The document discusses the history of pi, beginning with the Egyptians approximating it as 3.16 around 4000 years ago based on inscribing a square inside a circle. The Babylonians estimated it as 3.125 using a hexagon inscribed in a circle. Archimedes was the first to rigorously calculate pi between 3 and 3.14 by inscribing and circumscribing polygons with increasingly more sides inside and outside a circle. Modern calculation of pi has reached trillions of digits using computers and programs.

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History of pi

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History
 Calculate the area of a circle, the volume of a sphere, or the
surface area of a cone. But, what do we know about pi? We all
probably know that it is approximately 3.14, but what else do we
know? Well, there is an endless list of interesting facts about pi
that you should know.

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Egyptian method
 A definition of pi is: the circumference divided by the diameter of a circle. In other words, pi
is the ratio of the circumference of a circle to its diameter. You can find the value of pi
yourself when you measure around the edge of a circle and divide it by the measurement
across the circle.
 The existence of pi was known long ago–almost 4000 years ago. The Egyptians’
approximations of pi were indicated through the Rhind Papyrus: they had an approximation
of about 3.160. A translation of a part of the Rhind Papyrus states how the approximation
was derived. They hypothesised that the area of a square with a side length that is 8/9 of a
circle’s diameter would be equal to the area of the circle. Within a given circle in which the
diameter is 2rr, Egyptians took away 1/9 of its diameter, which leaves 16rr/9. They squared
the measurement to get the area of the square, which is 256r2r2/81. Comparing this
measurement with the formula of a circle’s area, Egyptians concluded that pi
was 256/81, which is about 3.160.

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Babylonian method
The Babylonians, on the other hand, had an estimate of about 3.125. Many Babylonian
tablets were found in Susa, and one of them shows a list of mathematical constants.
24/25 is one of the constants, and it conforms with the ratio of a perimeter of a hexagon,
with side length rr, inscribed in a circle of radius rr, to the circumference of the circle.
Equating 24/25 to the ratio of lengths 6rr/(2 πrπr) and rearranging gives pi as 25/8 or
3.125.

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Archimedes
 Archimedes of Syracuse was the first to attempt calculating pi. Through observation, he
hypothesised that perimeters of polygons drawn inside and outside of a circle would be
close to the circumference of the circle. Subsequently, he used Pythagoras Theorem to find
the areas of a polygon inscribed in a circle and a polygon within which the circle was
circumscribed. Archimedes could successfully find the limits of the circle’s area as it is bigger
than the area of the polygon inside the circle, but smaller than the area of the polygon
outside the circle. He started this rigorous calculation by drawing hexagons. He first drew a
circle with a diameter of 1 and used the diameter and the fact that an angle of a hexagon is
60 degrees to find the measurements for each of the sides of the inner and outer hexagons.
He figured out that the perimeter of the inscribed hexagon is 3, and the perimeter of the
circumscribed hexagon is about 3.46. He concluded that pi would be bigger than 3 but
smaller than 3.46. Archimedes used polygons with bigger numbers of sides so that he could
get a better approximation for pi. He ultimately used polygons with 96 sides and utilised
these limits driven from calculation to support that pi is between 3.1408 and 3.14285.

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Modern approaches
 William Jones in 1706 first used the modern symbol for pi, ππ. The
reason why ππ was chosen instead of other Greek letters was that,
in Greek, is pronounced like the alphabet ‘p,’ which stands for the
perimeter. The symbol was later popularised in 1737 by Leonhard
Euler. Since then, pi has been widely used in diverse fields by
engineers, physicists, architects, and designers… anywhere that
mathematics is involved! Some people suggest that the Great
Pyramid at Giza was built based on pi because the ratio of the
perimeter to the height of the pyramid is strikingly close to pi. In
fact, the Pyramid has an elevation of 280 cubits and a perimeter of
1760 cubits, and the ratio of the perimeter to the height is 6.285714,
which is approximately two times pi.

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Continued
 Since pi is a non-repeating decimal, attempts to calculate pi up to as many
digits as possible also have some history. The most accurate calculation of
pi before computers was one done in 1945, by D.F. Ferguson. He calculated
pi up to 620 digits. This was the most accurate and lengthy calculation since,
before Ferguson, William Shanks was known to have calculated pi up to 707
digits–in which only 527 digits were correct. Later on, pi was calculated up
to trillions of digits with the help of digital programs. To list a couple of
people who made world records by carrying out the most extended
calculation of pi, Takahashi Kanada calculated pi up to 206,158,430,000
digits with a Hitachi SR8000 in 1999, and Shigeru Kondo calculated up to 10
trillion digits with Alexander Yee’s y-cruncher program in 2011.

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 Thank YOU