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: $3.14159265358979323846264338327950288419716939937510$ $58209749445923078164062862089986280348253421170679$ $82148086513282306647093844609550582231725359408128$ $48111745028410270193852110555964462294895493038196$ $44288109756659334461284756482337867831652712019091$ $45648566923460348610454326648213393607260249141273$ $72458700660631558817488152092096282925409171536436$ $\mathrm{7892590360011330530548820466521384146951941511609\; .\; .\; .}$ 
The following table gives a short list of common approximations used (and misused) for $\pi $, along with their inventor/discoverer.





Old Testament ($2500$ yrs ago) & State of Indiana ($1900$'s)  $\mathrm{3.00000000000...}$  $\mathrm{0.1415926535...}$ 

Egyptian value (Rhind Papyrus, $3600$ yrs ago)  $\mathrm{3.16049382716...}$  $\mathrm{0.0189011735...}$ 

Common modern approximation  $3.14$  $\mathrm{0.00159265...}$ 

Archimedes ($2500$ yrs ago)  $\mathrm{2857142857...}$  $\mathrm{0.00126449...}$ 

Tsu Ch'ungchih ($1500$ ago)  $\mathrm{3.141592920353...}$  $\mathrm{0.0000002667...}$ 

The 'Circle Squarers' ($8$th C)  $\mathrm{3.162277660168...}$  $\mathrm{0.0206850065...}$ 

Baskin Robbins  $\mathrm{3.14138065239139...}$  $\mathrm{0.000212001198...}$ 

Arctangent formula ($1671$)  $\mathrm{3.1415926535897932...}$  $\mathrm{0.0000000...}$Exact! 
After the discovery of arctan $x=x\frac{{x}^{3}}{3}+\frac{{x}^{5}}{5}\frac{{x}^{7}}{7}+\cdots $, 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.
http://seattlepi.nwsource.com/national/98912_pi07.shtml