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
Click here to introduce the idea of a paradox
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 an Activity with Paradox -- "Wrecked Angle"
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 + ...
Click here for a story about John von Neumann and Infinite Series
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
Click here for Trinity
Click here for Fleas
Click here for Infinity
Click here for Infinity
Click here for 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)
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