By displaying all the fractions in this way, potentially at least, we can construct
a one-dimensional list. If we start on the top row and move to the right at each
step we will never get to the second row. However, by choosing a devious zig-
zagging route, we can be successful. Starting at 1, the promised linear list
begins: 1, −1, ½, ⅓, −½, 2, −2, and follows the arrows. Every fraction, positive
or negative is somewhere in the linear list and conversely its position gives its
‘mate’ in the two-dimensional list of fractions. So we can conclude that the set of
fractions Q is countably infinite and write card(Q) =.
Listing the real numbers
While the set of fractions accounts for many elements on the real number line
there are also real numbers like , e and a which are not fractions. These are
the irrational numbers – they ‘fill in the gaps’ to give us the real number line R.
With the gaps filled in, the set R is referred to as the ‘continuum’. So, how
could we make a list of the real numbers? In a move of sheer brilliance, Cantor
showed that even an attempt to put the real numbers between 0 and 1 into a list
is doomed to failure. This will undoubtedly come as a shock to people who are
addicted to list-making, and they may indeed wonder how a set of numbers
cannot be written down one after another.