# 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) 3.142857142857... 0.00126449... 355/113 Tsu Ch'ung-chih (1500 ago) 3.141592920353... 0.0000002667... √10 The 'Circle Squarers' (8th C) 3.162277660168... 0.0206850065... 311/3 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 – (x3)/3 + (x5)/5 – (x7)/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 = 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 = 2 arctan(1/3) + arctan(1/7)

= 2/3 – 2/(3 · 33) + 2/(5 · 35) – . . . + 1/7 – 1/(3 · 73) + 1/(5 · 75) – . . .

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 Flashπ 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