# π GPS (Greater Precision Solutions)

In my previous blog post π Places, I reviewed special fractions that were very good approximations to the value of π to 2 and 6 decimal places: i.e. 22/7 and 355/113 respectively.

In this post, I would like to explore the more modern (since 1593) approximations based on infinite series.

Note: If the mathematics shown creates immediate “eye-glaze”, please skip to the Activities section, which is much more entertaining in comparison.

A Series sums up a finite or infinite collection of terms of a Sequence. If finite, there are a definite, bounded number of values that add up to a specific, measurable value. If infinite, there are an endless, unbounded number of values that add up to either a specific, measurable value (converge), or add up to an indefinite, unmeasurable value (diverge), which is denoted as infinity ().

Here are two examples of each:

#### Finite Series

##### First 10 Natural Numbers sum

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = sum_(n=1)^10 n=(10(10+1))/2=55     [1]

##### π expansion to 7 decimal places:

3 + 1/10 + 4/10^2 + 1/10^3 + 5/10^4 + 9/10^5 + 2/10^6 + 6/10^7 = 3.1415926                 [2]

#### Infinite Series (Convergent)

##### π decimal expansion:

3 + 1/10 + 4/10^2 + 1/10^3 + 5/10^4 + 9/10^5 + 2/10^6 + 6/10^7 + … =  π                      [3]

##### Geometric Series:

1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + … + 1/2^(n-1) + … = sum_(n=1)^oo 1/2^(n-1)=2            [4]

#### Infinite Series (Divergent)

##### Natural Number Series:

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + … = sum_(n=1)^oo n=oo                        [5]

##### Harmonic Series:

1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + … + 1/n + … = sum_(n=1)^oo 1/n=oo                 [6]

For those interested in mathematically (in)finite series, look at the following
List of Mathematical Series. Summed general expressions are given for series organized by sums of powers, power series, binomial coefficients, trigonometric and rational functions. At the top, there are subscripted letters and Greek letter functions (such a necessary annoyance) that are separately named and linked.

### Greater Decimal Places Approximators

Over the years, many mathematicians have tried to evaluate a finite number of terms in an infinite series to approximate π.

• Francois Viete in 1593, using an infinite product, calculated π to 9 decimal digits, by applying the Archimedes method of exhaustion to a polygon with bb 6 × 2^16 = 393216 sides and calculating the circumference (perimeter) when the diameter is 1. π =
2/[sqrt(1/2)*(sqrt[1/2+(1/2)*sqrt(1/2)])*(sqrt[1/2+(1/2)*sqrt{1/2+(1/2)*sqrt{1/2}]))*… [7]

Quite the eye-roller! Alternatively, this looks slightly less volatile, but is equivalent:

π ~~ 2^k*sqrt(2–sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+…))))))) ~~ 3.14157294   [8]

• James Gregory in 1671 published the arctangent (arctan) infinite series expansion and Gottfried W. Leibniz in 1682 published the specific case with x = 1 and based on

arctangent(1) = pi/4                [9]
Expanding this via the infinite series for arctangent(x), where bb | x | ≤ 1, we get:

arctangent(x)= x – x^3/3 + x^5/5 – x^7/7 + … = sum_(n=0)^oo (-1)^n*(x^(2n+1))/(2n+1)                [10]
• John Machin in 1706, using arctangent and the Gregory-Leibniz Infinite Series expansion for arctangent, as shown below, calculated it to 100 decimal digits.

π = 4∙[4∙arctangent(1/5) + arctangent(1/239)]                 [11]
π =                                                                                                 [12]
4*{ 4*[ 1/5 – 1/(3*5^3) + 1/(5*5^5) – 1/(7*5^7)+… ] + [ 1/239 – 1/(3*239^3) + 1/(5*239^5) – 1/(7*239^7)+… ] }
• Srinivasa Ramanujan, was a renowned Indian mathematician who made novel contributions to mathematics and to determining π during his short life (1887-1920). The first formula below gives π to 11 decimal places.

