Pi | Research & Encyclopedia Articles

Pi

Pi (or ) is the ratio of the circumference of a circle to its diameter. This ratio is the same for every circle--for instance, if you double the diameter of a circle, you double its circumference as well. The observation that this ratio is constant has been known so long that historians cannot say when it was first discovered. It was certainly known by the year 2000 b.c., when the Babylonians estimated its value at 3 1/8, and the Egyptians at 3.1605. Another early estimate of the value of pi, although a less accurate one, is found in the Old Testament, which puts the value at 3 in the following passage about the great temple of Solomon: "And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it about." (I Kings 7:23)

Although many early civilizations realized that the ratio of the circumference to the diameter is a constant, finding the exact value of that constant turned out to be difficult. Historians believe that the estimates of the Babylonians and Egyptians were calculated simply by measuring circles. The first theoretical estimate for the value of pi was made by the Greek mathematician Archimedes (287-212 b.c.), who estimated the circumference of a circle by comparing it to the perimeters of simpler shapes, such as hexagons, that were superscribed or inscribed in the circle. He came up with the estimate 223/71 < pi < 22/7. The average of those two bounds is 3.1418, which is within about 0.0002 of the actual value of pi.

To form this estimate, Archimedes considered shapes with as many as 96 sides. His calculation was a remarkable one, especially considering that in his day, much of the algebraic and trigonometric notation that we now rely on had not yet been developed. Over the centuries that followed, first Arab and then European scholars carried Archimedes's calculation further, using shapes with larger and larger numbers of sides. By 1600 the value of pi was known up to 35 decimal places.

Around this time, mathematicians began to explore the mathematics of infinite sums and products of numbers. They discovered that pi, although a number that arises in a purely geometric setting, can be expressed in strikingly elegant arithmetic ways, as an infinite sum or product. For instance, John Wallis (1616-1703) proved that pi=2x(2x2x4x4x6x6...)/(3x3x5x5x7x7...).

The mathematician James Gregory (1638-1675) proved the following formula, which is sometimes attributed to Leibniz: pi/4=1 - 1/3 + 1/5 - 1/7 +...

And the great mathematician Leonhard Euler (1707-1783) discovered the famous formula pi^2/6 = 1/1^2 + 1/2^2 + 1/3^2 + 1/4^2 +...

The discovery that pi could be expressed in terms of infinite series gave a new way to estimate its value: calculate larger and larger portions of an infinite series. With this new technique, by the start of the 18th century mathematicians were able to calculate pi to 100 decimal places. But they also realized that the calculation of the digits of pi is a task without an end: in 1766 Lambert proved that pi is not a rational number, so that the numbers that appear in its decimal form never fall into a repeating pattern. A century later, in 1882, Lindemann proved something even stronger: pi is transcendental--that is, it is not the solution of any polynomial equation with integer coefficients. Lindemann used this result to answer the classical question of "squaring the circle" in the negative: he showed that it is impossible to construct a square equal in area to a given circle using only a straightedge and compass.

In spite of the careful analysis to which pi had been subjected by this time, among non-mathematicians its nature still appeared to present some mysteries. In 1897, the House of Representatives of Indiana unanimously passed the following legislation: "Be it enacted by the General Assembly of the State of Indiana: It has been found that a circular area is to the square on a line equal to the quadrant of the circumference, as the area of an equilateral rectangle is to the square of one side." In other words, pi is equal to 4. Fortunately, the Senate of Indiana declined to adopt this "bill for an act introducing a new mathematical truth."

With the advent of supercomputing power, it has become possible to calculate pi to levels of precision that previously were inconceivable. The current record is 50 billion decimal places.