A Most Useful and Elusive Number


OF ALL the numbers used in mathematics, science, engineering, and daily life, few have received as much attention as pi (π). Pi “has fascinated the giants of science as well as amateurs around the world,” states the book Fractals for the Classroom. In fact, pi is regarded by some as one of the five most significant numbers in mathematics.

Pi represents the ratio of the circumference of a circle to its diameter. You can figure out the circumference of any circle, regardless of its size, by multiplying its diameter by pi. In 1706, English mathematician William Jones was the first to use the Greek letter π to designate this ratio, and it became popular after Swiss mathematician Leonhard Euler adopted it in 1737.

For many applications, using a value of 3.14159 for pi will be accurate enough. But pi can never be calculated exactly. Why not? Because it is an irrational number—that is, it cannot be written as a simple fraction. When written as a decimal, it simply goes on and on. In fact, it can be calculated to an infinite number of decimal places. Nevertheless, this has not deterred mathematicians from laboring tediously to calculate the value of pi to ever more decimal places.

It is not known who first realized that pi remains constant regardless of the size of the circle. But an accurate value of the elusive number has been sought since ancient times. The Babylonians approximated pi as 3 1/8 (3.125), and the Egyptians, slightly less accurately, as about 3.16. In the third century B.C.E., Greek mathematician Archimedes made perhaps the first scientific effort to compute it, arriving at a figure of about 3.14. By the year 200 C.E., it had been worked out to the equivalent of 3.1416, a figure that Chinese and Indian mathematicians had independently confirmed by the early sixth century C.E. Today, with the help of powerful computers, pi has been calculated to billions of decimal places. But as useful as pi has proved to be, notes Fractals for the Classroom, “it would be hard to find applications in scientific computing, where more than some 20 digits of [pi] are necessary.”

Pi shows up in formulas that are used in many fields—physics, electrical and electronic engineering, probability, structural design, and navigation, to name but a few. Just as there is no end to its digits, it seems that there is also no end to the number of practical applications for useful, elusive pi.