text: usually the terms "countable" and "uncountable" refer to a set with an infinite number of members.  in the context of an infinite number of elements, countable means "you can put them in a well-defined order".  mathematically this means you can define a one-to-one correspondence between the set's elements and the natural numbers.  \n\nso, the set of real numbers is uncountable, because if you go from 2.0 to 2.1, you have left out 2.05.. and if you instead go from 2.00 to 2.01, you have left out 2.005.. and  so on.\n\nanyway, rational numbers were shown to be countably infinite by georg cantor in 1891.  it was done so using his famous "diagonal argument".\n\nto do so, construct a table where you number the columns from left to right with the natural numbers 1, 2, 3, etc, and the rows from top to bottom the same way.  at each table element, write down a fraction where the numerator is the column number, and the denominator is the row number.  because this table has an infinite number of rows and an infinite number of columns, all of the rational numbers will be somewhere on it.\n\nnow, start in the upper left corner, which is 1/1.  take a step to the right (1/2), then down-left along the diagonal until you hit the edge (2/1).  go down (3/1), and then up-right along the diagonal (2/2, and then 1/3).  go right (1/4), and then down-left along the diagonal (2/3, 3/2, 4/1). go down.. and so on.  you keep zig-zagging along the diagonals and by doing so, you can hit every single rational number.  \n\nthe number of steps you have to take to get to a particular rational number defines its index, which means you can count them, which means the set is countably infinite.\n\nif it's hard to picture, you can see a diagram of what i'm talking about at the link below.