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Newton's Pi Approximation
This Demonstration gives Newton's approximation of
based on calculating the area of a semicircle using an integral.
The area
under the arc
(light blue) is equal to the area of the sector
minus the area of triangle
(light green), that is,
.
On the other hand, the semicircle has equation
, so
. Set
and
.
Therefore,
.
Contributed by:
Izidor Hafner
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Newton presented
to 16 decimal places using 20 terms of the binomial series [1, p. 177].
Reference
[1] W. Dunham,
Journey through Genius
, New York: Penguin Books, 1990 pp. 174–177.
RELATED LINKS
Gauss-Legendre Approximation of Pi
(
Wolfram Demonstrations Project
)
Pi
(
Wolfram
MathWorld
)
Vega's Calculation of Pi
(
Wolfram Demonstrations Project
)
Vega's Second Calculation of Pi
(
Wolfram Demonstrations Project
)
Euler's Estimate of Pi
(
Wolfram Demonstrations Project
)
PERMANENT CITATION
Izidor Hafner
"
Newton's Pi Approximation
"
http://demonstrations.wolfram.com/NewtonsPiApproximation/
Wolfram Demonstrations Project
Published: August 22, 2013
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