## Pi cannot be expressed accurately as a fraction

Pi or ?, the 16th letter of the Greek alphabet, is used to represent the most widely known mathematical constant. By definition, pi is the ratio of a circle’s circumference to its diameter. It occurs 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. It also involved to mathematical problems about the lengths of arcs or other curves, the areas of ellipses, sectors, and other curved surfaces, and the volumes of solids. In addition, pi is an important constant in researching and developing other scientific fields.     Pi is an irrational number, meaning it is not equal to any fraction. It is also a transcendental number which is not a root of a non-zero polynomial equation with integer or, equivalently, rational coefficients. In another word, pi will continue infinitely without any repetition or pattern. It also cannot be expressed accurately as a fraction and the decimal never ends. The history of pi is a very confusing one. No one knows exactly who discovered it; they just know assumptions and possible coincidences. The value of pi had known for nearly 4000 years from the ancient Babylonian and Egyptian records. The values were respectively 3.125 and 3.1605 which were not the bad approximation. The first recorded mathematical approach finding the right value was known Archimedes’ scheme by Greek mathematician Archimedes. He realized that a circle is really just a multi-sided object with an infinite number of sides. Starting with a triangle, he doubled the number of sides each time until he got 96. From there, he measured the total length and come up with the Estimate of pi value. His work was developed around 250 B.C. with the proving the range of pi’s value from 3.1208 to 3.1429. In the fifth century, a Chinese mathematician Zu Chongzhi was successfully discovered up to 7 decimal place1 of pi using Archimedean mean iteration. This result of accuracy was not archived until the15th century in the Western. In 1660, Issac Newton used the arcsin series to compute a 15 digits approximation of pi. In 1706, the symbols ? was represented for pi and introduced by a Welsh mathematician, William Jones. In 1782, Johann Lambert used a continued fraction expansion of e given by Euler in 1737 to prove that pi was an irrational number. It means the decimal digits of pi never end and do not have systematic repetition. From the 16th century, the approximation of pi was expanded with a lot of related research. In 1873, a British amateur mathematician, William Shanks used Machin’s formula to compute 707 digits, but only 527 were correct. In 1949, Reitwiesner and his colleagues applied Ferguson’s formula and computed 2037 digits of the pi approximation. A year after Yasumasa Kanada got sixteen million digits of pi, William Gosper computed up to seventeen million digits using Ramanujan’s series in 1988. By the dawn of the 21st century, the computation of pi has begun to stall out. Thanks to various modern scientific developments, supercomputers and application of several series and schemes, the scientists and mathematician have computed the pi value with up to 22.4 trillion decimal digits in 2016.    Today, pi can be used widely in navigation, engineering, architecture, agriculture and many other scientific fields. For example, electrical engineers used pi to solve problems for electrical applications. Statisticians use pi to track population dynamics. Medicine benefits from pi to study the structure of the eyes. In addition, biochemists see pi to understand the structure and function of DNA. Aircraft designers use it to calculate areas of the skin of the aircraft. The pi is helping mathematicians to find the mysteries behind the circle and the arc and curve. Besides that, there is a relationship between pi and music. In the early 1990s, a musician Lars Erickson composed the “Pi Symphony”. He assigned each numeral 0-9 to a musical note and played the notes corresponding to pi’s digits. Pi is a rational number with finite decimal expansion. If pi values are translated to the musical notes as Lars Erickson did, we will have an endless song. On the other side, any combinations of numbers can be found from the pi digits. Thus, we totally have abilities to find a part of our favorite song or even a whole song in the endless pi song. In another word, the endless pi song is a lord of songs which is a combination of all the songs in the world.    Pi is one of the essential numbers in all of mathematics. The exploration for the full value of pi is consisted of all man’s history, from the ancient civilization to the present day. Mathematicians and scientists have searched for ways to find the extent of pi over centuries. Even though the significance of pi has not yet fully known even to this day, it involves an important role in almost all fields of science and mathematics.

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