Pi

            Definition: Pi is the ratio of the circumference, C , to the diameter, d , of any circle.  The ratio is the same for any circle. 

            The symbol for Pi is π .

            Pi is an irrational number which means it does not have an exact fraction or decimal equivalent.  In algebra, the most commonly used approximations are 22 7 and 3.14 .  It is important that these values do not equal π .

            Pi is an exact theoretical value; the quest for a precise decimal approximation for it has been going on for thousands of years, but even the modest 15 -place accuracy, 3.141592653589793 , was not known until 1593 !  Today, with the use of supercomputers the accuracy can be calculated to millions of decimal places.  Below are the first 400 .

 In:
 N[Pi, 400]
 Out:
 
 3.14159265358979323846264338327950288419716939937510

 
   58209749445923078164062862089986280348253421170679


    82148086513282306647093844609550582231725359408128

 
   48111745028410270193852110555964462294895493038196

  
   44288109756659334461284756482337867831652712019091

  
   45648566923460348610454326648213393607260249141273


   72458700660631558817488152092096282925409171536436


    7892590360011330530548820466521384146951941511609 . . .

Here's some history of π , for advanced and/or interested students:

The following table gives a short list of common approximations used (and misused) for π , along with their inventor/discoverer.

Fraction or Expression

Origin and rough date

decimal value

error from true π

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 approximation 3.14 0.00159265...

22 7
Archimedes ( 2500 yrs ago) 2857142857... 0.00126449...

355 13
Tsu Ch'ung-chih ( 1500 ago) 3.141592920353... 0.0000002667...

10
The 'Circle Squarers' ( 8 th C) 3.162277660168... 0.0206850065...

31 1 3
Baskin Robbins 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

arctan1= π 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 π .

Now if you notice that

π 4 =2arctan( 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 π 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 π is transcendental, meaning not the root of any polynomial equation. This means π squared is not 10 , and π 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 π envelope.

News Flash π 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

http://seattlepi.nwsource.com/national/98912_pi07.shtml