Re: Rational Numbers/Irrational Numbers

In article <ZpqdnWcZMaLsPSbYnZ2dnUVZ_veinZ2d@xxxxxxxxxxxx>,
"David T. Ashley" <dta@xxxxxxxx> wrote:

"Dave Seaman" <dseaman@xxxxxxxxxxxx> wrote in message
news:epfl09$7cg$2@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
On Sat, 27 Jan 2007 01:28:41 -0500, David T. Ashley wrote:
<davidmarcus@xxxxxxxxxxxx> wrote in message
news:1169874202.106385.326130@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
On Jan 26, 11:36 pm, "David T. Ashley" <d...@xxxxxxxx> wrote:
"Leo" <newsdon...@xxxxxxxxxxx> wrote in
messagenews:1169780763.086460.114690@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

Which set has more numbers, the set of rational numbers or the set of
irrational numbers?
Well, the set of irrational numbers has at least twice as many elements
as
the set of rational numbers.

Think about the following functions:

f(x) = PI + x
g(x) = PI + PI + x

Every rational number x can be paired with at least two irrationals.

So, I'm going to go with "irrational" as being bigger.

Right answer. Wrong reason. The rationals are countable. The
irrationals are uncountable. The rationals have Lebesgue measure zero.
The irrationals in [0,1] have Lebesgue measure 1.

I was just clowning around ... but OK, let's explore your logic.

In order for me to have stated the "wrong" reason, there has to be a
counterexample where my test fits but where there are not "more".

Please provide a counterexample.

Consider one of the standard proofs that the rationals are countable. We
define a mapping f: N -> Q+ by listing the positive rationals in the
following order:

1/1, 1/2, 2/1, 3/1, 2/2, 1/3, 1/4, 2/3, ...

I'm not seeing the algorithm behind the sequence above. Could you state it
explicitly?

List all the positive rationals having a given sum of numerator and
denominator in order of size, listing such groupongs in order of
increasing sum.

The concept I understand (the mapping), just I don't see how the sequence
above is generated.

I might be having a slow day.

Thanks.
.

Relevant Pages

  • Re: A basic question on real numbers
    ... Irrationals are limits of convergent sequences of rationals. ... In other words, do reals have numbers from only N, Q ... cauchy sequence of rational numbers. ...
    (sci.math)
  • Re: abundance of irrationals
    ... And this sequence includes neither all rationals nor all irrationals. ... we get an alternating sequence ...
    (sci.math)
  • Re: A basic question on real numbers
    ... Irrationals are limits of convergent sequences of rationals. ... number is the limit of a sequence of rationals. ...
    (sci.math)
  • Re: A basic question on real numbers
    ... Irrationals are limits of convergent sequences of rationals. ... cauchy sequence of rational numbers. ...
    (sci.math)
  • Re: infinitely many nns = infinite nns?
    ... every two irrationals, it would seem that the rationals and the ... Possibly at first glance, but that assumes that the number of rationals between two irrationals is the same as the number of irrationals between two rationals, which is distinctly not the case. ... There are never two or more reals "in a row" without others between them, regardless of their rationality or lack therof. ... Now, if we want to say that we are limited by potential infinity when mentally examining a structure with actually infinitely many elements, I would certainly agree that we cannot mentally examine the elements individually, but in the limit, where actual infinity either CAN be used, or at least predicted, there cannot be a y between ANY two x's. ...
    (sci.logic)