The cool days are sparse in November, but they are generally more “special” than having all even numbered day, month, year. This week, tomorrow is an Addition Day:  11 + 3 = 14.

One thing you can do when you write checks tomorrow is put 11+3=14 in the date line, rather than 11/03/14. The bank doesn’t seem to care!

This week has the month’s last even numbered, multiples-of-two dates – 10/26, 10/28, and 10/30.

That is all for this week, but count your blessings because next week we jump into sparse November.

There are three days this week which are all multiples of two – even numbered month, day, year –  10/20/14, 10/22/14, and 10/24/14.

Next Saturday is also a minus subtract day, as 10 – 24 = -14.

A fun day occurs for high school chemistry students on 10/23 – Mole day.  A mole is a huge number for measuring molecular weight; 6.02 times 10 to the 23^rd power(6.02 * 10^23).  It would seem like June 2 would be more appropriate, but high school teachers noted that many schools are out for the summer.  They may have been jealous of the math teachers who had pi day in March, or maybe they just wanted a better opportunity for solid and fun teaching on a vital topic.

This week has a plethora of cool dates!  It is always fun to use the word “plethora.”

Being an even numbered month (10) in an even numbered year (14), there are four dates whose components are all multiples of two: 10/12, 10/14, 10/16, and 10/18.

And the half-back day for the month is next Saturday, 10/18/14.  Go up 8 from 10 to 18 and come back halfway (4) to 14.

The REALLY special date is today. 10/12/14 is a sequence of 2’s… appropriate in this even-happy month, and the sequence is made of consecutive even numbers!  The next similar event is next year on 11/13/15.  Then 12/14/16, which will be the last time this century.

A return to evenness.  Three days this week have even numbers for month, day, and year: 10/6/14, 10/8/14, and 10/10/14.

That’s it.

When October arrives later this week, we return to an even-numbered month where every other day has its month, day, and year all divisible by two. This week Thursday and Saturday are 10/2/14 and 10/4/14.

And, for variety, Saturday is also an Addition day.  10 + 4 = 14.

September only had three special dates.  Fortunately, the three dates are in three separate weeks.  We are honored to entertain 9/23/14 THIS week.  It is a Minus Subtract day since 9 – 23 = -14.

If you waded  through last week’s discussions of levels of infinity, then this week’s Half-Back day (9/19/14) will seem very simple.  Remember that Half-Back Days are named after the retirees who move from northern states (like Michigan, Pennsylvania, New York, and Wisconsin) to Florida, but then find the climate too warm and move half-way back to more temperate states like Tennessee and North Carolina. 

With that in mind, see how you can count 10 from 9 to 19 (month to day) and then count 5 – half-way – back from 19 to 14.

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.