Pi

*(Pi)
 
The constant pi, denoted*, is a real number and found everywhere in nature.  The constant pi is known to be irrational and transcendental number. The precise pi is determined through the ages and its decimal is never ending.  In nowadays the value of pi is known to more than one trillion places via the methods of the Gregory-Leibnize series, Maclaurin series and Machin-like formulas.  Most recently, the new formula called BBP-type series is used to compute the nth binary or hexadecimal digit of pi without having to compute the preceding n –1 digits.  Below are the BBP-type series with the base of 16 and 4096.

 

The formulas for the BBP-type with the base of 16 are
 
  •  .   (BBP-Type - this series was discovered by Bailey-Borwein-Plouffe in 1995.)
 
 

 

  • .

 

 
The BBP-type series formulas are found in the base of 4096 that causes the infinite series converges extraordinarily rapidly, namely

 

  •  .

 (posted May 29, 2006)

 
  • .

(posted July 4, 2006)

 
(Click here to view these series in full window size without the left/right menu bar.) 

 
Other infinite series of pi are found as showing in the following:
 
  •  

 

  

 

.

 
or
  •    

 

 

.

 
 
 
  •    

 

 

 

 

 

 

 

.

 
(June 20, 2005)
 
 
 
 
 
 
 
 
 
 
 
 
.
 
 
 
  • .

 

(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, 07/04/2006, from Series Math Study Resource.
Hyperlink: http://seriesmathstudy.com/sms/pi1.

The mathematical rules of the universe are visible to men in the form of beauty.

(John Michel)