

I think the two are equivalent. Jeffrey is merely saying that you can
represent the number in a computer program (e.g. as a String), not
nescessarily that you could compute it's value. For example, you can
represent Chaitin's constant as the string "Chaitain's constant" (or by its
one character symbol if you have unicode support) on a computer with finite
memory, even though actually computing the value of Chaitin's constant would
be equivalent to solving the halting problem.

http://en.wikipedia.org/wiki/Chaitin's_constant

- Oliver

 

[Oliver Wong]

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.

I learned this stuff by reading Roger Penrose's "The Emperor's New
Mind". It's a long, thick book, but all the math stuff lies in the first
1/3rd of it. The rest of the book is equally fascinating as well, of course.

http://www.amazon.com/gp/product/0140145346/103-2566881-0495862?v=glance&n=283155&v=glance

- Oliver

 

[Jeffrey Schwab]

Any irrational number that can
be clearly represented in a traditional mathematical formula or proof,
can also be represented in a computer program.

[snip]

You may be attempting to say something much weaker -- that anything
that can be expressed in mathematical notation could be formulated in
a Java String, e.g. as a TeX or PostScript program to typeset it.

I think the two are equivalent. Jeffrey is merely saying that you can
represent the number in a computer program (e.g. as a String), not
nescessarily that you could compute it's value. For example, you can
represent Chaitin's constant as the string "Chaitain's constant" (or by its
one character symbol if you have unicode support) on a computer with finite
memory, even though actually computing the value of Chaitin's constant would
be equivalent to solving the halting problem.

http://en.wikipedia.org/wiki/Chaitin's_constant

Thanks, that's what I was trying to say.

 

[Jeffrey Schwab]

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.

 

[Roedy Green]

Georg Kantor,

ITYM Cantor.

I think he spelled it Kantor, but it is often transliterated into
Cantor, singer in a synagogue, in English.

Sort of like how Joe Green wrote Italian operas.

 

[Roedy Green]

Probability theory as I studied it in school was approached a very
different way.

Advanced probability theory you would hardly recognise as related to
the probability of card games. It is all about integrating infinities.
I have no idea what practical use it might ever have.

 

[Roedy Green]

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!

Markov processes are ones that have state but no memory of history.
The are like generalisations of finite state automata. There is a
probability function that maps the previous state onto the next.

One of the strange results is that under some not that stringent
conditions, if the probability of a state occurring is non-zero, no
matter how vanishingly small that probability, if you wait long enough
it will happen infinitely often.