proof that the rationals are countable
Suppose we have a rational number in lowest terms with . Define the “height” of this number as . For example, , , and Note that the set of numbers with a given height is finite. The rationals can now be partitioned into classes by height, and the numbers in each class can be ordered by way of increasing numerators. Thus it is possible to assign a natural number to each of the rationals by starting with and progressing through classes of increasing heights. This assignment constitutes a bijection between and and proves that is countable.
|Title||proof that the rationals are countable|
|Date of creation||2013-03-22 11:59:53|
|Last modified on||2013-03-22 11:59:53|
|Last modified by||alozano (2414)|