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 17

^{th}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 527

^{th}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.

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