# 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 $\pi$.

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 $\frac{22}{7}$ and $3.14$.  It is important that these values do not equal $\pi$.

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 $\pi$, for advanced and/or interested students:

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

 Fraction or Expression Origin and rough date decimal value error from true $\pi$ $3$ Old Testament ($2500$ yrs ago) & State of Indiana ($1900$'s) $3.00000000000...$ $0.1415926535...$ $\frac{256}{81}$ Egyptian value (Rhind Papyrus, $3600$ yrs ago) $3.16049382716...$ $0.0189011735...$ $3.14$ Common modern approximation $3.14$ $0.00159265...$ $\frac{22}{7}$ Archimedes ($2500$ yrs ago) $2857142857...$ $0.00126449...$ $\frac{355}{13}$ Tsu Ch'ung-chih ($1500$ ago) $3.141592920353...$ $0.0000002667...$ $\sqrt{10}$ The 'Circle Squarers' ($8$th C) $3.162277660168...$ $0.0206850065...$ ${31}^{\frac{1}{3}}$ Baskin Robbins $3.14138065239139...$ $0.000212001198...$ $4-\frac{4}{3}+\frac{4}{5}-\frac{4}{7}+\cdots$ Arctangent formula ($1671$) $3.1415926535897932...$ $0.0000000...$Exact!

After the discovery of arctan , 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

$\mathrm{arctan}1=\frac{\pi }{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots$;

a great little formula but it takes thousands of terms just to get $3$ or $4$ places of accuracy for $\pi$.

Now if you notice that

$\begin{array}{l}\frac{\pi }{4}=2\mathrm{arctan}\left(\frac{1}{3}\right)+\mathrm{arctan}\left(\frac{1}{7}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{2}{3}-\frac{2}{3\cdot {3}^{3}}+\frac{2}{5\cdot {3}^{5}}-\cdots +\frac{1}{7}-\frac{1}{3\cdot {7}^{3}}+\frac{1}{5\cdot {7}^{5}}-\cdots \end{array}$

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 $\pi$ 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 $\frac{22}{7}$ and $\frac{355}{113}$.

In $1882$ Lindemann proved that $\pi$ 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$\pi$ 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