pi, in mathematics, the ratio of the circumference of a circle to its
diameter. The symbol for pi is . The ratio is the same for all circles and is
approximately 3.1416. It is of great importance in mathematics not only in
the measurement of the circle but also in more advanced mathematics in
connection with such topics as continued fractions, logarithms of imaginary
numbers, and periodic functions. Throughout the ages progressively more
accurate values have been found for {;} an early value was the Greek
approximation 3 1/7, found by considering the circle as the limit of a series
of regular polygons with an increasing number of sides inscribed in the
circle. About the mid-19th cent. its value was figured to 707 decimal places
and by the mid-20th cent. an electronic computer had calculated it to
100,000 digits. It would have taken a person working without error eight
hours a day on a desk calculator 30,000 years to make this {calculation;} it
took the computer eight hours. Although it has now been calculated to more
than 100,000,000 digits, the exact value of cannot be computed. It was
shown by the German mathematician Johann Lambert in 1770 that is
irrational and by Ferdinand Lindemann in 1882 that is {transcendental;}
i.e., cannot be the root of any algebraic equation with rational coefficients.
The important connection between and e, the base of natural logarithms,
was found by Leonhard Euler in the famous formula ei=-1, where i=.
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