Re: A basic question on real numbers


aone1504@xxxxxxxxx wrote:
Virgil wrote:
In article <1150182294.926919.36640@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
aone1504@xxxxxxxxx wrote:

I think I have not made myself clear. My question is from completeness
point of view. Rationals(Q) are generated by dividing whole numbers(N).
Irrationals are limits of convergent sequences of rationals. Let's
denote the set of irrationals obatained this way as IR1. Now, if we
write a convergent series of irrationals taken from this set IR1 only,
will the resulting value fall within IR1 or Q or N only. Or can it be
outside this set ? In other words, do reals have numbers from only N, Q
and IR1. Is there a proof for this.? I am not very clear whether the
proofs you have given establishes this.

Thanks very much.


Dave L. Renfro wrote:
aone1504@xxxxxxxxx wrote:

I have the following basic question:-

Irrational numbers can be obtained as the limit of
cauchy sequence of rational numbers.

Is there a proof that a convergent infinite series
having irrational terms converges to a number which
can always be obtained as the (above) limit of cauchy
sequence of rationls.

I would like to see the proof for this statement.

Let {x_n} converge to x, where each x_n is irrational.
For each n, choose a sequence {r_n_m} of rationals
that converges to x_n. Then the diagonalization of
these sequences of rationals -- namely, the sequence
r_1_1, r_2_2, r_3_3, ... -- is a sequence of rationals
that converges to x. This isn't immediately evident
(to a beginner, at least), but the proof also isn't
very difficult. More generally (same proof method),
a limit of limits is a limit, or in other language,
the (sequential) closure of a set B can be obtained
by (one application of) collecting together all the
limits of convergent sequences in B.

Dave L. Renfro

There are proofs that if one repeats the Cauchy sequence construction
based on sequences of real rather than rationals, nothing new is added.

Similarly for the Dedekind cut construction.

In other words, repeating the process adds nothing.

Is this what you were asking about?

It looks like this is what I want. Let me attempt once more to express
my question:-

As you have mentioned here, I am also talking about the process of
repetition on the set obtained in each successive step. From Integers,
ancients discovered rationals using division. They got a new set( the
set of rationals). At this point, (pls. correct me if I am wrong )
there was no need to have the concept of "Real numbers". Rationals can
be defined without knowing that there is a bigger set called Reals.
Later mathematicians used cauchy sequences on rationals and thus
discovered Irrational numbers. They showed that these sequences
converge to some value(not a rational) which lie between 2 rationals
and thus classified them as irrational numbers. Now, they got a bigger
set, which has N, Q and Irrationals. At this point, again my question
is, is there a need to have the concept of Reals to understand the
numbers we got so far ?. But of course they have found out these
numbers only with the initial assumption that there can be numbers in
between any two they got so far.

In each of these steps, some operation or sequence of operations was
used to get something new. At this point, I am trying to figure out
whether there can be some operation or sequence of operations that
would give me a new set of numbers which I have not obtained so far.

So, one question is:-- If we apply cauchy sequence on the set we got
till now, do we get some thing outside this set. ?

If this is not the case, is it a futile exercise to try to find out
numbers beyond this, even by some other means (ie., some new operations
not known to me as of now) or is it worth attempting ?

Thanks to each one of you.

--Murali

As the phrase "real number line" suggests, our intuitions about
the real number system are based on thinking of a line as made
up of points. It is quite mysterious that something continuous
like a line consists of a collection of individual points, which are
discrete in nature.

Constructing points at regularly spaced intervals on the line, as
tantamount to the integers, is a first step, quickly followed by
arbitrary regular subdivisions of those intervals, which is like the
construction of all rationals. But that's not all...

The existence of irrationals was known to Greek mathematicians
before Euclid, e.g. the irrationality of sqrt(2) is often thought to
have been discovered (and suppressed?) by the Pythagoreans.

So there's a big jump between the knowledge of irrationals
and the modern constructions of the real numbers by Cauchy
and Dedekind in the 19th century.

The result that every Cauchy sequence of real numbers
converges to a real number is known as "completeness"
of the real numbers (with the usual metric topology).

It is possible to extend the structure of the real numbers
as in reifying infinitesimals; cf. Robinson's nonstandard
analysis and other approaches. However in some sense
anything new we add to the real numbers does take us
"outside" the real line, right?

Also note that the algebraic closure of the rational field
produces only countably many "new" numbers, where
the topological completion (real numbers) produces an
uncountable infinity of irrationals.


regards, chip

.

Relevant Pages

  • Re: Peano curve is wrong
    ... The set of all rationals, ... sequence if there exists an integer N such that for any positive, ... It is in the system of reals. ... In 1890, Peano gave a space-filling curve, named ...
    (sci.math)
  • Re: Cantor Confusion
    ... least appears to totally disconnect the set of those remaining. ... sequence of reals with each term less than all terms of a strictly ... This is not the case for rationals. ...
    (sci.math)
  • Re: Galileos Paradox and the Project of the Reals
    ... The way you get reals by a neverending process of generating digits ... and which thereby constructs a sequence of nested intervals whose ... It isn't *already* in the rationals you started with. ...
    (sci.math)
  • Re: abundance of irrationals
    ... > And this sequence includes neither all rationals nor all irrationals. ... we get an alternating sequence ...
    (sci.math)
  • Re: abundance of irrationals
    ... And this sequence includes neither all rationals nor all irrationals. ... we get an alternating sequence ...
    (sci.math)