Yesterday we saw that whole numbers and fractions are both “countable,” so set theory says there is the same number of fractions as whole numbers.  It sure doesn’t look like it on a number line, but infinity is a big place; it goes on so far that those who count each fraction can get caught up, somewhere out there!

Now consider this – there are SETS of numbers bigger than the countable Aleph Null.

Fractions are called rational numbers, not because they make sense but because they are the RATIO of two integers.  There are other numbers, called irrational because they cannot be expressed as a fraction. PI is a famous example – 3.14159…  the decimal representation goes on and on without ever repeating. 

Suppose the owner of yesterday’s hotel decides to remodel and so clears everyone out of his infinite room inn.  On the day he reopens, an infinite number of people show up, each with a unique irrational number -an endless decimal representation which never repeats – on their t-shirt.  Does he have enough beds for all the new customers?

The proprietor tells them to get in a line, and starts assigning them rooms 1,2,3,4,5…  We can show the hotel with a countably infinite number of beds cannot hold all the irrational numbers if we can construct an irrational number that won’t have a bed. Here’s how:

Suppose all the people in the line got a bed.  Go to each bed and look at the number. Now build a number by taking the first digit from bed 1, the second digit from bed 2, the third digit from bed 3, etc. Then add one to each digit of the new number.  This makes a number that is not the number in bed 1 or bed 2 or bed 3 or any other bed.  And we just constructed an irrational without a place to lay his head.

So, there are more irrational numbers than there are rational numbers. Another way to say it is that the irrational set of numbers is a bigger infinity than the rational set of numbers.

BOOM!