Pi

Definition

Pi is an irrational and transcendental (proved by Lindeman in 1882) 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 $\frac{355}{113}$ .

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 $\pi$ .

One formula for pi is $4\left( \sum_{i = 0}^\infty (-1)^i \left(\frac{1}{2n+1}\right)\right)$ . This can be computed to the desired degree of accuracy.

Other interesting properties

Letting $\theta = \pi$ in the Euler formula gives $e^{\pi i} + 1 = 0$ , which is considered to be one of the most beautiful results in mathematics since it involves five of the greatest constants: e, pi, i, 1, and 0.

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Pi

Contents

Definition

Approximating pi

Other interesting properties

See Also