Now take the second number. Cover it with a carpet of size 1/2. Take the third number and cover it with a carpet of size 1/4, and so forth. Go on and on to infinity; since every rational number is on the seating chart, every rational number will eventually be covered by a carpet. What is the total size of the carpets? It’s our old friend, the Achilles sum. Adding up the size of the carpets, we see 1 + 1/2 + 1/4 + 1/8 + . . . + 1/2n goes to 2 as n goes to infinity. So we can cover the infinite cohorts of rational numbers in the number line with a set of carpets, and the total size of the
...more
