Pi is perhaps the most important mathematical constant. It appears in various formulas throughout math and science in fields as diverse as physics, statistics, and sociology. Although pi is defined in terms of the geometry of a circle, most applications of this number do not directly involve circles.
Since ancient times, people have been fascinated by pi. This is hardly a surprise, since the circle is one of the most basic, but nevertheless fascinating, geometric figures. Pi is defined as the ratio of the circumference to the diameter of a circle. (Any circle will work, since all circles are similar.) Rounded to 10 decimal places, its value is 3.1415926536.
Part of what makes pi fascinating is that it appears in several other formulas involving circles or spheres. For instance, the area of a circle is equal to pi times the square of its radius. Further, the surface area of a sphere is equal to 4 pi times the square of its radius, and its volume is equal to 4/3 pi times the cube of its radius. In fact, the formulas for the content of all higher dimensional analogs of the sphere also involve pi.
As mentioned, pi also appears in many formulas not directly involving circles or spheres. For instance, the periods of all the trigonometric functions are either equal to pi or 2 pi. Although trig functions may be defined in terms of a circle, they are usually used in contexts not directly involving circles. Another place pi is widely used is in the normal distribution, which is commonly used in statistics, whose formula involves the square root of pi.
The computation of pi has a long and fascinating history. Some of the most elaborate mathematical methods have been used in devising various formulas for pi. By the late 19th century, its value had been computed by hand to several hundred decimal places. Since the dawn of the computer age in the mid-20th century, the number of calculated digits of pi has skyrocketed. Since 2002, its value has been known to over a trillion decimal places – enough to fill a large library!
Part of the reason some mathematicians are fascinated with calculating so many digits of pi is in order to look for patterns in its digits. So far, no obvious ones have been found. It has been conjectured that pi is a normal number, meaning that every finite pattern of digits in every base occurs infinitely often in pi with the same frequency which would be expected if the digits were random.
In 1995, an amazing formula was found for pi, which allows one to compute hexadecimal (base 16) digits of pi without having to compute any previous digits. This formula was used in 2000 to compute the quadrillionth (10^15th) hexadecimal digit of pi, which happens to be 0. Several similar formulas have since been discovered, some in other bases, but none in base 10 have yet been found.