Re: Extrapolate divisor & dividend from quotient?

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), and sometimes there is not even a rule to generate the
expansion (these are the trascendentals? anyway, in this case, that
the expansion "expresses" the number, seems to me acceptable in mere
line of principle...). Also, this definition of real numbers reminds
me about randomness, or chaos. Is that what you actually mean by not
having all the rules?

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.)

-LV



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

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