the joy of pi - winthrop universityfaculty.winthrop.edu/abernathyk/pi day.pdf · search pi 2. it...
TRANSCRIPT
![Page 1: The Joy of Pi - Winthrop Universityfaculty.winthrop.edu/abernathyk/Pi Day.pdf · Search Pi 2. It All Began... \And he made a molten sea, ten cubits from the one brim to the other:](https://reader034.vdocuments.mx/reader034/viewer/2022042306/5ed249d657a90c76e074c6d3/html5/thumbnails/1.jpg)
The Joy of Pi
Kristen Kobylus Abernathy
Winthrop University
3.14
1
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Obsession with Pi
People with time on their hands
Pi in Song
Search Pi
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It All Began...
“And he made a molten sea, ten cubits from the one brim to the other: itwas round all about, and his height was five cubits: and a line of thirtycubits did compass it about.” (I Kings 7, 23)
This verse refers to an object built for the great temple of Solomon, builtaround 950 BC, and gives π = 3.
However, the first trace of a calculation of π can be found in the EgyptianRhind Papyrus, which is dated about 1650 BC and gives
π = 4(89
)2= 3.16.
3
![Page 4: The Joy of Pi - Winthrop Universityfaculty.winthrop.edu/abernathyk/Pi Day.pdf · Search Pi 2. It All Began... \And he made a molten sea, ten cubits from the one brim to the other:](https://reader034.vdocuments.mx/reader034/viewer/2022042306/5ed249d657a90c76e074c6d3/html5/thumbnails/4.jpg)
It All Began...
“And he made a molten sea, ten cubits from the one brim to the other: itwas round all about, and his height was five cubits: and a line of thirtycubits did compass it about.” (I Kings 7, 23)
This verse refers to an object built for the great temple of Solomon, builtaround 950 BC, and gives π = 3.
However, the first trace of a calculation of π can be found in the EgyptianRhind Papyrus, which is dated about 1650 BC and gives
π = 4(89
)2= 3.16.
3
![Page 5: The Joy of Pi - Winthrop Universityfaculty.winthrop.edu/abernathyk/Pi Day.pdf · Search Pi 2. It All Began... \And he made a molten sea, ten cubits from the one brim to the other:](https://reader034.vdocuments.mx/reader034/viewer/2022042306/5ed249d657a90c76e074c6d3/html5/thumbnails/5.jpg)
It All Began...
“And he made a molten sea, ten cubits from the one brim to the other: itwas round all about, and his height was five cubits: and a line of thirtycubits did compass it about.” (I Kings 7, 23)
This verse refers to an object built for the great temple of Solomon, builtaround 950 BC, and gives π = 3.
However, the first trace of a calculation of π can be found in the EgyptianRhind Papyrus, which is dated about 1650 BC and gives
π = 4(89
)2= 3.16.
3
![Page 6: The Joy of Pi - Winthrop Universityfaculty.winthrop.edu/abernathyk/Pi Day.pdf · Search Pi 2. It All Began... \And he made a molten sea, ten cubits from the one brim to the other:](https://reader034.vdocuments.mx/reader034/viewer/2022042306/5ed249d657a90c76e074c6d3/html5/thumbnails/6.jpg)
A Little History
Archimedes (287 - 212 BC) is credited with being the first to calculate πtheoretically. He obtained the approximation
223
71< π <
22
7.
Euler adopted the symbol π in 1737 (after it was introduced by WilliamJones in 1706) and it quickly became standard notation.
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![Page 7: The Joy of Pi - Winthrop Universityfaculty.winthrop.edu/abernathyk/Pi Day.pdf · Search Pi 2. It All Began... \And he made a molten sea, ten cubits from the one brim to the other:](https://reader034.vdocuments.mx/reader034/viewer/2022042306/5ed249d657a90c76e074c6d3/html5/thumbnails/7.jpg)
A Little History
Archimedes (287 - 212 BC) is credited with being the first to calculate πtheoretically. He obtained the approximation
223
71< π <
22
7.
Euler adopted the symbol π in 1737 (after it was introduced by WilliamJones in 1706) and it quickly became standard notation.
4
![Page 8: The Joy of Pi - Winthrop Universityfaculty.winthrop.edu/abernathyk/Pi Day.pdf · Search Pi 2. It All Began... \And he made a molten sea, ten cubits from the one brim to the other:](https://reader034.vdocuments.mx/reader034/viewer/2022042306/5ed249d657a90c76e074c6d3/html5/thumbnails/8.jpg)
Buffon’s Needle Experiment
If we have a uniform grid of parallel lines, unit distance apart and if wedrop a needle of length k < 1 on the grid, the probability that the needlefalls across a line is 2k
π .
