Rationals Dense but Sparse

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 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 N-th 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 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. <https://www.math.hmc.edu/funfacts>.

Fun Fact suggested by:
Lesley Ward 

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