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I was careless again.
==Countability and density==
==Countability and density==
An important theoretical property of the rationals is that they are countable. That is, the entire set of rational numbers can be put into a one-to-one correspondence with the integers. This is surprising at fist glance, since the integers are "sparse" while the rationals seem to fill out the real line. To see this correspondence, we need to list all rational numbers in some order. List the rationals (that is, the rationals in which the fraction has been fully reduced) that have the sum of their numerators and denominators equal to one. 0/1 is the only such. Then follow those with the rationals that have the sum of their numerators and denominators equal to two. 1/1 is the only such, since 0/2 is not reduced. Follow those with the rationals that have the sum of their numerators and denominators equal to three. They are 1/2 and 2/1. Continue without end.
An important theoretical property of the rationals is that they are countable. That is, the entire set of rational numbers can be put into a one-to-one correspondence with the integers. This is surprising at first glance, since the integers are "sparse" while the rationals seem to fill out the real line. To see this correspondence, we need to list all rational numbers in some order. List the rationals (that is, the rationals in which the fraction has been fully reduced) that have the sum of their numerators and denominators equal to one. 0/1 is the only such. Then follow those with the rationals that have the sum of their numerators and denominators equal to two. 1/1 is the only such, since 0/2 is not reduced. Follow those with the rationals that have the sum of their numerators and denominators equal to three. They are 1/2 and 2/1. Continue without end.
A more subtle theoretical property is that the rationals comprise a countable [[dense set|dense subset]] of the reals. This is used in some advanced theorems of topology.
A more subtle theoretical property is that the rationals comprise a countable [[dense set|dense subset]] of the reals. This is used in some advanced theorems of topology.