Transcript The Golden Rectangle - Biblical Christian World View

Zeno’s Paradoxes
by James D. Nickel
Copyright  2007
www.biblicalchristianworldview.net
Zeno of Elea
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ca. 490 BC – 430 BC.
A pre-Socratic Greek philosopher.
Zeno's paradoxes have puzzled,
challenged, influenced, inspired,
infuriated, and amused philosophers,
mathematicians, physicists and school
children for over two millennia.
The most famous are the so-called
"arguments against motion" described by
Aristotle (384-322 BC), in his Physics.
Dichotomy
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Aristotle “That which is in locomotion must
arrive at the half-way stage before it
arrives at the goal.”
In other words, to reach a goal, you first
must arrive at the half-way mark.
Before you get to the half-way mark, you
must arrive at the quarter mark.
Before you get to the quarter mark, you
must arrive at the eighth mark.
Before you get the the eighth mark, you
must arrive at the sixteenth mark.
Conclusion
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Carried ad infinitum, you cannot
even start!
Another perspective
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Every time I take a step, I get halfway closer to my goal.
I make it to the half-way mark with
my first step.
With my second step I make it to the
quarter mark.
With my third I make it to the eighth
mark.
Will I ever reach my goal?
My Starting Point
My Goal
Step 1
1/2
My Starting Point
My Goal
Step 2
1/2 + 1/4 =
3/4
My Starting Point
My Goal
Step 3
1/2 + 1/4 + 1/8 =
7/8
My Starting Point
My Goal
Step 4
1/2 + 1/4 + 1/8 + 1/16 =
15/16
My Starting Point
My Goal
Step 5
1/2 + 1/4 + 1/8 + 1/16 + 1/32 =
31/32
My Starting Point
My Goal
Step 6
1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 =
63/64
My Starting Point
My Goal
Step 7
1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 1/128 =
127/128
Will I Ever Reach My Goal?
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According to Zeno, it would take an
infinite number of steps to reach my
goal.
To him, this was impossible.
What do you think?
Consider this Infinite Sum
1 1 1 1 1
     ...  ?
2 4 8 16 32
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Draw a square 8 by 8 on graph
paper.
Let the area of the square be 1
square unit.
Now, color in half of it (1/2 - the first
number in the series).
Area of the first term
Sum of the first two terms
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Next, color an additional one-fourth
(1/4 is the second number in the
series).
What is the total area?
3
1 1 42 6 3
 
 
2 4
8
8 4
4
Area of the first and second terms
Sum of the first three terms
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Next, color an additional one-eighth
(1/8 - the third number in the
series).
What is the total area?
7
3 1 24  4 28 7
 

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4 8
32
32 8
8
Area of the first three terms
Sum of the first four terms
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Next, color an additional onesixteenth (1/16 - the fourth number
in the series).
What is the total area?
15
7 1 112  8 120 15
 
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8 16
128
128 16
16
Area of the first four terms
Continue the Process
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Go on and color in 1/16 and 1/32,
the 5th and 6th terms.
Questions
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What is happening to the partial
sums?
Are the sums getting bigger,
smaller, or staying the same size?
Answer: Getting bigger (ever so
slightly).
Questions
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Are the numbers you are adding
each time getting bigger or
smaller?
Answer: Getting smaller (ever so
slightly).
Questions
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Will the square ever get filled as we
keep on going?
Answer: As long as you stop at
some point, there will always be a
tiny bit unfilled.
Questions
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What is the sum getting close to (as
a threshold)?
Answer: 1
Questions
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What is the smallest number that
each term in the series gets closer
and closer to (as a threshold)?
Answer: 0
Mathematical Analysis
1 1 1 1 1
     ...  1
2 4 8 16 32
by James D. Nickel
Copyright  2007
www.biblicalchristianworldview.net