Submitted by admin on Tue, 03/17/2009 - 4:23pm
(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 ofhas 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 ofwithout needing to calculate the preceding (n-1) digits. Below are examples of BBP-like series with bases of 16 and 4096.
, 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 ofhas 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 ofwithout 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)
(Click here to view in full window.)
3. Other infinite series, which yield results connecting to pi, are shown below:
.
or
.
.
(June 20, 2005)
.-
.
(05/29/2006)
(07/04/2006 - update)
(03/17/2009 - update)
More Related Series
- Other BBP-type series.
- BBP-type series in Connection to one of the Roots of Quartic Equation.
- BBP-type series in connection ln2 and ln3.
In-Text or Website Citation
Tue N. Vu, Pi, from Series Math Study Resource.
Hyperlink: http://seriesmathstudy.com/sms/pi1.
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