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A sample from this course: π
There are many connections that can be made.
Circle
' 3.1459
transcendental #
Irrational #
√
2
circumference
area
Taylor series
geometry
sin(π) = 0
1 radian = π/180
o
degrees
Greek letter
real #
rational #
uncountably infinite
countably infinite
algebraic #
transcendental
normal #
world record
Archimedes
squaring the circle
proof
1
Definition of π: For any circle,
Cir cumf er ence
π=
Diameter
Perimeter in Greek: περιµετ ρoς
Ratio same for all circles, and slightly more than 3, known to
ancient Egyptian, Babylonian, Indian and Greek geometers.
Archimedes (287-212 BC) 223/71 < π < 22/7
wikipedia
2
An applet
http://upload.wikimedia.org/wikipedia/commons/2/2a/Piunrolled-720.gif
3
Trigonometry
Sine equals opposite over hypotenuse
Tangent equals opposite over adjacent
π
π
tan( ) = 1, arctan(1) = , π = 4 arctan(1), sin(π) = 0
4
4
4
Infinite series.
In principle can calculate π to any desired accuracy by
adding sufficiently many terms.
The Taylor series for arctan(x) centered about x = 0 is:
x5
x7
x9
x3
+
−
+
− ···
arctan(x) = x −
3
5
7
9
1
1
1
1
arctan(1) = 1 − + − + − · · ·
3
5
7
9
4
4
4
4
π = 4 − + − + − ···
3
5
7
9
5
6
The Error in Using Taylor Polynomials.
4
4
4
4
π = 4 − + − + − ···
3
5
7
9
How many terms do you need to get 5 decimal places of
accuracy? We know by Taylor’s Theorem that as you use
more terms in the approximation, the error goes to zero.
The error terms:
Using P7 (1) gives 2.895 with error E7 ≤ 0.247
Using P21 (1) gives 3.041 with error E21 ≤ 0.100
Using P51 (1) gives 3.182 with error E51 ≤ 0.040
7
Another way to approximate π
From Calculus - Integration.
Z
1
π=2
p
1 − x 2 dx
−1
8
By the definition of the definite integral.
Z
b
f (x) dx = lim
a
n→∞
9
n
X
i=1
f (xi )∆x
By the definition of the definite integral.
b
Z
f (x) dx = lim
a
Z
1
π=2
−1
n→∞
n
X
f (xi )∆x
i=1
n q
p
X
2
2
2
1 − x dx ' 2 ·
1 − xi ·
n
i=1
9
By the definition of the definite integral.
b
Z
f (x) dx = lim
n→∞
a
Z
1
π=2
−1
n
X
f (xi )∆x
i=1
n q
p
X
2
2
2
1 − x dx ' 2 ·
1 − xi ·
n
i=1
n
sum
error
En
7
2.96
≤ 0.182
≤ 0.247
21
3.07
≤ 0.072
≤ 0.100
51
3.13
≤ 0.012
≤ 0.040
9
3.14159265358979323846264338327950288419716939
9375105820974944592307816406286208998628034825
3421170679821480865132823066470938446095505822
3172535940812848111745028410270193852110555964
4622948954930381964428810975665933446128475648
2337867831652712019091456485669234603486104543
2664821339360726024914127372458700660631558817
4881520920962829254091715364367892590360011330
5305488204665213841469519415116097892590360011
3305305488204665213841469519415116094330572703
6575959195309218611738193261179310511854807446
2379962749567351885752724891227938183011949129
8336733624406566430860213949463952247371907021
7986094370277053921717629317675238467481846766
9405132000568127145263560827785771342757789609
1736371787214684409012249534301465495853710507
10
9227968925892354201995611212902196086403441815
9813629774771309960518707211349999998372978049
9510597317328160963185950244594553469083026425
2230825334468503526193118817101000313783875288
6587533208381420617177669147303598253490428755
4687311595628638823537875937519577818577805321
7122680661300192787661119590921642019893809525
7201065485863278865936153381827968230301952035
3018529689957736225994138912497217752834791315
1557485724245415069595082953311686172785588907
5098381754637464939319255060400927701671139009
8488240128583616035637076601047101819429555961
9894676783744944825537977472684710404753464620
80466842590694912933136770289891521047521...
http://www.eveandersson.com/pi/digits/1000000.txt?
11
Number systems
π is a real number.
π is an irrational number.
Proof (College Junior level math).
π is a transcendental number.
Proof (College Junior or Senior level math).
π is an infinite, non-repeating decimal.
12
The circle cannot be squared.
To square the circle of radius r ,
we need to solve
s2
=
πr 2 ,
or s =
√
π · r.
The rules of ruler and compass construction the Greeks had
laid out enabled them only to construct finite combinations
of sums, differences, products, quotients, and square roots
of lengths of given segments. It follows that every length
that can be constructed is an algebr aic number , a number
that can be the solution to a polynomial equation with
integer coefficients.
In 1882, Ferdinand Lindemann proved that π is a
tr anscendental number, not an algebr aic number.
13
Applications
• it occurs in the normalization of the normal distribution,
• in the distribution of primes,
• in the construction of numbers which are very close to
integers (the Ramanujan constant), and
• in the probability that a pin dropped on a set of parallel
lines intersects a line (Buffon’s needle problem).
• Pi also appears as the average ratio of the actual length
and the direct distance between source and mouth in a
meandering river (Stlum 1996, Singh 1997).
14
Pi jokes
The formula for the volume of a cylinder leads to the
mathematical joke: ”What is the volume of a pizza of
thickness a and radius z ?”
15
Pi jokes
The formula for the volume of a cylinder leads to the
mathematical joke: ”What is the volume of a pizza of
thickness a and radius z ?”
Answer: pi z z a. This result is sometimes known as the
second pizza theorem.
15
Pi jokes
The formula for the volume of a cylinder leads to the
mathematical joke: ”What is the volume of a pizza of
thickness a and radius z ?”
Answer: pi z z a. This result is sometimes known as the
second pizza theorem.
What is the area of a circle of radius R?
15
Pi jokes
The formula for the volume of a cylinder leads to the
mathematical joke: ”What is the volume of a pizza of
thickness a and radius z ?”
Answer: pi z z a. This result is sometimes known as the
second pizza theorem.
What is the area of a circle of radius R?
Pi “R” Squared.
15
Pi jokes
The formula for the volume of a cylinder leads to the
mathematical joke: ”What is the volume of a pizza of
thickness a and radius z ?”
Answer: pi z z a. This result is sometimes known as the
second pizza theorem.
What is the area of a circle of radius R?
Pi “R” Squared.
No - Pie are round!
15
Sample questions for teachers.
1. Estimate π using perimeters of regular polygons.
2. Use transformations to find the area of an ellipse.
3. Show that each of the following numbers are algebraic.
√
− 12,
1+
√
3,
√
2+
√
3,
1 − 51/3
4. Show that the set of algebraic numbers is countably
infinite.
5. Show that the open interval (0, 1) of real numbers
between 0 and 1 is not countably infinite.
Questions taken from: Mathematics for High School
Teachers, An Advanced Perspective. By Usiskin, Peressini,
Marchisotto, and Stanley. Prentice Hall. 2003.
16
Discussion questions
1. What are some benefits of teachers having this kind of
preparation?
2. What kind of preparation should teachers of teachers
have?
3. What other experiences should teachers in training
have?
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