What is Pi?

Pi is the 16th letter of the Greek alphabet, but it is best known as the symbol used to name a mathematical relationship: the ratio between the circumference of a circle and its diameter. As such, it is a mathematical constant and has many uses. Most obviously, it can be used to calculate the circumference of a circle from its diameter and vice-versa. Other examples are the formulae to find the area of a circle and the volume of a sphere. It is often represented by the Greek form of the letter, π, and is commonly given the value 3.14; however, this is only an approximation, and the number has some fascinating properties.

The precise value of pi cannot be stated. No fraction is exactly equivalent to it and when it is expressed as a decimal, there are an infinite number of digits after the decimal point. Therefore, whenever it is required for a calculation, an approximation has to be used. The value employed depends on how precise the calculation needs to be.

For some purposes, 3.14 is acceptable, while for others, a value correct to, say, eight decimal places — 3.14159265 — might be required. No calculations require a value accurate to more than 40 decimal places. Many people have used computers to perform record-breaking calculations of the value of π; as of 2013, it has been calculated to 10 trillion decimal places. There is, however, no conceivable application that requires such an accurate value.

Although pi is defined in terms of the diameter of a circle, in mathematical formulae, it is normally the radius, represented by "r," that is used, so that the formula for the circumference of a circle is 2πr, or radius multiplied by π times two. Other common mathematical formulae using π include the following:

The constant is also used extensively in

, statistics and

.

Pi is an irrational number, which means that it cannot be expressed as a ratio, or fraction, involving two integers, such as 2/5 or 7/3. Some fractions are close approximations, for example, 355/113 gives the number correct to 6 decimal places; however, an exact value cannot be obtained this way. When irrational numbers are expressed as decimals, the digits after the decimal point form an infinite, non-repeating, sequence.

It is also a transcendental number, meaning that it cannot be a root, or solution, to any algebraic equation with rational coefficients. The coefficients of an equation are simply the numbers that prefix the symbols; where there is no numeric prefix, the coefficient is 1. For example, in the equation 3x + y = 0, the coefficients of x and y are 3 and 1, respectively. The fact that pi is transcendental is proof that the ancient problem of “squaring the circle” — constructing a square with the same area as a circle using only a straight edge and a compass — is insoluble.

The sequence of digits after the decimal point appears to be random. Many attempts have been made to find patterns within this number, but all have failed. The randomness has not been proved but, as of 2013, the sequence, as far as it has been calculated, passes all tests for it.

The ancient Babylonians and ancient Egyptians both used rough approximations of π, calculating values a little over 3.1. Archimedes, the ancient Greek mathematician, found that the value lay between 223/71 and 22/7. Pi was found to be irrational in 1770 by the German mathematician Johann Lambert, and in 1882, the physicist Ferdinand Lindemann showed that it is a transcendental number. In more recent years, the value has been calculated to an ever-increasing number of decimal places — a trend that looks set to continue with the growth of computing power.

If the sequence of digits after the decimal point in π is random, this means, since it is also infinite, that every conceivable sequence of numbers, no matter how long or unlikely, must occur somewhere in the series. In fact, each must occur an infinite number of times. The digits can be used to represent other characters, such as letters of the alphabet and punctuation marks. In this way, every conceivable sequence of characters could, in theory, be found within pi by searching through a sufficient number of digits. These sequences would include the complete works of Shakespeare, all known mathematics textbooks and this article, as well as an infinity of books that have not yet been written.

To find anything meaningful more than just a few characters long would, however, require the calculation of pi to an unimaginable number of decimal places, many orders of magnitude greater than the current record. As of 2013, it is possible for anyone, using a simple online program, to search for character strings in the first four billion digits of π. The likelihood of finding a character sequence of a given length is easily calculated. For example, the probability of finding a given ten-character sequence in the first four billion digits of pi is 0.0003%.

So far, nothing that appears meaningful has been found in pi. There is, however, a sequence of six consecutive 9s, starting at the 762^nd digit. This is known as the Feynman point and is named after the physicist Richard Feynman. Its probability of occurring so early in the sequence is 0.0685%; nevertheless, it is thought to be simply a freak occurrence.

Many people have succeeded in memorizing π to a huge number of decimal places. As of 2013, the record is considered to be 67,890. The date 14 March (also written 3/14) has been designated “Pi Day” in the USA, with various pi-related activities taking place. Music based on this constant has been created and novels have been written where the word lengths are the digits of π in the correct sequence.