Re: Extrapolate divisor & dividend from quotient?

In article
<d9e722f5-28cf-4fd9-824f-dc38e176571f@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
julio@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.

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.

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.

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.

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.

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

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