Figure 1

How many rational numbers are there? Yes, infinitely
many, I hear you say. But how large is that infinity?
Are there just as many rational numbers as integers?
Well, this requires us to be precise about "just
as many". A mathematician would say that set A has
"just as many" objects as set B if the objects in A and
B can be put into onetoone correspondence with each
other. Seem plausible?
Using this definition we can show that lots of infinite
sets have the same "size" or cardinality.
For instance the even integers can be placed in 11
correspondence with the odd integers, using N>N+1.
Perhaps surprising is that the set of all integers
and the set of even integers have the same cardinality,
via N>2N.
The natural numbers {1, 2, 3, ...} and the integers
have the same cardinality, because the integers can be
"listed" in the order {0, 1, 1, 2, 2, 3, 3, ...} and
this ordering gives the correspondence with the natural
numbers. Any set with the same cardinality as the
natural numbers is called a countable set.
The rationals seem even more densely populated in the real
line than the integers, but it is possible show that
they are countable! The 11 correspondence
is given by drawing a double array of rationals
(as in the Figure) and then listing them in the order
given by snaking diagonally through the array to
hit every one.
Presentation Suggestions:
Draw arrows to indicated onetoone correspondence.
Draw a picture to illustrate this.
Possibly split this one into several fun facts, over
successive days. Ask students if they think there are
infinite sets that are not countable?
Then follow the next day with the Fun Fact Cantor Diagonalization.
The Math Behind the Fact:
Mathematicians try to make precise definitions that
correspond to our usual intuition, but help to resolve
issues when intuition fails.
Cantor tried to make precise the notion of the size of
an infinite set.
How to Cite this Page:
Su, Francis E., et al. "How many Rationals?."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