Various people have tried to calculate π by throwing needles. The mostremarkable result was that of Lazzerini (1901), who made 34080 tossesand got
π =335
113= 3.1415929.
5
![Page 9: The Joy of Pi - Winthrop Universityfaculty.winthrop.edu/abernathyk/Pi Day.pdf · Search Pi 2. It All Began... \And he made a molten sea, ten cubits from the one brim to the other:](https://reader034.vdocuments.mx/reader034/viewer/2022042306/5ed249d657a90c76e074c6d3/html5/thumbnails/9.jpg)
Buffon’s Needle Experiment
If we have a uniform grid of parallel lines, unit distance apart and if wedrop a needle of length k < 1 on the grid, the probability that the needlefalls across a line is 2k
π .Various people have tried to calculate π by throwing needles. The mostremarkable result was that of Lazzerini (1901), who made 34080 tossesand got
π =335
113= 3.1415929.
5
![Page 10: The Joy of Pi - Winthrop Universityfaculty.winthrop.edu/abernathyk/Pi Day.pdf · Search Pi 2. It All Began... \And he made a molten sea, ten cubits from the one brim to the other:](https://reader034.vdocuments.mx/reader034/viewer/2022042306/5ed249d657a90c76e074c6d3/html5/thumbnails/10.jpg)
Some open questions about the number π
Does each of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 each occur infinitelyoften in π?
Brouwer’s question: In the decimal expansion of π, is there a placewhere a thousand consecutive digits are all zero?
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Where does our distance formula for R2 come from?
By Pythagoras’ Theorem,
d2 = (x2 − x1)2 + (y2 − y1)2.
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Where does our distance formula for R2 come from?
By Pythagoras’ Theorem,
d2 = (x2 − x1)2 + (y2 − y1)2.
7
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What does this have to do with the unit circle?
The unit circle is all points (x , y) that are one-unit distance from theorigin; ie,
1 =√
(x − 0)2 + (y − 0)2
1 = x2 + y2
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What does this have to do with the unit circle?
The unit circle is all points (x , y) that are one-unit distance from theorigin; ie,
1 =√
(x − 0)2 + (y − 0)2
1 = x2 + y2
8
![Page 15: The Joy of Pi - Winthrop Universityfaculty.winthrop.edu/abernathyk/Pi Day.pdf · Search Pi 2. It All Began... \And he made a molten sea, ten cubits from the one brim to the other:](https://reader034.vdocuments.mx/reader034/viewer/2022042306/5ed249d657a90c76e074c6d3/html5/thumbnails/15.jpg)
What’s in a norm?
Given a vector space V over a subfield F of the complex numbers, a normon V is a function ‖ · ‖ : V → F with the following properties:For all a ∈ F and u, v ∈ V ,
‖v‖ ≥ 0 and ‖v‖ = 0 iff v = 0;
‖av‖ = |a|‖v‖;‖u + v‖ ≤ ‖u‖+ ‖v‖ (Triangle Inequality).
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What’s in a norm?
Given a vector space V over a subfield F of the complex numbers, a normon V is a function ‖ · ‖ : V → F with the following properties:For all a ∈ F and u, v ∈ V ,
‖v‖ ≥ 0 and ‖v‖ = 0 iff v = 0;
‖av‖ = |a|‖v‖;‖u + v‖ ≤ ‖u‖+ ‖v‖ (Triangle Inequality).
9
![Page 17: The Joy of Pi - Winthrop Universityfaculty.winthrop.edu/abernathyk/Pi Day.pdf · Search Pi 2. It All Began... \And he made a molten sea, ten cubits from the one brim to the other:](https://reader034.vdocuments.mx/reader034/viewer/2022042306/5ed249d657a90c76e074c6d3/html5/thumbnails/17.jpg)
What’s in a norm?
Given a vector space V over a subfield F of the complex numbers, a normon V is a function ‖ · ‖ : V → F with the following properties:For all a ∈ F and u, v ∈ V ,
‖v‖ ≥ 0 and ‖v‖ = 0 iff v = 0;
‖av‖ = |a|‖v‖;
‖u + v‖ ≤ ‖u‖+ ‖v‖ (Triangle Inequality).
9
![Page 18: The Joy of Pi - Winthrop Universityfaculty.winthrop.edu/abernathyk/Pi Day.pdf · Search Pi 2. It All Began... \And he made a molten sea, ten cubits from the one brim to the other:](https://reader034.vdocuments.mx/reader034/viewer/2022042306/5ed249d657a90c76e074c6d3/html5/thumbnails/18.jpg)
What’s in a norm?
Given a vector space V over a subfield F of the complex numbers, a normon V is a function ‖ · ‖ : V → F with the following properties:For all a ∈ F and u, v ∈ V ,
‖v‖ ≥ 0 and ‖v‖ = 0 iff v = 0;
‖av‖ = |a|‖v‖;‖u + v‖ ≤ ‖u‖+ ‖v‖ (Triangle Inequality).
