Well we all know that between any two real numbers there is
a rational. Mathematicians like to say that the rationals
are dense in the real line... what this means is that
any open set will contain some rational.
So they are "everywhere" in the line, aren't they?
Well, it depends on what you mean by "everywhere".
One could argue that the rationals
are pretty sparsely populated in the reals:
I claim that you can cover the rationals by a set whose
"length" is arbitrarily small. In other words, give me a
string of any positive length, no matter how short, and I
will be able to cover all the rationals with it!
Since the rationals are countable, I can run
through them sequentially, one by one. Take the string,
cut it in half, and cover the first rational with it.
Then take what's left of the string, cut it in half,
and use that to cover the 2nd rational. Continue
in this fashion, taking what's
left of the string, cutting it in half, and using
that to cover the Nth rational.
When complete, all the rationals will be covered!
So the rationals are dense but also "sparse"!
Presentation Suggestions:
Some students may object that this procedure will take
infinitely long. Counter by saying that if the first
covering takes 1sec, the 2nd covering takes 1/2sec,
the 3rd takes 1/4sec, etc., that you will
finish in 2 seconds. (Of course, you could also just
explain that you'll do the cutting and covering all at once.)
The Math Behind the Fact:
A mathematician would say a "sparse" set
(as we've defined it here) is a measure zero set.
It may be worth mentioning that the irrationals are
also dense, but unlike the rationals,
they are not "sparse" or measure zero.
This fact emphasizes that rationals
and irrationals are really quite different even though
you can find a rational between any two irrationals,
and an irrational between any two rationals!
Measure zero sets do not need to be countable;
an example of a measure zero set that is not is
a Cantor Set.
How to Cite this Page:
Su, Francis E., et al. "Rationals Dense but Sparse."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
