Oscar Veliz | Computing π: Machin-like formula @OscarVeliz | Uploaded 4 years ago | Updated 6 seconds ago
Machin-like formulae are used to find many decimal places of pi, although why they work can seem confusing. This lesson shows how the Taylor Series of arctangent is used to compute π, and the power of combining it with Machin's approach, as well as covering the history of Gregory, Leibniz, Euler, and Machin. Example code: github.com/osveliz/numerical-veliz/tree/master/src/series
Chapters
00:00 - Intro
00:11 - Taylor Series
00:20 - Unit Circle & Solve for π
00:57 - Gregory
01:07 - Gregory-Leibniz Series
01:33 - Leibniz
02:27 - arctan(1)
02:48 - Moving a
03:40 - Increasing n
04:11 - Getting Close
04:44 - Nearer to zero
05:07 - Adding arctangents example
05:28 - Arctangent Trick
05:51 - Euler
06:06 - Euler's Equation for π
06:26 - Deriving Euler's Equation
07:00 - Machin intro
07:15 - Machin-like formula
08:00 - John Machin
08:22 - Demo Code & Story Time
09:40 - 10,000 digits of π
10:08 - Beckmann's thoughts on higher digits
10:27 - Oscar's Notes
11:00 - Thank You
Suggested Viewing:
Origin of Taylor Series - youtu.be/KJYauVgXA4E
Newton Fractals - youtu.be/MWD2A0Vg2V0
Reference Links:
"James Gregory Tercentenary memorial volume" - catalog.hathitrust.org/Record/000438471
"A History of Pi" by Petr Beckmann books.google.com/books?id=XqqUUSyz138C&dq=a+history+of+pi
Leibniz's π paper - books.google.com/books?id=E7MasYIsMKQC&pg=PA41
"On the Leibnizian quadrature of the circle" - http://ac.inf.elte.hu/Vol_004_1983/075.pdf
"The Discovery of the Series Formula for π by Leibniz, Gregory and Nilakantha" - tandfonline.com/doi/pdf/10.1080/0025570X.1990.11977541
"How Euler Did Even More" by C. Edward Sandifer books.google.com/books?id=3c6iBQAAQBAJ&dq=how+euler+did+even+more
Euler's π paper E705 - http://eulerarchive.maa.org/pages/E705.html
"John Machin and Robert Simson on Inverse-tangent Series for π" jstor.org/stable/41133896
"Synopsis palmariorum matheseos" by William Jones - books.google.com/books/about/Synopsis_palmariorum_matheseos.html?id=eZPRxQEACAAJ
CLISP source code for π - github.com/clisp-lang/clisp/blob/520f02b36acbb49a9d0c93870f3dc4cb3d00e6d9/src/realtran.d
#pi #PiDay #π
Machin-like formulae are used to find many decimal places of pi, although why they work can seem confusing. This lesson shows how the Taylor Series of arctangent is used to compute π, and the power of combining it with Machin's approach, as well as covering the history of Gregory, Leibniz, Euler, and Machin. Example code: github.com/osveliz/numerical-veliz/tree/master/src/series
Chapters
00:00 - Intro
00:11 - Taylor Series
00:20 - Unit Circle & Solve for π
00:57 - Gregory
01:07 - Gregory-Leibniz Series
01:33 - Leibniz
02:27 - arctan(1)
02:48 - Moving a
03:40 - Increasing n
04:11 - Getting Close
04:44 - Nearer to zero
05:07 - Adding arctangents example
05:28 - Arctangent Trick
05:51 - Euler
06:06 - Euler's Equation for π
06:26 - Deriving Euler's Equation
07:00 - Machin intro
07:15 - Machin-like formula
08:00 - John Machin
08:22 - Demo Code & Story Time
09:40 - 10,000 digits of π
10:08 - Beckmann's thoughts on higher digits
10:27 - Oscar's Notes
11:00 - Thank You
Suggested Viewing:
Origin of Taylor Series - youtu.be/KJYauVgXA4E
Newton Fractals - youtu.be/MWD2A0Vg2V0
Reference Links:
"James Gregory Tercentenary memorial volume" - catalog.hathitrust.org/Record/000438471
"A History of Pi" by Petr Beckmann books.google.com/books?id=XqqUUSyz138C&dq=a+history+of+pi
Leibniz's π paper - books.google.com/books?id=E7MasYIsMKQC&pg=PA41
"On the Leibnizian quadrature of the circle" - http://ac.inf.elte.hu/Vol_004_1983/075.pdf
"The Discovery of the Series Formula for π by Leibniz, Gregory and Nilakantha" - tandfonline.com/doi/pdf/10.1080/0025570X.1990.11977541
"How Euler Did Even More" by C. Edward Sandifer books.google.com/books?id=3c6iBQAAQBAJ&dq=how+euler+did+even+more
Euler's π paper E705 - http://eulerarchive.maa.org/pages/E705.html
"John Machin and Robert Simson on Inverse-tangent Series for π" jstor.org/stable/41133896
"Synopsis palmariorum matheseos" by William Jones - books.google.com/books/about/Synopsis_palmariorum_matheseos.html?id=eZPRxQEACAAJ
CLISP source code for π - github.com/clisp-lang/clisp/blob/520f02b36acbb49a9d0c93870f3dc4cb3d00e6d9/src/realtran.d
#pi #PiDay #π
