Re: Extrapolate divisor & dividend from quotient?

Hmm, now I am confused :)

On 7 Jul, 08:24, Gerry Myerson <ge...@xxxxxxxxxxxxxxxxxxxxxxxxx>
wrote:
In article
<d9e722f5-28cf-4fd9-824f-dc38e1765...@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
 ju...@xxxxxxxxxxxxx wrote:
On 7 Jul, 04:30, Gerry Myerson <ge...@xxxxxxxxxxxxxxxxxxxxxxxxx>
wrote:
In article
<2b85964e-f4be-4c4a-b288-4af80df91...@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
 ju...@xxxxxxxxxxxxx wrote:
I'll take the chance for a couple of questions that where in my mind
anyway:

Can all irrational numbers be written down (at least in principle) as
infinite decimal expansions? (I might ask: can all numbers in R be
expressed or at least thought of as decimal periodic expansions with a
period that can be infinite? Is there anything in the domain of
"numbers" that escapes this definitions?)

For instance, Phi is an irrational number whose decimal expansion can
be easily expressed with a rule for a sequence, or otherwise as a
continuous fraction, or a nested radical, etc. Are all real numbers
like this?

Every real number has a decimal expression, and every dedcimal
expression represents a real number.

Not every real number can be expressed by a rule - there are
too many real numbers, and not enough rules.

Thank you for your answer, very interesting.

Just to be sure, if I get you right: any real number can be expressed
as a decimal number whose expansion can have infinite period
(irrationals),

No, there's no such thing as an infinite period, unless you're
using this as a funny way to say, no period.


Funny? I'd say I have found it in most books and articles: no period
is equivalent to infinite period, and then one just has to talk in
terms of periodic expansions, where the various classes of real
numbers (rationals, irrationals, etc.) can be qualified by the
specific character of their periodic expansion. BTW, I didn't mention
it, but there is also Kolmogorov complexity behind the hood of my
anyway basic questions.



and sometimes there is not even a rule to generate the
expansion (these are the trascendentals?

No, there are transcendentals whose decimal expansions are generated
by simple rules, e.g., .110001000000000000000001000...
where there's a 1 at place n-factorial.


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).



anyway, in this case, that
the expansion "expresses" the number, seems to me acceptable in mere
line of principle...).

I don't know what that means.


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).



Also, this definition of real numbers reminds
me about randomness, or chaos. Is that what you actually mean by not
having all the rules?

No. I mean what I said. There are countably many rules,
and uncountably many reals. Simple rules can produce
randomness, and simple rules can produce chaos.


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"?



Another crucial question in any case comes: if there are irrationals
whose decimal expansion cannot be expressed by any rule, how do we
identify and/or distinguish them? (Maybe an example might help me.)

We don't. We can't. There are more numbers than there are
names for numbers.


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... OTOH, I
think back about the class of random numbers, that is those numbers
that cannot be expressed by any rule...

(In any case, thanks for your patience. I understand my questions/
objections must be elementar under many respects.)

-LV


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

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