pi
The symbol $\pi $ was first introduced by William Jones [7, 8] in 1706 to denote the ratio between the perimeter^{} and the diameter^{} on any given circle. In other words, dividing the perimeter of any circle by its diameter always gives the same answer, and this number is defined to be $\pi $. A 12digit approximation of $\pi $ is given by $3.14159265358\mathrm{\dots}$
Over human history there were many attempts to calculate this number precisely. One of the oldest approximations appears in the Rhind Papyrus (circa 1650 B.C.) where a geometrical construction is given where ${(16/9)}^{2}=3.1604\mathrm{\dots}$ is used as an approximation to $\pi $ although this was not explicitly mentioned.
It wasn’t until the Greeks that there were systematical attempts to calculate $\pi $. Archimedes [1], in the third century B.C. used regular polygons^{} inscribed^{} and circumscribed^{} to a circle to approximate $\pi $: the more sides a polygon^{} has, the closer to the circle it becomes and therefore the ratio between the polygon’s area between the square of the radius yields approximations to $\pi $. Using this method he showed that $$ $$.
Around the world there were also attempts to calculate $\pi $. Brahmagupta [1] gave the value of $\sqrt{10}=3.16227\mathrm{\dots}$ using a method similar^{} to Archimedes’. Chinese mathematician Tsu ChungChih (ca. 500 A.D.) gave the approximation $355/113=3.141592920\mathrm{\dots}$.
Later, during the renaissance, Leonardo de Pisa (Fibonacci) [1] used 96sideed regular polygons to find the approximation $864/275=3.141818\mathrm{\dots}$
For centuries, variations on Archimedes’ method were the only tool known, but Viète [1] gave in 1593 the formula
$$\frac{2}{\pi}=\sqrt{\frac{1}{2}}\sqrt{\frac{1}{2}+\frac{1}{2}\sqrt{\frac{1}{2}}}\sqrt{\frac{1}{2}+\frac{1}{2}\sqrt{\frac{1}{2}+\frac{1}{2}\sqrt{\frac{1}{2}}}}\mathrm{\cdots}$$ 
which was the first analytical expression for $\pi $ involving infinite summations or products. Later with the advent of calculus many of these formulas were discovered. Some examples are Wallis’ [1] formula:
$$\frac{\pi}{2}=\frac{2}{1}\cdot \frac{2}{3}\cdot \frac{4}{3}\cdot \frac{4}{5}\cdot \frac{6}{5}\mathrm{\cdots}$$ 
and Leibniz’s formula,
$$\frac{\pi}{4}=1\frac{1}{3}+\frac{1}{5}\frac{1}{7}+\frac{1}{9}\frac{1}{11}+\mathrm{\cdots},$$ 
obtained by developing $\mathrm{arctan}(\pi /4)$ using power series, and with some more advanced techniques,
$$\pi =\sqrt{6\zeta (2)},$$ 
found by determining the value of the Riemann Zeta function^{} at $s=2$ (http://planetmath.org/ValueOfTheRiemannZetaFunctionAtS2).
The Leibniz expression provides an alternate way to define $\pi $ (namely 4 times the limit of the series) and it is one of the formal ways to define $\pi $ when studying analysis in order to avoid the geometrical definition.
It is known that $\pi $ is not a rational number^{} (quotient of two integers). Moreover, $\pi $ is not algebraic over the rationals (that is, it is a transcendental number^{}). This means that no polynomial^{} with rational coefficients can have $\pi $ as a root. Its irrationality implies that its decimal expansion (or any integer base for that matter) is not finite nor periodic.
References
 1 The MacTutor History of Mathematics archive, http://wwwgroups.dcs.stand.ac.uk/ history/BiogIndex.htmlThe MacTutor History of Mathematics Archive

2
The ubiquitous $\pi $ [Dario Castellanos] Mathematics Magazine Vol 61, No. 2. April 1988. Mathematical Association of America.

3
A history of pi. [O’Connor and Robertson]
http://wwwgroups.dcs.stand.ac.uk/~history/HistTopics/Pi_through_the_ages.html

4
Pi chronology. [O’Connor and Robertson]
http://wwwgroups.dcs.stand.ac.uk/~history/HistTopics/Pi_chronology.html

5
Asian contributions to Mathematics. [Ramesh Gangolli]
http://www.pps.k12.or.us/deptsc/mcme/beasma.pdf

6
Archimedes’ approximation of pi. [Chuck Lindsey]
http://itech.fgcu.edu/faculty/clindsey/mhf4404/archimedes/archimedes.html
 7 N. Higham, Handbook of writing for the mathematical sciences, Society for Industrial and Applied Mathematics, 1998.
 8 The MacTutor History of Mathematics archive, http://wwwgap.dcs.stand.ac.uk/ history/Mathematicians/Jones.html William Jones
Title  pi 
Canonical name  Pi 
Date of creation  20130322 11:51:41 
Last modified on  20130322 11:51:41 
Owner  mathcam (2727) 
Last modified by  mathcam (2727) 
Numerical id  27 
Author  mathcam (2727) 
Entry type  Definition 
Classification  msc 01A40 
Classification  msc 01A32 
Classification  msc 01A25 
Classification  msc 01A20 
Classification  msc 01A16 
Classification  msc 5100 
Classification  msc 1100 
Classification  msc 22A22 
Classification  msc 46L05 
Classification  msc 8200 
Classification  msc 8300 
Classification  msc 8100 
Related topic  RegularPolygon 
Related topic  Limit 
Related topic  Diameter 
Related topic  TranscedentalNumber 
Related topic  AlgebraicNumber 
Related topic  ExtensionMathbbRmathbbQIsNotFinite 
Related topic  WallisFormulae 