Divide the number 1 by the numbers 1 through 10 consecutively. What conjectures can you make about rational numbers when represented as decimals?
If we think of division as a repeated subtraction, can you explain why it is impossible to divide by 0?
In a hotel with an infinite number of rooms and a counting number assigned to each, there is a "No Vacancy" sign outside. A traveler comes in and asks for a room for the night. How does the staff accommodate this traveler?
Think of the traveler as one more element to add to a countably infinite set. Is the new set also countably infinite? Close Tip
In a hotel with an infinite number of rooms and a counting number assigned to each, there is a "No Vacancy" sign outside. An infinite marching band -- one where each member has a unique number on his or her uniform -- enters and asks for a room for the night for each musician. How does the hotel staff accommodate everyone?
Think of the band and the rooms as two infinite sets to be added together. What kind of set do you get? How can you put this new set into one-to-one correspondence with the counting numbers? Close Tip
There's an infinite chain of infinite hotels, each with a unique address on the street. All of them are full. But one night, very suddenly, all but one of them go out of business! How does the one remaining hotel accommodate all the stranded guests?
Think of this as adding together an infinite number of infinite sets. This will be similar to putting rational numbers into one-to-one correspondence with the counting numbers. Close Tip
Read History and Transfinite Numbers: Counting Infinite Sets.