Quote of the Day: 
     "Perhaps the greatest paradox of all is that there are paradoxes 
      in mathematics." -- Kasner and Newman

Objectives:
    The student will be able to determine the sum of an infinite 
	geometric series.
    The student will be able to identify a paradox.     
	
1. Collect Homework Assignment.

2. Paradoxes 

    Examples of paradoxes               
	(1) Please ignore this notice.
	(2) This is not a sentence.
	(3) All rules have exceptions.
	(4) Never say never.
	(5) It is forbidden to forbid.

     Hymn #283  Presbyterian Hymnal
        We are not free when we're confined
        To every wish that sweeps the mind.
        But free when freely we accept
        The sacred bounds that must be kept.
            
     Other paradoxes:
	(1) Groucho Marx – 
		"I would never belong to any club that would 
                  have me as a member."

 	(2) Prayer of St. Francis of Assisi
                "For it is by giving that we receive;
                 It is in pardoning that we are pardoned;
                 And it is in dying that we are born 
                    to eternal life."

	(3) Socrates: "What Plato is about to say is false."
		 Plato:  "Socrates has just spoken truly."

	(4) Piece of paper:
		On one side it reads, 
                    "The sentence on the other side of this 
                       paper is false."

		On the other side it reads, 
                     "The sentence on the other side of this 
                      paper is true."

	(5) A man is about to be executed.  
                He can make one statement.
		If he tells the truth, he will be electrocuted.
		If he tells a lie, he will be hanged.
			
		He says, "I will be hanged."
		They had to let him go.

              Click here for a Comic Strip on Paradoxes

3. Infinite Geometric Series

       What is the sum of  1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ... ?
       Draw a square to illustrate (see Student Math Notes NCTM March 1983)
              

        Use the formula for the sum of an infinite geometric sum : 

                 



       Examples:  Determine the sum of   1 + 2/3 + 4/9 + …
                  Determine the sum of   12 + 3 + 3/4 + …

4. Zeno's Paradoxes

      1. The Greek warrior Achilles is racing a slow tortoise that has
         been given a head start. But every time Achilles reaches the 
         point where the tortoise was, the tortoise has moved to a new 
         position ahead of him. And when he runs to that position, the 
         tortoise has moved again, and again, and again. For each and 
         every new point Achilles reaches, the tortoise has been there 
         and moved on. Hence the tortoise is always ahead in the race 
         and so Achilles can never win. Or can he? 

		Achilles races a tortoise that has a 100-meter head
		start. Achilles travels 10 meters/second, and the tortoise 
		travels only 1 meter/second. Yet Achilles can
		never catch the tortoise because, when he has traveled
		the 100 meters, which was the tortoise's original head
		start, the tortoise will have traveled 10 meters and still
		be ahead. When Achilles has traveled those 10 meters,
		the tortoise Will still be 1 meter ahead. After Achilles
		travels that 1 meter, the tortoise will still be ahead by
		0.1 meter. This process goes on without end, so
		Achilles can never catch the tortoise.

		I then ask students to respond to the following
		prompts:
		1. Use basic algebra to find the time when Achilles
			does, in fact, catch the tortoise.
		2. Explain why the argument in the paragraph
			above is misleading. How does it mislead the
			reader (and momentarily make some of us think
			that maybe Achilles will not catch the tortoise)?


     2. An arrow is shot into the air. But does it ever move? At each 
        and every instant the arrow is at some particular point standing 
        still. But if it is always standing still at each and every 
        point, how can it be moving?

     3. When the bell rings, you walk towards the door of your calculus 
        class in the following manner: First, you walk half-way there; 
        then you half of that distance; then you go half of the remaining 
        distance;  and so on.  You'll NEVER get out of calculus class!!!!!

      Explanation:  The Greeks couldn't comprehend that an infinite sum
                    of numbers could have a finite sum! 

        This was used in the 1994 movie called I.Q. where Walter Matthau as 
        Albert Einstein plays matchmaker for his niece played by Meg Ryan. 
        Meg Ryan attempts to explain to Tim Robbins why she can't dance
        with him: she can only walk half the distance between them and then 
        half again and half again and she will never reach him.

5. Bungee Jump Problem – The Winchester Star April 18, 1992
	In 1992, at Shiley Acres in Bunker Hill, West Virginia, they opened 
        a Bungee Jump where people would jump from a platform 150 feet in 
        the air.  They would free fall 120 feet, then recoil 80% of the fall 
        in the opposite direction. 
        How far would a person travel in the air before coming to rest (if 
         they keep recoiling 80% each time)?

6. Lottery
     A mathematician organizes a lottery in which the prize is an infinite amount of money. 
     When the winning ticket is drawn, and the jubilant winner comes to claim his prize, 
     the mathematician explains the mode of payment: 
     "1 dollar now, 1/2 dollar next week, 1/3 dollar the week after that..." 

7. Poems about Infinity

8. Monkeys on Typewriters
   "If one puts an infinite number of monkeys in front of typewriters and lets them clap away,
    There is a certainty that one of them will come out with an exact version of the
    'Iliad," writes Nassim Nicholas Taleb in a recent book, "Fooled By Randomness."

   The monkey typist story is an old one (what's a typewriter?) and the key word is infinite.

9. Assignment: 
             Worksheets on Infinity (Student Math Notes NCTM March 1983)
             p. 37 (33a,b,c,e, 37, 39, 42, 47)
             p. 63 (12, 16, 20, 24, 38, 39)
             p. 650 (17, 19)