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The Mathematics of 
is more or less to study mathematics itself, because the quest for an accurate value for has occupied the imaginations of men (and
women) for longer than nearly any similar problem.
The Babylonians believed to be equal to 3 1/8 (within 0.53%). The Egyptians "improved" on this — they used 4 * (8/9)^2 (within 0.60%). Biblical
references (such as they are) usually assign the value of 3. These values may be an accident of recorded history, since all one need do is measure the
distance across an inscribed circle, and compare that measurement with the distance around the perimeter (the circumference).
While Europe apparently slept, in 500 A.D. the Hindus on the Indian subcontinent had a much better estimate of = 3 177/1250, or 3.1416 (within
0.002%). This is consistent with what we know of their mathematical and astronomical skills. The Chinese had arrived at similarly accurate results,
leading one to the unavoidable conclusion that Europeans were, to put it simply, behind.
There are some historical issues having to do with , such as: is it irrational, i.e. is it a number than cannot be expressed as the ratio of two integers,
and is it transcendental, i.e. is it a number that cannot be the root of an algebraic equation. Both these questions have been answered: is both
irrational and transcendental.
This result has some interesting side effects. Irrational numbers never "round off" — no matter how accurately one calculates them, there are always
more digits to resolve. Mathematicians have been trying to calculate , to increasing degrees of accuracy, for centuries.
In the computer era, the key to a successful calculation of is to have an efficient equation. Some equations, such as the author's, are simple but very inefficient (meaning they take a long time to calculate a significant number of places):
Another equation for looks much messier but is, in fact, very much more efficient:
The second equation is efficient because the arctangent calculation can be broken down into a series of relatively fast, well-behaved operations. This
equation was used by the author to calculate to 10,000 places — on an Apple II! In more recent times, and with the availability of programs such as
Mathematica, using a PC one can calculate to a million places without great stress, although some patience may be required.
One may ask, if is irrational, then are the individual digits themselves random, i.e. do their frequency of occurrence and sequence show no pattern?
The answer is yes, the digits of are a good source of pseudorandom numbers (pseudorandom in the sense that you can recreate the sequence by
returning to the starting point for the series). has a special place in our lives only because the surfaces and space around us are relatively flat. In the same way, the sum
of the internal angles of a triangle equals 180 degrees only if the triangle is drawn on a flat surface. Now imagine approaching a black hole, a place
where space-time curvature becomes extreme. As one approaches the black hole's event horizon, the sum of the triangle's angles will increase beyond
180 degrees, and the measured value for will increase. In fact, according to current theory, a black hole is thought to have a finite circumference but
a radius of zero. In this sense, then, the search for with greater and greater precision is an affirmation of a world with Euclidean (i.e. flat) geometry,
a world different from the one we occupy.
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