Re: Extrapolate divisor & dividend from quotient?

In article
<2acd62ba-2d71-4a6c-b98f-6628f6c093d3@xxxxxxxxxxxxxxxxxxxxxxxxxx>,
julio@xxxxxxxxxxxxx wrote:

Then what makes an irrational a "trascendental"? And what do we call
something that not even a rule can express? (I am basically asking
about some "standard" classification, if anything like that exists).

A number is algebraic if it is a root of a polynomial with integer
coefficients. A number is transcendental if it is not algebraic.

I don't know if there is a standard name for those numbers
for which no rule calculates their decimal expansion. I suppose
these might be called uncomputable numbers, although one
would always have to qualify this by saying the numbers are
uncomputable with respect to some particular formally specified system.

The point I was trying to make there is: if there are such numbers
that cannot be expressed by any rule, then it sounds to me improper to
say that such numbers can be "expressed" at all; actually, although
not in the context of this thread, I would go on and question their
existence (I mean, in order to learn more!). My observation above was:
that such numbers "have a decimal expansion" at all, seems to me just
a matter of stipulation (like the "existence" of such a thing as
Infinity, I guess).

A decimal expansion is just a sequence of digits. Standard operating
procedure in mathematics is to accept the existence of sequences
of digits, even those sequences that cannot be described by any
finite rule. There are mathematicians who reject this and insist that
only computable reals have any meaning. If you find this a congenial
thought, you might look up the work of Errett Bishop. It's a legitimate
philosophical position. It isn't the way most mathematicians have chosen
to go.

That simple rules can produce chaos I can understand, although, as to
the intrinsic complexity, that chaos is still "compressible" back to
the generating rule, so that I am still confused. In any case, as to
simple rules can produce randomness, that escapes me entirely: how can
it be? Isn't randomness the very absence of any kind of "rule"?

Well, the first thing you have to do is get a useful definition
of "random." This is harder than it looks. You might want to see
the effort Knuth puts into it in the 2nd volume of The Art Of
Computer Programming.

Let's take a very simple definition of random: every digit appears,
in the limit, with frequency one-tenth; every block of two digits,
with frequency one in a hundred; in general, every block of k digits
with frequency one in ten-to-the-k. Then the number
..1234567891011121314151617181920212223242526....
is random and is generated by a very simple rule.

Now this definition of random may strike you as inadequate.
Unfortunately, tightening the definition as much as you propose,
to where randomness is the absence of any rule whatsoever,
Knuth shows that under a reasonable interpretation this doesn't
leave any random sequences at all. So, some compromise is
necessary. But read Knuth, he says it all far better than I can.

So, if I get you right, you are saying that there are some numbers
which we cannot identify nor distinguish or otherwise name, and all we
know is that they must be there. In other words, we have got such non-
empty class of "non-expressible numbers", whose elements cannot be
told apart. I bet this must have to do with Cantor-Goedel...

As Arturo has said, this has nothing to do with Godel, just with
Cantor's notion of different sizes of infinity.

By the way, all three versions of your message reached my server.
Sometimes one just has to be patient.

--
Gerry Myerson (gerry@xxxxxxxxxxxxxxx) (i -> u for email)
.

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