Understanding Pi and Squaring the Circle

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Understanding Pi and Squaring the Circle
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The History of p and Squaring the Circle

• Egypt 1650 BC

Cut off 1/9 of a diameter and construct a
square upon the remainder. This has the same area
as the circle
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(No Transcript)
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Area of Circle Area of Square
pr2 (8/9d)2 (8/9.2r)2
256/81r2
p 256/81
8/9 d
3.16049
8/9 d
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The History of p and Squaring the Circle

• Egypt 1650 BC

The Bible 6th Century BC
Also he made sea of 10 cubits from brim to brim,
round in compass, and five cubits the height
thereof and a line of thirty cubits did compass
round it 1, Kings 723
p 3010 3
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The History of p and Squaring the Circle

• Egypt 1650 BC

The Bible 6th Century BC
Ancient Greeks 4th Century BC
Antiphon
Archimedes
Bryson
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310/71 lt p lt 31/7
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The History of p and Squaring the Circle
François Viète 1579
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The History of p and Squaring the Circle
Gregorys Series
Gottfried Wilhelm Leibniz (1646-1716)
arctan(1) p 1 1 1 1 ...
4 3 5 7
Gregory-Leibniz Series
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The History of p and Squaring the Circle
p/4 5arctan(1/7) 2arctan(3/79)
p/4 2arctan(1/3) arctan(1/7)
p2/6 (22/(22-1))x(32/(32-1))x(52/(52-1))x
p4/90 1/14 1/24 1/34
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SQUARING THE CIRCLE (properly introduced!)

• Construct a Square with exactly the same area as
  a Circle

Use only a ruler and compass
Use only a finite number of steps
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SQUARING THE CIRCLE (properly introduced!)
Archimedes
r
2pr
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History of p and Squaring the Circle

• p is irrational (Lambert 1761)

If were rational, then cos( ) could be
irrational, however cos( ) -1
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History of p and Squaring the Circle

• p is irrational (1761)

v2
1
1
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History of p and Squaring the Circle

• p is transcendental (Lindemann 1882)

If Ai,bi are algebraic with Ai 0 and bi all
different
A1eb1 A2eb2 .. Anebn 0
Eulers Identity
eip 1 0
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The Circle Squarers
ps only position in mathematics is its
relation to infinite series and p has no relation
to the circle
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The Circle Squarers
Lindemann proclaimed that squaring the circle is
impossible, but Lindemanns proof is misleading
for it uses numbers which are approximate in
themselves!
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The Circle Squarers
The Legal Value of Pi
Edwin J Goodwin, Indiana 1888
supernaturally taught the exact measurement of
the circlein due confirmation of scriptural
promises
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The Digits of Pi

• The Chudnovsky Brothers

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The Digits of Pi
No repeating pattern (Lambert 1768)
Fewer 7s in the first 600 digits
ABCDEFGHIJKLMNOPQRSTUVWXYZ
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3
1
4
1
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The Future of Pi
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THE END

• Any Questions ???