There are many numbers in mathematics, and some are more important than others. One of the most widely used constants throughout mathematics is the number pi. Pi is denoted by the Greek letter π. Despite the origins of pi in the subject of geometry, this number has applications throughout mathematics, and even shows up in the subjects of statistics and probability. Pi has even gained cultural recognition and its own holiday, with the celebration of Pi Day activities around the world.

### The Value of Pi

This number is defined to be the ratio of a circle’s circumference to its diameter. The value of pi is slightly greater than three, which means that every circle in the universe has a circumference with length is a little more than three times its diameter. More precisely, pi has a decimal representation that begins 3.14159265. . . This is only part of the decimal expansion of pi.

### Facts About Pi

Pi has many fascinating and unusual features, including:

**Pi is an irrational real number.**This means that pi cannot be expressed as a fraction*a/b*where*a*and*b*are both integers. Although the numbers 22/7 and 355/113 are helpful to*estimate*pi, neither of these fractions are the true value of pi.- Because pi is an irrational number,
**its decimal expansion never terminates and never repeats.**There are some open problems concerning this decimal expansion, such as does every possible string of digits show up somewhere in the decimal expansion of pi? If every possible string does appear, then your cell phone number is somewhere in the expansion of pi (but so is everyone else’s).

**Pi is a transcendental number.**This means that pi is not the zero of a polynomial with integer coefficients. This fact is important when exploring more advanced features of pi.**Pi is important geometrically**, and not just because it relates the circumference and diameter of a circle. This number also shows up in the formula for the area of a circle. The area of a circle of radius*r*is*A*= pi*r*^{2}. The number pi is used in other geometric formulas, such as the surface area and volume of a sphere, the volume of a cone and the volume of a cylinder with circular base.

**Pi has a habit of showing up where it's least expected.**For one of many, many examples of this, consider the infinite sum 1 + 1/4 + 1/9 + 1/16 + 1/25 + . . . This sum converges to the value pi^{2}/6.

### Pi in Statistics and Probability

Pi makes surprising appearances throughout mathematics, and some of these appearances are in the subjects of probability and statistics. The formula for the standard normal distribution, also known as the bell curve, features the number pi as a constant of normalization. In other words, dividing by an expression involving pi allows us to say that the area under the curve is truly equal to one. Pi is part of the formulas for other probability distributions as well.

Another surprising occurrence of pi in probability is the **Buffon needle throwing experiment**. In the 18th century Georges-Louis Leclerc, Comte de Buffon posed a question concerning the probability of dropping needles: Start with a floor with planks of wood of a uniform width in which the lines between each of the planks are parallel to one another. Now, take a needle with a length shorter than the distance between the planks. If one drops a needle on the floor at random, what is the probability that the needle will land on a line between two of the wood planks?

It turns out, the probability that the needle does land on a line between two planks is (twice the length of needle)/(length between the planks times pi). Yet again, an unexpected pi appears.