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!

We will use the opportunity of no cool number dates on my calendar this week to explain a little bit about infinity.

It turns out there are multiple distinguishable SETS of infinite numbers!

The easiest sets to identify are all those that are countable.  That is, they can be described and, if given enough time and space, placed in order 1,2,3,4,5,…  So whole numbers are a countably infinite set. 

There are famous stories that help one understand other countable infinite sets (called Aleph Null sets by the man who first used set theory to describe them).  The stories are called Hilbert Hotel stories, named for the other man who made them popular.

Suppose there is a hotel with an infinite number of beds, numbered 1,2,3,4,5… with each bed occupied. One day a new customer shows up and asks for a bed.  The owner says “No Problem,” gets on the intercom, and tells all the residents to shift – “Move from your current bed to the next higher numbered bed. One goes to two, two goes to three, etc.”  Then the owner puts the new guy in bed #1.

SO…. Aleph Null (countably infinite  set) + 1 = Aleph Null (countably infinite set)

Now suppose a bus with an infinite number of passengers arrives, all desiring a bed.  The hotel owner says, “Great, I have a place for every one of you!”  He uses his intercom to alert all the current residents, “Please look at your bed number.  Double it, and go to that bed number.  Thank you.”   Then the owner tells all the bus passengers, “Get in order and look at your number.  Each of you will double your number and subtract one.”  All the previous clients are in even numbered beds and all the new passengers are in odd numbered beds.  Everyone has a place to sleep.

SO… Aleph Null  + Aleph Null = Aleph Null

 Then, an infinite number of busses arrive, each with an infinite number of passengers!  Where will they sleep?  “Easy,” says the hotel owner.  Once again he tells all the current residents to double their bed number and go to their new even-numbered  crib to free up all odd-numbered beds.  Then he tells the new arrivals to get in line.  “Everyone look at your bus number and seat number and get in order like this: 

Bus 1, Seat 1  (all those whose bus and seat numbers add to 2)

Bus 1, Seat 2  (all those whose bus and seat add to 3)

Bus 2, Seat 1

Bus 1, Seat 3  (all those whose bus and seat add to 4)

Bus 2, Seat 2

Bus 3, Seat 1

Etc.”

Once in order, the next new person is given the next odd number and goes to that bed.

SO….. Aleph Null times Aleph Null = Aleph Null!

AND… check it out… think of all those bus and seat numbers as numerators and denominators of fractions.  There are just as many whole numbers as there are fractions!

BOOM!  That is the sound of your mind blowing.