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 one-to-one 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 1-1 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 1-1 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 one-to-one 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. <https://www.math.hmc.edu/funfacts>.
Fun Fact suggested by:
Arthur Benjamin