Lining Up the Fractions

Fractions feel denser than whole numbers, so it seems impossible to count them. Cantor found a way. Picture a grid: the row is the top number, the column is the bottom number, so cell (p, q) holds the fraction p/q.

Now trace a zigzag along the diagonals: (1,1), (2,1), (1,2), (3,1), (2,2), (1,3), ... This single path eventually reaches every positive fraction. Skip repeats like 2/2 (it's just 1), and you have one endless ordered list. A list is exactly what "countable" means — so the fractions are the same size as the whole numbers after all.