Pi

*(Pi)
 
 
 
The constant pi, denoted as*, is a real number that appears throughout nature. It is known to be both irrational and transcendental, meaning its exact value cannot be expressed as a simple fraction, and it is not a solution to any non-constant polynomial equation with rational coefficients. The value of*has been calculated with increasing precision over the centuries, and its decimal representation is infinite and non-repeating. Today,*is known to over one trillion decimal places, thanks to methods such as the Gregory-Leibniz series, Maclaurin series, and Machin-like formulas. More recently, the BBP-like series has been developed, allowing the computation of the nth binary or hexadecimal digit of*without needing to calculate the preceding (n-1) digits. Below are examples of BBP-like series with bases of 16 and 4096.
 
1. The formulas for the BBP-like series with the base of 16 are
 

(This BBP-like series was discovered by Bailey-Borwein-Plouffe in 1995.)

 

 

 

2. The formulas, BBP-like series, have been discovered in the base of 4096, cause the infinite series to converge extraordinarily rapidly, namely

 (posted May 29, 2006)

 

(posted July 4, 2006)

 
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3. Other infinite series, which yield results connecting to pi, are shown below:
 

  

 

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or

 

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(June 20, 2005)
 
 
 
 
 
 
 
 
 
 
 
 
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(05/29/2006)
(07/04/2006 - update)
(03/17/2009 - update)

 

More Related Series 

 

Main SMS

 

In-Text or Website Citation
Tue N. Vu, Pi, from Series Math Study Resource.
Hyperlink: http://seriesmathstudy.com/sms/pi1.