π = root(4)[81+(19^2/22)] = 3.14159265262                        [13]
The following iterative formula was due to Jonathan and Peter Borwein based on Ramanujan’s formulas:

π = [9801/(2*sqrt(2))]*[1/(sum_(n=0)^oo {([(4n)!]/(n!)^4)*[(1103 + 26390n)/(4*99)^(4n)]}]]        [14]
Ramanujan’s integration formula for π is also quite innovative:

(pi/2)^3 = int_0^oo [(log x)^2/(1+x^2)]dx        [15]
which can easily be solved for π.
• David and Gregory Chudnovsky in 1989 in New York City, created their own supercomputer (named m zero) to compute 1 billion decimal places of π from their home. It operated at 100 billion calculations per minute for almost a week.
In 1997, they moved to Polytechnic Institute of Brooklyn (my Alma Mater and now part of New York University), creating the Institute for Mathematics and Advanced Supercomputing. By then, their computer worked for a week (with a better algorithm) to compute up to 8 billion decimal places for π.

Their π calculation is based on:

1/pi = 12*sum_(n=0)^oo (-1)^n*{([(6n)!]/[(n!)^3*(3n)!])*[(13591409 + 545140134n)/(640320^(3n+3/2))]}  [16]
• Yasumasa Kanada holds the 2002 record for bb 1.2411 × 10^12 decimal places for π. His computer programs (written in Fortran and C) were based on K. Takano’s 1982 arctangent formula [17] below and ran on a Hitachi SR8000/MPP with 144 nodes. The computations were carried out in hexadecimal (base 16) arithmetic and converted at the end to decimal for maximal efficiency. Further details about the computation are found on Yasumasa Kanada 2002 π Summary Page.

π ~~                               [17]
48*arctan(1/49) +128*arctan(1/57) – 20*arctan (1/239) + 48*arctan(1/110443)
This took 400 hours (hexadecimal computing) + bb 23 1/3 hours (conversion).
It was verified with F.C.M. Stoemer’s 1896 arctangent formula:

π ~~                               [18]
176*arctan(1/57) +28*arctan(1/239) – 48*arctan (1/682) + 96*arctan(1/12943)
The verification formula [18] took 157.067 hours (hexadecimal computing) + 21.53 hours (conversion) to evaluate and compare.
• As of October 8, 2014, an anonymous mathematician named “houkouonchi” completed computing and verifying π to a total of bb 1.33 × 10^13 decimal places using the Chudnovsky formula [16]. It took his program 208 days to compute and 182 hours to verify on an 2 x Xeon E5-4650L @ 2.6 GHz. See:
Validation of π computation.

## π Activities To Try:

#### Entertaining Websites:

Within the mathisfun.com website, there is an activity to guide you to find the approximate value of π. Look at π approximation

Many people are emotionally attached to the numbers in their life. Their social security number (e.g. 423456789), their birthday (02022015), their 7-digit telephone number (3234777).

There is a great website, called the Pi Search Page that lets you enter your special number and it reports where, in the 200 million decimal places of π that it is located, how many times it shows up (including not at all) and how long it took to search (typically in bb 1/5 of a second).

Go to π Search Page and interact with it.

[Updated to add:] To see 10,000, 100,000 or 1 million digits of bb pi in all their glory in a downloaded file, a related site called digits of Pi; lets you view 10,000 places or access the download links.

Note that the probability that your birthday number string (8 digits long) will be embedded in the first 200 Million decimal places of π is equal to bb (1 – 1/e^2) which is just below 86.47%.

There is a World Ranking List of people who have memorized some number of digits of π and recited them. Please view π World Ranking List

##### Books worth reading:
• Blatner, David (1997). The Joy of π . London, UK. Walker/Bloomsbury Books.The website Joy of π has a set of links to many π oriented pages including: π mysteries, music and π, memorizing π digits, having fun and enjoying weird aspects of π.

David Blatner wrote about the statistical distribution of digits in the first million decimal places of π which are comprised of:
 99959  0's 99758  1's 100026 2's 100229 3's 100230 4's 100359 5's 99548  6's 99800  7's 99985  8's 100106 9's 
For these data, I computed the following statistics: the mean (average) is 100000; the median (middle) is 100005.5; the standard deviation is 247.41 and the range of the data is 811, with the digit 5 being most frequent and the digit 6 being least frequent.

• Beckmann, Petr (1971). A History of π (pi). New York, NY. St. Martin’s PressThis impressive source book traces the history of the constant and of the mathematicians that sought greater and greater π precision. There are many illustrations of artifacts and notations used to demonstrate the mathematics involved.

This completes the technical and non-technical tour of π. Please let me know if I’ve missed anything that could be further explored.