March 14 is officially Pi Day in recognition of the first three significant digits of , a transcendental number whose decimal form begins and goes on forever.

**Definition**

Pi is defined as the ratio of a circle’s circumference to its diameter .

It’s geometrically defined. That’s it.

**Appearances**

But pi is an ambitious number, not content with the confines of geometry. Consider the following series that has no apparent geometric meaning:

This is the sum of the reciprocals of the squares. Surprisingly, it has the curious value

Leonhard Euler discovered the above equation in 1735. Another interesting occurrence of pi is in Euler’s identity:

where is Euler’s number 2.71828… and the square root of negative one.

In mathematics we also have the Gaussian integral

which is essentially the equation modeling the standard distribution.

Another useful occurrence of the square root of pi is in the Stirling approximation to the factorial function . The approximation states

Pi also manages to show up in physics, largely due to the geometrical nature of our universe. For instance, the Heisenberg Uncertainty Principle says it is impossible to know both the position and momentum of a particle with high degrees of certainty. If we assign be the uncertainty in position, the uncertainty in momentum, and Plank’s constant, then we have the Heisenberg Uncertainty Principle:

The above is by no means a comprehensive list of usages of pi as a mathematical constant—there are countless more.

**Other Pi**

The capital pi () is used in mathematics to denote a product, as sigma () denotes a sum. For example, we could say

A slightly more sophisticated product might be

Combine this with the Stirling approximation, and we have both upper and lower case pi in one statement!

The lower case pi is also used as the prime counting function , which is the number of primes less than or equal to . The explicit formula for this function is too complicated to put here. Actually, why not?

where is defined as

Here, is the Möbius function, is the logarithmic integral, and indicates the enumeration of all nontrivial zeros of the Riemann zeta function.

Hap-pi Pi Day!

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