The set of rational numbers is countable
- and easy proof
A set is countable if you can count its elements. Of course if the
set is finite, you can easily count its elements. If the set is infinite,
being countable means that you are able to put the elements of the set in order
just like natural numbers are in order. Yet in other words, it means you
are able to put the elements of the set into a 'standing line' where each one
has a 'waiting number', but the 'line' can still continue to
infinity.
In mathematical terms, a set is countable either if it is finite, or it is
infinite and you can find a one-to-one correspondence between the elements of
the set and the set of natural numbers. Well the infinite case is the same
as giving the elements of the set a waiting number in an infinite line...
And here is how you can order rational numbers (fractions in other words)
into such a 'waiting line'. It's just positive fractions, but after you
have these ordered, you could just slip each negative fraction after the
corresponding positive one in the line, and put zero leading the crowd. I like this proof because it is so simple and intuitive yet convincing.
The numbers in red/blue table cells are not part of the proof but just show
you how the fractions are formed. You start at 1/1 which is 1, and follow
the arrows. You will encounter equivalent fractions, which will be skipped
over.
If you think about it, all possible fractions will be in the list. For
example 145/8793 will be in the table at the intersection of the 145th row and
8793rd column, and will eventually get listed in the 'waiting line'.
Cantor's
diagonal argument is a proof devised by Georg Cantor to demonstrate that the real numbers are not countably
infinite - or in other words, a proof that the real numbers are uncountable.
February 2010 newsletter
