Difference between revisions of "Pi"

(Definition)
(Definition)
Line 1: Line 1:
 
== Definition ==
 
== Definition ==
'''Pi''' is an [[irrational]] number denoted by the greek letter <math>\displaystyle \pi </math>.  It is the ratio of a circle's circumference, or perimeter, to its diameter.  It is roughly equal to 3.141592653.  The number pi is one of the most important [[constant]]s in all of mathematics and appears in some of the most surprising places, such as in the sum <math>\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}</math>.  The fractional approximation for pi (not exact) can be <math>\frac{22}{7}</math>.  An exact formula for pi is <math>4\left( \sum_{i = 0}^\infty (-1)^i \left(\frac{1}{2n+1}\right)\right) </math>.
+
'''Pi''' is an [[irrational]] number denoted by the greek letter <math>\displaystyle \pi </math>.  It is the ratio of a circle's circumference, or perimeter, to its diameter.  It is roughly equal to 3.141592653.  The number pi is one of the most important [[constant]]s in all of mathematics and appears in some of the most surprising places, such as in the sum <math>\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}</math>.  Some common fractional approximations for pi are <math>\frac{22}{7}</math> or <math>\frac{355}{113}</math>.  An exact formula for pi is <math>4\left( \sum_{i = 0}^\infty (-1)^i \left(\frac{1}{2n+1}\right)\right) </math>.
  
 
The number pi often shows up in problems in [[number theory]], particularly [[algebraic number theory]]. For example, many [[class number]] formulae involve pi.
 
The number pi often shows up in problems in [[number theory]], particularly [[algebraic number theory]]. For example, many [[class number]] formulae involve pi.

Revision as of 13:58, 5 July 2006

Definition

Pi is an irrational number denoted by the greek letter $\displaystyle \pi$. It is the ratio of a circle's circumference, or perimeter, to its diameter. It is roughly equal to 3.141592653. The number pi is one of the most important constants in all of mathematics and appears in some of the most surprising places, such as in the sum $\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}$. Some common fractional approximations for pi are $\frac{22}{7}$ or $\frac{355}{113}$. An exact formula for pi is $4\left( \sum_{i = 0}^\infty (-1)^i \left(\frac{1}{2n+1}\right)\right)$.

The number pi often shows up in problems in number theory, particularly algebraic number theory. For example, many class number formulae involve pi.

Approximating pi

$\pi$ can be calculated in several ways, and it can also be approximated. One way to approximate $\pi$ is to inscribe a unit circle in a square of side length 2. Using a computer, random points are placed inside the square. Because the area of the circle is $\pi$, and the area of the square is 4, the ratio of the amount of points inside the circle to the total number of points approximates $\frac{\pi}{4}$. This can simply be multiplied by 4 to approximate $\pi$.

See Also