the pi page 11/02 (also see my circle page!)
(c) 2002-07 by Dan Bach (and the entire www.dansmath.com empire!)
    Basic Definition:
 
The CIRCUMFERENCE C (dist. around) of any circle DIVIDED
BY its DIAMETER D = 2r always gives the same ratio: Pi = = C / D.
(A CIRCLE is the set of all points at a distance r from the center.)

* Many of the facts on this page are from The Joy Of , a wonderful book by David Blatner; visit joyofpi.com
 
Pi ( ) is an exact, theoretical value; the quest for a precise decimal approximation for it (3.141592653589793...)
has been going on for thousands of years, but even this modest 15-place accuracy wasn't known until 1593!*
 
Today it's a simple command typed into an off-the-shelf program (like the awesome Mathematica) to get thousands of
decimal places; here are the first 400 (ok, it's only 399 after the decimal place; I added the . . . myself):

 In: N[Pi, 400]   

 Out: 3.14159265358979323846264338327950288419716939937510
       58209749445923078164062862089986280348253421170679
       82148086513282306647093844609550582231725359408128
       48111745028410270193852110555964462294895493038196
       44288109756659334461284756482337867831652712019091
       45648566923460348610454326648213393607260249141273
       72458700660631558817488152092096282925409171536436
       7892590360011330530548820466521384146951941511609 . . .

 

      Want to see to a million decimal places? Click here! (From Duane Bailey!)
 
 
Approximating through the ages
Here is a short list of common approximations used (and misused) for , along with their inventor/discoverer.*

Fraction or Expression

Origin and rough date

decimal value

error from true

3
Old Testament (2500 yrs ago)
& State of Indiana (1900's)
3.00000000000... 0.1415926535...

256/81
Egyptian value
(Rhind Papyrus, 3600 yrs ago)
3.16049382716... 0.0189011735...

3.14
 Common modern approx. 3.14 0.00159265...

22/7
 Archimedes (2500 yrs ago) 3.142857142857... 0.00126449...

355/113
 Tsu Ch'ung-chih (1500 ago) 3.141592920353... 0.0000002667...

 10
 The 'Circle Squarers' (8thC) 3.162277660168... 0.0206850065...

(31)^(1/3)
 Baskin Robbins (root beer?) 3.14138065239139... 0.000212001198...

4 - 4/3 + 4/5 - 4/7 + . . .
 Arctangent formula (1671)
3.1415926535897932...
0.0000000... Exact!
After the discovery of arctan x = x - (x^3)/3 + (x^5)/5 - (x^7)/7 + . . . , there was a rush of computation,
all with slates, chalk, pens, parchment, sticks, sand; no pocket calculators or even slide rules at first.
 
The most obvious series is if x = 1 ; then arctan 1 = /4 = 1 - 1/3 + 1/5 - 1/7 + . . . ; a great little formula
but it takes thousands of terms just to get 3 or 4 places of accuracy for . Now if you notice that
/4 = 2 arctan(1/3) + arctan(1/7) = 2/3 - 2/(3*3^3) + 2/(5*3^5) - . . . + 1/7 - 1/(3*7^3) + 1/(5*7^5) - . . .
you do have to deal with two series, but they will "converge" much faster to the exact (irrational) value of pi.
 
Fractions are normal things to try, but since is irrational (not the ratio of two integers), you'll never get
it exactly that way. Lambert proved* the irrationality of pi in 1761. So that rules out 22/7 and 355/113.
 
In 1882 Lindemann proved* that is transcendental, meaning not the root of any polynomial equation.
This means pi squared is not 10, and pi cubed is not 31. But those are some close calls!
 
These days there are people like the Chudnovsky brothers who calculate to literally billions of places;
the computational theory alone is pushing the quality, and not only the size, of the pi envelope.*

 

 
News Flash - has now been calculated out to 1.24 TRILLION decimal places, in about 400
hours of computing time, by Professor Yasumasa Kanada and a team of mathematicians, using
a Hitachi supercomputer. The previous record, set by Kanada in 1999, was 206 billion places.
Read the article in the Seattle "Post-Intelligencer" or "PI" at www.seattlepi.com Thanks Lorenzo!
 
A "series of formulas" for:
11/07 - There are lots of series and other ways of approximating pi and its powers.
Here are a few; you should recognize the first one; (the F's are the Fibonacci numbers):
It's fun to write out a few terms of each series; I dare you to draw a pie chart for the second one!
 

 I have several more of these tucked away, but that's enough for now.
 
 
For more on the subject, search at www.google.com for 'pi' . . .
* much of the above is from "The Joy Of " by David Blatner; visit joyofpi.com
 
 
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