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## Important Facts About Pi

Pi is perhaps the most important mathematical constant. It appears in various formulas in mathematics 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 not a surprise, because 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 example, the area of a circle is pi times the square of its radius. Also, the area of a sphere is 4 pi times the square of its radius, and its volume is 4/3 pi times the cube of its radius. In fact, the formulas for the contents of all higher-dimensional analogs of the sphere also imply pi.

As mentioned, pi also appears in many formulas not directly involving circles or spheres. For example, the periods of all trigonometric functions are either equal to pi or 2 pi. Although trigonometric functions can be defined in terms of a circle, they are generally used in contexts not directly involving circles. Another place where pi is widely used is in the normal distribution, which is commonly used in statistics, the formula of which involves the square root of pi.

The calculation of pi has a long and fascinating history. Some of the most elaborate mathematical methods have been used to devise various formulas for pi. By the end of the 19th century, its value had been calculated by hand to several hundred decimal places. Since the dawn of the computer age in the mid-twentieth century, the number of calculated digits of pi has exploded. 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 to look for patterns in its digits. So far, no evidence has 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 as one would expect if the digits were random.

In 1995, an amazing formula was found for pi, which makes it possible to calculate the hexadecimal digits (base 16) of pi without having to calculate the preceding digits. This formula was used in 2000 to calculate 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.

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