9
![Page 19: The Joy of Pi - Winthrop Universityfaculty.winthrop.edu/abernathyk/Pi Day.pdf · Search Pi 2. It All Began... \And he made a molten sea, ten cubits from the one brim to the other:](https://reader034.vdocuments.mx/reader034/viewer/2022042306/5ed249d657a90c76e074c6d3/html5/thumbnails/19.jpg)
A couple examples
The absolute value on the space of real numbers is a norm.
The Euclidean norm for Rn is
‖ < x1, x2, · · · , xn > ‖ =√
x21 + x22 + · · ·+ x2n .
The Taxicab norm (or Manhattan norm) for Rn is
‖ < x1, x2, · · · , xn > ‖1 = |x1|+ |x2|+ · · · |xn|.
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A couple examples
The absolute value on the space of real numbers is a norm.
The Euclidean norm for Rn is
‖ < x1, x2, · · · , xn > ‖ =√
x21 + x22 + · · ·+ x2n .
The Taxicab norm (or Manhattan norm) for Rn is
‖ < x1, x2, · · · , xn > ‖1 = |x1|+ |x2|+ · · · |xn|.
10
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A couple examples
The absolute value on the space of real numbers is a norm.
The Euclidean norm for Rn is
‖ < x1, x2, · · · , xn > ‖ =√
x21 + x22 + · · ·+ x2n .
The Taxicab norm (or Manhattan norm) for Rn is
‖ < x1, x2, · · · , xn > ‖1 = |x1|+ |x2|+ · · · |xn|.
10
![Page 22: The Joy of Pi - Winthrop Universityfaculty.winthrop.edu/abernathyk/Pi Day.pdf · Search Pi 2. It All Began... \And he made a molten sea, ten cubits from the one brim to the other:](https://reader034.vdocuments.mx/reader034/viewer/2022042306/5ed249d657a90c76e074c6d3/html5/thumbnails/22.jpg)
A couple examples
The absolute value on the space of real numbers is a norm.
The Euclidean norm for Rn is
‖ < x1, x2, · · · , xn > ‖ =√
x21 + x22 + · · ·+ x2n .
The Taxicab norm (or Manhattan norm) for Rn is
‖ < x1, x2, · · · , xn > ‖1 = |x1|+ |x2|+ · · · |xn|.10
![Page 23: The Joy of Pi - Winthrop Universityfaculty.winthrop.edu/abernathyk/Pi Day.pdf · Search Pi 2. It All Began... \And he made a molten sea, ten cubits from the one brim to the other:](https://reader034.vdocuments.mx/reader034/viewer/2022042306/5ed249d657a90c76e074c6d3/html5/thumbnails/23.jpg)
Back to our idea of distance in R2
What if we used the Taxicab norm (instead of the Euclidean norm) tocompute the distance between two points?
d((x1, y1), (x2, y2)) = |x2 − x1|+ |y2 − y1|
“Unit circle”
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Back to our idea of distance in R2
What if we used the Taxicab norm (instead of the Euclidean norm) tocompute the distance between two points?
d((x1, y1), (x2, y2)) = |x2 − x1|+ |y2 − y1|
“Unit circle”
11
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To infinity and beyond!
Let’s define “p-distance” as
dp((x1, y1), (x2, y2)) = (|x2 − x1|p + |y2 − y1|p)1/p = ‖x2 − x1‖pwhere x1 = (x1, y1) and x2 = (x2, y2). We can now visualize what the unitball (ie, the set of all points of distance 1 from the origin) looks like underthese different norms in R2.
Notice as p gets larger, the unit ball is approaching
We define
d∞((x1, y1), (x2, y2)) = max{|x2 − x1|, |y2 − y1|}.
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To infinity and beyond!
Let’s define “p-distance” as
dp((x1, y1), (x2, y2)) = (|x2 − x1|p + |y2 − y1|p)1/p = ‖x2 − x1‖pwhere x1 = (x1, y1) and x2 = (x2, y2). We can now visualize what the unitball (ie, the set of all points of distance 1 from the origin) looks like underthese different norms in R2.Notice as p gets larger, the unit ball is approaching
We define
d∞((x1, y1), (x2, y2)) = max{|x2 − x1|, |y2 − y1|}.
12
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To infinity and beyond!
