Monday, March 25, 2013

The Somewhat Secret Life of Pi

It became a secret by default but is the overtly pragmatic outlook of contemporary mathematics educators causing undue mysticism of the “life” of pi? 

By: Ringo Bones 

Most contemporary mathematics educators discuss pi only as far as the ratio of the circumference of a circle to its diameter. Though it may allow students to solve the area of a given circle and other related engineering problems, but should today’s mathematics educators be “more adventurous” to have an extensive discussion on the somewhat secret historic life of pi? 

The symbol of pi that used the Greek letter was first employed in its present meaning by William Jones in 1707 and its adoption by Leonhard Euler in 1737 brought it into general use. In far earlier times, the determination of the value of pi and the calculation of the area of a circle of a given diameter were recognized early as closely related problems. 

The mathematicians of antiquity had learned how to calculate precisely the areas of plane figures bounded by straight lines, such as the rectangle, parallelogram and triangles. The circle, however, eluded the mathematicians of antiquity’s methods, although many approximations for its area were developed. It remained for the mathematicians of the last 150 years to disclose the real nature of the problem. 

The ancient Chinese and the ancient Hebrews seem to have taken pi as equal to 3. In the Ahmes Papyrus of ancient Egypt back in 1650 BC, the area of a circle is found by subtracting from the diameter 1/9 of its length and squaring the remainder. This is equivalent to taking pi as 256/81 or 3.1605. 

In ancient Greece, the problem took the form of requiring the construction of a square, by straight edge and compass alone, whose area equals that of a given circle – i.e. squaring the circle. Back around 225 BC, Archimedes approximated the area of a circle by comparing the areas of regular inscribed and circumscribed polygons of a large number of sides. Using polygons of 96 sides, Archimedes found that the ratio of the circumference of a circle to its diameter lies between 3 1/7 and 3 10/71. After the time of Archimedes, 3 1/7 and the square root of 10 were taken as a satisfactory approximation of pi. An approximation equivalent to the now familiar decimal 3.1416 dates from before 200 AD. 

After mathematicians invented decimals and promoted its widespread use, the calculation of pi was carried out to 35 decimal places by Ludolph van Ceulen (1540 – 1610). In Germany, pi is sometimes called the Lundolphian Number in his honor. 

Toward the end of the 17th Century, new methods of analysis were developed that made it possible to express pi as the sum or product of an infinite number of terms. One of the earliest of such expressions was formulated by Franciscus Vieta (1540 – 1603). In 1655, John Wallis (1616 – 1703) and in 1658 William Brouncker and Leibniz in 1673 published their various expressions of series to compute the value of pi.
By the use of such series, the value of pi may be computed to any desired number of decimal places. By 1873, William Shanks had carried the computation of the value of pi to 707 decimal places; in 1946, however, it was discovered that Shanks’ vas incorrect after the 527th decimal place. With the advent of mechanical – and later electronic – computing machines, more extensive calculations of the value of pi became feasible. Even as far back as 1958, pi was already accurately calculated to 10,000 decimal places using the most advanced electronic computers of the time. 

In our more pragmatic contemporary engineering practices computing the value of pi beyond 10 decimal places offer little practical value. To ten places, pi is given as 3.1415926536 – this is of sufficient accuracy to permit computing the circumference of a sphere the size of the planet Earth with an error less than 1/8 of an inch. Out to about 40 decimal places, pi would give the circumference of the entire visible universe with an error imperceptible even with an electron microscope. Thus these extensive calculations have no practical value and serve chiefly to show the power of modern methods of computation. 

Though in the field of number theory, the value of pi manifest itself in an exiting way that can sometimes only appreciated by pure mathematics enthusiasts. In 1761, Johann Lambert showed that pi is irrational – i.e. cannot be expressed as the ratio of two integers. While in 1794 Andre Marie Legendre made the proof a rigorous one, thus finally establishing that the value of pi is not a repeating decimal. 

Joseph Liouville in 1884 proved the existence of transcendental numbers – i.e. numbers that cannot be expressed as roots of algebraic equations with rational coefficients. Charles Hermite in 1873 proved that e – the base of natural logarithms – was such a number. Using this information and Euler’s theorem that e raised to the power of pi times the square root of negative one is equal to zero, Ferdinand Lindermann proved in 1882 that pi is transcendental thus assuring the impossibility of squaring the circle. 

The number pi enters into the measurement of many more curved plane figures other than the circle. Thus the area of an ellipse is given in terms of half the length of its axes as A = pi (a) (b). It also appears in many of the relations of mathematics, physics and engineering other than those directly concerned with areas or lengths of arcs.