Steven Bogart, a mathematics instructor at Georgia Perimeter College, provides the following explanation for Pi:

 

Pi - which is written as the Greek letter for p, or - is the ratio of the circumference of any circle to the diameter of that circle. Regardless of the circle's size, this ratio will always equal pi. In decimal form, the value of pi is approximately 3.14. But pi is an irrational number, meaning that its decimal form neither ends (like 1/4 = 0.25) nor becomes repetitive (like 1/6 = 0.166666...). To only 18 decimal places, pi is 3.141592653589793238. 

It is useful to have shorthand for this ratio of circumference to diameter. According to Petr Beckmann's A History of Pi, the Greek letter was first used for this purpose by William Jones in 1706, probably as an abbreviation of periphery, and became standard mathematical notation roughly 30 years later.

Pi is most commonly used in certain computations regarding circles. Pi not only relates circumference and diameter. Amazingly, it also connects the diameter or radius of a circle with the area of that circle by the formula: the area is equal to pi times the radius squared. Additionally, pi shows up often unexpectedly in many mathematical situations. For example, the sum of the infinite series

The importance of pi has been recognized for at least 4,000 years. A History of Pi notes that by 2000 B.C., "the Babylonians and the Egyptians (at least) were aware of the existence and significance of the constant ," recognizing that every circle has the same ratio of circumference to diameter. Both the Babylonians and Egyptians had rough numerical approximations to the value of pi, and later mathematicians in ancient Greece, particularly Archimedes, improved on those approximations. By the start of the 20th century, about 500 digits of pi were known. With computation advances, Pi fanatics have recently been able to calculate Pi to 12.1 trillion digits!

Why do mathematicians care so much about pi?

According to Steven Strogatz, professor of mathematics at Cornell, in his March 13, 2015 article in the New Yorker, the beauty of pi, in part, is that it puts infinity within reach. Even young children get this. The digits of pi never end and never show a pattern. They go on forever, seemingly at random—except that they can’t possibly be random, because they embody the order inherent in a perfect circle. This tension between order and randomness is one of the most tantalizing aspects of pi".

Pi touches infinity in other ways. For example, there are astonishing formulas in which an endless procession of smaller and smaller numbers adds up to pi. One of the earliest such infinite series to be discovered says that pi equals four times the sum 1 – + – + – + ⋯. The appearance of this formula alone is cause for celebration. It connects all odd numbers to pi, thereby also linking number theory to circles and geometry. In this way, pi joins two seemingly separate mathematical universes, like a cosmic wormhole.

But there’s still more to pi. After all, other famous irrational numbers, like e (the base of natural logarithms) and the square root of two, bridge different areas of mathematics, and they, too, have never-ending, seemingly random sequences of digits.

What distinguishes pi from all other numbers is its connection to cycles. For those of us interested in the applications of mathematics to the real world, this makes pi indispensable. Whenever we think about rhythms—processes that repeat periodically, with a fixed tempo, like a pulsing heart or a planet orbiting the sun—we inevitably encounter pi. There it is in the formula for a Fourier series:

Strogatz_Fourier_60H

That series is an all-encompassing representation of any process, x(t), that repeats every T units of time. The building blocks of the formula are pi and the sine and cosine functions from trigonometry. Through the Fourier series, pi appears in the math that describes the gentle breathing of a baby and the circadian rhythms of sleep and wakefulness that govern our bodies. When structural engineers need to design buildings to withstand earthquakes, pi always shows up in their calculations. Pi is inescapable because cycles are the temporal cousins of circles; they are to time as circles are to space. Pi is at the heart of both.

For this reason, pi is intimately associated with waves, from the ebb and flow of the ocean’s tides to the electromagnetic waves that let us communicate wirelessly. At a deeper level, pi appears in both the statement of Heisenberg’s uncertainty principle and the Schrödinger wave equation, which capture the fundamental behavior of atoms and subatomic particles. In short, pi is woven into our descriptions of the innermost workings of the universe.

It is for these reasons, and probably just an excuse to eat pie, that we love and celebrate Pi Day!