Let’s define “p-distance” as
dp((x1, y1), (x2, y2)) = (|x2 − x1|p + |y2 − y1|p)1/p = ‖x2 − x1‖pwhere x1 = (x1, y1) and x2 = (x2, y2). We can now visualize what the unitball (ie, the set of all points of distance 1 from the origin) looks like underthese different norms in R2.Notice as p gets larger, the unit ball is approaching
We define
d∞((x1, y1), (x2, y2)) = max{|x2 − x1|, |y2 − y1|}.12
![Page 28: The Joy of Pi - Winthrop Universityfaculty.winthrop.edu/abernathyk/Pi Day.pdf · Search Pi 2. It All Began... \And he made a molten sea, ten cubits from the one brim to the other:](https://reader034.vdocuments.mx/reader034/viewer/2022042306/5ed249d657a90c76e074c6d3/html5/thumbnails/28.jpg)
Pi equals what?!
π is defined as π = C2r where C represents the circumference of a circle,
and r is the circle’s radius. What happens to the value of π as we changehow we calculate distance?
For example, in the 1-norm,
d((1, 0), (0, 1))1 = |0− 1|+ |1− 0| = 2 so C = 8 and r = 1 which givesπ = 4!
13
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Pi equals what?!
π is defined as π = C2r where C represents the circumference of a circle,
and r is the circle’s radius. What happens to the value of π as we changehow we calculate distance?For example, in the 1-norm,
d((1, 0), (0, 1))1 = |0− 1|+ |1− 0| = 2 so C = 8 and r = 1 which gives
π = 4!
13
![Page 30: The Joy of Pi - Winthrop Universityfaculty.winthrop.edu/abernathyk/Pi Day.pdf · Search Pi 2. It All Began... \And he made a molten sea, ten cubits from the one brim to the other:](https://reader034.vdocuments.mx/reader034/viewer/2022042306/5ed249d657a90c76e074c6d3/html5/thumbnails/30.jpg)
Pi equals what?!
π is defined as π = C2r where C represents the circumference of a circle,
and r is the circle’s radius. What happens to the value of π as we changehow we calculate distance?For example, in the 1-norm,
d((1, 0), (0, 1))1 = |0− 1|+ |1− 0| = 2 so C = 8 and r = 1 which givesπ = 4!
13
![Page 31: The Joy of Pi - Winthrop Universityfaculty.winthrop.edu/abernathyk/Pi Day.pdf · Search Pi 2. It All Began... \And he made a molten sea, ten cubits from the one brim to the other:](https://reader034.vdocuments.mx/reader034/viewer/2022042306/5ed249d657a90c76e074c6d3/html5/thumbnails/31.jpg)
What?!
From Calculus, we can compute the length of the curve y = f (t) fromt = a to t = b using a formula for arc length:
L =
∫ b
a
√1 + [f ′(t)]2dt.
We can calculate what π would be in any norm by adjusting our formulafor arc length:
π = 2 ∗∫ 1
0
(1 +
(1
p(1− tp)(1−p)/p(−pt)
)p)(1/p)
dt
14
![Page 32: The Joy of Pi - Winthrop Universityfaculty.winthrop.edu/abernathyk/Pi Day.pdf · Search Pi 2. It All Began... \And he made a molten sea, ten cubits from the one brim to the other:](https://reader034.vdocuments.mx/reader034/viewer/2022042306/5ed249d657a90c76e074c6d3/html5/thumbnails/32.jpg)
What?!
From Calculus, we can compute the length of the curve y = f (t) fromt = a to t = b using a formula for arc length:
L =
∫ b
a
√1 + [f ′(t)]2dt.
We can calculate what π would be in any norm by adjusting our formulafor arc length:
π = 2 ∗∫ 1
0
(1 +
(1
p(1− tp)(1−p)/p(−pt)
)p)(1/p)
dt
14
![Page 33: The Joy of Pi - Winthrop Universityfaculty.winthrop.edu/abernathyk/Pi Day.pdf · Search Pi 2. It All Began... \And he made a molten sea, ten cubits from the one brim to the other:](https://reader034.vdocuments.mx/reader034/viewer/2022042306/5ed249d657a90c76e074c6d3/html5/thumbnails/33.jpg)
What?!
From Calculus, we can compute the length of the curve y = f (t) fromt = a to t = b using a formula for arc length:
L =
∫ b
a
√1 + [f ′(t)]2dt.
We can calculate what π would be in any norm by adjusting our formulafor arc length:
π = 2 ∗∫ 1
0
(1 +
(1
p(1− tp)(1−p)/p(−pt)
)p)(1/p)
dt
14
![Page 34: The Joy of Pi - Winthrop Universityfaculty.winthrop.edu/abernathyk/Pi Day.pdf · Search Pi 2. It All Began... \And he made a molten sea, ten cubits from the one brim to the other:](https://reader034.vdocuments.mx/reader034/viewer/2022042306/5ed249d657a90c76e074c6d3/html5/thumbnails/34.jpg)
Thank you for having me!!
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